UNIT 1
WHAT IS LOGIC?
Objectives
Introduction
1.1. Definition, Subject Matter and Purpose of Logic
1.2. Value of Logic
1.3. The Three Laws of Thought/Logic
1.4. Brief History of Logic
Objectives
Dear Students! The main objective of this chapter is to give you a general introduction to
the meaning and concerns of Logic. After studying this unit you will be able to:
 define logic
 know its subject matter
 know the three laws of logic
 understand the importance of studying logic
Introduction
Dear Students! Under this chapter you will study the meaning, subject matter, and purpose
of logic. This will be followed by explanation of the relevance or importance of studying
logic; that is, the benefit you get by studying logic. The three laws of logic you will study
next are the fundamental principles of all thinking.
1.1. Definition, a subject matter and purpose of logic
Logic may be defined as the science that evaluates arguments. It is one of the major branches
of philosophy that is concerned with formulation of principles of correct thinking.
It
formulates the rules and principles that help to determine correctness or incorrectness of any
thinking or reasoning.
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The word logic comes from Greek word “logos", which means “word”, “speech”, “reason”,
“meaning” or “thought”. The proper subject matter of logic is in fact thought or thinking.
That is, logic is about, or concerned with, thought/thinking. More specifically, it deals with
correctness and incorrectness of thinking or reasoning.
The purpose of logic is, then, to develop methods and techniques that allow us to distinguish
correct thinking or argument from incorrect one. It provides us with objective, universal, and
necessary (i.e. binding or irresistible) rules and principles which serve as a criteria or
standard for evaluating thinking /argument.
Logic is also defined as the science that evaluates arguments. As you will study in the next
chapter, in logic argument or arguing is reasoning /thinking process, or marshalling
(arranging) evidences or reasons to justify the truth of any claim.
1.2 Value of studying logic
Dear learner, why do we study logic? What is the relevance/relation of logic to your carrier
or profession? Or, what are the benefits you are expected to gain from your study of logic?
This section gives you answers to these questions.
As a science which directs the operations of our mind (intellect) in the attainment of truth,
logic schools us in reasoning correctly with facility, and in detecting and avoiding error. It
removes the original dimness of the mind’s eye; it strengthens and perfects its vision; it
enable it to look out into the world right forward, steadily and truly; it gives the mind
clearness, accuracy, precision; it enables it to use words aright, to understand what it says, to
conceive justly what it thinks about, to abstract, compare, analyze, divide, define, and reason
correctly.
Furthermore, the importance of the study of Logic is derived from its undeniable claim to
universal dominion over the minds of humans. It is said that: No one can ever think correctly
unless he/she thinks logically. No one can ever judge aright unless his/her judgment is one,
which Logic can approve. No one can arrive at well-grounded conclusions unless he/she
argues in conformity with the laws of Logic.
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The significance of studying logic, then, is that it develops the skills needed to construct
sound arguments of ours and to evaluate the arguments of others. Logic aims at cultivation of
habits of correct reasoning and the development of sensitivity for clear and accurate use of
language; increase in confidence that we are making sense when we criticize the
views/arguments of others and when we advance our own views. In brief, the tools of logic
can help you refine your ability to think in a careful, precise knowledgeable way; understand
and apply a variety of argumentative strategies.
1.4. The three laws of logic/thought
Logic, as the science of evaluation of arguments and correct thinking, is concerned with the
Laws of Thought, which guarantee consistency. Every reasoning /argument must rest upon
judgments which are clear and self-evident.
The Three Laws of Logic thought are: Law of Identity, Law of Non-contradiction, and Law
of the Excluded Middle. These laws are fundamental and self-evident truth; when we say
they are fundamental or basic laws we mean that they are universal laws or principles by
which all other claims (truths) may be shown to be true. They are, in other words,
foundations of all other truths. By self-evident we mean they do not need further evidence for
their truth. They cannot be denied without self-contradiction.
The Law of Identity
It states that "Something is what it is”, “If anything is A, it is A”, or “A rose is a rose”.
]
The Law of Non-contradiction
States those contradictions are never true. It is formulated as: “ ‘A’ cannot both be and not be
‘A’”. “It is impossible for the same thing both to be and not to be at the same time and with
the same respect”. Notice that this law applies the phrase “at the same time and with the
same respect” without which the law will be meaningless. When applied in reasoning, or
thought, it means, “No proposition can be both true and false “or applied to reality”. “A tree
cannot be both oak and not-oak.”
The Law of the Excluded Middle
It states that for any given statement, either it or its negation is true. It reads “Everything
must be either ‘A’ or not ‘A’”, “‘A’ is either B or not B”. In thought: “Between affirmation
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and denial there is no middle course”, “Between two contradictories middle is excluded”;
“one of the two contradictories must be true”. In other words, there is no middle thing
possible between “being” and “not being”, it is either the one or the other.
1.5 A brief history of logic
Historically, ancient Greek philosophers, also known as Pre-Socratic philosophers, used logic
widely. However, Aristotle (384-322 B.C.) is credited with the title Father of Logic because
he was the first Greek thinker to devise the rules and principles of correct thinking in his
work known as Organon.
There are many types of logic. One of these was Aristotle’s logic, which is the foundation for
all subsequent types of logic. Aristotle’s logic is basically deductive logic and named as
Traditional Syllogistic, or Aristotelian logic or syllogistic logic.
Based on Aristotle’s work of logic the subsequent philosophers and mathematicians
contributed a lot to the development of logic. G.W.Leibniz (1646-1716), German
philosopher, developed a symbolic logic or calculus, as a result of which he is credited with
the title Father of symbolic logic.
Symbolic logic (also known as Modern logic
Mathematical logic) consists of prepositional logic and predicate/ quantified logic. On the
other had, the development of Inductive logic is associated with British Philosopher John
Start Mill (1806-1873).
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Exercise 1
1. Which of the following is not among the benefits you gain from your study of logic?
A. The ability to construct sound argument of your own
B. The skill to evaluate arguments of others
C. The ability to communicate ideas effectively, clearly, and meaningfully
D. Sensitivity to detect and avoid errors in thinking
E. None of the above
2. Which of the following is not among the nature of the laws of thinking that logicians
formulate? They are;
A. Object
B. Universal
C. Necessary
D. Subjective
E. None of the above
3. The philosopher who is regarded as the father of logic is ________________.
4. "A is A" is an example of the law of Excluded Middle. (True, False)
Check Your Progress Exercise 1.
1. E None of the above
2. D. Subjective
3. Aristotle
4. False (it is an example of the law of Identity
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UNIT 2
BASIC CONCEPTS
2.0.
Objectives
2.1.
Introduction
2.2.
Argument, Premises, and Conclusion
2.3.
Recognizing Arguments in Passage
2.3.1. Passage Lacking Inferential Claim
2.3.2. Conditional Statements
2.3.3. Explanations
2.0.
2.4.
Deductive and Inductive Arguments
2.5.
Recognizing Deductive and Inductive Arguments
2.6.
Validity, Truth, Soundness, Strength and Cogency
2.7.
Summary
2.8.
Answers to Check Your Progress Exercise
Objectives
Dear learner! The basic aim of this unit is to introduce you to the basic concepts in logic
that help you to evaluate arguments. At the end of the unit you are expected to be able to:
 define argument
 identify the premises and conclusion of an arguments
 distinguish passages that contain argument from those do not.
 distinguish deductive and inductive arguments
 explain the meaning of valid, sound, strong and cogent argument.
2.1.
Introduction
Dear learner! in the previous chapter we defined logic as the science that evaluates
arguments. This makes “argument” the most basic concept in logic. Hence, it is important
that you have good understanding of “argument”. To help you in this, the present unit
introduces you to the meaning, components and types of arguments.
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2.2. Argument, Premise and Conclusion
An argument is defined as a group of statements one or more of (the premises) which is
claimed to provide support for, or reasons to believe, one of the others( conclusions). The
statement which is claimed to follow from others is called the conclusion to the argument
whereas the statement(s) from which the conclusion follows one known on the premise(s) of
the argument. Premises are, in other words, the reasons or evidences for accepting the
conclusion. Premises and conclusion are therefore the two components of an argument.
Hence, an argument could be defined as a group of statements that consists of at least one
premise and a conclusion.
Consider the following examples of an argument:
1. it is wrong to kill a human being
2. abortion is killing a human being
3. therefore, abortion is wrong
In the argument, the first two statements give reasons for accepting the third and they are
said to be premises of the argument, and the third statement is called the argument’s
conclusion.
Inference: A mere sequence of ideas and statements, even if true, does not constitute an
argument. Here is a case in point:
“All cows are animals; and some trees are oaks; so water is composed of hydrogen
and oxygen”.
All the statements are true but the conclusion has no logical relation or connection with the
other statements. Hence this group of statements is not an argument. In order to exist an
argument there should be an inference. Inference is the movement of thought by which we
reach a conclusion from premises; it is the reasoning process used to produce an argument or
a mental operation where we pass from evidence to a truth called the conclusion.
Statements: We defined an argument as a group of statements, but what is a statement? A
statement is a sentence that is either true or false. It is truth-claim, or a sentence that has
truth-value.
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Note that all statements are sentences but not all sentences are statements. For example:
Questions (“what time is it?”), proposals (“Let’s sleep at the foot of a tree today”),
suggestions (“I suggest that you travel by train”), command (“Go out”), and exclamations
(Oh!), are sentences but not statements, for they do not have truth-values.
Identifying the Premises and Conclusion of an argument
One of the most important tasks in the analysis of arguments is being able to distinguish
premises from conclusion. If what is thought to be a conclusion is really a premise, and vice
versa, the subsequent analysis cannot possibly be correct.
Frequently, arguments contain certain indicator words that provide clues in identifying
premises and conclusion. Some typical conclusion indicators are
Therefore
hence
whence
Wherefore
thus
so
Accordingly
consequently
it follows that
We may conclude
we may infer
implies that
Entails that
it must be that
as a result
Whenever a statement follows one of these indicators, it can usually be identified as the
conclusion. By process of elimination the other statements in the argument are the premises.
If an argument does not contain a conclusion indicator, it may contain a premise indicator.
Some typical premise indicators are
Since
in that
seeing that
As indicated by
may be inferred from for the reason that
Because
as
inasmuch as
For
given that
owing to
A statement following one of these indicators can usually be identified as a premise.
Sometimes an argument contains no indicators. When this occurs, the reader/listener must
ask himself or herself such questions as: What single statement is claimed (implicitly) to
follow from the others? What is the arguer trying to prove? What is the main point in the
passage? The answers to these questions would point to the conclusion.
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Recognizing Arguments In Passages
Not all passages contain arguments. Because logic deals with arguments, it is important to
be able to distinguish passages that contain arguments from those that do not. In general, a
passage contains an argument if it purports to prove something; if it does not do so, it does no
contain an argument. Two conditions must be fulfilled for a passage to purport to prove
something:
At least one of the statements must claim to present evidence or reasons. (This is
called factual claim)
There must be claim that the alleged evidence or reasons supports or implies
something that is, a claim that something follows from the alleged evidence. (This is
called inferential claim).
Assist your further in distinguishing passages that contains arguments from those that do not,
let us now investigate some typical kinds of non-arguments. These include several forms of
expression that lack an inferential claim, conditional statements, and explanations.
2.3.1 Passages Lacking Inferential Claim
Passages lacking an inferential claim contain statements that could be premises or
conclusions (or both), but what is missing is a claim that a reasoning process is being
expressed; or that potential premises support a conclusion or that a potential conclusion
follows from premises. The following all typical examples of passages that are not argument
because they lack inferential claim.
Warnings (such as “Watch out that you don’t slip on the ice”) and pieces of advice (such as
“I suggest you take accounting during your first semester”) are kinds of discourse aimed at
modifying someone’s behavior.
Hence, a passage contains argument if and only if it has both “inferential claim” and “factual
claim”
Statements of belief or opinion are expressions of what someone happens to believe or think
at a certain time.
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Loosely associated statements may be about the same general subject, but they lack a claim
that one of them is proved by the others.
A report consists of a group of statements that convey information about some situation or
ever.
An expository passage is a kind of discourse that begins with a topic sentence followed by
one or more sentences that develop the topic sentence. If the objective is not to prove the
topic sentence but only to expand it or elaborate it, then there is no argument.
An illustration consists of a statement about a certain subject combined with a reference to
one or more specific instances intended to exemplify that statement. Illustrations are often
confused with arguments because many of them contain indicator words such as “thus”.
2.3.2 Conditional Statements
A conditional statement is one in the form: “if… then …”,
If air is removed from a solid closed container, then the container will weigh less
than it did.
Every conditional statement is made up of two component statements. The component
immediately following the “if” is called the antecedent, and the one following the “then” is
called the consequent. (Occasionally the word “then” is left out, and occasionally the order
of antecedent and consequent is reversed.)
Conditional statements are not arguments, because they fail to meet the criteria given earlier.
In an argument, at least one statement must claim to present evidence, and there must be a
claim that this evidence implies something. In a conditional statement, there is no claim that
either the antecedent or the consequent presents evidence. In other words, there is no
assertion that either the antecedent or the consequent is true.
2.3.3 Explanations
An explanation consists of a statement or group of statements intended to shed light on some
phenomenon that is usually accepted as a matter of fact.
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Every explanation is composed of two distinct components: the explanandum and the
explanans. The explanandum is the statement that describes the event or phenomenon to be
explained, and the explanans is the statement or group of statements that purports to do the
explaining.
Explanations are sometimes mistaken for arguments because they often contain the indicator
word “because.” Yet explanations are not arguments for the following reason: In an
explanation, the explanans is intended to show why something is the case, whereas in an
argument the premises are intended to prove that something is the case.
2.3 Deductive and Inductive Arguments
Arguments are either deductive or inductive. These are the two major types of arguments.
Deductive argument is an argument in which it is claimed that if the premises are true, then
the conclusion must be true. In this type of argument the conclusion is claimed to follow
necessarily from the premises. In deductive reasoning the word “necessarily” is stressed and
it is means, “given the truth of the premises, it is absolutely impossible for the conclusion to
be false.”
Inductive argument is one in which it is claimed that if the premises are true, then it is
probable that the conclusion is true. It is the matter of probability, not necessity. In
inductive arguments the premises are claimed to suggest the conclusion but not to guarantee
it. They do not imply the conclusion; the conclusion contains information, which is not
contained in the premises.
In other words, the premises are claimed to support the
conclusion not conclusively but with some degree of probability.
Most students are misled by the phrase: “A deductive argument/reasoning is that proceeds
from general statement to the particular statement, while inductive argument is that proceeds
from the particular statement to the general one”. This fact should not be used as a standard
criterion or method for distinguishing induction from deduction. The reason is that there are
deductive arguments that proceed from the general to general, from the particular to the
particular, and from the particular to the general, as well as from the general to the particular.
And there are inductive arguments that do the same.
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2.4 Recognizing Deductive and Inductive Argument.
How do we know whether an argument is deductive or inductive? What are the criteria or
factors that bear upon our decision as to whether an argument is deductive or inductive?
To determining any argument as deductive or inductive considers the following three factors;
1. The nature of link between premises and conclusion
2. The occurrence of special indicator words; and
3. The form of argumentation the speaker/arguer uses the arguer or speaker uses.
The first factor, “the nature of link between premises and conclusion”, means whether the
connection between premises and conclusion is a matter of “necessity” or “probability”.
Hence if the conclusion actually follows necessarily from the premises, then the argument is
deductive, not inductive. Conversely, if the conclusion does not follow necessarily from the
premises, but does follow probably, then it is best to consider the argument inductive.
The second factor, “special indicators”, means there are key words or phrases that belong
either to inductive or deductive arguments. If, in drawing a conclusion, the arguer uses
words such as: “probably or improbable”, “plausible or implausible”, “likely or unlikely”,
“reasonable to conclude”, then the argument is inductive. On the other hand, if the speaker
uses words such as: “necessarily”, “certainly”, “absolutely”, or “definitely”, then the
evaluation may want to consider the argument deductive.
The word “must” serves as inductive or deductive indicator based on the context. Let us see
the following argument:
The road is wet; it must have been rained.
The above argument is inductive and, thus, “Must” serves as probable or inductive indicator.
The third factor, “the form of argumentation used” means that if the arguer’s reasoning
depends on: purely arithmetic or geometric, or argument from definition, or syllogistic way
of thinking, then we consider these forms of reasoning are typically deductive.
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Examples:
a.
Purely arithmetic/geometric reasoning:“Abebe placed three oranges and five bananas into a paper bas; so, the
bas
contained a total of eight pieces of fruit”.
Note that it does not include statistics.
b. Argument from definition: means a reasoning in which the conclusion is claimed to
depend merely on the definition of some word or phrase used in premise or
conclusion.
- “Since some men are bachelors, it follows that they are
unmarried males”.
- “This figure is a square; accordingly, the figure has four sides”.
Note that there negation lead to self-contradiction
All the above arguments are deductive because their conclusions follow necessarily from the
premises.
On the other hand, the following form of argumentation belongs to inductive arguments.
A. Prediction: - The argument in which the premises deal with some known
event in the present or past, and the conclusion moves beyond this event to some
event in the future. Example.
“Because certain meteorological phenomena have been observe to develop clouds
over the highlands of Harar, a heavy rainfall will occur at Dire-Dawa in the
afternoon.”
B. Inductive generalization: - Proceeds from the knowledge of a selected sample to
some claim about the whole group. Because the members of the sample have a
certain characteristic, it is argued that all the members of the group have that same
characteristic. Example
“Abebe selected three oranges from a sack and they were tasty. Therefore, all the
oranges from that sack are tasty”.
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C. Argument from analogy: - Reasoning depending on the existence of similarity or
analogy between two things; and due to the existence of the similarity a certain
condition affects the familiar (or better-known) thing or situation is concluded to
affect the similar (lesser-known) situation. Example
“Because the water being pumped from lake “X” has a high calcium content, it
follows that most likely the water being pumped from nearby lakes also have a high
calcium content.”
D. Argument from authority: is reasoning in which the conclusion rests
on a statement made by some presumed authority or witness.
Example
“Ato Tolosa committed the murder because an eyewitness testified to that effect
under oath.” Since the eyewitness could be lying, such argument is essentially
probabilistic.
E. Argument based on signs: is a reasoning proceeding from the knowledge of a
certain sign (symbol) to knowledge of the thing or situation that the sign (natural or
manmade) symbolizes.
F. Casual inference: - A reasoning proceeding from knowledge of a cause to
knowledge of the effect or the vice-versa.
2.6 Validity, Truth-Soundness, Strength and Cogency
These are words or concepts used to evaluate arguments.
A valid argument is deductive argument in which its premises and conclusion are related to
each other in such a way that if the premises were true, it would be impossible for the
conclusion to be false. The conclusion necessarily follows from the premises. In other
words, a valid argument is a sort of trend, which takes the form “If you accept the premises
as true, then you must accept the conclusion as true as well, on the pain of selfcontradiction”. This means the truth of the conclusion is guaranteed by the truth of the
premises. In valid arguments the truth of the conclusion is implied or contained by premises.
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A deductive argument that is not valid is said to be invalid. An invalid argument is
deductive argument in which the conclusion does not follow necessarily from the premises.
This means, if the premises are assumed true, then it is possible that the conclusion be false.
Consider the following arguments:
a. All roses are flowers
b. All roses are plants
All flowers are plants
All flowers are plants
Therefore, all roses are flowers.
Therefore, all roses are
flowers
Both (a and b) are deductive arguments because their conclusions are claimed to follow
necessarily from the premises. However, everyone that claims should not be accepted; (a) is
valid argument whereas (b) is invalid argument. The reason why (a) is valid is because it is
impossible for the conclusion to be false given the premises true. In other words, in
argument (a) the conclusion follows necessarily from the premises. However, argument (b)
is invalid deductive argument because, even if the premises are true the conclusion will be
false; given its premises true, it is possible that the conclusion be false. Thus, the conclusion
of (b) argument doesn’t follow necessarily from the premises.
Note that there is no middle ground between valid and invalid; no arguments are said to be
“almost” valid or “almost” invalid. Do not judge arguments containing true statements as
valid and those containing false statements as invalid.
Moreover, there is no direct
relationship between truth and validity. That is, it is not necessary that either the premises or
the conclusion be true for an argument to be valid. A valid argument can have false premises
and false conclusion; false premises and true conclusion; true premises and true conclusion.
But it can never have true premises and false conclusion.
Validity is a concept for logical correctness or incorrectness, not factual correctness or
incorrectness (true or false). Validity means: it is impossible for the conclusion to be false
while the premises are all true! Validity concerns the structure or form of an argument, not
the content (subject matter) or true or false of the individual premises and conclusion.
A deductive argument is said to be SOUND when it has true premises and it is valid. If any
of these two conditions is missing, the argument is unsound. A sound argument is “good”
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deductive argument without qualification; or argument that is logically correct (valid) and
that is factually correct (or has all true statements).
A strong argument is inductive argument such that if the premises are assumed to be true,
then based on that assumption it is improbable that the conclusion is false; its premises make
the conclusion more probable, but do not guarantee its truth. It is, then, logically possible for
the conclusion of an inductive strong argument to be false even though all of its premises are
true. It is probable (but not necessary) that if the premises of strong inductive argument are
true, then the conclusion is true. For instance:
“Ninety percent of American males over 50 years of age could not run a mile in less
than 6 minutes. Thomas is an American male over 50 years age. So, Thomas cannot
run a mile in less than 6 minutes.
The premises of the above argument do not absolutely guarantee the truth of the conclusion.
Possibly Thomas belongs to that small percentage of American men over 50 who can run a
mile in less than 6 minutes. Nevertheless, it is probable that the conclusion of the argument
is true assuming its premises are true.
Note that if we change the above argument (systematically to clarify the concept of strength)
and replace “90%”, the arguments still remain all strong; in fact, it is even stronger. From
this we learn that Strength, unlike validity in deductive reasoning, is a matter of degree.
Suppose we replace “ninety” with “fifty-one”, the argument remains strong because the
conclusion remains slightly more probable than not. Of course, once we replace “ninety”
with “fifty-one”, the argument is of little value: The amount of support (premises) given to
the conclusion is scarcely worth mentioning!
But the important point to keep in mind is that because strength comes in degrees, we can
legitimately speak of inductive arguments that are slightly strong moderately strong, or very
strong. By contrast, it would make no sense to speak of “slightly valid” or “moderately
valid”, “almost valid” deductive arguments. Validity is an all-or-nothing affair!
Inductive argument which is not strong is said to be weak.
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A weak argument is inductive argument in which the conclusion does not follow probably
from the premises. Then, a weak argument has this essential feature: it is not likely that if its
premises are true, then its conclusion is true. To illustrate this consider the above-mentioned
reasoning. If we replace “ninety” with “fifty”, then it is just as likely, (given the premises),
that Tomas can run the mile in less than 6 minutes as that he cannot. So the argument is
weak. And, of course, the argument becomes progressively weaker as we replace “ninety”
with “forty”, “thirty”, and so on.
An inductive argument that is strong and has true premises is called a cogent argument; and
if either condition is missing, then the argument is uncogent. Cogent argument is good
inductive argument without qualification.
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2.5Summary
Logic is the science that evaluates arguments. Argument is an attempt to show that
something is true by providing evidence for it. Premise is the component of an argument
that is provided as an evidence or reason for accepting something as true whereas conclusion
is something claimed to be true given the premises.
Truth or falsity is the property of statements, but not of arguments. That is, arguments are
not said to be true or false.
Arguments are either deductive or inductive. Deductive arguments are evaluated as either
valid or invalid, or sound or unsound. Indicative arguments are evaluated as either strong or
weak, or cogent or uncogent.
True
Statements
Sound
False
Valid
Unsound
Deductive
Arguments
Invalid
(all unsound)
Inductive
Strong
Weak
Cogent
Uncogent
(all uncogent)
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Check Your Progress Exercise 2.
Part I. Say True or False
1. A deductively sound argument can have a false conclusion. _________
2. Some arguments are true. ____________________
3. Some premises are valid. __________________
4. All valid arguments have at least one false premise ____________
5. Every invalid argument is deductively unsound. _________________
6. If all of the premises of an argument are true, then it is deductively sound.
___________
7. “The defendant is not guilty of murder since he is insane”- is argument.
____________
Part II. Choose the Best Answer
1. “All birds are animals and no trees are cats. Therefore, no trees are animals”. Which
statement is correct? The passage:A) Contains no argument
D) A valid deductive argument
B) Has no inference
E) A and B
C) Has premises and conclusion
F) C and
2. “Scholars are like the roman emperor Nero. Nero played his violin while Rome burned.
Similarly, scholars play with ideas while the “flames” of greed, poverty, racism, and
violence threaten civilization. Nero was morally irresponsible. Hence, scholars are
morally irresponsible also.” This is an example of
A. Argument from authority
D. Argument from analogy
B. Inductive argument
E. B and D
C. Deductive argument
F. All except C
Part III. Identify the premises and conclusion.
“Frankly speaking, abortion is wrong, for abortion is a homicide and all homicides are
wrong.”
Premise 1. _________________________
Premise 2. _________________________
Conclusion ________________________
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Answers to Check Your Progress Exercise 2.
I
1. False 5. True
2. True 6. False
3. False 7. True
4. False
II
1. E
2. E
III
Premise 1. Abortion is a homicide
Premise 2. All homicides are wrong
Conclusion. Abortion is wrong
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UNIT 3
LANGUAGE :MEANING AND DEFINATIONS
Contents
3.0
Aims and Objectives
3.1.
Introduction
3.2.
Emotive and Cognitive meaning
3.3.
Intentional and Extensional Meaning
(Intension and Extension of terms)
3.4
The Meaning and Purpose of definitions
3.5
Classifications of Definition
3.5.1 Definitions Classified According to Purpose
3.5.2 Definitions Classified According to Methods
3.5.2.1 Extensional Definition
3.5.2.2 Intentional Definitions
3.6 Criteria for Evaluating Definitions
3.4.
Summary
3.0. AIMS and Objectives
Dear learner! the basic aim of this unit is to introduce you to different types of meanings of
terms that you will use in your study of definitions in the next unit. At the end of this unit
you are be able to:
 define the meaning of terms
 identify emotive and cognitive meaning
 distinguish intentional and extensional meaning
 be able to put terms in increasing and decreasing intension and extension
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 identify the different kinds of definitions
 distinguish the difference among the different kinds of definitions
 know the meaning and purposes of definition
 be able to identify the types of definitions
 know the criteria used to evaluate definitions
 have the skill to evaluate definitions
3.1. Introduction
Dear learner! the basic units of any ordinary language are words. Our main concern in this
unit, however, is not with words in general but with terms. The unit will present the
definition components, different types of meanings of terms. The first section of the unit will
be about emotive and cognitive meaning of terms. This will be followed by the description of
intentional and extensional meaning of terms. The first section of the unit will present the
meaning, purpose and components of definition. This will be followed by the presentation of
various types of definitions classified according to purpose and technique. Finally
in this
chapter, you will study the criteria used to evaluate definitions.
3.2. Emotive and Cognitive Meaning of Terms
Language serves various functions in our day-to-day lives.
For our purpose, two linguistic functions are particularly important: (1) to convey
information and (2) to express or evoke feelings. Consider, for example, the following
statements:
The death penalty, which is legal in thirty-six states, has been carried out most often
in Georgia; however, since 1977 Texas holds the record for the greatest number of
executions.
The death penalty is a cruel and inhuman form of punishment in which hapless
prisoners are dragged form their cells and summarily slaughtered only to satiate the
bloodlust of a vengeful public
The first statement is intended primarily to convey information; the second is intended, at
least in part, to express or evoke feelings. These statements accomplish their respective
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functions through the distinct kinds of terminology in which they are phrased. Terminology
that conveys information is said to have cognitive meaning, and terminology that expresses
or evokes feelings is said to have emotive meaning. Thus, in the first statement the words
“legal,” “thirty-six,” “most often,” “Georgia,” “record” and so on have primarily a cognitive
meaning, while in the second statement the words “cruel,” “inhuman,” “hapless,” “dragged,”
“slaughtered,” “bloodlust,” and “vengeful” have a strong emotive meaning. Of course, these
latter words have cognitive meaning as well.
“Cruel” means tending to hurt others,
“inhuman” means inappropriate for humans, “hapless” means unfortunate, and so on.
The emotively charged statement about the death penalty illustrates two important points.
The first is that statements of this sort usually have both cognitive meaning and emotive
meaning. Therefore, since logic is concerned chiefly with cognitive meaning, it is important
that we be able to distinguish and disengage the cognitive meaning of such statements from
the emotive meaning. The second point is that part of the cognitive meaning of such
statements is a value claim. For example, the statement about the death penalty asserts the
value claim that the death penalty is wrong or immoral. Indeed, such value claims are often
the most important part of the cognitive meaning of emotive statements. Thus, for the
purposes of logic, it is important that we be able to disengage the value claims of emotively
charged statements from the emotive meaning and treat these claims as separate statements.
These observations suggest the reason that people use emotive terminology as often as they
do: Value claims as such normally require evidence to support them. For example, the claim
that the death penalty is immoral cannot simply stand by itself. It cries out for reasons to
support it.
But when value claims are couched in emotive terminology, the emotive
“clothing” tends to obscure the fact that a value claim is being made, and it simultaneously
gives psychological momentum to that claim. As a result, readers and listeners are inclined
to swallow the value claim whole without any evidence. Furthermore, the intellectual
laziness of many speakers and writers, combined with their inability to supply supporting
reasons for their value claims, reinforces the desirability of couching such claims in emotive
terminology.
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Many people, for example, will refer to someone as “crazy,” “stupid,” or “weird” when they
want to express the claim that what that person is doing is bad or wrong and when they are
unable or unwilling to give reasons for this claim. Also, many people will refer to things or
situations as “awesome” or “gross” for the same reasons. Those who happen to be listening,
especially if they are friendly with the speaker, will often accept these claims without
hesitation.
Indeed, this technique of couching value claims in emotive terminology is so effective that in
some circles it has reached the level of a science. The world of advertising provides a prime
example. The objective of all advertising is to convey the message that the product being
advertised is good. On most occasions, however, it is not feasible to give evidence for this
claim, so the claim is phrased in terminology that elicits a favorable emotional response.
Another field where much attention is paid to emotive terminology are the military.
Language associated with military ventures often calls forth-negative emotions.
To
counteract this effect, military spokespersons are trained to describe those ventures in terms
that evoke a neutral response. For example, human targets are referred to as “soft targets,”
and, napalm, which is aimed at human targets, is called “soft ordnance.” Dropping bombs is
called “servicing a site,” saturation bombing is “terrain alteration,”
Let us now consider emotive terminology as it occurs in arguments. In arguments, emotive
terminology accomplishes basically the same function as emotive terminology in statements.
It allows the arguer to make value claims about the subject matter of the argument without
providing evidence, and it gives the argument a kind of steamroller quality by which it tends
to crush potential counter augments before the reader or listener has a chance to think of
them. This steamroller quality also tends to paralyze the logical thought processes of readers
or listeners so that they are not able to see illogical arguments in their true light. These
effects of emotive terminology can be avoided if the reader or listener will disengage the
value claims and other cognitive meanings from the emotive meaning of the language and re
express them as distinct premises.
Consider, for example, the following emotively charged argument taken from the letters to
the editor section of a newspaper:
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Now that we know that the rocks on the moon are similar to those in our back yard
and that tadpoles can exist in a weightless environment, and now that we have put the
rest of the world in order, can we concentrate on the problems here at home? Like
what makes people hungry and why is unemployment so elusive?
The conclusion of this argument is that our government should take money that has been
spent on the space program and on international police actions and redirect it to solving
domestic problems. The author minimizes the importance of the space program by covertly
suggesting that it amounts to nothing more than work on ordinary rocks and tadpoles (which,
by themselves are relatively insignificant), and he exaggerates the scope of the international
effort by covertly suggestion that it has solved every problem on earth but our own. Also,
the phrase “put… in order” suggests that the international effort has been no more important
than restoring order to a room in one’s house. We might rephrase the argument in emotively
neutral language, making the implicit suggestions and value claims explicit, as follows:
The space program has been confined to work on ordinary rocks and tadpoles.
Ordinary rocks and tadpoles are less important than domestic hunger and unemployment.
Our international efforts have restored order to every nation on
earth but our own.
These efforts have been directed to problems that are less demanding
than own domestic problems. Therefore, our government should redirect funds that have
been spent on these projects to solving our own domestic problems.
By restructuring the argument in this way, we can more easily evaluate the degree to which
the premises support the conclusion. Inspection of the premises reveals that the first, third,
and possibly fourth premises are false. Thus, the actual support provided by the premises is
less than what we might have first expect.
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3.3 Intentional and Extensional Meaning
(the intension and extension of terms)
A term is any word or arrangement of words that may serve as the subject of a statement.
Terms consist of proper names (e.g. Abebe), common names(e.g. animal), and descriptive
phrases(e.g. first president of the United States).
Words that are not terms include verbs, nonsubstantive adjectives, adverbs, prepositions,
conjunctions, and all nonsyntactic arrangements of words. The following words or phrases
are not terms; none can serve as the subject of a statement:
Dictatorial
moreover
Runs quickly
craves
Above and beyond
cabbages into again the forest
The last example is a nonsyntactic arrangement.
Words are usually considered to be symbols, and the entities they symbolize are usually
called meanings. Terms, being made up of words, are also symbols, but the meanings they
symbolize are of two kinds: intentional and extensional. The intentional meaning consists
of the qualities or attributes that the term connotes, and the extensional meaning consists of
the members of the class that the term denotes. These two kinds of meaning will provide the
basis for the definitional techniques developed in the next unit.
The intentional meaning is otherwise known as the intension or connotation, and the
extensional meaning is known as the extension or denotation. Thus, for example, the
intension (or connotation) of the term “cat” consists of the attributes of being furry, of having
four legs, of moving in a certain way, of emitting certain sounds, and so on, and the
extension (or denotation) consists of the cats themselves-all the cats in the universe.
Because terms symbolize meanings to individual persons, it is inevitable for subjective
elements to invade the notion of connotation. For example, to a cat lover, the connotation of
the word “cat” might include the attributes of being cuddly and adorable, while to one who
hates cats it might include those of being
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Obnoxious and disgusting. Because these subjective elements inevitably lead to confusion
when it comes to identifying the connotation of specific terms, logicians typically restrict the
meaning of “connotation” to what may be called conventional connotation.
The
conventional connotation of a term consists of the properties or attributes that the term
commonly connotes to the members of the community who speak the language in question.
Under this interpretation, the connotation of a term remains more or less the same from
person to person and from time to time.
The denotation of a term also typically remains the same from person to person, but it may
change with the passage of time. The denotation of “currently living cat,” for example, is
constantly fluctuating as some cats die and others and born. The denotation of the term
“cat,” on the other hand, is presumably constant because it denotes all cats, past, present, and
future.
Sometimes the denotation of a term can change radically with the passage of time. The terms
“currently living dodo bird” and “current king of France,” for example, at one time denoted
actually existing entities, but today all such entities have perished. Accordingly, these terms
now have what is called empty extension. They are said to denote the empty (or “null”)
class, the class that has no members. Other terms with empty extension include “unicorn,”
“leprechaun,” “gnome,” “elf,” and “griffin.” While these terms have empty extension,
however, thy do not have empty intension. “Currently living dodo bird” and “current king of
France,” as well as “unicorn,” “elf,” and “griffin,” connote a variety of intelligible attributes.
The fact that some terms have empty extension leads us to an important connection between
extension and intension, namely, that intension determines extension.
The intensional
meaning of a term serves as the criterion for deciding what the extension consists of.
Because we know the attributes connoted by the term “unicorn,” for example, we know that
the term has empty extension. That is, we know that there are no four-legged mammals
having a single straight horn projecting from their forehead. Similarly, the intension of the
word “cat” serves as the criterion for determining what is and what is not a member of the
class of cats.
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The distinction between intension and extension may be further illustrated by comparing the
way in which these concepts can be used to give order to random sequences of terms. Terms
may be put in the order of increasing intension, increasing extension, decreasing intension,
and decreasing extension. A series of terms is in the order of increasing intension when each
term in the series (except the first) is more specific than the one preceding it. (A term is
specific to the degree that it connotes more attributes.) The order of decreasing intension is
the reverse of that of increasing intension.
A series of terms is in the order of increasing extension when each term in the series (except
the first) denotes a class having more members than the class denoted by the term preceding
it.
In other words, the class size gets larger with each successive term.
Decreasing
extension is, of course, the reverse of this order. Examples:
Increasing intension: animal, mammal, feline, tiger
Increasing extension: tiger, feline, mammal, animal
Decreasing intension: tiger, feline, mammal, animal
Decreasing extension: animal, mammal, feline, tiger
Theses examples illustrate a fact pertaining to most such series: The order of increasing
intension is usually the same as that of decreasing extension. Conversely, the order of
decreasing intension is usually the same as that of increasing extension. There are some
exceptions, however. Consider the following series:
Unicorn; unicorn with blue eyes; unicorn with blue eyes and green horn, and
a weight of over 400 pounds
Each term in this series has empty extension; so, while the series exhibits the order of
increasing intension, it does not exhibit the order of decreasing extension. Here is another,
slightly different, example:
Living human being; living human being with a genetic code;
Living human being with a genetic code and a brain; living
human being with a genetic code, a brain, and a height of less
than 100 feet.
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In this series none of the terms has empty extension, but each term has exactly the same
extension as the others. Thus, while the intension increases with each successive term, once
again the extension does not decrease.
The aim of this unit is to make you familiar with different types of definition and definitions
techniques. At the end of the unit you are expected to:
3.4 The meaning and Purpose of Definitions
The best way to avoid vagueness, ambiguity and verbal disagreement is to define or limit
one’s terms.
Definition is explanation of the meaning of a word or phrase; it is a group of words that
assigns a meaning (or essence) to some word or group of words. Thus any definition has two
parts: definiendum and definiens.
Definiendum: is the word/group of words that is supposed to be defined; i.e. it is the word to
be defined.
Definiens: is the word or group of words that does the defining i.e. words that do the
defining.
3.5 Classification of Definitions
We classify definitions with respect to:
the purpose to be achieved, and the method
(techniques) used to achieve the purpose. Thus, we get the following types of definition.
3.5.1 Definitions Classified According to Purpose
There are six types of definitions when classified according to the purpose.
1. Reportive (Lexical) Definition:- is a type of definition in which its main purpose is
to report the conventional meaning of terms.
Reportive definition attempts to
explain/report usage of a term or phrase and it’s meaning in given language. Good
examples of reportive definition are all definitions given in dictionaries. Hence it is
possible to evaluate reportive definition as true or false. The purpose of a
lexical/reportive definition is, thus, to eliminate the ambiguity that may arise in a
given language usage.
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2. Stipulative Definition:- is one which assigns a meaning to a word for the first time;
and thus, its purpose is to introduce a term having no previous meaning. Then,
instead of reporting the usage of a word, stipulative definition specifies or stipulates
the meaning of the word. Note that this type of definition is neither true or false
because it is arbitrary giving a meaning to a word for the first time.
Stipulative definition is performed either when there is no existing term to express
what we wish to say or replace a more complex expression with a simpler one. This
involves either coining a new word or giving a new meaning to an old word –
reforming definition.
 Coining a new word: for example Scientists introduced the science of man as
“sociology”; a phrase “person married more than twice”, is stipulated new
word, as “multimarried”.
 Giving a new meaning to already existing term:
for example the word
“skyscraper” existed but later it was confusingly used; thus to reduce the
vagueness of this already existing word (“skyscraper”) people defined it to
mean “building over 500 feet tall”.
It is to bear in mind that stipulative definitions are useful only if they do not deceive,
confuse, or cause one’s listeners needless effort.
3. Precising Definition is a sort of definition in which its purpose is to reduce the
vagueness of a word. Once the vagueness has been reduced, one can reach a decision
as to the applicability of the word to a specific situation.
For example: the word “poor” is vague; when we want to give help or financial
assistance to poor people, how do we distinguish among the population those who are
poor from those who are not? Then, a précising definition would have to be supplied
specifying exactly “who is poor” and “who is not”. Thus the definition of “poor” is
“having an annual income of less than (let’s say) 1,000 birr and a net worth of less
than 10,000 birr.”
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Notice that a précising definition differs from stipulative definition in that the latter
involves a purely arbitrary assignment of meaning, whereas the assignment of
meaning in a précising definition is not at all arbitrary.
4. Theoretical Definition: is assigning meaning to a word in order to state a theory; it
provides the characterization of things denoted by the definiendum; it gives a
systematic way of viewing or understanding objects that suggest deductive
consequences and further investigation. Theoretical definition aims to dig out the
essence of an entity (thing) and know the laws that govern it. These types of
definition solve theoretical problems and bring about an increase and systematization
of knowledge.
5. Persuasive Definition: its purpose is to influence the attitudes of the listener; to
change the feelings of people. In other words, a persuasive definition produces
favorable or unfavorable attitudes toward what is denoted by the definiendum. In this
type of definition we observe emotionally changed words given as meaning to a term
or phrase. Thus, persuasive definition (as a synthesis) amounts to stipulative, lexical,
and possibly theoretical definitions. One word can be defined differently as opposing
pairs of definitional. For instance: “Abortion” is the ruthless killing of innocent human beings.
 “Abortion” is a safe and established surgical procedure where by a woman is
relieved of an unwanted burden.
6. Abbreviative Definition: a type of definition in which its purposes is to introduce a
shorter, more convenient expression to replace a longer, more difficult one. For
example the phrase “ Severe acute respiratory syndrome” is to be replaced as “
SARS” and “ 4X4X4X4X” as (using an exponents) “ 45 “ Abbreviate definitions do
not eliminate vagueness and ambiguity but they save our time and energy
3.5.2 Definition Classified According to Method /Techniques
The methods or techniques used to produce different types of definitions are classified in
terms of the two kinds of meanings, extensional meaning and intentional meaning, of terms
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you studied in the last unit. From the standpoint of extensional (denotative) method we get
the following types of extensional definitions: demonstrative, enumerative, and a definition
by subclass. What is extensional definition?
3.5.2.1 Extensional Definition
It is one that assigns a meaning to term by indicating the members of the class that the
definiendum denotes. There are at least three ways of indicating the members, and therefore
there three types of extensional definitions.
1. Demonstrative or ostensive definition: assign a meaning to a word by pointing to the
members. Here, when defining a term, we do not name or describe, but only indicate, for
example by pointing or nodding one’s head. These types of definition are said to be the
most primitive form of definition and they are also the most limited. To define or explain
the meaning of the wow “dog “ we simply utter the wow and pint at a dog (or picture of
dog); to explain what does the wow “ should music “ we play certain records. From this
we may conclude that ostensive /demonstrative definition is a sort of a nonlinguistic
method of defining.
2. Enumerative Definition: is one in which a term is defined by giving complete or partial
list of the objects denoted by it; assign a meaning to a work by naming (or enumerating )
the members of the class the term denotes .
For instance, complete list of the objects: the term “ Scandinavian Country “ is defined as
(naming “ Sweden, Norway, Denmark, or Iceland”. This is complete enumerative
definition. Let’s see the partial enumerative definition the class of insects, the class of
trees are finite but have too many members to enumerate. Thus, enumerative definitions
have the further limitation that it is often impractical, or impossible, to list everything
denoted by a term
3. A Definition by Subclass: is a type of extensional definition or methods that assigns a
meaning to a term by naming the subclass of the class by the term. (subclass “ means the
class, for instance “ X”, is a subclass for class “Y” given that every member of ”X” is a
member of “Y”. The class of dogs is a subclass of the class of animals). Definition by
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subclass also can be partial or complete, depending on whether the subclass named
include all the members of the class or only some of them. For instance:
 (Partial): “ Tree” means oak, acacia, maple, etc.
 (Complete):“Cetacean” means ether a whole, dolphin or a porpoise
 (Complete): “Fictional work” means either a poem, a play, a novel, or a short story
3.5.2.2Intentional (or Connotative) Definitions
An intentional definition is one that assigns a meaning to a word by indicating the qualities or
attributes that the word connotes. Because at least four strategies may be used to indicate the
attributes a word connotes, there are at least four kinds of intensional definitions:
synonymous definition, etymological definition, operational definition, and definition by
genus and difference.
A synonymous definition is one in which the de4finiens is a single word that connotes the
same attributes as the defines is a synonym of the word being defined. Example:
“Physician” means doctor.
“Intentional” means willful.
“Voracious” means ravenous.
“Observe” means see.
An etymological definition assigns a meaning to a word by disclosing the word’s ancestry in
both its own language and the languages. Most ordinary English words have ancestors either
in old or middle English or in some other language such as Greek, Latin, or French, and the
current English meaning (as well as spelling and pronunciation) is often closely tied to the
meaning (and spelling and pronunciation) of these ancestor words. For example, the English
word “license” is derived from the Latin verb licere, which means to be permitted, and the
English word “captain” derived from the Latin noun caput which means head.
An operational definition assigns a meaning to a word by specifying certain experimental
procedures that determine whether or not the word applies to a certain thing. Examples:
One substance is “harder than” another if and only if one cratches the
other when the two are rubbed together.
A subject has “brain
activity” if and only if an electroencephalo- Graph shows oscillations
when attached to the subject’s head. A “potential difference” exists
33
between two conductors if and only if a voltmeter shows a reading
when connected to the two conductors. Solution is an “acid” if and
only if litmus paper turns red hen dipped into it
A definition by genus and difference assigns a meaning to a term by identifying a genus
term and one or more difference words that, when combined, convey the meaning of the term
being defined. Definition by genus and difference is more generally applicable and achieves
more adequate results than any of the other kinds of intentional definition. To explain how it
works, we must first explain the meanings of the terms “genus,” “species,” and “specific
difference.”
In logic, “genus” and “species” have a somewhat different meaning than they have in
biology. In logic, “genus” simply means a relatively larger class, and “species” means a
subclass; “species” is the subclass of genus. It is a proper subset of the genus (In logic unlike
in biology, we may speak of the genus “ dog” and the species “ puppy” or “ young dog” of
the genus “ animal” and the species “ puppy” or “ young dog”, of the genus “ animal” and
the species “ dog”, or of the genus “ animal ‘ the species “ mammal”.) The “ difference”, (or
in Latin: the singular as differentae, and plural as differentia) is the attribute that
distinguishes the members of a given species from the members of other species in the same
genus. In other words, “ difference” is the characteristic which differentiates the species
being defined from other species . For example, suppose “sibling” is the genus and “sister”
is the species; then the difference is the attribute of being female, which distinguishes “
sisters” form the species “brother”, which also belongs to the genus “sibling”. And “dog” is
the genus and “puppy” is the species; and the difference is the attribute of “being young”,
which distinguishes puppies form other species in the same genus- for example, adult dogs.
Thus, the definition of “Puppy” means a young dog. Now, we can define “definition by
genus and difference” as: methods of defining by identifying an attribute that distinguish the
members of a species form other species in the same genus. It is a technique /method stating
a general class, (genus), which contains as a subclass, (species), the class to be defined, and
then stating the specific manner, (difference), in which members of the subclass differ form
other members of the general class. For example: “Triangle” means a polygon having, three
sides. (The term “Polygon” refers to a general class, which contain triangles as a subclass;
34
and the term “having three sides” refers to the specific manner in which triangle differs form
other polygons. Thus in the definition we saw genus: “polygon”, the species; “triangle”, and
differentia: “having three sides”). The relationship between genus, species, and differences is
shown in the accompanying diagram below, with the rectangle standing for classes.
GENUS (e.g. horse)
Species: Filly
Species:
Colt
Difference: Young
female
Difference: young
male
Species
Difference
Genus
“Triangle” means
three sides
polygon
“ Kitten”: means
young
“ Lake” means
arge
cat
and body of
water
3.6. Criteria for Evaluating Definitions
Thinking must have two very general characteristics: first, it must be clear; we think already
when we know what we are thinking abut; and second, it must be straight; straight thinking
is thinking that goes to its goal in an orderly and accurate way that moves step by step, each
step being connected with what has gone before. To the vagueness of words their ambiguity
must be added as a serious danger to straight or accurate thinking. The ambiguity and
vagueness of words may invalidate a reasoned discourse. Much of the best effort of human
thought must go, therefore, to delimit the vagueness of words and eliminate their ambiguity.
Note that vagueness can be reduces, but never completely eliminated. And to do this,
definition is the appropriate tool and plays a vital role in making our thoughts clear and
straight. Accordingly, it is necessary that we have a set of rules that we may use in
constructing definitions of our own and in evaluating any type of definition of others.
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Make sure that your listeners; readers have the experience of knowledge of the issue to be
explained.
a) Whenever possible, avoid vagueness and ambiguity.
b) You should avoid emotive terminologies or language.
c) You should disclose essential meaning of the word being defined. This means, express
the important attributes that distinguish it form others. For example: “Man” means
featherless biped or two-legged creature. Such definition fails to convey the essential
meaning of “man”: the capacity to reason and to use language.
d) Definition should be nether too broad nor too narrow. This means definition should be
proportionate - i.e., the scope (range or extent) of the defining concept should be equal to
the extent of the concept to be defined:
 defininendum = definiens .
e) You should not define a term in circular way. This error occurs when we try to define “
definiendum “ by means of “definiens” and definens was already defined by the
definiendum. For example: “Artist means a person who is well-qualified in art ”; and
“Art means a work performed by artist”.
f) Definitions should not be needlessly negative (when it can be affirmative). For example:
“Bachelor” means a man who is not married; “Bus” is a vehicle that is not a car. Notice,
however, that some words are intrinsically (by themselves) negative. For such
definitions, a negative definition is quite appropriate and possible. For instance,
“Darkness “ means the absence of light; “Bold” means the absence of hair.
g) You should not express your definition in figurative, obscure, vague, or ambiguous
language. ( definition is figurative if it involves metaphors or tends to paint a picture
instead of exposing the essential meaning of a term.) For instance: “Camel” means a ship
of a desert . (definition is obscure if its meaning is hidden.) You should not use too
technical language because you are concealing the essence of the thing.
h) Your definition should indicate the context to which the definiens refers to (pertains).
Whenever the definiendum is different things that mean different things in different
contexts, a reference to the context is important.
For instance, “strike”, (in baseball), means a pitch at which a batter swings and misses;
“strike” (in bowling), means the act of knocking down all the pins with the first ball of a
frame; “strike” (in fishing ), means a pull on a live made by a fish in taking the bait
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3.7 Summary
The other functions, language include: used to convey information and to express or
evoke feelings. Language fulfills these respective functions through distinctive types of
terminology.
Terminology that conveys information is said to have cognitive meaning; and terminology
that expresses or evokes feelings is said to have emotive meaning.
A term is any word or arrangement of words that may serve as the subject of a statement.
Terms consist of proper names, common names, and descriptive phrases.
Terms symbolize meanings of two kinds: intentional and extensional. Intentional meaning
of a term consists of the qualities or attributes that the term connotes. Extensional meaning
consists of the members of the class that the term denotes.
Definition is a group of words that assigns a meaning to some word or group of words.
Hence, every definition consists of two parts: definiendum and definiens. Definiendum is the
word to be defined whereas definiens is the part that does the defining.
Definitions are classified according to the purpose to be achieved and the method or
technique used to achieve the purpose.
Classification of definitions according to purpose gives us the following types of definitions;
stipulative, lexical reportive, précising, theoretical, and persuasive.
Classification of definitions according to methods gives us the following types:
A. Extensional definition, which consists of demonstrative, enumerative, and definition
by sub-class.
Intentional definition, which consists of synonymous, etymological, operational definition
and definition by genus and difference
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Check Your Progress Exercise 3.
Part I. Say True or False
_________1. Some terms cannot be defined extensionally because their extensions are
empty.
_________2. Since some terms have empty intension, they cannot be defined intentionally,
but be defined by extension.
_________3. The extension of a term consists of things to which the term applies.
_________4. The intension of a term consists of the attributes (properties) an object must
have in order to be included in the term’s extension.
_________5. To the extent that a sentence expresses emotions (feelings), it is said to have
cognitive meaning.
_________6. To the extent that a specific term conveys information, it is said to have
emotive meaning.
Check Your Progress Exercise 4.
I. Identify the rule(s) or criteria violated by the following definitions.
1. “Aves” means Ostrich. ____________________
2. “Penguin” means bird that can’t fly, but not an
____________________
3. “Red” means having a reddish color ____________________
ostrich,
for
example;
4. “Time is the greater container into which we pour our lives ____________________
5. “Monarchy” means a form of government in which the ruling power belongs to one
person. ____________________
6. “Cow” means animal that can give milk. ____________________
7. A “pencil” is a tool for writing. ____________________
8. “Blindness” means the absence of sight. ____________________
9. A “wealthy person” is one who has as money as Bill Gates or Alamudihn
____________________
II. Identify the species , difference, and genus in the following definition.
“Vixen “means female fox.
Species: ________________
Difference: ______________
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Genus: _______________
Answer to check your progress exercise
Check Your Progress Exercise 3
1. True
4. True
2. False
5. False
3. True
6. False
Check Your Progress Exercise 4.
I
1. Too narrow
2. Unnecessarily negative
3. Circular
4. Figurated
5. It is ok
6. Too broad
7. Too broad
8. It is ok
9. Unsuitable attribute
II
Species: vixen
Deference: Female
Genus: Fox
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UNIT 4
INFORMAL FALLACIES
Contents
4.0 Aims and Objectives
4.1 Introduction
4.2 Fallacies in General
4.2.1 Classifications of Fallacies
4.2.2 The Value of Studying Informal Fallacies
4.2.3 Classification of Informal Fallacies
4.3 Fallacies of Relevance
4.3.1 Appeal to Force (argumentum AD Baculum: Appeal to The “Stick")
4.3.2 Appeal to Pity (argumentum AD Misericordasim)
4.3.3 Appeal to The People (argumentum AD Populum)
4.3.4 Argument Against the Person (Argumentum AD Hominem)
4.3.5 Accident
4.3.6 Straw Man
4.3.7 Missing the Point
4.3.8 Red Herring
4.4 Fallacies of Weak Induction
4.4.1 Appeal to Unqualified Authority (Argumentum ad Verecundiam)
4.4.2 Appeal to Ignorance (Argumentum ad Ignorantiam)
4.4.3 Hasty Generalization (Converse Accident)
4.4.4 False Cause
4.4.5 Slippery Slope
4.4.6 Weak Analogy
4.5 Fallacies of Presumption
4.5.1 Begging the Question (Petitio Principii)
4.5.2 Complex question
4.5.3 False Dichotomy (False dilemma)
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4.5.4 Suppressed Evidence
4.6 Linguistic Fallacies
4.6.1 Equivocation
4.6.2 Amphiboly
4.6.3 Composition
4.7 Summary
4.8 Answer to Check Your Progress Exercise
4.0. Aims And Objectives
Dear leaner! The basic aim of this unit is to introduce you to informal fallacies. At the end
of the unit you are expected to:
 know the meaning of fallacy;
 distinguish the two major types (formal and informal) fallacies;
 be able to name the major and specific types of informal fallacies;
 understand or appreciate the importance of mastering fallacies;
 have the skill to detect or identify fallacies in the arguments of others; and
 have less tendency of committing/making fallacies yourself.
4.1. Introduction
Dear learner! in the previous unit you have studied the different types of definitions. The
knowledge you gained from that study serves you as a basis for your present unit. This is
because definitions are the best means to avoid fallacies.
The first section of the unit deals with meaning and classification of fallacies in general; and
the remaining four sections are concerned with informal fallacies in particular. In these latter
sections you will study four general types of informal fallacies each of which has several
specific types.
4.2 Fallacies in general
A fallacy is defect or error in argument that consists in something other than merely false
premises. Fallacies involve usually either a mistake in reasoning or the creation of some
illusion that makes a bad argument appear/seem good.
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Obviously wrong argument, or that which violates the rules of correct thinking plainly, is
usually not called fallacy. Fallacy (fallacious argument) is an argument, which is wrong or
bad but appears correct or good. Hence a fallacy is deceptive argument, an argument that
deceives the reader or listener, or the arguer himself.
Consider the following two arguments, for example.
1. Abebe is taller than Bekele.
Bekele is taller than Kebede.
Therefore, Kebede is the tallest of the three.
2. If it rains the street will be wet.
The street is wet
Therefore, it rained.
The first argument is wrong; and it is wrong obviously. Nobody, or very few if at all, would
be fooled into thinking it as correct. (For such plainly wrong arguments logicians use the
term paralogism instead of fallacy.)
As to the second argument, many people think at the first glance that it is correct argument;
or very few recognize that it is wrong. That is, to most people the conclusion appears to
follow necessarily given the two premises. Close investigation reveals however, that the
argument is wrong or that the conclusion does not follow necessarily from the premises. It is
such misleading or deceiving arguments that are called fallacies. The fallacy committed by
the present argument is specifically called the fallacy of affirming the consequent.
4.2.1 Classifications of Fallacies
Fallacies are classified (and also named) on the basis of the sources of their fallaciousness;
that is based on the reason why the arguments are regarded as fallacies. Accordingly,
fallacies are divided into two major types: formal and informal.
A formal fallacy is one that may be identified through mere inspection of the form or
structure of an argument. The source of fallaciousness of such an argument, or the reason
why it is regarded as fallacy, is violation of the rules of valid argument form. As a result,
formal fallacies are found only in deductive arguments that have clearly recognizable forms.
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(For the same reason formal fallacies will be studied in the context of categorical syllogism,
which is the best expression of deductive reasoning; the major concern of this unit is
informal fallacies.)
Informal fallacies are those that can be detected only through analysis of the content of the
argument. Such arguments are fallacious for reasons other than violation of the rules of
valid argument form. In other words, an argument may not violate any of the rules of
validity, and yet be fallacious. The fallacy committed by such an argument is informal
fallacy. The following argument is an example of the case:
All factories are plants
All plants are things that contain chlorophyll.
Therefore, all factories are things that contain chlorophyll.
This argument is in valid argument form; or it does not violate the rules of valid argument
form. Nevertheless the argument is invalid as it contains true premises and false conclusion.
Thus, the fallacy it commits is not formal but informal. And further analysis of the content
of the argument reveals that the problem lies in the way the world “plant” is used in the
argument. In sum, formal fallacies affect the form of arguments whereas informal fallacies
affect the content or subject matter of argument.
4.2.2. The Value of Studying Informal Fallacies
Dear learner! Could you explain why we study the informal fallacies.
Dear learner! you should required to understand the reason why people make informal
fallacies in order to understand the value of studying them. There are two explanations for
the reason why people make informal fallacies. Firstly, informal fallacies are frequently
backed by some motive on the part of the arguer to deceive the reader or listener. The arguer
may not have sufficient evidence to support a certain conclusion and as a result may attempt
to win its acceptance by resorting to a thick. The second reason is unintentional. That is, the
arguer may delude himself/ herself into thinking that she/he is presenting genuine evidence
when in fact she or he is not. Hence, people make informal fallacies because either they
want to deceive others or they are themselves deceived.
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These two reasons for making informal fallacies imply two major advantages one gains by
mastering informal fallacies. Firstly, one will easily detect or identify fallacies in the
arguments of others and therefore hardly deceived by others. Secondly, one will be sensitive
to the fallacies and therefore hardly commit them himself/herself. In other words, the
mastery of informal fallacies prevents many unnecessary blunders (foolish mistakes) in any
discussion whatsoever.
4.2.3. Classification of Informal Fallacies
Since the time of Aristotle, logicians have attempted to classify the various informal
fallacies. In all cases, the basis of the classifications is the source of fallaciousness. Thus, a
number of informal fallacies which have common characteristics (that is, the same reasons
for being regarded as fallacy) are grouped or classified under one general type of informal
fallacy. In the remaining section of this unit you will study four such general types of
informal fallacies each of which has several specific fallacies under it. The four major types
of informal fallacies you will study are: fallacies of relevance, Fallacies of weak induction;
Fallacies of presumption: and Linguistic fallacies
4.3. Fallacies of relevance
Fallacies of relevance are group of informal fallacies, which share the common
characteristics that the arguments in which they occur have premises that are logically
irrelevant to the conclusion. Yet the premises are relevant psychologically, so the conclusion
may seem to follow from the premises, even though it does not follow logically. In a good
argument the premises provide genuine evidence in support of the conclusion.
In an
argument that commits a fallacy of relevance, the connection between premises and
conclusion is emotional. To identify a fallacy of relevance, therefore, one must be able to
distinguish genuine evidence from various forms of emotional appeal.
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4.3.1 Appeal to Force (Argumentum AD Baculum: Appeal to The
“Stick")
The fallacy of appeal to force occurs whenever an arguer poses a conclusion to another
person and tells that person either implicitly or explicitly that some harm will come to him or
her if he or she does not accept the conclusion. The fallacy always involves a threat by the
arguer to the physical or psychological well-being to the listener or reader, who may be either
a single person or a group of persons. Obviously, such a threat is logically irrelevant to the
subject matter of the conclusion, so any argument base on such a procedure is fallacious.
The ad baculum fallacy often occurs when children argue with one another:
Child to playmate: “Mister Rogers” is the best show on TV; and if you don’t believe
it, I’m going to call my big brother over here and he’s going to beat you up.
But it occurs among adults as well:
Secretary to boss: I deserve a raise in salary for the coming year. After all, you
know how friendly I am with your wife, and I’m sure you wouldn’t want her to find
out what’s been going on between you and that sexpot client of yours.
The first example involves a physical threat, the second a psychological threat. While neither
threat provides any genuine evidence that the conclusion is true, both provide evidence that
someone might be injured. If the two types of evidence are confused with each other, both
arguer and listener may be deluded into thinking that the conclusion is supported by
evidence, when in fact it is not.
4.3.2. Appeal to Pity (Argumentum AD Misericordasim)
The fallacy of appeal to pity occurs whenever an arguer poses a conclusion and then attempts
to evoke pity from the reader or listener in an effort to get him or her to accept the
conclusion. Example:
Taxpayer to judge:
Your Honor, I admit that I declared thirteen children as
dependents on my tax return, even though I have only two, and I realize that this was
wrong. But if you find me guilty of tax evasion, my reputation will be ruined. I’ll
probably lose my job, my poor wife will not be able to have the operation that she
desperately needs, and my kids will starve. Surely I am not guilty.
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The conclusion of this argument is “Surely I am not guilty.” Obviously, the conclusion is not
logically relevant to the arguer’s set of pathetic circumstances, although it is psychologically
relevant. If the arguer succeeds in evoking pity from the listener or reader, the latter is liable
to exercise his or her desire to help the arguer by accepting the argument. In this way the
reader or listener may be fooled into accepting a conclusion that is not supported by any
evidence. The appeal to pity is quite common and is frequently used by students on their
instructors at exam time and by lawyers on behalf of their clients before judges and juries.
4.3.3 Appeal to The People (Argumentum AD Populum)
Nearly everyone wants to be loved, esteemed, admired, valued, recognized, and accepted by
others. The appeal to the people uses these desires to get the reader or listener to accept a
conclusion. Two approaches are involved, one of them direct, the other indirect.
The direct approach occurs when an arguer, addressing a large group of people, excites the
emotions and enthusiasm of the crowd to win acceptance for his conclusion. The objective is
to arouse a kind of mob mentality.
The direct approach is not limited to oral argumentation, of course; a similar effect can be
accomplished in writing. By employing such emotionally charged phraseology as “fighter of
communism” “champion of the free enterprise system,” and “defender of the workingman,”
polemicists can awaken the same kind of mob mentality as they would if they were speaking.
In the indirect approach the arguer directs his or her appeal not to the crowd as a whole but to
one or more individuals separately, focusing upon some aspect of their relationship to the
crowd. The indirect approach includes such specific forms as a bandwagon argument, the
appeal to vanity, and the appeal to snobbery. All the standard techniques of the advertising
industry. Here is an example of the bandwagon argument:
Of course you want to buy Zest toothpaste. Why, 90 percent of America brushes
with Zest.
The idea is that you will be left behind or left out of the group if you do not use the
product.
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The appeal to vanity often associates the product with a certain celebrity who is
admired and pursued, the idea being that you, too, will be admired and pursued if you
use it. Example:
Only the ultimate is fashion could complement the face of Christie Brinkley.
Spectrum sunglasses – for the beautiful people in the jet set.
And here is an example of the appeal to snobbery:
A rolls Royce is not for everyone. If you qualify as one of the select
few, this distinguished classic may be seen and driven at British
Motor Cars, Ltd. (By appointment only, please).
The indirect approach is also used by others besides advertisers:
Mother to child: You want to grow up and be just like wonder
Woman, don’t you? Then eat your liver and carrots.
4.3.4 Argument Against the Person (Argumentum AD Hominem)
This fallacy always involves two arguers. One of them advances (either directly or implicitly)
a certain argument, and the other then responds by directing his or her attention not to the first
person’s argument but to the first person himself. When this occurs, the second person is said
to commit an argument against the person. The argument against the person occurs in three
forms: the ad hominem abusive, the ad hominem circumstantial, and the tu quoque. In the ad
hominem abusive, the second person responds to the first person’s argument by verbally
abusing the first person. Example:
Poet Allen Ginsberg has argued in favor of abolishing censorship of pornographic
literature. But Ginsberg’s arguments are nothing but trash. Ginsberg, you know, is
a marijuana-smoking homosexual and a thoroughgoing advocate of the drug
culture.
Because Ginsberg’s being a marijuana-smoking homosexual and advocate of the drug culture
is irrelevant to whether the premises of his argument support the conclusion, this argument is
fallacious.
The ad hominem circumstantial begins the same way as the ad hominem abusive, but
instead of heaping verbal abuse on his or her opponent, the respondent attempts to discredit
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the opponent’s argument by alluding to certain circumstances that affect the opponent. By
doing so the respondent hopes to show that the opponent is predisposed to argue the way he
or she does and should therefore not be taken seriously. Here is an example:
Bill Gates has argued at length that Microsoft Corporation does not have a monopoly
on computer disc operating systems. But Gates is chief executive officer of Microsoft,
and he desperately wants to avoid antitrust action against his company. Therefore, we
should ignore Gates’s arguments.
The author of this passage ignores the substance of Gates’s argument and attempts instead to
discredit it by calling attention to certain circumstance that affect Gates- namely, the fact that
he is chief executive officer of Microsoft. The fact that Gates happens to be affected by these
circumstances, however, is irrelevant to whether his premises support a conclusion. The ad
hominem circumstantial is easy to recognize because it always takes this form: “Of course
Mr. X argues this way; just look at the circumstances that affect him.”
The tu quoque (“you too”) fallacy begins the same way as the other two varieties of the ad
hominem argument, except that the second arguer attempts to make the first appear to be
hypocritical or arguing in bad faith. The second arguer usually accomplishes this by citing
features in the life or behavior of the first arguer that conflict with the latter’s conclusion. In
effect, the second arguer says. “How dare you argue that I should stop doing X; why, you do
(or have done) X yourself. “Example:
Child to parent: Your argument that I should stop stealing candy from the corner
store is no good. You told me yourself just a week ago that you, too, stole candy
when you were a kid.
Obviously, whether the parent stole candy is irrelevant to whether the parent’s premises
support the conclusion that the child should not steal candy.
4.3.5 Accident
The fallacy of accident is committed when a general rule is applied to a specific case it was
not intended to cover. Typically, the general rule is cited (either directly or implicitly) in the
premises and then wrongly applied to the specific case mentioned in the conclusion. Because
of the “accidental” features of the specific case, the general rule does not fit. Two examples:
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Freedom of speech is a constitutionally guaranteed right. Therefore, John Q.
Radical should not be arrested for his speech that incited the riot last week.
Property should be returned to its rightful owner. That drunken sailor who is
starting a fight with his opponents at the pool table lent you his 45-caliber pistol,
and now the wants it back. Therefore, you should return it to him now.
The right of freedom of speech has its limits, as does the rule that property be returned to its
rightful owner. These rules are obviously misapplied in the above circumstances. The
arguments therefore commit the fallacy of accident.
4.3.6 Straw Man
The straw man fallacy is committed when an arguer distorts an opponent’s argument for the
purpose of more easily attacking it, demolishes the distorted argument, and then concludes
that the opponent’s real argument has been demolished. By so doing, the arguer is said to
have set up a straw man and knocked it down, only to conclude that the real man (opposing
argument) has been knocked down as well. Example:
Mr. Goldberg has argued against prayer in the public schools. Obviously Mr.
Goldberg advocates atheism. But atheism is what they use to have in Russia.
Atheism leads to the suppression of all religions and the replacement of God by
an omnipotent state. Is that what we want for this country? I hardly think so.
Clearly Mr. Goldberg’s argument is nonsense.
4.3.7 Missing the Point
This fallacy occurs when the premises of an argument support one particular conclusion, but
then a different conclusion, often vaguely related to the correct conclusion, is drawn.
Whenever one suspect that such a fallacy is being committed, he or she should be able to
identify the correct conclusion, the conclusion that the premises logically imply.
This
conclusion must be significantly different from the conclusion that is actually drawn.
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Examples:
Crimes of the theft and robbery have been increasing at an alarming rate lately.
The conclusion is obvious: we must reinstate the death penalty immediately.
Abuse of the welfare system is rampant nowadays. Our only alternative is to
abolish the system altogether.
4.3.8 Red Herring
The red herring fallacy is committed when the arguer diverts the attention of the reader or
listener by changing the subject to some totally different issue. He or she then finishes by
either drawing a conclusion about this different issue or by merely presuming that some
conclusion has been established. By so doing, the arguer purports to have won the argument.
An example:
The consumers digest reports that Sylvania light bulbs last longer than GE bulbs.
But do you realize that GE is this country’s major manufacturer of nuclear
weapons? The social cost of GE’s irresponsible behavior has been tremendous.
Among other things, we are left with thousands of tons of nuclear waste with
nowhere to put it. Obviously, the consumers digest is wrong.
4.4 Fallacies of weak induction
The fallacies of weak induction occur not because the premises are logically
irrelevant to the conclusion, as is the case with the eight fallacies of relevance, but
because the connection between premises and conclusion is not strong enough to
support the conclusion. In each of the following fallacies, the premises provide
some evidence in support of the conclusion, but the evidence is not nearly good
enough to cause a reasonable person to believe the conclusion. Like the fallacies of
relevance, however, the fallacies of weak induction often involve emotional grounds
for believing the conclusion.
4.4.1 Appeal to Unqualified Authority (Argumentum ad
Verecundiam)
The appeal to unqualified authority fallacy is a variety of the argument from authority and
occurs when the cited authority or witness is not trustworthy. There are several reasons why
an authority or witness might not be trustworthy.
50
1. The person might lack the requisite expertise,
2. Might be biased or prejudiced,
3. Might have a motive to lie or disseminate “misinformation,”
4. Might lack the requisite ability to perceive or recall. The following
example illustrate these reasons…
4.4.2. Appeal to Ignorance (Argumentum ad Ignorantiam)
When the premises of an argument state that nothing has been proved one way or the other
about something, and the conclusion then makes a definite assertion about that thing, the
argument commits an appeal to ignorance. The issue usually involves something that is
incapable of being proved or something that has not yet been proved. Example:
People have been trying for centuries to provide conclusive evidence for the
claims of astrology, and no one has ever succeeded. Therefore, we must
conclude that astrology is a lot of nonsense.
Conversely, the following argument commits the same fallacy.
People have been trying for centuries to disprove the claims of astrology,
and no one has ever succeeded. Therefore, we must conclude that the claims
of astrology are true.
There is an exception to the appeal to ignorance, which relates to courtroom procedure.
According to the law, a person is presumed innocent until proven guilty. If the prosecutor in
a criminal trial fails to prove the guilt of the defendant beyond reasonable doubt, counsel for
the defense may justifiably argue that his or her client is not guilty. Example:
Members of the jury, you have heard the prosecution present its case against
the defendant. Nothing, however, has been proved beyond a reasonable
doubt. Therefore, under the law, the defendant is not guilty.
This argument commits no fallacy because “not guilty” means, in the legal sense, that “guilt
beyond a reasonable doubt has not been proved”. The defendant may indeed have
51
committed the crime of which he or she is accused, but if the prosecutor fails to prove guilt
beyond a reasonable doubt, the defendant is considered “not guilty.”
4.4.3 Hasty Generalization (Converse Accident)
Hasty generalization is a fallacy that affects inductive generalizations. In Block 1 we saw
that an inductive generalization is an argument that draws a conclusion about all members of
a group from evidence that pertains to a selected sample. The fallacy occurs when the
sample is either not sufficient or representative of the group; that is, if the sample is either
too small or not randomly selected. Here are two examples:
Hasty generalization is otherwise called “converse accident” because it proceeds in a
direction opposite to that of accident. Whereas accident proceeds from the general to the
particular, converse accident moves from the particular to the general. The premises cite
some characteristic affecting one or more a typical instances of a certain class, and the
conclusion then applies that characteristic to all members of the class.
4.4.4 False Cause
The fallacy of false cause occurs whenever the link between premises and conclusion depends
on some imagined causal connection that probably does not exist. Whenever an argument is
suspected of committing the false cause fallacy, the reader or listener should be able to say
that the conclusion depends on the supposition that X causes Y, whereas X probably does not
cause Y at all.
4.4.5 Slippery Slope
It occurs when the conclusion of an argument rests upon an alleged chain reaction and there
is not sufficient reason to think that the chain reaction will actually take place. Here is and
example
Immediate steps should be taken to outlaw pornography once and for all.
The continued manufacture and sale of pornographic material will almost
certainly lead to an increase in sex-related crimes such as rape and incest.
This in turn will gradually erode the moral fabric of society and result in an
52
increase in crimes of all sorts. Eventually a complete disintegration of law
and order will occur, leading in the end to the total collapse of cilization
Because there is not good reason to think that the here failure to outlaw pornography will
result in all these dire consequences, this argument is fallacious.
4.4.6 Weak Analogy
This fallacy affects inductive arguments from analogy. As we saw in unit 2, an argument
from analogy is one in which the conclusion depends on the existence of an analogy, or
similarity, between two things or situations. The fallacy of weak analogy is committed
when the analogy is not strong enough to support the conclusion that is drawn. Example:
Kebede’s new car is bright blue, has leather upholstery, and gets excellent
gas mileage. Bekele’s new car is also bright blue and has leather
upholstery. Therefore, it probably gets excellent gas mileage, too.
Because the color of a car and the choice of upholstery have nothing to do with gasoline
consumption, this argument is fallacious. It is also called “jumping to conclusion”, because
the arguer rushes to a conclusion without sufficient observation or reflection.
4.5 Fallacies of Presumption
The fallacies of presumption include begging the question, complex question, false
dichotomy, and suppressed evidence. These fallacies arise not because the premises are
irrelevant to the conclusion or provided insufficient reason for believing the conclusion but
because the premises presume what they purport to prove.
4.5.1 Begging the Question (Petitio Principii)
Begging the question occurs when an arguer uses some form of phraseology that tends to
conceal the questionably true character of a premise. If the reader or listener is deceived into
thinking that the key premise is true, he or she will accept the argument as sound, when in fact
it may not be. Two requirements must be met for this fallacy to occur:
1. The argument must be valid.
2. Some form of phraseology must be used to conceal the questionably true
character of a key premise.
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The kind of phraseology used varies from argument to argument, but it often involves using
the conclusion to support the questionable premise. One way of accomplishing this is to
phrase the argument so that the premise and conclusion say the same thing in two slightly
different ways. Example:
Capital punishment is justified for the crimes of murder and kidnapping because it is
quite legitimate and appropriate that someone be put to death for having committed
such hateful and inhuman acts.
4.5.2 Complex Question
The fallacy of complex question is committed when a single question that is really two (or
more) questions is asked and a single answer is then applied to both questions. Example.
Have you stopped cheating on exams?
Where did you hide the cookies you stole?
4.5.3.
False Dichotomy (False dilemma)
The fallacy of false dichotomy (otherwise called “false bifurcation” and the “either-or
fallacy”) is committed when one premise of an argument is an “either… or … “(disjunctive)
statement that presents two alternatives as if they were jointly exhaustive (as if no third
alternative were possible).
Either you buy only American-made products or you don’t
deserve to be
called a loyal American. Yesterday you bought a new Toyota. It’s therefore
clear that you don’t deserve to be called a loyal American.
4.5.4 Suppressed Evidence
Unit 2 explained that a cogent argument is an inductive argument with good reasoning and
true premises. The requirement of true premises includes the proviso that the premises not
ignore some important piece of evidence that outweighs the presented evidence and entails a
very different conclusion. If an inductive argument does indeed ignore evidence, then the
argument commits the fallacy of suppressed evidence. Consider, for example, the following
argument:
54
Most dogs are friendly and pose no threat to people who pet them.
Therefore, it would be safe to pet the little dog that is approaching us now.
If the arguer ignores the fact that the little dog is excited and foaming at the mouth (which
suggests rabies,) then the argument commits a suppressed evidence fallacy. This fallacy is
classified as a fallacy of presumption because it works by creating the presumption that the
premises are both true and complete when in fact they are not.
4.6 Linguistic fallacies
4.6.1 Equivocation
The fallacy of equivocation occurs when the conclusion of an argument depends on the fact
that one or more words are used, either explicitly or implicitly, in two different senses in the
argument. Such arguments are either invalid or have a false premise, and in either case they
are unsound.
(a)
All contemporary philosophers should take account of recent advances in symbolic
logic.
Plato and Parmenides are contemporary philosophers. Therefore, Plato and
Parmenides should take into account the recent developments in symbolic logic.
(b)
The end of a thing is its perfection; death is the end of life; hence, death is the
perfection of life.
The first argument would be valid if we were to interpret the term ‘contemporary
philosophers’ as having the same meaning in both propositions. However, we would
normally interpret the first occurrence as meaning ‘philosophers living today’ and the
second as ‘philosophers living at the same time. Consequently, the argument is fallacious.
This particular type of equivocation is known as the fallacy of four terms.
4.6.2 Amphiboly
The fallacy of amphiboly occurs when the arguer misinterprets a statement that is ambiguous
owing to some structural defect and proceeds to draw a conclusion based on this faulty
interpretation. The original statement is usually asserted by someone other than the arguer,
and the structural defect is usually a mistake in grammar or punctuation- a missing comma, a
55
dangling modifier, an ambiguous antecedent of a pronoun, or some other careless arrangement
of words.
Because of this defect, the statement may be understood in two clearly
distinguishable ways. The arguer typically selects the unintended interpretation and proceeds
to draw a conclusion based upon it. Here are some examples:
John told Henry that he had made a mistake. If follows that
John has at least the courage to admit his own mistakes.
Professor Johnson said that he will give a lecture about heart
Failure in the biology lecture hall. It must be the case that a number of
heart failures have occurred there recently.
Two areas where cases of amphiboly cause serious problems involve contracts and wills.
The drafters of these documents often express their intentions in terms of ambiguous
statements, and alternate interpretations of these statements then lead to different
conclusion. Example:
Mrs. Hart stated in her will, “I leave my 500-carat diamond necklace and my
pet chinchilla to Alice and Theresa. “Therefore, we conclude that Alice gets
the necklace and Theresa gets the chinchilla.
Mr. James signed a contract that reads. “In exchange for painting my
house, I promise to pay David $5000 nor and give him my new Cadillac only
if he finishes the job by May 1.” Therefore, since David did not finish until
May 10, it follows that he gets neither the $5000 nor the Cadillac.
4.6.3 Composition
The fallacy of composition is committed when the conclusion of an argument depends on
the erroneous transference of an attribute from the parts of something onto the whole. In
other words, the fallacy occurs when it is argued that because the parts have a certain
attribute, it follows that the whole has that attribute too and the situation is such that the
attribute in question cannot be legitimately transferred from parts to whole. Example:
Each player on this basketball team is an excellent athlete.
Therefore, the team as a whole is excellent.
Each atom in this piece of chalk is invisible. Therefore, the piece of
chalk is invisible.
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4.6.4 Division
The fallacy of division is the exact reverse of composition. As composition goes from parts
to whole, division goes from whole to parts. The fallacy is committed when the conclusion
of an argument depends on the erroneous transference of an attribute from a whole (or a
class) onto its parts (or members). Examples:
Salt is a nonpoisonous compound. Therefore, its component elements, sodium
and chlorine, are nonpoisonous
4.7 Summary
Appeal to force: arguer threatens reader/listener.
Appeal to pity: Arguer elicits pity from reader/listener.
Appeal to the people (direct): arguer arouses mob mentality.
Appeal to the people (indirect): Arguer appeals to reader/listener’s desire for security, love,
respect, etc.
Argument against the person (abusive): arguer verbally abuses other arguer.
Argument against the person (Circumstantial): arguer presents other arguer as predisposed to
argue this way.
Argument against the person (tu quoque): arguer presents other arguer as hypocrite.
Accident: General rule is applied to a specific case it was not intended to cover.
Straw man: Arguer distorts opponent’s argument and then at tacks the distorted argument.
Missing the point: Arguer draws conclusion different from that supported by premises.
Red herring: Arguer leads reader/listener off track
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Check Your Progress Exercise 5.
Identify the fallacies of relevance, weak induction, presumption, and linguistic fallacies
committed by the following arguments. If no fallacy is committed, write “no fallacy”.
1. Either the government imposes price controls on the cost of prescription drugs, or the
pharmaceutical companies will continue to reap huge profits. Therefore, price controls
must be imposed, because we cannot tolerate these huge profits any longer.
2. Of course you should eat Wheaties. Wheaties is the breakfast of champions, you know.
3. No one has ever proved that the human fetus is not completely human. Therefore,
abortion is morally wrong.
4. California condors are rapidly disappearing. This bird is a California condor. Therefore,
this bird should disappear any minute now.
5. Ronald Reagan maintained a hard line against the Soviet bloc countries throughout his
eight years as president. Then, shortly after he left office communism began to
disappear from Eastern Europe. Obviously Reagan’s policy was affective.
6. Pope Jhon Paul II has stated that artificial insemination of women is immoral. We can
only conclude that this practice is indeed immoral.
7. Molecules are in constant random motion. The statue of Liberty is composed of
molecules. Therefore, the statue of liberty is in constant random motion.
8. Either we dismantle the pentagon entirely, or we will continue to waste trillions of
dollars on a useless military machine. We certainly can not afford to waste the money,
so we should dismantle the pentagon entirely.
9. Humanitarian groups have argued recently about the need for housing for the poor and
homeless. Unfortunately, these high-density housing projects have been trid in the past
and have failed. In no time they turn into ghettos with astronomical rates of crime and
delinquency. Chicago’s Cabrini-Green is a prime example. Clearly, these humanitarian
arguments are not what they seem.
10. A Crust of bread is better than nothing. Nothing is better than true love. Therefore, a
crust of bread is better than true love.
11. A line is composed of points. Points have no length. Therefore, a line has no length.
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12. Raising a child is like growing a tree. Sometimes violent things, such as cutting off
branches, have to be done to force the tree to grow straight. Similarly, corporal
punishment must sometimes be infected on consider to force them to develop in
properly.
13. The book of Mormon is true because it was written by Joseph smith. Joseph Smith
wrote the truth because he was divinely inspired. We know that Joseph Smith was
divinely inspired because the Book of Mormon says that he was, and the Book of
Mormon is true.
14. What goes up must come down. The price of gold has been going up for months.
Therefore, it will surely come down soon.
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Check Your Progress Exercise 5.
1. Weak analogy
2. Equivocation
3. Appeal to the people
4. Appeal to ignorance
5. Division
6. False cause
7. Appeal to unqualified authority
8. Composition
9. False dichotomy (Bifurcation)
10.Adhomineim (argument against the person) circumstantional
11. Equivocation
12.Composition
13. Straw man
14. Begging the question
15. False dichotomy (Bifurcation)
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UNIT 5
THE MEANING AND COMPONENTS OF CATEGORICAL
PROPOSITIONS
Contents
6.0. Aims and Objectives
5.1 Introduction
5.2 The Meaning of Categorical Proposition
5.3 The Components of Categorical Proposition
5.4 Quality, Quantity and Distribution
5.4.1 Attributes of Categorical Proposition
5.4.2 Distribution
5.5 Standard Form of Categorical Proposition
5.6 Translating Ordinary Statements into Standard Categorical
Propositions
5.7 Summary
5.0. Aims and Objectives
Dear learner! the very aim of this unit is to show fundamentally the basic unit of categorical
syllogism. Dear learner, at the end of this unit you will be able to:
 explain the meaning of categorical propositions.
 identify the components of categorical syllogism
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5.1. Introduction
Dear learner! in order to understand categorical arguments, we must first understand
categorical propositions. In a wide range of cases, validity depends on the relationship
between categories, sets or classes of things a categorical proposition relates. In this unit will
take a deeper look at what of categorical propositions. Finally, we will develop techniques
fro translating ordinary categorical propositions into a standard form.
5.2. The meaning of categorical proposition
A categorical proposition is a proposition or statement that relates two classes, sets, or
categories. The two classes, sets, or categories that are related by a categorical proposition
are denoted by the “subject term” and “predicate term”. A categorical proposition expresses
the relations between the subject term and the predicate term.
A categorical proposition asserts that either all or part of the class denoted by the subject
term is included in or excluded from the class denoted by the predicate term. Here are some
examples of categorical propositions:
1. all human being are rational beings
2. no plants are animals
3. some flowers are roses
4. some animals are not domestic animals
In the above example, statements (1) say that every member of the class of human beings is a
member of the class of rational beings. Statements (2) say that the classes of plants and the
class of animals have no common members. Statement (3) says that some (i.e., at least one)
member of the class of flowers is the member of the class of roses. Statement (4) says that
some (i.e., at least one) of the members of the class of animals are not the members of the
class of domestic animals.
5.3. The Components of Categorical Propositions
A standard form categorical proposition has (see section 1.5) four component parts, to wit,
quantifier, subject term, copula and predicate term.
Consider the following example:
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All philosophers are persons who are lovers of wisdom.
This standard form is analyzed as follows:
Quantifier: All
Subject term: Philosophers Copula: Are Predicate term: Persons who are lovers of wisdom.
Quantifiers:- They specify how much of the subject class is included in or excluded from
the predicate class.
The words “All,” “No,” and “some” are called quantifiers.
The
quantifier “All” and “No” asserts that the whole class of the subject term included in and
excluded from the predicate class respectively. They assert about the whole class of the
subject class. And the quantifier “some” asserts that at least one member of the subject class
is included in (if the proposition is affirmative) or excluded from (if the proposition is
negative) the predicate class. What should be noted here is that the meaning of the quantifier
“some” in formal deductive logic is restricted only to the meaning of what is given in the
context (“some” means at least one). “Some x are y” does not necessarily mean “some x are
not y” and vice versa. This is because it does not reveal the same truth-value in all respects.
For example, “some flowers are roses” is true when it is interpreted as implying “some
flowers are not roses”. Lily is a flower that is not rose. But “some roses are flowers” is
false when it is interpreted as implying “some roses are not flowers”. “Some roses are
flowers” means at least one rose is flower and the statement asserts nothing about the other
roses. There are exactly three forms of quantifiers: All, No and some.
In a standard categorical proposition the subject term comes next to the qualifier and the
predicate term next to the copula. The terms in a categorical proposition denote classes. The
terms of a standard categorical propositions must include a noun or pronoun that denotes a
class. Let us take an example: All humans are rational. Here a noun must be added in the
predicate term. Accordingly, it should be:
All humans are rational beings
Copula:- They link (or couple) the subject term with the predicate term, and consist of
either of the words “are” or “are not”. Here are some examples that have no such copulas as
“are” and “are not”:
a. All fish swim
b. All criminals should be punished
c. Everyone is happy
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The above examples are not in a standard form of a categorical proposition. To change into a
standard form you rewrite in the following way:
Example (a). Rewrite: All fish are swimmers
Example (b). Rewrite: All criminals are people who should be
Punished.
Example (c). Rewrite: All persons are happy persons
5.4. Quality, Quantity and Distribution
5.4.1 Attributes of a Categorical Proposition
 Quality- An attribute of a categorical proposition that affirms or denies class
membership. It affirms or denies whether the subject class is included in or excluded
from the predicate class.
The quality of categorical proposition is either
AFFIRMATIVE or NEGATIVE. Accordingly,
Affirmative quality: Affirms membership of the subject class
1. All S are P
2. Some S are P
Negative Quality: Denies membership of subject class
1. No S are P
2. Some S are not P.
A categorical proposition has no “qualifier”. In universal propositions- All S are P and No S
are P- the quality is determined by the quantifier. The quantifier “All” is for universal
affirmative statements and “No” for universal negative standard form of propositions. In
particular propositions: Some S are P and some S are not P- the quality is determined by the
copula. As much as the copula “are” is for affirmative proposition, “are not” is for the
negative proposition.
All S are P- affirmative statement
No S are P- negative statement
Some S are P- affirmative statement
Some S are not P- negative statement
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 Quantity- An attribute of a categorical proposition that makes a claim about the
whole or part of the member of the class denoted by the subject term. The quantity of
a categorical proposition is either Universal or particular: A categorical proposition
has “quantifiers”. “All,” “No,” and “Some” are quantifiers.
All S are P- Universal statement
No S are P- Universal statement
Some S are P- Particular statement
Some S are not P- Particular statement
5.4.2. Distribution
Distribution is an attribute of the subject and predicate terms of a categorical proposition.
The term colloquial propositions are either distributed or undistributed. A term is distributed
if and only if the proposition makes an assertion about each and every member of the class
denoted by the term. If a statement does not assert something about each and every member
of the class denoted by the term, then the term is undistributed.
Whether a term is distributed or undistributed may be represented using the following
diagrams
1.
An “A” proposition (All S are P)The subject term (S) is contained in the
P
S
Predicate class (P). The statement asserts that
each and every member of the subject class (S)
is included in the predicate class (P) and, thus,
the subject class (S) is distributed and pre
predicate class (P) is undistributed. The two
circles should overlap exactly if the subject and predicate terms were terms denoting
identical classes. But the above proposition (A proposition) does not assert that the predicate
class denotes the class that is exactly denoted by the subject class. There may be some
members of the predicate class that are out side of the subject class.
2.
An “E” proposition (No S are P)
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The S circle is completely outside the P circle
S
and the statement asserts that each and every
P
member of the subject
class is excluded from the predicate class and the whole member of the predicate class is also
excluded from the subject class. There is a claim about each and every member of both
terms. Accordingly, Both terms are distributed.
3. An ‘I’ proposition (Some S are P)
As the asterisk indicates, at least, one member of the
subject term (S) is also the member of the predicate
*
S
class (P). The statement does not make a claim whether
P
the whole member of the subject term (s) is also the
member of the predicate term (P) or the whole member
of the predicate class (P) is also the member of the
subject
class.
Accordingly,
both
forms
are
undistributed.
4. An “O” proposition (some S are hot P)
As the asterisk indicates, at least, one member of the
subject term (S) is excluded from the predicate term. It
*S
is not claimed about the whole member of the subject
P
class (S) and, thus, there members of the subject class
(S) may or may not be outside the predicate class.
(P). But the statement asserts that the whole member of the predicate class is excluded from
at least one member of the subject class (S) stated another way, the statement makes a class
about each every member of the predicate class.
Accordingly, the subject term is
undistributed but the predicate term is distributed.
Based on the above diagrams, a categorical proposition may or may not have distributed
terms depending on the kind, of proposition. A term is distributed if and only if the
proposition makes a claim about each and every member of the class that the form denotes.
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5.5. Standard forms of Categorical Propositions
There are four different standard forms of categorical propositions. We can summarize the
four standard forms as follows:
Categorical Proposition
Form
Universal Affirmative
All S are P
Universal Negative
No S are P
Particular Affirmative
Some S are P
Particular Negative
Some S are not P
The letter S stands for the subject term, the letter P stands for the predicate term. When a
categorical proposition is in a standard form the quantifier comes first, the subject term
second, the copula third, and the predicate term last. For example, the letter “S” is the
subject term and “P” the predicate term in the standard form “All S are P.” What should be
noted here is subject term does not mean, “subject” in a grammar because “subject” in a
grammar includes “quantifier” and “subject term”.
Predicate term does not mean,
“predicate” in a grammar because “predicate” in a grammar includes “copula” and “predicate
term”.
5.6. Translating Ordinary Statements into Standard Categorical
Propositions
The universal affirmatives can be expressed in various ways in ordinary English. For
example, there are many ways of saying, “all cats are mammals” in ordinary English:
Every cat is a mammal
Each cat is a mammal
Cats are mammals
Any cat is a mammal
If anything is a cat, then it is a mammal
Things are cats only if they are mammals
To put any of these statements in to a standard form, we simply write, “all cats are
mammals”.
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The universal negative can be expressed in various ways in ordinary English. Here are some
ways of expressing a standard form “no plants are animals”.
Nothing that is a plant is an animal.
All plants are non-animals (take special note)
If anything is a plant, then it is not an animal
Nothing is a plant unless it is not an animal
To put any of these statements in to a standard form, we simply write, “no plants are
animals”.
The particular negative statements also take a number of stylistic variants. The proposition
“some roses are flowers” can be expressed in each of the following ways:
There are roses that are flowers
At least one rose is a flower
There exists a rose that is a flower
Something is both rose and flower
To put any of these statements in to a standard form, we simply write, “some roses are
plants”.
Finally, the standard form of particular negative statement, “some flowers are not lily” can be
expressed as follows:
At least one flower is not a lily
Not all flowers are lily
Not every flower is a lily
Something is a flower but not a lily
There is a flower that is not a lily
To put any of these statements in to a standard form, we simply write, “some flowers are not
lily”.
Each of the four standard forms of categorical proposition commonly be designated by
letters. The affirmative propositions are represented by the letters derived from the first two
vowels of the Latin word AffIrmo (Meaning, “I affirm”)
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“A” represents the universal affirmative: All S are P.
“I” represents the particular affirmative: Some S are P.
The negative propositions are represented by the letters derived from the first two vowels of
the Latin word Nego (meaning, “I deny”)
“E” represents the Universal negative: No S are P
“O” represents the particular negative: Some S are not P.
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5.7 Summary
Proposition
Letters
Quality
Quantity
Distributed Term
All A are B
A
Affirmative
Universal
Subject term
No A are B
E
Negative
Universal
Both terms
Some A are B
I
Affirmative
Particular
None of the terms
Some A are not B
O
Negative
Particular
Predicate term
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Check Your Progress Exercise 6.
I. In the following categorical propositions identify the quantifier, subject term, copula and
predicate term
1. No persons who live in rural area are persons who appreciate deforestation.
2. Some preachers who are in tolerant of other beliefs are not television evangelists
3. Some artificial hearts are mechanisms that are prone to failure.
4. All humans over the age or 70 are elderly persons.
II. Determine whether the following statements as universal affirmative, universal negative,
particular affirmative or particular negative. If any statement is not in a standard form,
rewrite it in the standard categorical form and then make the determination.
1. No diamonds are emeralds.
2. All criminals are non-saints.
3. Each patriotic Ethiopian loves justice.
4. Only domestic animals are dogs.
5. Not every animal that can fly is a bird.
6. No people who are unlucky are happy.
7. Things are birds only if they have feathers.
8. Something is a flower but not a rose.
9. There exists a mountain that is beautiful.
10. If anything is a bad tempered person, then it is harmful.
11. Nothing is a bird unless it is not a female.
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UNIT 6
VENN DIAGRAMS, TRADITIONAL SQUARE OF
OPPOSITION, AND MODERN SQUARE OF
PROPOSITION
Contents
6.0 Aims and Objectives
6.1 Introduction
6.2 Venn Diagram
6.3 Immediate Inferences
6.3.1 Conversion
6.3.2 Obversion
6.3.3 Contraposition
6.4 Modern Square of Opposition
6.5 Summary
6.0. Aims and Objective
Dear learner! The aim of this unit is to explore thoroughly a system of logical categorical
propositions worked out by the traditional Aristotelian logic and the modern Boolean logic.
Therefore at the end of this unit you will have the knowledge and understandings of:
 The Venn diagram
 Traditional square of opposition
 Immediate inference
 Modern Square of Opposition
6.1. Introduction
Dear leaner! In the previous unit we saw statement forms, attribute of terms and meanings
of the four kinds of categorical propositions. In this unit, we will refer to the logical
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relationships between standard form of categorical propositions having the same subject and
predicate terms as corresponding statements.
6.2. Venn diagram
Dear learner, in this section, we will learn how to diagram the four basic forms of
categorical propositions. The English logician John Venn developed the system of diagrams
to represent the information a categorical propositions expresses. These diagrams have come
to be known as Venn diagrams.
A Venn diagram is an arrangement of overlapping circles in which each circle represents the
class denoted by a term in a categorical proposition. Because a categorical proposition has
exactly two terms, the Venn diagram for a single categorical proposition consists of two
overlapping circles. Each circle stands for one of the terms in the proposition. The area of
overlap of the two circles stand for those things that belong to both classes. Let’s see the
following overlapping circles.
1
S
2
3
4
P
In the above overlapping circles, the letter “S” represents the subject term and the letter “P”
represents the predicate terms of a categorical proposition. The number “1” (i.e., the area to
the left of the overlapping portion) stands for the class of “S” that are not “P”. The number
“3” (i.e., the area to the right of the overlapping portion) stands for the class of “P” that are
not “S”. The number “2” (i.e., the overlapping portion) stands for those things that stand for
both “S” and “P” class that is, those things that are both “S” and “P”. The area marked “4” is
the area outside both circles. (i.e., anything in this area is neither “S” nor “P”)
In order to represent the information expressed by a categorical proposition using Venn
diagram, we use an “X” to show that an area contains at least one object and to show that an
area is empty we shade it in. If an area does not contain an “X” and is not shaded in, we
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simply have no information about it (i.e., nothing is known about that area). Lets begin with
the universal affirmative, such as the form All S are P.
This diagram shows that S has no members that
are not P, or, in other words, the members of
class S are members of class P. Notice that this
diagram says, “if there are any “S”, then they
P
S
are the members of “P””.
The universal negative, such as the form No S are P:
This diagram shows that the area of overlap
between the two circles is empty; it says that
nothing belongs that are both ‘S’ and ‘P’; S has
S
P
no member that is P, or in other words, the
member of the class “S are excluded from the
member of the “P”
Notice that this diagram does not say that there is any S, nor does it say that there is any P.
The particular affirmative, such as the form Some S are P looks like this:
This diagram says that S and P have at least one
x
member in common. It asserts that there exists
at least one S that is a P.
S
P
The particular negative have the form Some S are not P and it looks like this
This diagram says that S has at least one
existing member that does not belong to P.
X
P
S
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According to the preceding diagrams, shading is always used to represent the content of
universal proposition (A and E). The mark “X” is always used to represent particular
statements (I and O).
Because there is no “X” in the diagram that represent universal propositions, they say
nothing about existence. Those parts that contain no marks at all indicate that something
might exist or they might be completely empty.
6.3. Traditional square of opposition
It is the Aristotelian traditions that offer the logical relations between statements having the
same subject and predicate terms.
The Aristotelian standpoint, unlike Boolean (modern) standpoint, which is neutral to
existence for the universal statements, assumes that the subject terms of A and E propositions
denote at least one existing thing.
Both traditions affirm existence for particular
propositions.
Contraries
A
F
E
F
Sub-alternation
T
Sub-alternation
I
O
T
Sub- contraries
1. Contradictory
Contradictory statements:
1. Both cannot be true
2. Both can not be false
3. If the one is true, the other must be false
4. If the one is false, the other must be true.
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A and O propositions are contradictories and
E and I propositions are also contradictories.
Example:
“All cats are animals” is a given true A proposition, based on this the
corresponding O proposition, “Some cats are not animals,” is false. The truth value of an E
and I proposition is logically undetermined.
The same reasoning works for the
corresponding E and I propositions.
Contrary
Form Aristotelian standpoint, the corresponding A and e propositions are contraries
Contrary Propositions
1. Both can not be true
2. Both can be false
3. At least the one is false.
Contraries express only partial opposition. If a is false, the corresponding E proposition is
logically undetermined and vise-versa.
Examples:
a) My car is a red color and my car is white colors are contraries. This is because
1. Both cannot be true for one thing cannot be white and black at the same time.
2. Both can be false for it might be pink in color (i.e., other colors other than white
and/ or black)
3. If my car is indeed a red color, then it cannot be white. (at east the one is false)
b) All cats are animals and No cats are animals are contraries because the first statement is
true while the other is false.
c) All animals are cats and No animals are cats are contraries because both are false
propositions.
Note that if one of the propositions is False the truth value of the other proposition is
logically undetermined. It is logically undetermined because its truth value varies depending
on the context/subject matter of the proposition.
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Sub-alternation
According to Aristotelian standpoint, if an A proposition is given true, based on the subalternation relation, the corresponding I proposition is true and if an I proposition is given
false. The corresponding A proposition is false. Analogous reasoning prevails for the subalternation relation between E and I propositions.
If an A proposition is false, the truth value of the corresponding I statement is logically
undetermined and if an I proposition is true, the truth value of the corresponding A
proposition is logically undetermined.
Similar reasoning works for the sub-alternation
relation between E and I propositions.
Sub-contrary
From Aristotelian standpoint, the corresponding I and O statements are sub-contraries.
Sub-contrary Propositions
1. both can not be false
2. both can be true
3. at least the one is true.
Sub contraries express only partial opposition. If I is true, the corresponding O statement is
logically undetermined.
Examples.
“Some dogs are collies” and “some dogs are not collies” are subcontraries.
6.4 Immediate Inference
6.4.1. Conversion
Simply interchanging it’s subject and predicate terms forms the converse of a standard form
of categorical statements. Conversion is the inference from the categorical statement to its
converse. It is valid for E and I statements, but it is not in general valid for A and O
statements.
For example,
The following arguments are valid:
1. No plants are animals. So, no animals are plants.
2. Some plants are trees. So, some trees are plants.
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The following arguments are invalid:
3. All human beings are mortals. So, All mortals are human beings.
4. Some flowers are not roses. So, Some roses are not flowers.
Arguments of example 3 and 4 move from a true premise to a false conclusion, and hence are
invalid. Each commits the fallacy of illicit conversion
Statements
Converse
A All S are P
All P are S (not in general valid)
E No S are P
No P are S (Valid)
I some S are P
Some P are S (Valid)
O some S are not P
Some P are not S (not in general valid)
6.4.2. Ob Part version
The obverse of a statement is formed by
a) Changing its quality without changing its quantity
b) Replacing the predicate term with its term complement (see below)
Note that the term complement is the word or phrase that denotes the class complement. For
instance, the term complement of “dogs” is “non-dogs” which denotes the class containing
everything that is not a dog. And the term complement of “non dogs” is simply “dogs”,
which denote a class containing everything that is not a non-dog.
Examples:
A statement:
All trees are plants.
Step 1. No trees are plants (changing the quality)
Step 2. No trees are non-plants – obverse
Let us now obverse the obverse A statement.
No trees are non-plants
Step 1. All trees are non-plants.
Step 2. All trees are plants – obverse
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An E statement:
No trees are animals
Step 1. All trees are animals
Step 2. All trees are non-animals. – Obverse
Let us now obverse the obversed E statement
All trees are non-animals
Step 1. No trees are non-animal
No trees are animals
An I statement
Some cats are animals.
Step 1. Some cats are not animal
Step 2. Some cats are not non-animals – Obverse
Let us now obverse the obversed I statement
Some cats are not non-animals
Step 1. Some cats are non-animals
Step 2. Some cats are animals
An O statement
Some Lions are not plants
Step 1. Some Lions are plants
Step 2. Some Lions are non-plants – obverse
Let us now obverse the obversed O statement
Some Lions are non-plants
Step 1. Some Lions are not non-plant
Step 2. Some Lions are not plants
Obversion is the inference from a categorical statement to its obverse.
In the above examples we saw that overted A,E, I and O statements have the same truth value
with the original statements respectively. Therefore, obversion is always valid.
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6.4.3. Contraposition
The contrapositive of a statement is formed in three ways. The one is formed by (a)
replacing the subject term with the term complement of its predicate term and
(b) replacing the predicate term with the term complement of its subject ter.
For example.
All cats are feline
Step 1. All non-feline are cats.
Step 2. All non-feline are non-cats. – contrapositive
The other way is formed by (a) switching the subject term and the predicate term (i.e.,
conversion) and (b) replacing the subject term and the predicate term with their term
complements. For exampleAll cats are feline
Step 1- All feline are cats
Step 2- All non-cats are non-feline.
Contraposition is the inference from a statement to its contrapositive. It is valid for A and O
statements. The following argument is valid:
1) All roses are flowers. So, All non-flowers are non-roses.
Let us now consider the inference from an A statement to its contrapositive. It is the third
form.
All roses are flowers.
Step 1. No roses are non-flowers [abverse of A statement]
Step 2. No non-flowers are roses [converse of Step 1]
Step 3. All non-flowers are non-roses [obverse of step 2]
Let us now proff the validity of the contrapositive of an A and an O statements:
A statement
All Lions are carnivorous animals.
All non-carnivorous animals are non-Lions contrapositive.
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Step 1. The obverse of an A contrapositive statement:
All non-carniverous animals are non-Lions.
No non-carniverous animals are Lions
Step 2. The converse of step 1.
No non-carnivorous animals are Lions.
No Lions are non-carniverous animals.
Step 3. The obverse of step 2
No Lions are non carnivorous
All Lions are carnivorous.
O statement
Some Cows are not carnivorous.
Some non-carnivorous are not non cows contra positive.
Step 1. The obverse of an O contra positive statement.
Some non-carnivorous are not non cows
Some non-carnivorous are cows
Step 2. The converse of step 1
Some non –carnivorous are cows
Some cows are not non-conversions
Step 3. The obverse of Step 3
Some cows are not non-carnivorous
Some cows are carnivorous.
Accordingly, an A and O statements have the same truth-value with their respective
contrapositives. Contraposition is not valid in general for E and I statements. For example
the following argument is invalid:
Some numbers are odd numbers
So, some non-odd numbers are non-numbers. It is invalid because its premise is true,
but its conclusion is false. It commits the fallacy of illicit contrapositive.
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To see how the four categorical statements relate to their converse, compare the following
sets of Venn diagrams
Statement form
Converse
All A and B
All B are A
A
B
B
A
No A are B
A
No B are A
B
Some A arex B
A
A
Some B are A
B
A
B
x
B
xSome A are not B
Some B are noxt A
A
A
B
B
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Having diagrammed the statements and their respective converses, we can see how they
relate. In the Diagrams for the e and I statements, are we see that they are identical to that of
their respective converses. The E statement and its converse and the I statement and its
converse are logically equivalent. Two statements are said to be logically equivalent when
they necessarily have the same truth value (i.e. the same meaning.)
Statement form
Obverse
All A are B
No A are non B
B
A
B
A
No A are B
All A are not B
A
B
A
B
Some A are B
Some A are not non B
x
x
A
B
x
Some A are not B
A
B
A
B
x
Some A are not non-B
A
B
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The above diagrams for A,E, I and O statements reveal that they are identical to the diagrams
of their obverses respectively.
Let us now see how the obverse diagrams are drawing
“No A are non-B” asserts that All member of A are outside non-B.
Because non-B
designates everything outside B, All members of A are inside B. This means that the area
where A overlaps non-B is empty. This is represented by shading the left-hand part of the
circle.
“All A are non-B” asserts that No members of A is outside non-B. This means No members
of A are inside B. Thus represented by shading the area where the two circles overlap.
“Some A are not non-B” asserts that at least one member of A is outside non-B. That is at
least one member of A is not inside non-B. Thus, at least one member of A is inside B, So
‘X’ is placed in the area where A and B overlap.
Finally, “some A are non-B” asserts that at least one member of A is inside non-B. This
means at least one member of A is outside B. This is represented by placing ‘X’ in the left
hand of the circle.
Statement form
Contrapositive
All A are B
A
All non-B are non A
B
A
B
“All non-B are non A” asserts that no member of non-B are outside non-A. This means no
member of non-B is inside A. This, in its turn, means all members of A is inside B.
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No A are B
A
No B are non-A
B
A
B
“No non B are non-A” asserts that No member of non-B are inside non-A. Then, the area
where non-B overlaps non-A is empty. This is the area outside both circles, so we shade this
area.
Some A are B
Some non-B are non-A
X
x
A
A
B
B
‘Some non-B are non-A’ asserts that something exists in the area where non-B overlaps nona. This is the area outside both circles, so place ‘X’ is this area.
Some A are not B
A X
Some non-B are not non-A.
B
X
A
B
‘Some non-B are not non-A’ asserts that at least one member of non-B is out side non-a.
This means at least one member of non-B is inside a, so we place ‘X’ is the are where non-B
overlaps A.
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The above diagrams for A and O propositions reveal that they are identical to the diagrams of
their contra positive.
6.5. Modern Square of Opposition
The modern square of opposition is an arrangement of lines that illustrate logically necessary
relations among categorical proposition. The modern square depends on the interpretations
of categorical propositions based on the logician George Boole. Boole developed an
interpretation that is neutral from existence. Unlike Aristotle, interpretation (which is found
in the coming section), it is not assumed of that members of the subject term of the universal
propositions actually exist. For both Aristotelian and Boolean interpretations, in particular
statements assume that at least one member of the subject class exists.
The diagram, which arises from the modern (or Boolean) interpretation of categorical
proposition, is represented as follows:
A
E
I
O
Let’s compare the diagram for the “A” proposition with the diagram with the O proposition
and the diagram for the E proposition with the diagram with the I proposition.
A= All S are P
S
O= Some S are not P
P
It asserts that the left-hand
Mark of the S circle is empty.
S
X
P
It asserts that the left-hand mark of
S circle is not empty
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Based on their respective assertions, we see that they are the exact opposite of each other. As
a result, their corresponding statements are said to contradict each other.
E= No S are B
I= No S are P
X
It asserts that the area where
The two circles overlap is empty.
It asserts that the area where the two
Circles overlap is not empty.
Based on their respective assertions we see that their corresponding statements contradict
each other.
Modern square of opposition, thus, represent the diagram that shows the relationships of the
mutually contradictory pairs of proposition.
Accordingly, corresponding A and O
propositions and E and I propositions are contradictories.
The statements are contradictories if they cannot both be true and they cannot both be false.
Stated another way, if one is false, the other must be true. Thus, if a certain A proposition is
true, the corresponding O proposition must be False. Similarly, if a certain E proposition is
true, the corresponding I proposition must be False.
Nevertheless, given the truth-value of A or O proposition, logic cannot determine the truthvalue of An E or I proposition. They are said to have logically undetermined truth-value.
Analogously, given the truth-value of E or I propositions, nothing can be logically
determined about the truth-value of an A or O propositions.
The modern square of opposition can provide the basis for evaluating certain arguments as
valid or invalid. Here is an example of a valid argument:
C. All doges are collics. There fore, it is false that some dogs are not collies.
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This argument is called immediate inference. This is because the conclusion is drawn from a
single premises. In such arguments, there is no transition in thought from the premise to
another and then to the conclusion.
The above argument is valid because, based on the modern square, if an A proposition is
true, then it follows necessarily that the corresponding O proposition is false.
Here is an example of invalid argument:
All human being are rational beings.
Therefore, it is true that No human being are rational binges.
So as to evaluate the above arguments using Venn diagram, one has to see whether the
information expressed by the conclusion diagram is also expressed is the premise diagram.
All dogs are collies. Therefore it is false that some dogs are not collies
The argument is the example takes the form.
All A are B. Therefore
It is false that some A are not B
Let’s draw a Venn diagram for the premise first and the conclusion last.
Premise
A
Conclusion
B
A
All A are B
B
It is false that some A are not B
The conclusion expresses what is expressed by the premise and, thus, it is valid.
All human beings are rational beings. Therefore it is true that no human beings are rational
beings.
This argument takes the form: All A are B so, it is true that no A are B.
A
B
Premise- All A are B
A
B
Conclusion- it is true that No A are B.
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In this diagram we see that the conclusion expresses that which is not given by the premise,
and, thus, it is invalid.
Finally, it should be realized that the modern square of opposition and the Venn diagram
technique for evaluating arguments are applicable regardless of whether the terms in a
particular argument denote actually existing things or not.
6.6 SUMMARY
I. Valid inferences According to Both Aristotelian and Modern Logicians
1. All S are P. So, it is false that some S are not P.
2. Some S are not P. so, it is false that all S are P.
3. No s are P. so, it is false that some S are P.
4. Some s are P. So, it is false that no S are P.
5. It is false that all S are P. So, some S are not P.
6. It is false that no S are P. so, some S are P.
7. It is false that some S are P. So, no S are P.
8. It is false that some S are not P. S, all S are P.
(Note:
The above inferences are all based on the fact that if two statements are
contradictories, then if one is true, the other must be false; and if one is false, the other must
be true.)
II. Valid inferences According to Aristotelians but Not Modern Logicians
1. All S are P. so, some S are P.
2. No S are P. So, some S are not P.
(Note:
At issue here is whether A and E
3. It is false that some S are P. So, some S statements have existential import.)
are not P.
4. It is false that some S are not P. So, some (Note: At issue here whether corresponding I
S are P.
and O statements are subsontraries.)
5. All S are P. so, it is false that no S are P.
6. No s are P. So, it is false that all S are P.
(Note: At issue here is whether corresponding
A and E statements are contraries.)
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III Conversion
A All S are P. So, all P are S. (not in general valid)
E No S are P. So, no P are S (valid)
I some S are P. so, Some P are S (Valid)
O Some S are not P. So, Some P are not S (not in general valid )
Obversion
A All S are P. So, no S are non P (Valid)
E No S are P. so, all S are non P (Valid)
I Some S are P. So, some S are not non P (valid)
O Some S are not P. So, Some S are non P (valid)
Contraposition
A All S are P. So, all non P are non S (valid)
E No S are P. So, no non P are non S (not in general valid)
I Some S are P. So, some non P are non S (not in general valid)
O Some S are non P. So, some non P are not non S (valid)
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Check Your Progress Exercises 7
I. Determine whether the following pair of statement are contradictories, contraries, sub
contraries or sub alternates ?
1. All lovers are happy people./ some lovers are not happy people.
2. Some artists are painters./ some artists are not painters.
3. All bureaucrats are mystics. /No bureaucrats are mystics.
4. All politicians are honest. /Some politicians are honest.
5. No soldiers are saints. /All soldiers are saints.
II. Which of the following are valid arguments according to Aristotelian logic? Which are
not valid according to Aristotelian logic? Which are not valid according to modern logic?
1. All vampires are dangerous creatures. So, some vampires are dangerous creatures.
2. No angels are mortal beings. Hence, some angels are not mortal beings.
3. No karate experts are wimps. Hence, it is false that all karate experts are wimps.
4. No nerds are snobs. Therefore, no snobs are nerds.
5. No doctors are astrologers. So, no non-astrologers are non-doctors
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UNIT 7
CATEGORICAL SYLLOGISMS
Contents
7.0 Aims and Objectives
7.1 Introduction
7.2 A Categorical Syllogism
7.3 Valid forms of Categorical Syllogisms
7.4 Rules and Fallacies
7.5 Summary
7.0. Aims and objectives
Drear learner! The aim of this unit is to examine syllogistic forms for establishing the
validity and invalidity of categorical syllogisms. Therefore at the end of this unit you should
be able to:
 explain categorical syllogism
 valid forms of categorical syllogism
7.0. Aims and objectives
Drear learner! The aim of this unit is to examine syllogistic forms for establishing the
validity and invalidity of categorical syllogisms. Therefore at the end of this unit you should
be able to:
 explain categorical syllogism
 valid forms of categorical syllogism
7.1. Introduction
Dear learner! in unit two, we looked into how to determine the validity or invalidity of
immediate inferences based on the logical relationships between standard forms of
categorical propositions. In this unit, we will see how to determine the mood and figure of a
categorical syllogism and also its form. Besides, we will see those syllogistic rules that may
serve as a means of identifying valid and invalid categorical syllogisms.
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7.2. A Categorical syllogism
Categorical syllogism is a syllogism consisting of three categorical propositions and
containing a total of three different terms, each of which appears twice in distinct
propositions. Let us see an example of a categorical syllogism
All humans are rational beings.
Some animals are not rational beings.
Therefore, some animals are not humans.
The three terms are: humans, rational being and animals. The term “humans” is the Major
term. The major term, by definition, is the predicate term of the conclusion. The term
“animals” is the minor term. The minor term, by definition, is the subject term of the
conclusion. The term “rational beings” is the middle term. The middle term is the one that
occurs once in each premise and does not occur in the conclusion.
The premise that contains the major term is the major premise and the premise that contains
the minor term is the minor premise. Thus, in the above categorical syllogism the major
premise is “All humans are rational being,” and the minor premise is “some animals are not
rational beings.”
A categorical syllogism is in a standard form when the following conditions are met:
1. It is build up with a standard form of categorical propositions.
2. Each term occurs twice in a distinct proposition.
3. Each term reveals the same meaning throughout the argument.
4. The major premise is listed first, the minor premise second and the conclusion
last.
Syllogisms differ according to their mood and figure. The mood of a syllogism (in a
standard form) is determined by the kinds of categorical proposition (A, E, I, O) comprising it.
Let us see the following example:
All bats are mammals
No birds are mammals.
So, No bats are birds.
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Here, the mood is EAE. You see the above argument is not in a standard form. In a standard
form the major premise must come first, the minor premise second and the conclusion last.
The figure of a syllogism (in a standard form) is determined by the position (placement) of
the middle term. There are four possible figures, which can be diagrammed as follows:
Figure 1 [First figure]
M
- P
S-
M
Figure 2 [Second Figure]
P-
M
M
S -
So, S-P
So, S-P
Figure 3 [Third Figure]
Figure 4 [Fourth Figure]
M
M
-P
-S
So, S-P
P-
M
-M
S
So, S-P
The letter within the box represents the middle term. In the first Figure, the middle term is
the subject term and predicate term of the major and minor premises respectively. In the
second figure, the middle term is the predicate term of the premises. In the third figure, the
middle term is the subject term of the premises. In the fourth Figure, the middle term is the
predicate term and the subject term of the major and minor premises respectively.
It should be noted that prior you embark on passing judgment on the mood and the figure of a
syllogism, make sure that the syllogism is in a standard form.
All bats are mammals
No birds are mammals
So, no bats are birds
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This example should be rewrite in the following form: to make it in a standard form:
No birds are mammals
All bats are mammals
So, no bats are birds
Accordingly, the mood is EAE and it is figure 2. Because the form of a syllogism is
completely specified by its mood and figure, the form is designated as EAE-4.
Validity is determined by mood and Figure.
How many possible different moods are there? Because there are four kinds of categorical
propositions (A, E, I, O] and there are three categorical propositions in a syllogism, therefore
43= 4x4x4 = 64 possible moods. And because there are four different Figures, there are
256(64x4) different kinds of categorical syllogisms.
7.3. Valid Forms of Categorical Syllogisms
Out of the different kinds of categorical syllogisms, only the following 15 forms are
unconditionally valid; they are valid from Both ancient (Aristotlean) and modern (Boolean]
stand points.
Unconditionally Valid
Figure 1
Figure 2
AAA
EAE
Figure 3
IAI
Figure 4
AEE
EAE
AEE
AII
IAI
AII
EIO
OAO
EIO
EIO
AOO
EIO
If the Aristotelian stand point is taken alone, and additional nine forms are valid if the
required condition stated below is fulfilled.
Conditionally Valid
Figure 1
Figure 2
AAI
AEO
Figure 3
AAI
EAO
EAO
EAO
Figure 4
AEO
EAO
AA I
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Note that all the forms in the additional set of nine involve an inference from two universal
premises to a particular conclusion. Accordingly, such forms are valid only if universal
affirmatives and universal negatives have existential import.
The modern logicians deny the validity of the above nine forms because it leads to the
conclusion that the corresponding A and O statements are not contradictories (and to the
conclusion that E and I statements are not contradictories).
7.4. Rules and Fallacies
Valid syllogistic forms are identified through a set of rules. Let us now see those rules.
Rule-1 The middle term must be distributed at least once.
All tigures are feline
All cats are feline
So, All cats are tigers.
Hence, the middle term, “feline,” is not distributed. Therefore, this argument commits what
is called the fallacy of undistributed middle. It is invalid because it has true premises and
false conclusion.
The distribution of the middle term matters because the middle term has to serve as a middle
ground between the other terms. In an undistributed middle term, it is possible that the major
and minor terms relate to different parts of the middle term. And this does not provide a
guaranteed link between the minor and major term.
Let “M,” “P,” and “S” designate the middle, the major and the minor terms.
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Now Let us see the above argument by drawing Venn diagram for each premise and the
conclusion.
All tigers are feline
P
M
All cats are feline
So, All cats are tigers
S
M
S
P
Because the information contained in the conclusion is not contained in the premise
diagrams, the argument is invalid.
Rule-2- A term distributed in the conclusion must be distributed in a premise.
Example:
All birds are animals
No bats are birds
So, no bats are animals.
Here, both the major and minor terms are distributed in the conclusion but only the minor
term is distributed in the premise. The argument commits the fallacy of illicit major. This is
because the major term is undistributed in the premise. It is invalid because it has true
premises and false conclusion.
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All squares are rectangles. All squares
are figures Therefore, All figures are rectangles.
Here, only the minor term is distributed in the conclusion but it is undistributed in the
premise. Therefore, the argument commits the fallacy of illicit minor. It is invalid because it
has true premises and false conclusion.
The term distributed in the conclusion must be distributed in premise for the syllogism to be
valid. Because the conclusion asserts about each and every member of the class denoted by
the term while the premise do not say something about the whole member of the class, the
conclusion “goes beyond” the information contained in the premise. The conclusion says
something, which is not warranted by the premises. Therefore, the argument is invalid.
Now let us see the above argument by drawing Venn diagram for each premise and the
M
P
conclusion.
All Square are rectangles
M
S
All Squares are figures
So, All figures are rectangle
Because the information contained in the conclusion diagram is not contained in the premise
diagrams, the argument is invalid.
Rule-3. Two negative premises are not allowed.
No fish are mammals
Some dogs are not fish
S
P
So, some dogs are not mammals
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This argument is invalid because it has true premises and false conclusion. The defect is
attributed to the fact that it has two negative premises.
In this argument because the major class is separate wholly from the middle class, and the
minor term is separate partially from the middle class, nothing is said about the relation
between the minor term and major term. Therefore, one is not guaranteed to claim with
necessity the relation between the minor and the major terms.
Now Let us see the above argument by drawing Venn diagram for each premise and the
conclusion.
No fish are mammals
M
P
Some dogs are not fish
X
S
M
So, some dogs are not mammals
X
S
P
Because the information contained in the conclusion diagram is not contained in the premise
diagrams, the argument is invalid.
The above argument commits the fallacy of exclusive premises.
Rule 4- A negative premises requires a negative conclusion, and a negative
Conclusion requires a negative premise.
No figers are wolves.
Some felines are tigers
So, some felines are wolves.
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This argument is invalid because it has true premises and false conclusion. The defect is
attributed to the fact that it draws an affirmative conclusion from a negative premise. Let us
now see these arguments using Venn diagram.
No tigers are wolves
M
P
Some felines are tigers
So, some felines are wolves
x
S
M
x
S
P
The conclusion contains information that is not contained in the premises. Therefore, it is
invalid and the argument commits the fallacy of drawing an affirmative conclusion from a
negative premise.
Rule 5 also tells us the invalidity of the following form of argument:
All collies are dogs
Some animals are collies
So, some dogs are not animals
It is plainly invalid because it has true premises and false conclusion. The argument commits
fallacy of drawing negative conclusion from affirmative premises. The conclusion asserts
class separation while the premises assert class inclusion.
Rule 4 and rule 5 can also be formulated as:
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 Any syllogism having exactly one negative statement is invalid
 The number of negative premise must be equal to the number of negative conclusion
Based on the above rules, it is possible to see the invalid form of a syllogism with two
particular premises. If both premises are particular affirmative statements, and none of the
terms are distributed in a particular affirmative statement, the argument violets rule 2. If one
of the particular statements in the syllogism is negative, then based on rule 5 a negative
conclusion will be followed. If the conclusion is, either a particular negative statement, or a
universal negative statement, then at least two (for the particular negative) and three (for the
universal negative) terms must be distributed in the premises (i.e., based on rule 2 and 3).
The premise with particular statements distributes none of the terms if they are affirmative
statements. It distributes only one term if one of its statement is negative. Thus, no valid
syllogism can have two particular premises.
Rule5. If both premises are universal, the conclusion cannot be particular
All mammals are animals
All unicorns are Mammals
Some, unicorns are animals
Based on Boolean standpoint the above argument is invalid.
According to Boolean
standpoint universal statements make no assertion about existence, while particular
statements do. Thus, the conclusion contains more information that the premises, and the
syllogism is invalid. It commits an existential fallacy. But this same argument is valid if and
only if the subject term of the universal statements denote one existing thing and the
argument breaks no other rules. Because the subject of a Universal statement denotes at least
one existing thing and the syllogism breaks no other rules, such a syllogism is valid from
Aristotelian standpoint.
From Boolean standpoint a syllogism that breaks any of the above rules is invalid. If one is
justified in adopting the Aristotelian standpoint, there is no such thing as an existential
fallacy, and the syllogism is valid if it breaks no other rules.
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Short cut method for providing unconditionally valid syllogisms:
Let us suppose we are given the following valid syllogisms:
A) With an A proposition for its conclusion.
 A Universal affirmative
 Only its subject term is distributed (i.e., the minor term is distributed.)
Based on the syllogistic rules:

The premises must be affirmative

No two particular premises are allowed

The minor term must be distributed in the premise at least once.

The middle term must be distributed at least once.
Based on the above information

The premises must distribute at least two terms.

Both premises must be universal; otherwise, the rule of distribution is
violated.

In a standard form of syllogism the major premise comes first, the minor
premise second and the conclusion last. Thus, in the first premise only middle
term is distributed and in the second premise the minor term is distributed.

Thus, it takes the form:
All M are P
All S are M
Therefore, All S are P
Accordingly, the figure is figure 1.
The mood is AAA.
The form of the syllogism is AAA-1
This is the only unconditionally valid form we have if the conclusion of
the syllogism is an A proposition.
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1. With an E proposition as a conclusion.
i. An E proposition
1. A universal negative
2. Both terms are distributed (i.e. both the minor, and the major
terms.)
Based on the syllogistic rules:
3. One of the two premises must be negative
4. No two particular premises are allowed
5. The minor term must be distribute in the premise at least once
6. Based on the above information at least three terms must be
distributed in the premises
7. Two negative premises are not allowed
The two premises must be, universal negative and, affirmative statements..
Thus it takes the form:
1. All P are M
2. No S are M
No S are M
All P are M
Therefore, No S are P
Therefore, No P are S
3. All P are M
4. No M are S
No M are S
All P are M
Therefore, No P are S
Therefore, No S are P
It should be noted that two of the above arguments are results of the operation of
conversion. The conversion of an E proposition is equivalent with the original one (they
have the same truth value.) The figures, moods and Forms of the above arguments are as
follows:
Argument 1
Figure 2
Mood AEE
Form AEE –2
Argument 2
Figure  2
Mood  EAE
Form  EAE –2
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Argument 3
Argument 4
Figure  4
Figure1
Mood  AEE
Mood  EAE
Form  AEE – 4
Form  EAE –1
There are the only unconditionally valid forms we have if the conclusion of the syllogism
is an E statement.
C) With an”I” proposition as a conclusion
- Particular affirmative
- None of the terms are distributed
- No two particular premises are allowed
- Based on the above rules no two universal affirmative premises are
allowed
(neutral of existence for the universal statements).
- The minor term must be distributed at least once.
- The premises must be all affirmative.
Based on the above information:
- The premise must distribute at least one term (i.e., at least the middle term)
- Because two particular statements are allowed, it is made of one universal and one
particular affirmative premises
Thus it takes the form:
1) All m are P
Some M are S
So, Some S are P
3) Some M are S
2. All M are P
Some S are M
So, Some S are P
4. Some S are M
All M are P
All M are P
So, Some P are S
So, Some P are S
It should be noted that two of the above arguments are results of the operation of conversion.
The conversion of an “I” proposition is equivalent with the original one. (They have the
same truth-value.) The figures, moods and forms of the above arguments are as follows:
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Argument 1
Argument 2
Fig 3
Fig 1
Mood  AII
Mood  AII
Form  AII – 3
Form  AII – 1
Argument 3
Argument 4
Fig  3
Fig  4
Mood  IAI
Mood  IAI
Form  IAI –3
Form  IAI – 4
There are the only unconditionally valid forms we have if the conclusion of the syllogism is
an “I” statement.
4. With an “O” proposition as a conclusion.
 Particular negative
 Only the predicate term (i.e., the major term) is distributed.
 The minor term must be distributed at least once.
 Based on the syllogistic rules one of the two premises must be negative
 Based on the above information two particular premises are not allowed.
 Two negative premises are not allowed
 At least two terms must be distributed
 The premises of the argument are made of one universal negative and one particular
affirmative or one universal affirmative and on one particular negative statement.
Thus it takes the form:
1) No P are M
2) No M are P
Some M are S
Some M are S
So, Some S are not P
So, Some S are not P
3) No M are P
4) No P are M
Some S are M
Some S are M
So, Some S are not P
So, Some S are not P
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It should be noted that operation conversion is applied to both e and I statements. Converted
E and I statements have the same truth value with their original statements respectively.
5) All P are M
6) Some M are not S
Some S are not M
All M are P
Therefore, some S are not P
Therefore, some P are not
Theses are the only unconditionally valid forms if the conclusion of the syllogism is an O
proposition.
Argument 1
Argument 2
Argument 3
Fig  4
Fig 3
Fig  1
Mood  EIO
Mood  EIO
Form  EIO – 4
Form  EIO – 3
Argument 4
Fig  2
Mood  AEI
Form  EIO – 2
Argument 5
Fig  2
Mood  AOO
Form  AOO – 2
Mood  EIO
Form  EIO –1
Argument 6
Fig  2
Mood  OAO
Form  OAO-2
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7.5 Summary
On the forms of the unconditionally valid syllogisms in respect to the type of the propositions
of their conclusion:
1) Universal affirmative (A statement) conclusion.
Valid Form:
a) AAA-1
2) For a universal negative (E statement) conclusion.
Valid forms:
a) AEE-2
b) EAE-2
c) AEE-4
d) EAE-1
3) For a particular affirmative (I statement) conclusion
Valid forms:
a) AII-3
b) AII-1
c) IAI-3
d) IAI-4
4) For a particular negative (O statement) conclusion.
Valid Forms:
a) EIO-4
b) EIO-3
c) EIO-1
d) EIO-2
e) AOO-2
f) OAO-3
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Check Your Progress Exercise 8.
Specify the mood and figure the following forms. Then determine the validity and invalidity
of the syllogisms. Bear in mind that the letters “M”, “S”, and “P” stands for the middle, the
subject and the predicate terms respectively.
1. No P are M. No M are S. So, no S are P.
2. All P are M. All S are M. so, all S are P.
3. All P are M. Some S are not M. So, some S are not P.
4. Some P are not M. Some S are not M. So, some S are not P.
5. Some M are not P. All S are M. so, some S are not P
6. All M are P. All S are M. so, some S are P.
7. Some M are P. Some M are S. So, some S are P.
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UNIT 8
PROPOSITIONAL LOGIC
Contents
8.0 Aims and Objectives
8.1 Introduction
8.2 Symbols and Translations
8.3 Truth Functions
8.3.1 Introduction
8.3.2 Objective
8.4 Truth Table for Propositions
8.4.1 Introduction
8.4.2 Aims and Objectives
8.5 Argument Forms Fallacies
8.5.1 Introduction
8.5.2 Objectives
8.6 Answer to Check Your Progress Exercise
8.0 Aims and Objectives
Dear leaner! the basic aim of this unit is to introduce you to prepositional logic. At the end
of the unit you are expected to:
 understand the meaning of prepositional logic and its difference from
syllogistic arguments
 understand the logical operators
 be able to symbolize prepositional and prepositional arguments and,
 be able to determine validity of prepositional arguments.
8.1. Introduction
Dear learner! In the previous units we have seen that the validity of a deductive argument is
purely a function of its form. By knowing the form of an argument, we can often tell
immediately whether it is valid or invalid. Unfortunately, however, ordinary linguistic usage
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often obscures the form of an argument. To dispel this obscurity, logic introduces various
simplifying procedures.
Letters were used to represent the terms in a syllogism, and
techniques were developed to reduce syllogisms, and techniques were developed to reduce
syllogisms to what is called standard form. In this unit, form recognition is facilitated
through the introduction of special symbols called operations, or connectives.
When
arguments are expressed in terms of these symbols, determining validity often becomes a
matter of mere visual inspection.
In the two previous units, the fundamental elements were terms. In propositional logic,
however, the fundamental elements are whole statements (or propositions). Statements are
represented by letters, and these letters are then combined by means of the operators to form
more complex symbolic representations.
8.2 Symbols and Translation
To understand the symbolic representation used in propositional logic, it is necessary to
distinguish what are called simple statements from compound statements.
A simple
statement is one that does not contain any other statement as a component. Her are some
examples.
Butane is a hydrogen compound.
James Joyce wrote Ulysses.
Parakeets are colorful birds.
The monk seal is threatened with extinction.
Any convenient upper-case letter may be selected to represent each statement. Thus B might
be selected to represent the first, J the second, P the third, and M the fourth. As will be
explained shortly, lower-case letters are reserved for use as statement variables.
A compound statement is one that contains at least one simple statement as a component.
Here are some examples:
It is not the case that Emily Bronte wrote Jane Eyre.
The Boston Symphony will perform Mozart, and the Cleveland symphony will
perform Strauss.
Either private air traffic will be restricted or mid-air collisions will continue.
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If IBM introduces a new line, then Apple will also.
The Giants will win if and only if their defense holds up.
Using letters to stand for the simple statements, these compound statements may be
represented as follows:
It is not the case that E.
B and C
Either P or M.
If I then A.
G if and only if D.
Expressions like “it is not the case that,” “and,” “or,” “if…then …,” and “if and only if” may
be translated by logical operators. These are the operators:
Operator
Name
Logical function
Used to translate

tilde
negation
not, it is not the case that

dot
conjunction
and, also, moreover

wedge
disjunction
or, unless

horseshoe
implication
if … then …, only if

triple bar
equivalence
if and only if
When we use the operators to translate the previous examples of compound statements, the
results are as follows:
It is not the case that E
E
B and C
BC
Either P or M
PM
If I then A
IA
G if and only if D
GD
The statement E is called a negation. The statement B  C is called a conjunctive statement
(or a conjunction), and the statement P  M is called a disjunctive statement (or a
disjunction); in the conjunctive statement, the components B and C are called conjuncts, and
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in the disjunctive statement the comp0onents P and M are called disjuncts. The statement I
 A is called a conditional statement, and its components are called antecedent (I) and
consequent (A). Lastly, G  D is called a biconditional statement.
Let us now use the logical operators to translate additional English statements.
Negation (~)
The tilde symbol is used to translate any negated simple proposition:
State Farm is not like a good neighbor.
S
It is not the case that State farm is like a good neighbor
S
It is false that State Farm is like a good neighbor
S
He is hardly a person of integrity = ~ H.
It is not the case that Tola has soft spots for her = ~ T.
Points worth recalling about tilde:
1.
It negates only the unit that immediately follows it.
2.
The “unit” by light of logical function is understood as or represented by a:
-
Single upper case letter that comes next to the very sign itself; for example ~P,
~J, ~K etc
-
Parenthesis. Any expression within a pair of parentheses is understood as a unit:
~(pM)
-
Brackets. Any expression inside a pair of brackets is taken to be a unit: ~[L. (T
V S)]
-
Brackets. all expressions contained and fastened or included in by braces are to
be considered as a unit.
* ~ {P[F  (J V Q)}
*~ {[M(K  P)] V Z}
*~ {W V [U  (E  B)]}
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Conjunction ()
The dot operator can translate (immediately and effectively) such conjunctive or conjoining
word or words and expressions as: and, but still, yet, although, though, even though,
nevertheless, nonetheless etc.
For instance:
1. He adores her and she worships him = H.S
2. Kebede hates even to touch it with a barge pole but she a springs
up at it. K.S
3. Maradona and Pele are world class footballers = M.P
Be on the watch!! You cannot translate the following compound statement just the way you
did the above example
4. Maradona and Pele are classmates. This statement cannot by
symbolized as = M.P
You may say why; here is the explanation: A compound statement can only be symbolized
as in example 3 above, provided statement. Let’s take again our previous example:
Eg. 3 “Maradona and Pelle are world class footballers” can be translated as
M.P; because = This compound statement is logically equivalent to:
“Maradona is a world class footballer”.
“Pelle is a world class footballer”.
On the contrary, the compound statement: “Maradona and Pelle are classmates” is not
equivalent to:
a. Maradona is a classmate.
b. Pelle is a classmate.
As you can the last two simple statements are not only equivalent to the compound statement,
but also inspid(not interesting, not attractive) and inane(silly).
Last but not least, each simple statement in a compound conjunctive statement is
named conjunct. For example:
1= (above)= H.S  S.H= they are logically equivalent
2= (above)= K.S  S.K= they are logically equivalent.
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Disjunction (V)
The wedge sign translates word/s and expressions such as:
a. Or
b. Unless
c. If not
Let’s see the application of the above three disjunctives
a. Either you train hard or you will lose the game:
b. You will lose the game unless you train hard.
c. If you don’t (do not) train hard, you will lose the game.
Note: The word either:
a. serves to introduce disjunctive statements.
b. plays an indicator-role
c. tells us before hand where we must put the parenthesis, brackets, comas. That is
why it is sometimes called a word with punctuation role and nothing more.
Sample Exercise
Example: 1. Mamo will win or his coach will get mad= M  H (please read this as M or H)
2. Either Mamo will win or his coach will get mad= M  H
3. Unless Mamo will win, his coach will get mad= M  H or (~MH)
Example 1=(above)= MVH is logically equivalent to H  M
Example 2=(above)= MVH is logically equivalent to H  M
Note: The principle of commutativity is very well applicable to disjunction as
much as it is to conjunction.
Note: Each component part of a disjunctive statement is known to go by the
name disjunction.
Implication ()
The horseshoe translates such expressions and words as: “If---then---“, “implies that”,
“entails that” etc.
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In translating conditional compound statements we must always keep in mind that
(remembering):
1. The conditional statement is composed of two parts, namely antecedent
and
consequent.
Tips about the ANECEDENT of a conditional statement:
 It follows more often than not the word “IF”
 It as well comes after the following words or expressions: In case, given that,
provided that, on the condition that, on the ground that, is sufficient condition etc.
 It always comes first. Even if, at times antecedent indicator words, such as the ones
we just touched upon above, spear in the middle of a given conditional statement, it
should be borne in mind that are arrangement must immediately be made in such a
way that the antecedent appears first and the consequence afterwards.
Tips about the consequent of a conditional statement

The consequent of any conditional statement comes after the antecedent.

The consequent is indicated by such consequent indicators as: then, only if,
implies, entails, follows that, is a necessary condition etc.
Note: In symbolizing a conditional statement the order of antecedent and consequent is sine
qua non; is indispensable, is of decisive import. Simply stated antecedent stands first
or must come first and then the consequent. That is why it is stated pat: “P  Q” is
not logically equivalent to “Q  P”.
- In symbolizing conditional statements the one principle that finds no room what so ever
here is commutatively.
The horseshoe symbol very well translates statements that are expressed in terms of
sufficient and necessary conditions. Whatever stands as sufficient condition fall right on
under the antecedent and necessary conditions? Whatever stands as sufficient condition fall
right on under the antecedent and necessary condition under the consequent?
Stated
differently, this sign helps us know quite easily what is to be taken as sufficient condition and
what as necessary. The reason is simple to be uncovered: all that, which comes before the
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horseshoe sign is automatically a sufficient condition and what comes after the horseshoe
symbol a necessary condition.
Samples of Conditional Statements
a. P  Q
b. ~P  Q
c. ~P  ~Q
d. P  ~Q
e. P  (A  Q)
f. P [Q . (T  J)]
g. ~P  {U  [T . (M  B)]}
Note: No matter how complicated the expression within parentheses, brackets or
braces appear, they still can’t get an inch beyond being a unit. Our unit in all the
above cases is the consequent of a given conditional statement.
Let’s symbolize the following statements
1. If Haile wins then it will add new zest to the already existing taste of heroism
in athletics = H  I
2. Tamy goes down the drain only if Kete fails to reprim and him in time K  T
3. Derartus is right provided that Brhane’s complaint is groundless
=BD
4. M if B = B  M
5. M only if B = M  B
6. M if not B= ~ B  M
Biconditional ()
(Material equivalence)
The “triple bar” connective translates the following words or expressions.
a. Is equivalent to
b. If and only if
c. Is both the sufficient and necessary condition for.
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Samples of Symbolized Bi conditional Statement
1. B  C
2. B  ~C
3. ~ B  ~C
4. [M.B  (T  G)]  Z
5. J  {L [M . (D  C)]}
6. (A  B)  (C . D)
Tips about Biconditional statements:

The principle of commutatively holds good as far as Material equivalence
is concerned.
Eg. B  C is equal to C  B
B  C and C  B are statements that can be conveniently concised and correctly
represented at once by material equivalence as follows: B  C
In other words (B  C). (B  C) can be translated as B  C.
Mixed compound statements
Compound or molecular statements can be formed using either one sign once or more times.
In this case symbolizing such statements is somewhat a child’s play.
But then, we face a stark problem staring us deep in our analytical eyes; while attempting to
symbolize statements I named “Mixed compound statements”. Such compound statements
are argument by the use of tow or more different connectives or symbols. Here comes the
need to know and thereby limit the:
 range of operation of each connective
 the component part of the mixed molecular statement
Which falls under the impact of a given operator. What is considered as a way out, as
regards the problem we just touched upon above, is to us parentheses, brackets and braces for
those mixed compound statements which have, in their translated form, more than two
letters.
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Take a good look at the following translated statement:
A.T  B = this statement is open to two versions.
(A.T)  B and
A. (T  B)
What do we have to rely on so that we can for sure know where to place the parentheses?
The answer to the above question has it to say that: we have hints or clues to rely on in
determining the correct placement of parenthesis. These clues are more often than presented
in the form of:

Commas

Semicolons and such words as,

Either

Both
To make it congenial for understanding, let’s translate the following mixed compound
statements.
1. Ethiopia exports coffee and Kenya exports tea, Uganda exports wheat. (E. K)  U
2. Ethiopia exports coffee, and Kenya exports tea or Uganda exports wheat. E. (K  U)
Whenever the word “BOTH” is preceded by the word “NOT” the tilde sign stays outside the
expression within parenthesis.
A. Not both Mercury and Saturn are jovial planets.
a  ~(M . S)

Whenever words “both” and “not” are separated by one or more words,
the tilde sign is given to each statement a piece.
B. Both Mercury and Saturn are not Jovial Planets.
b  ~M . ~S
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
Whenever words “not” and “either” stand one after the other with “not”
preceding and “either” following, the tilde sign should be written outside
(to the left) of the expression inside the parentheses.
Eg. C. Not either Derartu or Haile are astronauts.
C . ~(D  H)

If words “either” and “not” are separated by a word or group of words,
dispense the tilde sign a piece to each letter.
Eg. d. Either Tana or Ashenge is not a rift valley lake.
Eg. D. ~T  ~A
Note: “Not either” and “neither” are logically equivalent. Thus whatever goes
for “Not either” also goes for “Neither”
8.3 Truth Functions
8.3.1 Introduction
Have you ever asked yourself the following question? Not to worry. I will answer it for you
What is a truth function?
- A truth function is a logical operation, which deals with factors that determine the truth
value (True/False) of statements.
- A truth function is a system, which shows the determinants of the truth-value of a given
statement.
8.3.2. Objectives
This sub-heading is intended to show clearly the following objectives.
a. The truth-value of a given molecular statement is determined by the truth-values of its
atomic components.
b. As far as truth-value is concerned, what of elements determines what of the whole?
c. The truth or falsity of a compound statement is contingent to (a function of) the truth
or falsity of its atomic components.
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Key concepts
Before we take each connective, for further clarification of the foregoing definitions, we need
to deal with the following set of concepts and words.
-
Statement variable
-
Statement form
-
Substitution instance
Statement variable refers to any lower case (small) letter used to represent any statement of
our choosing, be it atomic or compound. Eg. a, b, c, d,…
Statement Form is a combination of a connective and a statement variable.
Eg. ~p, p, q, p  q etc.
Substitution instance of a given statement form points to any statement, which has the same
form as the one under discussion. This idea can further and better be grasped when we figure
on the fact that different statements can have the same statement form. Hence the different
statements we are talking about 12 g ht now are said to be substitution instances of that one
and same form.
Now let’s see how the truth-value of the compound statement/s is a function of the truthvalues of its/their component parts.
1. Negation
2. Conjunction
P
~P
T
F
F
T
N.B. This table clearly establishes the fact that the form of a
statement, exactly the way validity, determines truth
p
T
T
F
F
q
T
F
T
F
P.q
T
F
F
F
functions and validity are determined.
2. Disjunction
P
T
F
F
q
T
F
T
F
3. Implication
Pq
T
T
T
F
p
T
T
F
F
q
T
F
T
F
Pq
T
F
T
T
Bi conditional
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(Material equivalence)
P
T
T
F
F
Pq
T
F
F
T
q
T
F
T
F
Sample Exercises
Compute the truth-values of the following statements when M, Z and B are true and H, A, T
are false.
2. (M  Z)  H
to handle this question first of all remember to write the truth value of each atomic statement
right below each letter- Thus we have this:
(M  Z)  H
(T  T)  F
T
 F
F
3. = (M . Z)  (H . A)
(M . z)  (H . A)
T.T  F.F
= (T . T)  (F . F)
T  F
F
4. = ~(B  ~M)  ~Z
= ~(B  ~M)  ~Z
T
T
T
= ~(T  ~T)  ~Z
= ~(T)
 ~T
~T  ~T
F  F
T
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8.4. Truth Table for Proposition
8.4.1. Introduction
Constructing truth tables for propositions can easily be done once we understand the what- of
the following key terms and concepts:

Truth Table

Line

Number of line. Once we are clear with these concepts, we can grasp fully the nature
of truth tables.
The major objectives of this sub-heading are to show very lucidly:
a. Truth Table is a special arrangement of truth-values, which clearly
established the fact that the truth-value of a compound statement (proposition) is a
function of the truth-values of its atomic components.
Truth Table is a diagram showing how the truth-values of molecular statements is determined
by the truth values of its atomic (simple) parts.
b. A line is part and parcel of a truth table which shows one (among
many) possible combination of truth values for a given proposition.
c. “Number of line” refers to the number of all possible combinations of
truth values in a given proposition. Therefore to know the number of all possible
combinations of truth values a given compound proposition, we can rely on the
formula: L = 2n, where “L” stands for the number of lines (possible combinations
of truth values in a given proposition) and “n” stands for the number of different
atomic (simple) propositions in a given compound proposition.
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8.5 Summary
To put it in a nut-shell:
 L = 2n
 Total number of lines in a truth table is equal to 2n
 Total number of possible combinations of truth values is equal to 2n
Note: “n” stands for the number of different atomic propositions contained in a given
compound proposition.
Summarizing Chart
No. of different
The formula
simple proposition
No of lines in
truth table
1
2n
21 = 2
2
2n
22 = 4
3
2n
23 = 8
Once we are clear with the cardinal points adduced above, we can glide on to constructing
truth tables for any compound proposition.
Sample Exercises
1. (G  ~I)  I
 First: Before you jump to the task of constructing a truth table for this compound
propositon, I strongly advise you to determine the number of lines you should
draw (have) in the truth table
 Second: To determine the number of lines, help yourself to the formula L = 2n
 Third: To make use of this formula, you should know the number of different
simple proposition contained in the compound proposition (G v ~I)  I.
 Forth: Mind you, it is said that “the number of DIFFERENT simple proposition.
I capitalized the word “different” because it is the key to know the number of
atomic proposition contained in a given compound proposition. Recall that you
are not asked to tell the number of similar atomic statements. A
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 Fifth: All similar atomic statements (no matter what their number maybe) count
as one atomic statement.
 Six: If there are two identical upper case letters where one of them is negated and
the other is not, it still doesn’t matter as regards the
 number of different atomic propositions. They count as one; and not as two.
Now let’s go down to business:
a. What is the number simple but different atomic proposition compound
proposition : (G  -I)  I?
 The answer is two; namely the “G” statement and the “I” statement
b. What is the number of lines (possible combination of truth values in the above
compound proposition)?
 The answer can be found by using the formula 2 = 2n
 We already knew “n”, which is 2 +
 Thus 2 = 2n
2 = 22
2=4
(G  ~I)  I
T
T
T
T
F
F
F
T
T
F
F
F
Assigning truth value, like we did above, is a task that calls for understanding the following
points:
1. What is the number of lines? Four (4)
2. Once you know the number of lines in a given compound proposition, you are
advised to divide it by two. 42 = 2
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3. Thus, we have to assign 2 “True” or “Ts” first and two (2) False or “Fs” later on to
the “G” atomic statement. (See the table above).
4. We get here an “I” statement. Once we know how to assign truth values to “I”
statement, we can duplicate the same type of truth values to the other I (no matter
what the number of “I” statements is).
5. Go to point no 2 and you will learn that the QUOTIENT is 2 without hesitation divide
the already obtained quotient by 2. The result [which in this case is 1 (one)] will tell
you how the truth values should be written under the second atomic statement (which
in our case is “T”). In other words, you should write one (1) True and one (1) False
alternately until the end of the line below the “I” statement. (see the above table).
6. How many connectives do we have in this proposition? There: V, ~, ; the horseshoe
is major operator
Start from the tilde:
(G  ~I)  I
T
F
T
T
T
F
F
F
T
F
T
F
Then go to the disjunction.
(G  ~I)  I
T
T
F
T
T
T
T
T
F
F
F
F
F
T
T
F
T
T
F
F
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Finally compute the truth values for the horseshoe.
(G  ~I)  I
T
T
F
T
T
T
T
T
T
F
F
F
F
F
F
T
T
T
F
T
T
T
F
F
For purposes of Clarity let’s compute the truth-values of the following compound
proposition.
(A . A)  ~B
T
T T F F T
T
T T T T F
F
F F T F T
F
F F T T F
Here we have three operators:
 . (Conjunction)
 ~ (Tilde)
  (horseshoe)
The horseshoe being the Major
operator
Based on the reading, (arrangement of truth values), we have in
the final (the one in bold rectangle) section of the truth table or under the major operator,
which in this case is, we can classify statements into three categories or groups:
1. Tautologies (logically true)
2. Self-contradictory (logically false)
3. Contingent
1. A compound statement is called tautologous or logically true when it is TRUE
irrespective of the truth-values of its atomic components.
2. A compound statement is said to be tautologous if the truth values under the major
operator are all TRUE.
3. A compound statement is called self-contradictory or logically false if it is FALSE
regardless of the truth-values of its atomic components.
4. A compound statement is said to be logically false if the truth values under the major
connective or symbol or operator are all FALSE.
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5. A compound statement is said to be contingent if its truth value varies depending on the
truth values of its atomic components.
6. A compound statement is said to be contingent when the truth values under the major
operator show that at least one true and at least one false.
Therefore, we can classify our compound statement (see the truth table above) as
contingent.
Finally we can claim the veracity of the following two points:
7. Truth tables show how the truth-values of compound statements are determined by the
content (truth values) of their atomic components. Eg. All contingent statements.
8. Truth tables also show how the truth values of compound statements are determined
solely by their forms.
Eg. All tautologies & self contradictory statements
9. Truth tables also help us to compare or to relate two propositions.
Comparing statements
Comparing two propositions renders three different types of relations possible. They are:
1. Logically equivalent
2. Contradictory
3. Consistent
 Two propositions are said to be logically equivalent when they have the same truthvalues irrespective of the truth-values of their atomic components.
 Differently stated, without considering either the similarity or difference of the truth
values of their atomic component, if two propositions have the same truth value, then
we car rightly name them logically equivalent statements
 Last but not least, if the truth-values of two propositions under their major operators are
the same, these proposition are undoubtedly equivalent.
 Two propositions are said to be contradictory when their truth-values are opposites (no
matter the nature of the truth values of their atomic components).
 If the set of truth-values of two propositions under their major connectives are
opposites, the two proposition are beyond a morsel of doubt, contradictory.
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 While relating or comparing two or more propositions, we can end up in calling them
CONSISTENT if there is at least one line on which the statements turn out to be true.
 The relation of two or more propositions is named consistent when there is at least one
line, in the respective truth tables, on which the entire truth values (the pertinent ones)
happen to be TRUE.
Logically Equivalent Statements
~ ~ K
KL
T
T
T
F
T
T
F
T
T
F
F
T
F
F
F
T
F
T
T
F
T
T
T
F
F
T
F
T
F
T
T
F
Contradictory Propositions
KL
K.~L
T
T
T
T
F
F
T
T
F
F
T
T
T
F
F
T
T
F
F
F
T
F
T
F
F
F
T
F
Consistent Statements
KL
S.L
T
T
T
T
T
F
F
T
T
F
F
F
T
T
T
T
F
F
F
F
T
F
F
F
Here, there is one time (the 1st line) in each truth table where the truth-value under the major
operators of both propositions happens to be TRUE.
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Exercise I
Use truth tables to determine whether the following symbolized statements are
tautologies, self-contradictor or contingent.
1.
2.
(S  R) . (S . ~R)
(B  B)  B
T
T
F
T
T
F
T
F
T
F
T
T
T
T
F
F
F
T
T
F
T
F
Contingent
F
F
F
F
T
F
F
T
T
T
T
F
F
F
F
T
F
F
F
F
(M  P) v (P  M)
T
T
T
T
T
T
Self-contradictory (All false)
T
F
F
F
T
T
Necessarily false.
T
F
F
F
T
F
F
T
T
T
F
T
F
T
T
T
This is a proverbial case of a TAUTOLOGOUS proposition.
Do the rest for yourself
4. (H  I )  (H . ~I)
5. [(C  D) . ~C] ~  ~D
6. [X  (R  F)]  [(X R)  F]
7. [(H  N). (T  N)]  [(H  T)  N]
8. {[(G . H)  N] . [(G  N)  P]}  (H P)
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Check Your Progress Exercise
Exercise II
Use truth tables to determine whether the following pairs of symbolized statements are
logically equivalent, contradictory, or consistent.
1
.
1.
~K  L
F
T
T
T
F
T
T
F
T
F
T
T
T
F
T
F
K ~ L
T
T
F
F
F
T
T
T
F
T
F
T
T
F
T
F
They are consistent. Because on the 2nd and 3rd lines of each truth table the column under the
main operators turns out TRUE.
R  ~S
R  ~S
2.
2
T
T
F
T
T
T
T
F
F
F
F
T
F
T
T
F
T
F
T
F
F
F
T
F
F
T
F7
T
T
F
T
F
They are contradictory. There is no line in the columns under each major operator where the
truth-values are the same. Each line, under the major connective shows opposite truthvalues.
3. ~A  X
3. (X . ~A)  (~X . A)
4. G  ~H
4. (H . G)  (~G . ~H)
5. M  (K  P)
5. (K . M)  P
6. P. (R  S)
6. (S  P) . (R  P)
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86 Argument Forms Fallacies
8.6.1 Introduction
What do you know about the number and nature of argument forms and fallacies?
Broadly speaking, there are six valid and two invalid argument forms we need discuss
under this unit,
The six valid argument forms are:
1. Disjunctive syllogism (DS)
Pq
= a disjunctive premise
~P
= the 2nd premise denies one of the disjuncts of the 1st premise
q
= the conclusion asserting the remaining disjunct of the first
premise
2. Modus Ponens (MP)
P  q = A conditional premise
P
q
= The 2nd premise affirming the antecedent of the 1st premise
= The conclusion affirming the consequent of the 1st premise
3. Modus tollens (MT)
Pq
= A conditional premise
~P
= The 2nd premise denying the consequent of
~P
= The conclusion denying the antecedent of the 1st premise.
4. Pure Hypothetical syllogism (HS)
Pq
= A conditional premise
qr
= Another conditional premise
Pr
= A conclusion denying the antecedent of the 1st premise
5. Constructive Dilemma (CD)
Horns of the dilemma the conjunctive premise
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(P  q) . (r S) = A conjunctive premise consisting 2 conditional statements
P  r
= A disjunctive premise affirming the antecedents of each
qs
conditional component of the 1st premise.
= A disjunctive condition affirming the consequents of the conditionals
6. Destructive Dilemma (DD)
(p  q) . (r  s)
(~q  ~s)
~p  ~r
The two invalid argument Forms are:
1. p  q
q
The Fallacy of Affirming the Consequent (AC)
p
2. p  q
~p
The Fallacy of Denying the Antecedent (DA)
q
8.6.2. Objectives
Quintessentiaes (the most essential part) of these Arguments Forms we have got to keep
enshrined in our minds. In other words, the objective of this sub-heading is to show that:
a.
An argument form is a special arrangement of lower case letters and operators.
b.
It is not always true that, any arrangement of small letters (statement variables) and
connectives turns out to be a valid argument form
c.
If the argument form is a valid one it can always satisfy the truth table test.
d.
If the argument form is valid, and if the statement variables are uniformly replaced or
substituted by statements, it will always result in a valid argument.
e.
The order of the premises doesn’t affect the form of the argument
How to convert written Statements into symbolized argument forms and check them for
validity?
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Keep the following points in mind:
1. Use upper case letters
2. See the symbolized argument if it fits one of the aforementioned six valid argument
forms
3. P v q is equivalent to q v q
4. P is equivalent to ~~P
Check Your Progress Exercise 9.
I. Identify the forms of the following symbolized arguments
1. N  C
2.
SF
~C
F  ~L
~N
S  ~L
3. ~N
~N  T
T
4. M  ~B
~M
~B
5. ~B  ~L
G  ~B
II. Symbolize the following arguments and see if thy are Valid or not
1. If both the Yankees and the Bluejays win their next game, then the A’s will finish with
the most strikeouts if and only if the Indians finish with the most RBI’s. The Orioles will
finish with the highest batting average unless both the Yankees and the Bluejays win their
next game. The Orioles will not finish with the highest batting average. Therefore, the
A’s will finish with the most strikeouts if and only if the Indians finish with the most
RBI’s. (Y, B, A, I, O)
2. If the fact that the Tigers have the highest team batting average implies that the Royals do
not win the most games, then the White Sox will finish with the most base hits only if the
Brewers do not finish with the most stolen bases. The Tigers will have the highest team
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batting average only if the White Sox finish with the most vase hits. If the White Sox
finish with the most base hits, then the Royals will not win the most games. Therefore,
the Tigers will have the highest team batting average only if the Brewers do not finish
with the most stolen bases. (T, R, W, B)
3. If the Padres finish last only if the Giants make the playoffs, then if the Giants make the
playoffs, the Expos will finish with the most errors. The Expos will finish with the most
errors unless the Padres’ finishing last implies that the Giants make the playoffs. The
Expos will not finish with the most errors. Therefore, the padres will not finish last. (P,
G, E)
4. Either the Cubs will finish with the most homeruns or if the Dodgers lead in defense the
Phillies will have the most base hits. The Braves will not finish with the most double
plays unless the Phillies’ having the most base hits implies that the Astros do not get the
most shutouts. The Cubs will not finish with the most homeruns. In addition, it is not the
case that the Braves will not finish with the most double plays. Therefore, the Astros will
not get the most shutouts if the Dodgers lead in defense. (C, D, P, B, A,)
5. Either the Rangers will not earn the highest batting average, or the Angels will have the
most complete games unless the Twins do not lead in homeruns. The Angels will have
the most complete games if the Rangers do not earn the highest batting average. But the
Angels will not have the most complete games. Therefore, the Twins will not lead in
homeruns. (R, A, T)
6. If the Red Sox lose their fifth game, then the Orioles will move into first place. Either the
Mariners will be rained out or the Orioles will not move into first place. The Mariners
will not be rained out. Therefore, the Red Sox will not lose their fifth game. (R,O,M).
7. Either the Cardinals and the Astros will lose their next game or the Pirates and the
Dodgers will make the playoffs. If the Mets’ not getting the most base hits implies that
the Expos finish with the most homeruns, then it is not the case that both the Cardinals
and the Astros will lose their next game. The Braves will have the lowest team ERA if
the mets do not get the most base hits.
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8.7 Answers to Check Your Progress Exercises
Check Your Progress Exercise 6.
1. No people who are unlucky are happy people, universal negative
2. No criminals are saints, universal negative.
3. All patriotic Ethiopia are Ethiopians who love justice, universal affirmative.
4. All dogs are domestic animals, universal affirmative.
5. Some animals that can fly are not birds, particular negative,
6. Universal negative.
7. All birds are feathered things, universal affirmative.
8. Some flowers are not roses, particular negative.
9. Some mountains are beautiful things, particular affirmative.
10. All bad tempered persons are harmful persons, universal affirmative.
11. No birds are felines, universal negative.
Check Your Progress Exercise 7.
I. 1. Contradictories
2. Sub-contraries
3. Contraries
4. Sub alternates
5. Contraries
II. 1. Valid according to Aristotelians; not valid according to modern logicians.
2. Valid according to Aristotelians; not valid according to modern logicians.
3. Valid according to Aristotelians; not valid according to modern logicians.
4. Valid according to Aristotelians; and modern logicians
5. Not valid according to Aristotelians; not valid according to modern logicians
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Check Your Progress Exercise 8.
1. EEE. Forth figure violates Rule 4: invalid
2. AAA. Second figure violates Rule 2: Invalid
3. AOO. Second figure. Satisfies all time rules. Valid.
4. OOO. Second figure. Violates rule 4:invalid.
5. OAO. First figure. Violates rule 2: invalid
6. AAI. First figure. Violates rule 5 : invalid
7. I I I.Third figure. Violates rule 2: invalid
Check Your Progress Exercise 9.
I. 1. (MT)
4. (DS)
2. (HS)
5. (HS)
3. (MP)
II. 1. RCO
MV2O
2M
2R
Model Examination Questions
Part I. Translate the following statements into symbolic form.
1. Austria embargoes steel imports and Belgium does not develop nuclear weapons.
2. Either Canada curtails grain exports or Denmark decreases military spending.
3. Both Austria and Belgium embargo steel imports.
4. If England increases oil production, then France drops out of NATO.
5. England increases oil production if France drops out of NATO.
6. England increases oil production if and only if France drops out of NATO.
7. England increases oil production only if France drops out of NATO.
8. France’s dropping out of NATO implies that England increases oil production.
9. Austria does not embargo steel imports unless Belgium develops nuclear weapons.
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10. Canada’s curtailing grain exports entails that both Denmark decreases military spending
and England increases oil production.
11. Austria embargoes steel imports and either Belgium develops nuclear weapons or Canada
curtails grain export.
12. Either Austria embargoes steel imports and Belgium develops nuclear weapons or
Canada curtails grain exports.
13. Not both Canada and Denmark decrease military spending.
14. Canada and Denmark both do not decrease military spending.
15. Either England or France does not drop out of NATO.
16. Not either England or France drops out of NATO.
17. Neither England nor France drops out of NATO.
18. If Austria embargoes steel imports, then if Belgium develops nuclear weapons, then
Canada curtails grain exports.
19. If Austria’s embargoing steel imports implies that Belgium develops nuclear weapons,
then Canada curtails grain exports.
20. Greece subsidizes olive exports if and only if neither Hungary devalues its currency not
Israel settles with the Arabs.
21. If Jordan promotes educational reform, then either Kuwait blockades the Persian Gulf or
Lebanon kicks out foreign national.
22. Either Hungary devalues its currency or if Israel settles with the Arabs then Kuwait
blockades the Persian Gulf.
23. If either Greece subsidizes olive exports or Hungary devalues its currency, then neither
Kuwait blockades the Persian Gulf nor Lebanon kicks out foreign nationals.
24. If both Greece subsidizes olive exports and Hungary devalues its currency, then both
Kuwait and Lebanon do not blockade the Persian Gulf.
25. Jordan promotes educational reform, and Greece subsidizes olive exports or Hungary
devalues its currency.
26. Jordan promotes educational reform and Greece subsidizes olive exports, or Hungary
devalues its currency.
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27. Greece and Israel subsidize olive exports unless Hungary and Jordan value their
currency.
28. If Hungary devalues its currency, then Israel settles with the Arabs; nevertheless, it is
false that Lebanon kicks out foreign nationals.
29. Jordan promotes educational reform or Kuwait blockades the Persian Gulf but they do
not both do so.
30. Israel settles with the Arabs; however, if Jordan promotes educational reform, then either
Kuwait blockades the Persian Gulf and Lebanon kicks out foreign nationals.
31. Israel settles with the Arabs, then if Jordan promotes educational reform, then both
Kuwait blockades the Persian Gulf and Lebanon kicks out foreign nationals.
32. Kuwait’s blockading the Persian Gulf is a sufficient condition for Lebanon to kick out
foreign nationals.
33. Kuwait’s blockading the Persian Gulf is a necessary condition for Lebanon to kick out
foreign nationals.
34. Kuwait’s blockading the Persian Gulf is a sufficient and necessary condition for Lebanon
to kick out foreign nationals.
35. Austria’s embargoing steel imports is a sufficient condition for Belgium’s developing
nuclear weapons only if Greece’s subsidizing olive exports is a necessary condition for
Hungary’s devaluing its currency.
36. Both Canada’s curtailing grain exports and Denmark’s decreasing military spending is a
sufficient and necessary condition for either
Israel’s settling with the Arabs or Jordan’s promoting educational reform.
37. England increases oil production if France drops out of NATO, provided that Greece
subsidizes olive exports.
38. Austria embargoes steel imports on condition that Belgium develops nuclear weapons;
moreover, Greece and Israel subsidize olive exports only if Jordan’s promoting
educational reform is a sufficient condition for Kuwait’s not blockading the Persian Gulf.
40. If both Canada’s curtailing grain exports and Denmark’s not developing nuclear
weapons is a sufficient and necessary condition for England to increase oil production,
then neither Hungary promotes educational refo not Israel settles with the Arabs.
138
Part II. Determine whether the following statements are basically negations,
conjunctions, disjunctions, implication, or equivalences.
1. ~(A  M) . ~(C  E)
2. (G . ~P)  ~ (H  ~W)
3.
~[P . (S  K)]
4.
~(K. ~O)  (R  ~B)
5. (M . B)  ~[E  ~(C  I)]
6. ~[(P . ~R)  (~E  F)]
7. ~[(S  L) . M]  (C  N)
8. [~F  (N . U)]  ~H
9. E. [(FA)  (~G  H)]
10. ~[(X  T) . (N  F)]  (KL)
Part III. Write the following molecular statements in symbolic form, then use your
knowledge of the historical events referred to by the atomic statements to
determine the truth-value of the molecular statements.
1. It is not the case that Hitler ran the Third Reich.
2. Nixon resigned the presidency and Lincoln wrote the Gettysburg Address.
3. Caesar Governed China or Lindbergh crossed the Atlantic.
4. Hitler ran the Third Reich and Nixon did not resign the presidency.
5. Edison invented the telephone or Magellan did not sail around the world.
6. Alexander the Great conquered America if Napoleon ruled France.
7. Washington was assassinated only if Edison invented the telephone.
8. Lincoln wrote the Gettysburg Address if and only if Caesar governed China.
9. It is not the case that either Alexander the Great conquered America or Washington was
assassinated.
10. If Hitler ran the Third Reich, then either Magellan sailed around the world or Einstein
discovered aspirin.
139
11. Either Lindbergh crossed the Atlantic and Edison invented the telephone or both Nixon
resigned the presidency and it is false that Edison invented the telephone.
12. Lincoln’s having written the Gettysburg Address is a sufficient condition for Alexander
the Great’s having conquered America if and only if Washington’s being assassinated is
a necessary condition for Magellan’s having sailed around the world.
13. Both Hitler ran the Third Reich and Lindbergh crossed the Atlantic if neither Einstein
discovered aspirin not Caesar governed China.
14. It is not the case the Magellan sailed around the world unless both Nixon resigned the
presidency and Edison invented the telephone.
15. Magellan sailed around the world, and Lincoln’s having written the Gettysburg Address
implies that either Washington was assassinated or Alexander the Great conquered
America.
Part IV. Determine the truth-values of the following symbolized statements. Let A,B,
and C be true; X,Y, and Z, false. Circle your answer.
1. A . X
2. B . ~Y
3. X  ~Y
4. ~C  Z
5. B  ~Z
6. Y  ~A
7. ~X Z
8. B  Y
9. ~C  Z
10. ~(A . ~Z)
11. ~B  (YA)
12. A  ~(Z  ~Y)
13. (A. Y)  (~Z . C)
14. ~(X  ~B) . (~Y  A)
15. (Y C) . ~( B ~X)
140
16. (C  ~A) (Y  Z)
17. ~(A . ~C)  (~ X  B)
18. ~[(B  ~C) . (X  ~Z)]
19. ~[~ (X  C)  ~ (B  Z)]
20. (X  Z)  [( B  ~X) . ~(C  ~A)]
21. [(~X  Z)  (~C  B)] . [(~X . A)  (~Y . Z)]
22. ~[(A  X)  (Z  Y)]  [(~Y  B) . (Z  C)]
23. [(B . ~C)  (X . ~Y)]  ~[(Y . ~X)  (A . ~Z)]
24. ~{~[(C  ~B) .(Z  ~A)] . ~[~ (B  Y) . (~X  Z)]}
25. (Z  C)  {[(~X  B)  (C  Y)]  [(Z  X)  (~Y  Z)]}
Part V.
When possible, determine the truth-values of the following symbolized
statements. Let A and B be true, Y and Z false. P and Q have unknown
truth-value. If the truth-value of the statement cannot be determined, write
“undetermined.”
1. A  P
2. Q  Z
3. Q . Y
4. Q . A
5. P  B
6. Z  Q
7. A  P
8. P  ~P
9. (P  A)  Z
10. (P  A)  (Q  B)
11. (Q  B)  (A  Y)
12. ~(P  Y)  (Z  Q)
13. ~(Q . Y)  ~(Q  A)
14. [(Z  P)  P]  P
15. [Q(A  P)]  [(Q  B)  Y]
141
References
Hurely : A concise Introduction to Logic
142