Symbolic Logic – A Primer
Part I: Introduction
§2: Basic Concepts
SECTION 2
BASIC CONCEPTS
§2.1
§2.2
§2.3
§2.4
§2.5
§2.6
§2.7
§2.8
§2.9
§3.0
Getting Started.............................................................9
Object Language and Metalanguage.........................10
Propositions ...............................................................12
Arguments..................................................................20
Arguments and Corresponding Conditionals .............29
Valid and Invalid, A Closer Look................................32
Basic Definitions ...................................................32
Valid, Invalid, True, and False ..............................34
Two Foundational Issues ......................................38
Non-technical Proofs of Invalidity and Validity...........41
Establishing Invalidity ...........................................42
Establishing Validity ..............................................45
Wrap-up .....................................................................48
Exercises ...................................................................51
Quick Reference: Summary of Basic Concepts.........55
§2.1
GETTING STARTED
The first question you might ask in an introductory logic course is:
“What is logic?” At first, the answer is likely to seem perplexing because
answering the question requires some knowledge of logic. The key is that
logic, like other areas of knowledge, has its own specialized vocabulary.
Understanding some of logic’s basic vocabulary is necessary to answer
your question. Logic is the same as any other subject matter in this
respect – penetrating the essential vocabulary is the first, and possibly the
most important, step.
The purpose of Section 2 is to introduce and explain some of the
central concepts necessary for understanding logic. In order to get a foot
in the door, let us begin by saying that one of the main tasks of logic is the
analysis of arguments and the development of techniques for
distinguishing between good and bad arguments. Focusing on techniques
suggests a view of logic as a toolbox containing methods for assessing
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and constructing arguments. Certainly, this is a central part of logic.
However, as you progress in your study of the subject you will recognize
that logic also presents issues that go substantially beyond the notion of
logic as simply a toolbox.
An important word of caution is in order at this point. There is an
enormous difference between simply understanding a new concept and
being able to use it fluently. The basic vocabulary of logic introduced in
Section 2 is not difficult to understand. However, it is vital that you be able
to think using these terms without having to stop and ponder their
meaning. In short, they must become part of your working vocabulary, and
your use of them must be effortless and comfortable. Carefully reading
and studying Section 2 is necessary to achieve this goal, but competence
with the new vocabulary will emerge with continued use and practice.
§2.2
OBJECT LANGUAGE AND
METALANGUAGE
Logic investigates arguments, and arguments occur in language.
Because arguments occur in language, part of our task involves the study
of structures in a language. In talking about structures in language we also
make use of language. For example, books on English grammar contain
statements about English words and sentences, and those statements
may be written in English. Occasionally this may result in confusion
because statements about English are written in English. A simple
distinction is useful here and is best introduced by example.
(1)
(2)
Bison are four-legged animals.
Bison is a five-letter word.
Sentence (1) is true, but sentence (2) is false; (2) is false because bison
are animals not words and do not have any number of letters in them.
However, the intent of sentence (2) is clear. If sentence (2) is to be true,
then it must be understood to be making a statement about the name of a
kind of animal, not about an animal itself. And, that name is indeed a fiveletter word.
In order for a sentence to make a claim about an object, the object
itself cannot occur in the sentence. What does occur in the sentence is the
name of the object. If we now reconsider sentence (2) the problem is
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obvious. The first word in that sentence is the name of an animal. To
make an assertion about a word we need a name for the word. A common
convention for creating the name of a word is to enclose the word in single
quotation marks.1 Thus, the name of the first word in sentence (2) is
‘bison’. Using this convention, sentence (2) may be rewritten with the
desired results.
(3)
‘Bison’ is a five-letter word.
To construct the name of a word or phrase, enclose it in single quotation
marks. All of the following sentences are true.
(4)
(5)
(6)
‘Rose’ is a noun.
The first long word in many dictionaries is ‘aardvark’.
Texas is a very big state even though ‘Texas’ is short.
The same method is used to talk about a collection of words, a part of a
sentence, a complete sentence, or a collection of sentences – enclose the
expression to be named in single quotation marks.
(7) War and Peace contains many thousands of words even
though ‘War and Peace’ contains only three words.
(8) ‘My face in thine eye, thine in mine appears,
And true plain hearts do in the faces rest’
are words penned by John Donne.
The distinction that we are considering is frequently referred as the
difference between using and mentioning a word or phrase. A word or
phrase is used when it functions to make a claim about whatever it names
or describes. In sentence (1) the word ‘bison’ is used. A word or phrase is
mentioned when an assertion is being made regarding it. In sentence (4)
‘Rose’ is mentioned.
The distinction between using and mentioning a word or phrase is
usually clear from context, and the lack of single quotation marks, or a
similar device, is ordinarily not a problem. No one would misunderstand
the intent of sentence (2). However, when we introduce variables to
represent different objects and entertain some very abstract expressions
1
Other conventions are frequently encountered, e.g., italics, a distinctive font, underline,
bold print, and placing an expression on its own print-line to distinguish it from other
expressions.
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where precision is important, it is possible to hopelessly confuse issues if
the difference between use and mention is not clearly maintained.
In using language to talk about language, the terms ‘object
language’ and ‘metalanguage’ are frequently used to mark the distinction
between the language being discussed and the language used to conduct
the discussion. The object language is the language we are talking about
and the metalanguage is the language we use to talk about the object
language. Object language and metalanguage may be the same natural
language. In an English grammar book written in English, the object
language is English and the metalanguage is English. Although they are
the same natural language, their functions are logically distinct. Object
language and metalanguage may be different natural languages. For
example, in a textbook written to teach German to non-German speakers,
the object language is German, but the metalanguage may be English,
Spanish, or Chinese.
When we begin to study the particulars of elementary symbolic logic,
the logic itself will be presented as an artificial language. At that point, the
importance of the distinction between object language and metalanguage
will be clearer. In studying logic as a language, sometimes expressions
written in the language of logic will be used to solve problems, and at other
times the language of logic itself will be the subject of study. In that
situation, clarity regarding object language and metalanguage will be
essential.
§2.3
PROPOSITIONS
Arguments are composed of certain types of sentences called
propositions. Because arguments are intended to provide evidence for the
truth of their conclusions, not all types of sentences can function in
arguments. ‘Open the door’ is not a sentence one would expect to find
playing a role in an argument because sentences such as commands,
requests, and questions are neither true nor false. The sentences of
importance in arguments are those that are either true or false.
A proposition is a sentence (or a fragment of a sentence able to
function as a sentence), which is understood to have a fixed meaning and
is either true or false; in addition, the truth or falsity of a proposition holds
at all times and is independent of the proposition’s author. Admittedly, the
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§2: Basic Concepts
last characteristic is not abundantly clear; however, examples will clarify
the definition.
Suppose that George W. Bush in a news conference on July 4, 2008
stated “I am the President of the United Sates of America”. Given the
context that is provided, this statement is clearly true. But, the sentence as
quoted does not meet the conditions for being a proposition. If a
proposition is true we want it to be true no matter who states it. The
example is not true if anyone other than George W. Bush makes the
statement. Although the context makes it unnecessary to rewrite the
statement in question, it can easily be restated so that its truth does not
depend on specific person stating it, viz., ‘George W. Bush is the President
of the United Sates of America’. But this new version still fails as a
proposition because its truth is dependent on the time at which it is said.
When stated on July 4, 2008, it is true; however, if stated on March 4, 2009
it will be false. Earlier we stated that a true proposition should be true no
matter who states it; now we want to add that it should be true at any time
whatsoever. True propositions are, in this sense, timeless. This problem
can also be addressed by a simple restatement: ‘George W. Bush is the
President of the United Sates of America on July 4, 2008.’
Sentences in arguments routinely fail to meet the stringent definition
of proposition. However, it is rarely necessary to rewrite sentences to
make their truth conditions explicit because sentences usually occur in a
context we recognize and automatically incorporate into our understanding
of the sentence. On rare occasions, explicit re-writing may be necessary.
For example, suppose a complex discussion involving a number of people;
someone says “I am a Chief Financial Officer”, later someone else says “I
am not a Chief Financial Officer”. If there is enough noise in the
discussion and you lose track of who said what, then there is a chance of
misinterpreting the statements and concluding that they are inconsistent.
They obviously are not inconsistent unless said by the same person.
Propositions can be divided into three mutually exclusive and
exhaustive groups.
•
Analytic propositions (or logically true propositions)
•
Inconsistent propositions (or logically false propositions)
•
Synthetic propositions (or contingent prepositions)
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An analytic proposition is a proposition that is necessarily true, and
its truth depends only on the meanings of the words in the proposition.
The truth of an analytic proposition is not established by gathering data,
making observations, or conducting scientific experiments. All of the
following propositions are analytic.
(1)
(2)
(3)
(4)
All roses are roses.
The litmus paper is red or the litmus paper is not red.
All zithers are musical instruments.
England is north of Malta or England is not north of Malta.
An inconsistent proposition is a proposition that is necessarily false,
and its falsity depends only on the meaning of the words in the proposition.
An inconsistent proposition can also be defined as the negation of an
analytic proposition. All of the following are inconsistent propositions
(5)
(6)
(7)
No mammals are mammals.
The Mississippi river is the longest river in the United
States and the Mississippi river is not the longest river in
the United States.
All yawls are sloops.
A synthetic proposition is a proposition that is neither necessarily
true nor necessarily false. i.e., it is neither analytic nor inconsistent.
Synthetic propositions are sometimes called ‘contingent’ because their
truth depends on empirical circumstances. Making observations and
collecting data are methods relevant to establishing the truth or falsity of
synthetic propositions. The following are all examples of synthetic
propositions.
(8)
(9)
(10)
(11)
All dogs are friendly.
There are more dairy cows in Vermont than people.
Flowers are found in many homes.
Turnips are nourishing.
One additional frequently used term requires definition. Some
propositions are ‘possible propositions’ or ‘logically possible propositions’.
All true propositions are, of course, possible propositions. But some, false
propositions are also possible. For example, ‘Hudson is the capital of New
York’ is false, but it might have been true and in that sense the proposition
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is correctly labeled as ‘possible’. A possible proposition (or logically
possible proposition) is a proposition that is either analytic or synthetic.
This is of course equivalent to defining a logically possible proposition as
any proposition that is not inconsistent.
CHALLENGE
Analytic propositions are of central interest in logic. However,
logic is not simply a listing of specific analytic propositions. Logic,
like physics, is a general science. You will not find the statement ‘A
six ounce apple dropped from the 25th floor of the Empire State
Building on April 23rd, 1998 accelerates at 32 feet per second per
second’ listed in a physics book as a law of physics. It is too specific.
Physics is interested in a general rule covering all free-falling bodies.
In a similar manner, the logician is not particularly concerned with
individual analytic propositions, but with investigating general laws
regarding analytic propositions.
Before reading beyond this point, consider the problem of how
logic can attain the desired generality. In approaching this question,
it will be instructive to review some of your own answers to the
questions posed in §1 Beginning. Properly done your answer will
apply to both propositions and arguments.
Generality is achieved in logic by introducing the concept of form or
structure. Propositions (2) and (4) are clearly different, yet they have a
good deal in common. What they have in common is their form or
structure; they differ in content. One of the central tasks of logic is to
devise methods to reveal the form of propositions. No simple definition of
‘form’ will make the concept precise. In part, this is because there are
different ways of analyzing the form of a proposition. 2 Replacing a
proposition’s content words with variables and retaining the logical form
words will frequently reveal the logical framework of a proposition. Logical
form words shape the skeleton of propositions. Expressions such as ‘all’,
2
Notice that this uses the expression ‘the form of a proposition’. It is virtually impossible
to consistently avoid using the phrase ‘…the form of...’; however, this expression is
misleading in its singularity. The notion that all propositions have a single, unique form
is not defensible.
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‘none’, ‘some’, ‘most’, ‘if...then’, ‘and’, ‘but’, ‘only if’, ‘if and only if’, ‘or’,
‘unless’, ‘provided that’, ‘neither…nor’, and ‘not’ are a few examples of
logical form language found in propositions.
Replacing the content words and retaining the logical form words in
propositions (1) through (4) yields the following structures.3
(1.1)
(1.2)
(1.3)
(1.4)
All X are X
p or not p
All X are Z
q or not q
The letters ‘X’, ‘Z’, ‘p’, and ‘q’ in the above examples are variables.
Variables are of fundamental importance in achieving generality, and they
are best thought of as place-holders. Variables indicate a position within a
structure where appropriate expressions may be substituted. In our
examples, ‘X’, and ‘Z’ may be replaced by class names such as ‘apples’,
‘men over 6 feet tall’, and ‘positive integers’. While ‘X’ and ‘Z in (1.1) and
(1.3)’ cannot represent complete sentences, the occurrences of ‘p’ and ‘q’
in (1.2) and (1.4) can be replaced by complete sentences, but not by class
names. It is important that you are not misled by the apparently literal
nature of a variable. The form of ‘All logicians are fascinating people’, may
be represented by ‘All L are F’, ‘All Z are X’, ‘All D are B’, ‘All F are L’, and
‘All ☼ are ☺’. All of these represent the same logical form.
(1.1) through (4.1) are the forms of propositions (1) through (4), and
the forms succeed in showing that (2) and (4) have the same logical form.
Although (1.1) through (4.1) are not propositions, propositions may be
obtained from them by uniformly substituting appropriate linguistic
3
(1.1) and (1.3) are easily created by replacing words in (1) and (3) with variables, the
situation is more complex with regard to the relation of (2) and (4) to (2.1) and (4.1). In
these cases, we did more than replace a string of words with a variable. The problem
revolves around the word ‘not’. In symbolizing ‘the litmus paper is not red’ the ‘not’ was
removed and placed in front of the variable, i.e., ‘not p’. We have taken ‘the litmus paper
is not red’ to be equivalent to ‘it is false that the litmus paper is red’, and replaced ‘it is
false that’ with ‘not’. English does not have simple and invariable rules for the placement
of ‘not’. For example, consider how to express the negation of ‘all that glitters is gold’.
You may recall that we said logic would be presented as an artificial language. It is
precisely to avoid the vagaries and ambiguities of English (or any other natural
language) that logic is developed as a very simple and very precise artificial language.
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expressions for the variables. Expressions such as (1.1) through (4.1) are
propositional forms; a propositional form is the form of a proposition.
When is it correct to say that a proposition has a specific form? A
proposition exhibits a given form if that proposition can be obtained from
the form by uniformly substituting appropriate linguistic expressions in
place of the form’s variables; the resulting proposition is a substitution
instance of the propositional form. All propositions may be viewed as a
substitution instance of at least one propositional form. A proposition is a
substitution instance of a given propositional form if and only if it can be
obtained by uniform substitution from that form.
‘All cows are ducks’ and ‘All aardvarks are animals’ are both
substitution instances of propositional form (1.3). They are not substitution
instances of propositional form (1.1) because they cannot be obtained from
it by uniform substitution. Uniform substitution requires that when an
expression is substituted for a specific variable in a form, it must be
substituted for all occurrences of that variable in that form. We cannot
substitute different linguistic expressions for different occurrences of the
same variable. However, we can substitute the same linguistic expression
for different variables. For example, ‘All roses are roses’ can be obtained
by uniformly substituting into ‘All X are Y’ or ‘All X are X’.
Propositions (1) and (3) above are both analytic, although with an
important difference. It is not necessary to know what ‘rose’ means in
order to know that proposition (1) is true. It is only necessary to
understand ‘All...are...’ and to note that the subject and predicate terms in
the proposition are identical. For example, it is not necessary to know
what ‘yataghan’ means in order to recognize that ‘All yataghans are
yataghans’ is true.4 The truth of proposition (1) is a function of its logical
form words alone. The same is also true of propositions (2) and (4). It is
not necessary to know what litmus paper is, or where England and Malta
are located; it is only necessary to know how ‘or’ and ‘not’ function.
Propositions of this type are formally analytic propositions. A formally
analytic proposition is a proposition that is necessarily true simply by virtue
of its logical form words.
Unlike proposition (1), recognizing the analyticity of proposition (3)
requires knowing what ‘zither’ and ‘musical instruments’ mean in addition
to understanding the logical form words. Propositions of this type are
semantically analytic propositions. A semantically analytic proposition is a
4
Provided, of course, that ‘yataghan’ is a meaningful English word.
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proposition that is necessarily true by virtue of the meanings of its content
and logical form words.
We have already observed that logic is a general discipline, and
generality is achieved by concentrating on propositional forms rather than
individual propositions. Therefore, we will extend our notion of formal
analyticity to propositional forms. Propositional forms (1.1), (1.2) and (1.4)
are analytic propositional forms. A propositional form is an analytic
propositional form if and only if it is the form of a formally analytic
proposition.
There is another way to distinguish formally and semantically
analytic propositions, which will further clarify the relationship between
these types of proposition and their corresponding forms. Each of the
following propositions has the same form as proposition (2) and all of them
are analytic.
(12)
(13)
(14)
Everest is taller than K-2 or Everest is not taller than
K-2.
Spenser wrote The Faerie Queene or Spenser did not write
The Faerie Queene.
Montpelier is the capital of Vermont or Montpelier is not the
capital of Vermont.
Contrast the above three propositions with the following four propositions.
Each of the four has the same form as proposition (3), and proposition (3)
is semantically analytic.
(15)
(16)
(17)
(18)
All triangles are triangles.
All red apples are colored objects.
All umpires are men.
All peonies are fragrant.
Observe that (15) and (16) are analytic, but (17) and (18) are not. If a
proposition is formally analytic then all propositions of that form are
analytic, but this is not true of semantically analytic propositions.
A proposition is formally analytic if and only if every proposition of
the same form is analytic. A proposition is semantically analytic if and only
if it is analytic and not all propositions of that form are analytic.
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If a proposition is formally analytic, then all propositions of that form
are analytic, and if a group of formally analytic propositions all have the
same form, then they are substitution instances of that form. We now have
another, and perhaps clearer, way to define an analytic propositional form.
A propositional form is analytic if and only if all substitution instances of
that form are analytic.5
Following is a summary of some, but not all, of the central points
regarding propositions.
• A proposition is sentence (or a fragment of a sentence able
to function as a sentence), which is understood to have a
fixed meaning and is either true or false. The truth or
falsity of a proposition holds at all times and is independent
of the proposition’s author.
• A propositional form is the form of a proposition.
• A proposition is a substitution instance of a propositional
form if and only if it can be obtained by uniform substitution
from that form.
•••••
• An analytic proposition is a proposition that is necessarily
true.
• An inconsistent proposition is a proposition that is
necessarily false.
• A synthetic proposition is a proposition that is neither
necessarily true nor necessarily false.
•••••
• A proposition is formally analytic if and only if every
proposition of the same form is analytic.
• A proposition is semantically analytic if and only if it is
analytic and not all propositions of that form are analytic.
5
The concept of analyticity is considerably more intricate than has been suggested here.
Many questions have not been addressed. For example: Are all semantically analytic
propositions reducible to formally analytic propositions? What makes an analytic
proposition analytic? What is the fundamental nature of analyticity? Is analyticity a
viable concept? Philosophers have considered a number of theories to explain the
nature of analyticity and at present there is no generally agreed upon answer to these
fundamental questions.
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•••••
• A propositional form is analytic if and only if all substitution
instances of that form are analytic.
If you look carefully at the summary of major points and the more
detailed information preceding that summary, you may notice that there
are some interesting concepts and relationships that were not explicitly
defined. Before moving on to the section on arguments, try to answer the
questions in the challenge box.
CHALLENGE
1. Is there an inconsistent propositional form? If so, provide a
definition.
2. Is there a synthetic propositional form? If so, provide a
definition.
3. If you defined ‘inconsistent propositional form’, what is the
nature of the relationship of the form and its substitution
instances?
4. If you defined ‘synthetic propositional form’, what is the nature
of the relationship of the form and its substitution instances?
5. A distinction was made between formally analytic propositions
and semantically analytic propositions. Can a similar
distinction be applied to inconsistent propositions?
In this section we have developed one major dimension of the
subject matter of logic – logic is the study of analytic propositions and
analytic propositional forms. This undertaking has two important aspects.
First, logic formulates general procedures for identifying analytic
propositions and analytic propositional forms. Second, logic devises
techniques for the construction of analytic propositions and analytic
propositional forms.
§2.4
ARGUMENTS
An argument is an organization of propositions making a distinctive
claim. Specifically, in an argument the claim is made that some of the
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propositions in the argument provide evidence for the truth of another
proposition in the argument. The propositions in an argument fall into two
functionally different categories. The proposition(s) providing the evidence
are the premise(s) of the argument; the proposition supported by the
premises is the conclusion of the argument. No proposition by itself is
either a premise or a conclusion. Within an argument, a proposition may
function as a premise (or conclusion) only in relation to another that
functions as conclusion (or premise). And, of course, the same proposition
may be a premise in one argument and a conclusion in another argument.
There are two important characteristics of an argument. First, as
mentioned above, in an argument the claim is made that the truth of the
premises provides evidence for the truth of the conclusion. Second, the
premises in an argument are asserted as true. An example will help
explain the last point.
(1)
(2)
If the fuel injectors in John’s car are blocked, then his car
will not run.
The fuel injectors in John’s car are blocked. Therefore, his
car will not run.
Number (2) is an argument. In number (2) the proposition ‘The fuel
injectors in John’s car are blocked’ is asserted. No conditions are
attached; it is simply stated as true. This is not the case in number (1)
where no premise is asserted and no conclusion is drawn. Number (1) is a
single conditional sentence, and although it may be true, it does not follow
that the individual sentences that make up the ‘if…then’ conditional are
themselves true. Number (1) does not assert the proposition ‘The fuel
injectors in John’s car are blocked’. What is put forth as true in number (1)
is the single conditional sentence: ‘If the fuel injectors in John’s car are
blocked, then his car will not run’.
In short, arguments are to be distinguished from conditional
propositions.6 To say that an argument asserts its premises are true does
not mean that they are true, or are necessarily believed to be true by
whoever puts forth the argument. It does mean that in an argument the
premises are simply declared as true, and their truth is presented as
evidence for the truth of the conclusion. This does not prohibit a
conditional proposition from being a premise or a conclusion in an
6
Although arguments and conditional propositions are not the same, it is clear that there
is an important relationship between them. This relationship will be explained in section
2.5
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argument. For example, following is an argument whose first premise
happens to be a conditional proposition.
(3)
If the minimum airspeed at which the wing of the Skycar
generates lift is 85 miles per hour, then at airspeeds below
85 miles per hour the wing will stall. On its test flight, the
Skycar flew successfully at 75 miles per hour. Therefore,
the stall speed of the Skycar’s wing is not 85 miles per
hour.
Logic develops techniques for evaluating arguments. This
presupposes the ability to correctly identify an argument and differentiate
premises and conclusion. Although this sounds relatively simple, it is not.
The technical aspects of logic are, in a sense, easy to understand because
they are always precise and frequently mechanical. The identification of
arguments, premises, and conclusions is not a mechanical process. The
successful identification and analysis of arguments requires interpreting
language with a reasonable amount of sensitivity. There are no hard and
fast rules here, no methods to guarantee success, and no substitute for
practice.
Some logical form words are helpful in disentangling the premises
and conclusions in arguments. For example, ‘therefore’, ‘hence’,
‘consequently’, ‘so’, ‘it follows that’, ‘it must be true that’ and ‘this implies
that’ are some common words and phrases that may introduce the
conclusion in an argument. ‘Since’, ‘for’, ‘because’, and ‘for these reasons’
may introduce premises. However, the presence of these logical form
words does not necessarily mean that there is an argument present. For
example the following is not an argument.
(4)
Steve gave his wife flowers because she was sick.
It is not an argument because the truth of ‘she was sick’ is not offered as
evidence for the truth of ‘Steve gave his wife flowers’. In this case, the
word ‘because’ does not introduce a premise; it introduces an explanation
for why Steve gave his wife flowers.
In the arguments we have used so far, the premises were stated first
and the conclusion last. But, premises and conclusions may occur in any
order; the conclusion may be first, last, or in the middle of an argument.
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Careful reading and attention to context is the only way to identify the
premises and conclusion in an argument.
Arguments are generally divided into two classes: deductive and
inductive.7 This distinction reflects the relative strength of the evidence an
argument claims to exist between its premises and conclusion. A
deductive argument claims that its premises provide conclusive evidence
for its conclusion, ‘conclusive’ in the sense that it is not logically possible
for the premises to be true and the conclusion false. On the other hand,
an inductive argument puts forth a weaker claim. An inductive argument
alleges that its premises provide sufficient evidence for rendering the
conclusion probable for the purposes at hand. An inductive argument
claims that if the premises are true, the conclusion is probably true.
Because deductive and inductive arguments make such different
claims, the criteria used to evaluate them are not the same. An argument
interpreted deductively might be a bad deductive argument, while the
same argument interpreted inductively might be a good inductive
argument. In the following examples, (5) is deductive and (6) is inductive.
(5)
(6)
Since there are more than 365 people in the room and
there are only 365 days in a year, it must be the case that
there are two people in the room with the same birthday.
If Einstein is correct, the velocity of light in any given
medium is a constant. We know that the velocity of light in
air is a constant. And this evidence supports Einstein’s
claim.
It is possible to evaluate any argument using criteria appropriate to
deductive arguments. One could evaluate the presentation of evidence in
a courtroom using deductive criteria; however, doing so is insensitive to
the function of this type of argument. Most courtroom arguments are
probably best evaluated from an inductive standpoint.
We will focus on that part of logic concerned with the evaluation of
deductive arguments. From now on, when reference is made to
arguments it will be understood to mean deductive arguments, unless
explicitly stated to the contrary.
7
Some logicians have argued that not all arguments are deductive or inductive, i.e.,
deduction and induction are mutually exclusive, but not exhaustive categories of
arguments. See Stephen Barker, The Elements of Logic.
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A deductive argument claims that its premises furnish conclusive
evidence for its conclusion. The evaluation of deductive arguments
consists in ascertaining if that claim is correct. The words ‘valid’ and
‘invalid’ are used in evaluating the claim of deductive arguments.8 The
fundamental objective of the definition of ‘valid’ is to absolutely guarantee
that arguments constructed in accord with the rules of the logic will never
lead from true to false statements. In other words, validity is defined in a
as a truth-preserving relationship, .i.e., if you start with true propositions
and argue validly, the result must be a true proposition.
If a deductive argument succeeds in furnishing conclusive evidence
for its conclusion, then it is valid. A valid deductive argument is an
argument in which it is not logically possible for its premises to be true and
its conclusion false. A deductive argument is invalid if and only if it is not
valid. In other words, if the claim of conclusive evidence is not correct then
it is possible for the premises to be true and the conclusion false, and the
argument is an invalid deductive argument. A deductive argument is
invalid if and only if it is logically possible for its premises to be true and its
conclusion false.9
In section 2.3 a distinction was made between propositions and
propositional forms. This distinction introduced generality into the
discussion of propositions and logic. It also led to the observation of an
important relationship between formally analytic propositions and their
propositional forms. For similar reasons, we now distinguish between
arguments and argument forms. As you follow the discussion, you should
observe a very close parallel between the concepts used in the discussion
of propositions and those used for arguments.
An argument is composed of propositions, and the form of an
argument consists of the form of those propositions together with the
logical connections between those propositions. An argument form is the
form of an argument.
8
The terms ‘valid’ and ‘invalid’ have a broader use in ordinary English than in logic. In
our use, these terms are restricted to describing deductive arguments, and they are both
exhaustive and exclusive.
9
There is much more by way of both explanation and qualification that needs to be said
regarding the concepts of validity and invalidity and their relation to truth and falsity;
section 2.6 will address these issues in more detail.
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Following are examples of arguments (not necessarily valid) and
their corresponding argument forms. We adopt the convention of stating
the premises of the argument above the line and the conclusion below it.10
(7) If the sky is blue, it will be a
pleasant day. The sky is
blue.
∴It will be a pleasant day.
(7.1)
If p, then q
p
∴q
(8) No emus are wallabies.
∴No wallabies are emus.
(8.1) No X are Y
∴No Y are X
(9) All tomatoes are red.
∴All tomatoes are colored.
(9.1) All X are Y
∴All X are Z
(10) The picnic was a success or
the guests were unhappy.
The picnic was a success.
∴The guests were not unhappy.
(10.1) p or q
p
∴not q
(11) No chemists are fools.
Some illiterates are fools.
∴No chemists are illiterates.
(11.1) No Z are Y
Some X are Y
∴No Z are X
(12) Some cats are pets.
∴Most cats are pets.
(12.1) Some X are Y
∴Most X are Y
(7.1) through (12.1) are not arguments because they are not made
up of propositions. They are argument forms. Specifically, they are the
forms of arguments (7) through (12). We previously observed that specific
propositions are substitution instances of propositional forms. The same
relationship holds between arguments and argument forms. Arguments
are substitution instances of argument forms. Arguments (7) through (12)
are substitution instances of argument forms (7.1) through (12.1)
10
The symbol ‘∴’ is shorthand for ‘therefore’.
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respectively. An argument is a substitution instance of a given argument
form if and only if it can be obtained by uniform substitution from that form.
For example, by substituting the appropriate linguistic expressions for the
variables in (11.1) we may generate an infinite number of arguments all
having the same form as argument (11).
In the above examples, arguments (7) through (9) are valid, and (10)
through (12) are invalid. A careful reading of the arguments should make
this intuitively clear. The terms ‘valid’ and ‘invalid’, as they have been
defined, apply only to arguments. Since we wish to talk about argument
forms as well as arguments, these terms will be extended to cover
argument forms. An argument form is valid if and only if all substitution
instances of that form are valid arguments. An argument form is invalid if
and only if not all substitution instances of that form are valid, i.e., there is
at least one substitution instance of the argument form with true premises
and a false conclusion.
Although arguments (7), (8) and (9) are all valid, only argument
forms (7.1) and (8.1) are valid. At first glance, it may be surprising that
argument (9) is valid but its form is invalid. But not all substitution
instances of (9.1) are valid, and therefore, it is an invalid argument form.
For example, substitute ‘cats’ for ‘X’, ‘animals’ for ‘Y’ and ‘boats’ for ‘Z’.
The result is ‘all cats are animals; therefore, all cats are boats’, which is
obviously invalid. On the other hand, all substitution instances of (7.1) and
(8.1) are valid arguments, so these are valid argument forms.
While arguments (8) and (9) are both valid there is an important
difference between them. The validity of (8) does not depend on the
meaning of ‘emus’ and ‘wallabies’; it is only necessary to understand the
logical form words used in the argument. In argument (9), knowledge of
the logical form words alone is not sufficient to ascertain its validity; it is
also necessary to understand the meaning and relationship of ‘red’ and
‘colored’. Argument (8) is a formally valid argument. Argument (9) is a
semantically valid argument. A formally valid argument is an argument
whose validity is only a function of its logical form words. A semantically
valid argument is an argument whose validity is not only a function of its
logical form words.
The distinction between formally and semantically valid arguments
may also be defined by the relation of the arguments to their argument
forms. An argument is formally valid if and only if all arguments of that
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form are valid. An argument is semantically valid if and only if it is valid,
and not all arguments of the same form are valid.
The following bullets are a summary of the major points developed in
this section regarding arguments. The parallelism of language between
propositions and arguments should be obvious.
• An argument is an organization of propositions claiming
that the truth of some of its propositions (the premise or
premises) constitutes evidence for the truth of another
proposition in the argument (the conclusion).
• An argument form is the form or structure of an argument.
• An argument is a substitution instance of a given form if
and only if it can be obtained by uniform substitution from
that form.
•••••
• A deductive argument claims that it is not logically possible
for its premises to be true and its conclusion false.
• An inductive argument claims that if its premises are true,
then acceptable evidence has been provided establishing
the probable truth of the conclusion.
•••••
• A valid argument is one in which it is not logically possible
for the premises to be true and the conclusion false.
• An argument is formally valid if and only if all arguments of
that form are valid.
• An argument form is valid if and only if all substitution
instances of that form are valid arguments.
• An argument is semantically valid if and only if it is valid,
and not all arguments of the same form are valid.
•••••
• An invalid deductive argument is one in which it is logically
possible for the premises to be true and the conclusion
false.
• An argument form is invalid if and only if not all substitution
instances of that form are valid, i.e., there is at least one
substitution instance of the argument form with true
premises and a false conclusion.
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At the end of the previous section, logic was characterized as the
study of certain types of propositions and propositional forms. We may
now give a description of logic in terms of the study of arguments. Logic
is, in part, concerned with the analysis and evaluation of arguments.
Formal deductive logic is the examination of the principles of formal
validity. As was the case with propositions, this undertaking also has two
aspects. First, logic develops general procedures for identifying valid and
invalid arguments and argument forms. Second, logic develops
techniques for the construction of valid arguments and valid argument
forms.
In section 2.4, a footnote stated “Although arguments and
conditional propositions are not the same, it is clear that there is an
important relationship between them.” The nature of that relationship is
addressed in section 2.5. However, before you begin reading section 2.5,
try to answer the question posed in the following Challenge Box.
CHALLENGE
In §2.3 logic was characterized in terms of propositions, and in
§2.4 logic was characterized in terms of arguments. These two
approaches can be linked. The question is what is the connection
between propositions and arguments?
See if you can establish a relationship between an argument
and a specific proposition that would allow you to connect the validity
of the argument to a logical characteristic of the proposition.
If you succeed in connecting arguments and propositions, try
to do the same for argument forms and propositional forms. In other
words, is there a relationship between an argument form and a
propositional form that would allow you to connect the validity of the
argument form with a logical characteristic of the propositional form?
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§2.5
ARGUMENTS AND
CORRESPONDING CONDITIONAL PROPOSITIONS
We have developed two ostensibly different conceptions of logic.
From one perspective logic is concerned with propositions, and from
another it is concerned with arguments. Are these two viewpoints simply
different, or is there some link between the two? It should be apparent
from the structure of the preceding sections on propositions and
arguments that there is a very important connection between arguments
and propositions.
For every argument there is a specific proposition that bears an
important and distinctive relationship to the argument. This proposition is
the corresponding conditional proposition for the argument. A conditional
proposition is any proposition of the form ‘if …then…’.11 The
corresponding conditional for any deductive argument is the conditional
proposition whose antecedent is the conjunction of the premises in the
argument and whose consequent is the conclusion of the argument. 12
Following are examples of arguments paired with their
corresponding conditionals.
(1) Either Jones or Smith will (1.1) If either Jones or Smith will be
be the regular
the regular quarterback and
quarterback. Jones will
Jones will not be the regular
not be the regular
quarterback, then Smith will
Quarterback.
be the regular quarterback.
∴Smith will be the regular
quarterback.
(2) The ship on the horizon
is a Xebec.
∴The ship on the horizon
has three masts.
(2.1) If the ship on the horizon is a
Xebec, then the ship on the
horizon has three masts.
11
The proposition occurring between ‘if’ and ‘then’ is the antecedent of the conditional;
the proposition following ‘then’ is the consequent of the conditional.
12
The conjunction of the premises is formed by connecting all of the premises together
with ‘and’; that conjunction is true if and only if each conjoined premise is true. If there is
only one premise, then that single proposition is the antecedent in the corresponding
conditional.
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Symbolic Logic – A Primer
(3) Some bloodhounds are
excellent pets.
∴Some bloodhounds are
not excellent pets.
Part I: Introduction
§2: Basic Concepts
(3.1) If some bloodhounds are
excellent pets, then some
bloodhounds are not excellent
pets.
The concept of the corresponding conditional is readily extended
from arguments to argument forms. In the case of an argument form, the
corresponding conditional is constructed in the same way except that the
antecedent and consequent will be propositional forms rather than
propositions. Following are two examples of argument forms and their
corresponding conditional.
(4) All X are T
No Y are T
∴No Y are X
(4.1) If all X are T and no Y are T,
then no Y are X
(5) Some X are Y
∴Some X are not Y
(5.1) If some X are Y, then some X
are not Y
In general, given any argument, or argument form, with Pn premises
and C as conclusion,
P1
P.2
.
Pn-1
Pn
∴C
the corresponding conditional would be:
If P1 and P2 ... and Pn-1 and Pn, then C.
What exactly is the connection between an argument and its
corresponding conditional? An argument is valid if and only if the
statement ‘the premises are true and the conclusion false’, when applied to
the argument, is necessarily false. And it follows from this that an
argument is valid if and only if the statement ‘if the premises are true, then
the conclusion is true’, is necessarily true. But notice that the
corresponding conditional is essentially the statement ‘if the premises are
true, then the conclusion is true’. Consequently, if an argument is valid its
corresponding conditional is true. However, as we just observed, the
corresponding conditional for a valid argument is not “merely” true, but
necessarily true, and a necessarily true statement is analytic.
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We now have a fundamental connection between arguments and
their corresponding conditionals. An argument is valid if and only if the
corresponding conditional is analytic. An argument is invalid if and only if
the corresponding conditional is not analytic.
Conditionals (1.1) and (2.1) are both analytic, but notice that the first
is formally analytic, and the second is semantically analytic. The
arguments that correspond to these are both valid, but as you would
expect (1) is formally valid and (2) is semantically valid. This relationship
holds in general. An argument is formally valid if and only if the
corresponding conditional is formally analytic. An argument is semantically
valid if and only if the corresponding conditional is semantically analytic.
We may also link argument forms and their corresponding
conditional forms. Argument form (4) is valid and the corresponding
conditional (4.1) is an analytic propositional form. Argument form (5) is not
valid and the corresponding conditional (5.1) is not an analytic
propositional form. An argument form is valid if and only if the
corresponding conditional is an analytic propositional form.
Summarized below are some of the major points regarding
arguments and their corresponding conditional propositions.
• The corresponding conditional for any deductive argument is
the conditional proposition whose antecedent is the
conjunction of all of the premises in the argument and whose
consequent is the conclusion of the argument.
•••••
• An argument is valid if and only if its corresponding conditional
is analytic.
• An argument is formally valid if and only if its corresponding
conditional is formally analytic.
• An argument form is valid if and only if its corresponding
conditional is an analytic propositional form.
•••••
• An argument is semantically valid if and only if the
corresponding conditional is semantically analytic.
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§2.6
VALID AND INVALID, A CLOSER LOOK
In section 2.4, the concepts of validity and invalidity as applied to
deductive arguments were introduced. This section focuses in
considerable detail on those basic concepts, and it is divided into three
segments. First, the basic definitions; second, a more detailed look at how
valid and invalid as properties of deductive arguments relate to the truth or
falsity of the propositions in those arguments; and lastly, comments on
what may be termed “foundational” issues.13
Basic Definitions
A deductive argument is valid if and only if it is not
logically possible for the premises to be true and the
conclusion false.
A deductive argument is invalid if and only if it is
logically possible for the premises to be true and the
conclusion false. (Or, more concisely: A deductive
argument is invalid if and only if it is not valid.)
As a direct consequence of the definition of ‘valid’, it is necessarily
true that a valid argument with true premises will have a true conclusion. A
frequently encountered alternative formulation of this statement is: if a
valid deductive argument has true premises, then the conclusion is
necessarily true. This formulation is not correct due to the placement of
the key word ‘necessarily’. In a valid argument, what is necessary is the
relationship between the premises and the conclusion; it is not the case
that the conclusion must itself be necessarily true. For example, the
following argument is clearly valid: ‘There are ten apples on the desk;
13
§2.4 also placed boundaries on the type of arguments examined in formal
deductive logic. They are worth restating with some expansion. Unless otherwise
specified, our interest in arguments is now limited to deductive arguments and matters of
formal validity and invalidity. When the term ‘argument’ is used without qualification it
will mean ‘deductive argument’, ‘valid’ will mean ‘formally valid’, and ‘invalid’ will mean
‘formally invalid’. Parallel considerations hold for propositions; we are interested in
formally analytic propositions and analytic propositional forms.
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therefore, there are at least five apples on the desk’. What is necessarily
true is ‘If there are ten apples on the desk, then there are at least five
apples on the desk’. But, ‘there are at least five apples on the desk’ is not
necessarily true.
The fundamental objective behind the definition of ‘valid’ is that valid
inferences should guarantee that if an argument has true premises and is
valid the conclusion will be true. In short, the relationship of validity is
truth-preserving. Another way of expressing this basic point is to say that
truth is a hereditary property with respect to validity, i.e., if you start with
true propositions and validly deduce other propositions all of the derived
proposition will inherit the property of being true.
A number of essential points should be emphasized regarding the
definition of ‘valid’.
• The definition of ‘valid’ states that it is not logically possible
for a valid argument to have true premises and a false
conclusion.
• The definition of ‘valid’ implies that if a valid argument has
true premises, it must have a true conclusion.
• The definition of ‘valid’ implies that if a valid argument has
a false conclusion, not all of its premises can be true.
• The definition of ‘valid’ neither states nor implies that a
valid argument has true premises and a true conclusion.
• The definition of ‘valid’ neither states nor implies that if an
argument has true premises and a true conclusion, it is
valid.
Following are critical points that should be underscored concerning
the definition of ‘invalid’.
• The definition of ‘invalid’ asserts that it is logically possible
for an invalid argument to have true premises and a false
conclusion.
• The definition of ‘invalid’ does imply that if the premises of
an argument are true and the conclusion is false, then the
argument is invalid.
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• The definition of ‘invalid’ neither states nor implies that an
invalid argument must have true premises and a false
conclusion.
• The definition of ‘invalid’ neither states nor implies that if
the premises and conclusion of an argument are false, the
argument is invalid.
• The definition of ‘invalid’ neither states nor implies that if an
argument contains a false statement, it must be invalid.
Valid, Invalid, True, and False
There are four possible combinations of valid and invalid arguments
with premises that are either all true or all false.
(1)
(2)
(3)
(4)
A valid argument with all premises true.
A valid argument with all premises false.
An invalid argument with all premises true.
An invalid argument with all premises false.
To further clarify the concepts of validity and invalidity, we will take each of
these four possibilities and ask what may be correctly inferred regarding an
argument’s validity given knowledge of the truth-value of the propositions
in the argument.14 And conversely, what can be inferred about the truthvalue of an argument’s propositions given knowledge about its validity.
In investigating these four basic combinations, specific arguments
will be used to illustrate each situation. All of the examples will be created
by uniform substitution in two argument forms. Using your knowledge of
argument forms, you should examine the two forms and see how the
examples are arrived at by uniform substitution into the forms. The two
forms are:
VALID
All Y are Z
All X are Y
∴All X are Z
INVALID
All Z are Y
All X are Y
∴All X are Z
14
Propositions are either true or false. The truth-value of a proposition refers to which of
the two values a proposition possess.
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CHALLENGE
Consider each of the four combinations of true and false
premises occurring in valid and invalid arguments listed above. In
each case, see if you can determine the truth value of the argument’s
conclusion based only on the information about the truth value of the
argument’s premises and its validity.
To answer this question it will be helpful to construct
arguments that are substitution instances of the above two argument
forms.
(1) Valid argument with true premise
If an argument is valid and all its premises are true, then we know
that the conclusion is true; this follows immediately from the definition of
‘valid’. The following argument is valid, all of the premises are true, and
the conclusion is true.
All flowers are plants.
All petunias are flowers.
∴All petunias are plants.
(2) Valid argument with false premises
At first, you may find this case the most puzzling. It would seem
reasonable to suppose that if an argument is valid and the premises are
false, then the conclusion will likewise “follow” and be false. But this is not
correct. A valid argument with false premises may have either a true or a
false conclusion. The following two arguments are valid and all premises
are false, but in one the conclusion is true and in the other it is false.
All reptiles are flowers.
All roses are reptiles.
∴All roses are flowers.
All birds are cats.
All sailboats are birds.
∴All sailboats are cats.
(3) Invalid argument with true premises
If an argument is invalid and its premises are true, then it is not
possible to correctly infer the truth-value of the conclusion. From the
definition of ‘invalid’ it clearly follows that an invalid argument may have a
false conclusion, but it is not necessary for an invalid argument to have a
false conclusion. An invalid argument with true premises may have a true
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Symbolic Logic – A Primer
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or false conclusion. Both of the following arguments are invalid and both
have true premises, but in one the conclusion is true and in the other it is
false.
All insects are animals.
All butterflies are animals.
∴All butterflies are insects.
All ducks are animals.
All antelopes are animals.
∴ All antelopes are ducks.
(4) Invalid argument with false premises
If an argument is invalid and its premises are false it is not possible
to correctly infer the truth-value of the conclusion. The following
arguments are both invalid and all the premises are false, but in one the
conclusion is true and in the other it is false.
All balloons are catfish.
All snakes are catfish.
∴ All snakes are balloons.
All animals are mountains.
All aardvarks are mountains.
∴All aardvarks are animals.
In each of the above examples the premises are either all true or all
false. In (2), (3) and (4) above, the same results would follow even if the
premises were a mix of true and false propositions. This qualification does
not hold true in case (1). The comments in (1) are correct if and only if all
of the premises in the argument are true.15
15
With reference to the comments in (2), (3) and (4) a qualification is necessary. It is not
the case that each and every valid (or invalid) argument form must have substitution
instances of all of the various combinations of truth-values illustrated. For example the
form ‘some X are Y, therefore, some Y are X’ is valid. But there is no argument of this
form with a false premise and a true conclusion because the premise and conclusion
happen to be logically equivalent propositions. The form ‘p, therefore p and q’ is invalid.
But there is no substitution instance of this with a false premise and a true conclusion
because the conclusion of the argument happens to validly imply the premise of the
argument. However, the points made in (2), (3) and (4) still hold in general. That is,
there are valid arguments with false premises and a true conclusion, and invalid
arguments with false premises and a true conclusion, and so forth. If an argument form
is valid, then there can be no substitution instance of it with true premises and a false
conclusion. And, if an argument form is invalid, then there must be a substitution
instance with true premises and a false conclusion. The crucial point is that the
definition of ‘valid’ excludes one combination of truth-values, viz., true premises and a
false conclusion. The definition implies nothing about other combinations of truthvalues. On the other hand, the definition of ‘invalid’ requires the possibility of one
combination of truth-values, viz., true premises and a false conclusion. The definition
implies nothing about other combinations of truth-values.
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A clear and firm grasp of the following points is essential to
understanding the relation of truth-values and validity.
• If the truth-values of an argument’s premises and its
validity are known, then it is possible to correctly infer the
truth-value of the argument’s conclusion in only one case,
viz., when all premises are true and the argument is valid,
the conclusion must be true.
• If the truth-values of an argument’s premises and
conclusion are known, then it is possible to correctly
determine the question of validity in only one case, viz.,
when all premises are true and the conclusion is false, the
argument is invalid.16
• A correct inference can be made regarding the truth of at
least one the premises in an argument based on the
argument’s validity and the truth-value its conclusion, viz., if
an argument is valid and the conclusion is false, then at
least one premise must be false.
Valid arguments with true premises are of special interest. It is only
in this case that an argument provides a proof of the truth of its conclusion.
Valid arguments with true premises are called ‘sound’. Invalid arguments
and arguments whose premises are not all true are called ‘unsound’. A
sound argument proves the truth of its conclusion.17
• A sound argument is an argument that is valid and whose
premises are all true.
• An unsound argument is an argument that is invalid, or has
at least one false premise, or both.
Note that a sound argument is not defined as a valid argument with a true
conclusion. It is true that a sound argument must have a true conclusion;
because a sound argument must be valid and have true premises it follows
16
This is true as stated. However, the situation changes if we have stronger information
than the simple facts of the truth or falsity of the propositions in the argument.
Specifically, the situation changes if we know the premises are logically false
(inconsistent) or the conclusion logically true (analytic). This is a topic to be considered
later.
17
Strictly speaking, the determination of the soundness of an argument does not fall
within the province of logic. Ascertaining soundness requires extra-logical knowledge,
e.g., knowledge from the sciences or direct reporting.
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immediately that its conclusion must be true. However, it is possible to
have a valid argument with a true conclusion and false premises, and this
is not a sound argument.18
Two Foundational Issues
(1) Logical Possibility
The concept of validity is frequently explained by using terms such
as ‘possible’ and ‘impossible’. There are different meanings of ‘possible’,
or, as it is sometimes stated, there are different kinds of possibility.
Something may be technologically or practically possible in relation
to our current knowledge, technology, and skills. At the moment, no
automobile manufacturer markets a vehicle where the traditional steering
wheel is replaced by a joystick. Clearly, this can be done. It may be a bad
idea, but building such a vehicle is not dependent on new developments in
science and technology.
In contrast, consider the possibility of building a supersonic aircraft
capable of carrying 3,000 people. The construction of such an aircraft is
(presumably) not practically or technology possible. But, in some sense,
building such a craft is possible, and this example leads to the concept of
physical possibility. Something is physically possible if it is consistent with
the laws of nature as we know them. Our supersonic plane is consistent
with the laws of nature, so although it is not now practically possible, it is
physically possible. Clearly, everything that is practically possible is
physically possible.
Now consider the possibility of constructing a spaceship capable of
traveling from the Earth to Alpha Centauri B with a total travel time of one
hundred years. This is not practically possible, but it is consistent with the
laws of physics, so it is physically possible. Suppose we now change the
example to the possibility of building a faster version of the spaceship that
will make the journey in two years rather than one hundred years. Alpha
Centauri B is 4.3 light years from the Earth. Therefore, to make that
journey in two years requires traveling faster than the speed of light. But,
traveling faster than the speed of light is inconsistent with accepted
science, therefore it is not physically possible to build such a spaceship.
18
The definition of ‘sound’ (and ‘unsound’) used here is common. However, there are
other uses; some logicians define ‘sound’ and ‘valid’ equivalently.
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We now come to the concept of logical possibility. In explicating the
concepts of validity and invalidity it is logical possibility (and logical
necessity and impossibility) that underpin the explanation. Logical
possibility is a weaker sense of possibility than practical or physical
possibility, ‘weaker’ in the sense of being much more inclusive.
We have no problem in consistently conceiving of our Earth being
different than it is. For example, imagining the Earth in which the elevation
of Mt. Everest is 31,000 feet is quite straightforward. Now broaden the
subject to include the entire universe. The universe includes not just our
own solar system but also all galaxies and all existent objects no matter
where they are located. We can consistently conceive of universes that
are different from our universe. We can consistently conceive of universes
containing one more or one less rock than is found in our actual universe.
Just as we can conceive of a universe containing objects that are
not in our universe, we can conceive of a universe in which the speed of
light is faster than the speed of light in our universe, and in such a universe
our hypothetical spaceship is possible. But this is a very weak sense of
possible, and it amounts to asserting that one can, without logical
contradiction, conceive of a universe in which a spaceship is capable of
speeds not possible in the known universe.
A consistent possibility for a universe is usually referred to as a
‘possible universe’. Our actual universe is one of the infinitely many
possible universes. Given this approach we can define a number of core
terms. A true proposition is one that is true in our actual universe, e.g. the
capital of Maine is Augusta. A false proposition is one that is false in our
actual universe, e.g., the speed of light is 1,000 miles per second. A
logically possible proposition is one that is true in at least one possible
universe, e.g., the University of Nevada at Reno is located in Tonopah. A
logically true (analytic) proposition is one that is true in all possible
universes, e.g., Mars is larger than Pluto or Mars is not larger than Pluto.
A logically false (impossible) proposition is one that is not true in any
possible universe, e.g., John is 5 years older than Mary, and they are the
same age.
The above comments on possibility are not without their own
problems. For example, explaining the concept of a possible universe
assumes the notion of logical consistency, and to that extent the
explanation is circular. Additionally, the concept of possible worlds is itself
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subject to dispute. Although these issues are extremely important they are
beyond the scope of this discussion.
(2) A Question About Validity
The definition of ‘valid’ used here is found in virtually all elementary
logic textbooks. However, some logicians do not accept the definition.
The definition itself is equivalent to the conjunction of two ‘if…then…’
statements. There is no disagreement about (a), but (b) is another matter
entirely.
(a) If a deductive argument is valid, then it is not possible for the
premises to be true and the conclusion false.
and
(b) If it is not possible for the premises to be true and the
conclusion false in a deductive argument, then it is valid.
CHALLENGE
There are interesting and important consequences that follow
from the (b) alternative above. According some logicians, these
consequences are surprising but true, to others they are disturbing,
and for others they are patently false. The issues raised by the
consequences of (b) are fundamental questions in the philosophy of
logic. Using the following information see if you can identify these
peculiarities.
The concerns arise when arguments contain logically true or
logically false propositions. Think about the following two situations
and try to determine if they have any implications for the validity or
invalidity of the arguments.
1. Assume that an argument has an inconsistent (logically false)
premise, e.g., ‘Nitrogen is an inert gas and nitrogen is not an
inert gas’.
2. Assume that an argument has an analytic (logically true)
conclusion, e.g., ‘Nitrogen is an inert gas or nitrogen is not an
inert gas’.
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If the premises of an argument are inconsistent, then they cannot all
be true. If the premises of an argument cannot be true, then it immediately
follows that the argument cannot have true premises and a false
conclusion (since it cannot have true premises). Therefore, it is valid. For
example, it is an immediate consequence of the definition of ‘valid’ that the
argument ‘the earth is round and the earth is not round, therefore, Caesar
crossed the Rubicon’ is valid.
In a parallel manner, if the conclusion of an argument is logically true
(analytic), then it cannot be false. If the conclusion cannot be false, it
immediately follows that the argument cannot have true premises and a
false conclusion (since it cannot have a false conclusion). Therefore, it is
valid. For example, ‘Caesar crossed the Rubicon; therefore, the earth is
round or the earth is not round’ is valid.
Some philosophers find these consequences of the traditional
definition of valid to be preposterous. However, these seemingly
outlandish consequences are easily derivable from some very simple
argument forms that are difficult to reject. Although this is a topic for
another day, suffice it to say that it is much easier to object to the
consequences of the definition of ‘valid’ than it is to “fix” it.
§2.7
NON-TECHNICAL PROOFS OF INVALIDITY AND VALIDITY
As your study of logic continues, a variety of tools will be added to
your logic toolbox. These tools are relevant to problems such as
determining if a proposition, or propositional form, is logically true; if an
argument, or argument form, is valid; if propositions, or propositional
forms, are logically equivalent, inconsistent, contradictory, and so forth.
However, in ordinary day-to-day discussions it is sometimes
inappropriate and pointless to unleash your logical tools, viz., when the
tools will not be understood by your audience and are not easily and
quickly explainable. For example, suppose that someone proposes an
argument claiming to prove that the Supreme Court of the United States is
a legislative body and should be viewed as a part of the congress. You
assert that the proposed argument is invalid. If you are challenged to
prove your assertion it would be boorish in the extreme to reply, “The
argument is invalid because a truth tree constructed for the set of
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propositions comprised of the argument’s premises and the negation of its
conclusion contains an open path”.
So the inevitable question arises: are there any relatively simple
techniques available that can be used to establish invalidity or validity and
do not require formal training in logic? The answer must be ‘yes’ if for no
other reason than the one stated in Section 1; we commonly create and
correctly evaluate arguments without having studied logic.
The remainder of section 2.7 explains some simple, readily
accessible, and non-technical tools relevant to assessing arguments. The
methods are non-technical in the sense that they require no prior
knowledge of logic other than recognizing that the hallmark of good
deductive argument is that it can never lead from true premises to a false
conclusion.
ESTABLISHING INVALIDITY
CHALLENGE
Look at argument number 11 on page 25. This argument is
invalid, and it is possible for you to prove that it is invalid. The proof
is non-technical in the sense that no specialized knowledge of logic is
required other than the concepts explained on the preceding pages.
See if you can devise a way to prove the invalidity of the
argument in question. If you succeed, you will no longer have to take
it on faith that the argument is invalid because some else said so – a
far better position to be in for a student of logic. When you find a
solution, consider whether there are any limitations on the
applicability of your method.
There is a very simple technique available to prove formal invalidity.
In fact, we have previously used the technique a number of times, but it
was not specifically identified and named. The technique is based on the
fact that a single example showing that an argument can have true
premises and a false conclusion incontrovertibly proves invalidity. In
contrast, validity cannot be established by giving examples – no matter
how numerous the examples.
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Suppose that someone uses the following argument in the course of
a discussion.
Some professors are members of the Democratic Party.
Some members of the Democratic Party are members of
academic associations.
∴Some professors are members of academic associations.
Although this argument might appear valid it is not. One way of refuting
this argument is to construct another argument such as the following.
Some trees are tall objects.
Some tall objects are buildings.
∴Some trees are buildings.
This new argument shows that the original argument is not formally valid.
The reason for this is quite simple. If the first argument is formally valid,
then all arguments of that form must be valid arguments. The second
argument is identical to the first in form; however, the content of the
second argument was selected to yield true premises and a false
conclusion. Since the second argument has true premises and a false
conclusion it is invalid, and therefore, the first is not formally valid. In this
example we are treating validity as a function of the form of the argument.
This method of showing that an argument is not formally valid is called
‘refutation by logical analogy’.
In order to construct a refutation by logical analogy two things must
be accomplished, viz., construct an argument of the same form as the
argument to be refuted, and select the content to make the premises of the
new argument true and its conclusion false.19
19
In interpreting refutations by logical analogy two words of caution are necessary. First,
a successful refutation by logical analogy does not prove that the original argument is
invalid in an unqualified sense. The argument ‘grass is green, therefore, grass is
colored’ is valid. ‘Roses are red; therefore, roses are trains’ is invalid, but it is logically
analogous to the first argument. The second argument does constitute a refutation by
logical analogy of the first, but what it proves is that the first argument is not formally
valid. The first argument is, of course, semantically valid. Refutation by logical analogy
can only be used to establish formal invalidity. Secondly, to say that an argument has
been refuted by logical analogy does not mean that the conclusion or premises of the
argument have been refuted in the sense of being shown to be false. All of the
propositions in the argument may be true. The refutation proves simply that the
premises do not formally imply the conclusion.
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It is clear that if we wish to show that an argument form is invalid a
similar procedure is available. For example, the following argument form is
invalid.
If p then q
q
∴p
The argument form may be proven invalid by constructing a substitution
instance of it with true premises and a false conclusion. The following
argument demonstrates that the above form is invalid.
If someone is skiing on Mt. Rose, then Mt. Rose has snow.
Mt. Rose has snow.
∴ Someone is skiing on Mt. Rose.
When a substitution instance of an argument form is constructed to
prove that the form is invalid the instance is called a ‘counter example’. A
counter example to an argument form is a substitution instance of that form
with true premises and a false conclusion.
The procedures discussed above are interesting for a number of
reasons. First, they offer a simple illustration of the relation between
validity and structure. Second, refutations by logical analogy and counter
example are frequently useful in ordinary discourse, where appeal to
technical matters in logic would likely be pointless.
However, as a technical tool these techniques are not of great
interest to the logician. They suffer from a number of serious limitations.
Suppose you try to refute an argument by these methods and fail, what
follows? The argument might be valid, and therefore no refutation is
possible. Or, the argument might be invalid and you were not sufficiently
clever in constructing the refutation. Neither of these procedures, of
course, can be used to establish validity. If all the attempted counter
examples and logical analogies have true premises and a true conclusion,
this does not prove the original valid. In addition, although these
procedures work reasonably well with simple arguments their application to
complex arguments is frequently quite difficult. And lastly, the successful
construction of a counter example or a refutation by logical analogy
depends quite heavily on the individual’s wit and imagination. It would be
preferable to devise methods for dealing with arguments and argument
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forms that are mechanical and can be applied in a straightforward and
systematic manner.20
ESTABLISHING VALIDITY
CHALLENGE
In the previous challenge you devised a non-technical way of
establishing invalidity. In some cases, validity may be established by
systematically deriving the conclusion from the premises using
extremely simple argument patterns. The following argument is valid;
try to construct a series of simple arguments that prove its validity.
If Descartes is right, then the mind is immaterial. If the mind
and body are radically different types of entities, then minds
exist independently of bodies. If the mind is immaterial, then
mind and body are radically different type of entities. If minds
exist independently of bodies, then immortality is a
characteristic of mind but not of body. Descartes is right.
Therefore, immortality is a characteristic of mind but not of
body.
If you do succeed in proving the argument is valid, are there
any limitations on your procedure? Could it be used in the case of
other arguments? If there are limitations on your technique, do you
have any ideas for dealing with them?
In approaching the argument in the challenge frame, it will be useful to
rearrange the sequence of the premises. An argument’s validity is
independent of the order of premises; however, the order of the premises,
especially in a long argument, may result in critical relationships being
obscured and overlooked. Following is a rewrite of the original argument;
the premises are numbered for convenience.
20
The preceding challenge asked you to prove the invalidity of argument number 11 on
p. - 25 -; the argument is proven formally invalid by the following counter example: No
tigers are fish and some animals are fish; therefore, no tigers are animals.
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1. If Descartes is right, then the mind is immaterial.
2. Descartes is right.
3. If the mind is immaterial, then mind and body are radically
different type of entities.
4. If the mind and body are radically different type of entities,
then minds exist independently of bodies.
5. If minds exist independently of bodies, then immortality is a
characteristic of mind but not of body.
Therefore, immortality is a characteristic of mind but not of
body.
Appealing to a single extremely simple and common valid argument
form, the validity of the above argument can be demonstrated by deducing
the conclusion from the premises. The argument form in question states
that if a conditional sentence is true and if the antecedent of that sentence
is true, then the consequent must be true.
If P then Q
P
∴Q
The proof proceeds as follows. Premises 1 and 2 validly imply ‘the
mind is immaterial’; this newly derived conclusion can then be used as a
premise to support additional inferences. In this case, ‘the mind is
immaterial’ and premise 3 imply ‘mind and body are radically different type
of entities’. Premise 4 and ‘mind and body are radically different type of
entities’ imply ‘minds exist independently of bodies’, and the latter with
premise 5 imply the desired conclusion ‘immortality is a characteristic of
mind but not of body’.
The argument we wanted to prove valid is reasonably complex. Yet,
its validity is established by repetitive applications of a simple argument
pattern. In ordinary discourse where simple non-technical procedures are
particularly appropriate, this type of approach may be helpful. Of course,
the procedure used here to establish validity has obvious limitations;
clearly, we cannot be limited to the single argument form used above.
Consequently, if you use this procedure, your logic toolbox will have to
contain an extensive collection of simple valid argument forms.
There is another frequently encountered technique used to establish
validity that is applicable in simple situations and meets our current goals
of a non-technical proof. Before the technique is explained, you might
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want to consider if you can devise a procedure to prove the validity of the
argument in the following challenge box. In the preceding example we
were able to directly use a very simple argument form, (viz., if P then Q, P,
therefore, Q) to prove validity. This worked because one of the premises
was a conditional sentence and another premise was the antecedent of the
conditional, and this set the stage for repeated uses of the argument form.
However, in the following argument each of the premises is a conditional
sentence, which means that the previous technique will not work, or to be
more exact, it will not work in exactly the same way. One more clue: you
can prove the validity of the following argument by appealing to the same
argument form as used in the previous example. The key to the problem is
to think carefully about the specific propositions you want to use in your
proof.
CHALLENGE
If the Burning Man Festival becomes a national icon, then
attendance at the Festival will soar. If the Black Rock Desert suffers
irreparable damage, then the Burning Festival will not become a
national icon. If attendance at the Festival soars, then the Black
Rock Desert will suffer irreparable damage. Therefore, the Burning
Man Festival will not become a national icon.
To start let us rewrite the argument and number the premises for ease of
reference.
1. If the Burning Man Festival becomes a national icon, then
attendance at the Festival will soar.
2. If attendance at the Festival soars, then the Black Rock Desert
will suffer irreparable damage.
3. If the Black Rock Desert suffers irreparable damage, then the
Burning Festival will not become a national icon.
Therefore, the Burning Man Festival will not become a
national icon.
The method we will use to prove this argument valid is called an
‘indirect proof’ or a ‘reductio ad absurdum’. This method assumes that the
argument in question is invalid, and then draws inferences based on that
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assumption. If we can demonstrate that the assumption of invalidity is in
fact inconsistent, then the argument in question must be valid.
1. Assume that the argument is invalid, i.e., assume that the
premises are true and the conclusion is false.
2. If the conclusion ‘The Burning Man Festival will not become a
national icon’ is false, then ‘The Burning Man Festival will
become a national icon’ is true.
3. ‘The Burning Man Festival will become a national icon’ and
premise #1 implies ‘Attendance at the Festival will soar’.
4. ‘Attendance at the Festival will soar’ with premise #2 implies ‘The
Black Rock Desert will suffer irreparable damage’.
5. ‘The Black Rock Desert will suffer irreparable damage’ with
premise #3 implies ‘The Burning Man Festival will not become a
national icon’; but the latter is inconsistent with assumption that
the argument has a false conclusion and true premises;
consequently, the argument must be valid.
In short, we began with the assumption that the argument was invalid and
derived a contradiction, viz., ‘The Burning Man Festival will become a
national icon’ (2 above), and ‘The Burning Man Festival will not become a
national icon’ (5 above). This is impossible, consequently, the assumption
of invalidity is impossible and the argument is valid.
The non-technical techniques for proving invalidity and validity are
reasonably simple and are useful in informal settings, albeit their utility is
quite limited. In addition, in order to attain an applicable, non-technical,
and simple process a number of things were left unsaid. For example, in
the last illustration from ‘The Burning Man Festival will not become a
national icon is false’, we inferred ‘The Burning Man Festival will become a
national icon’. Needless to say there is much more to be said about these
procedures. Nevertheless, even with their limited applicability, they are
valuable.
§2.8
WRAP-UP
Up to this point, we have developed some of the basic vocabulary of
logic. Using that vocabulary, formal deductive logic has been described in
two different ways. First, logic was portrayed as the study of certain types
of propositions, and second, it was characterized as the study of
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arguments. We have also seen that the relation between arguments and
conditional sentences ties these two perspectives together.
There is another important characteristic of logic running through
much of the previous material, although it was not specifically mentioned.
We talked about logic as a tool for assessing arguments and propositions.
However, we also posed questions about logic itself. Logic presents these
two different dimensions. Learning to use the tools in logic’s toolbox to
assess arguments is one thing, but it is quite another matter to wonder if
the toolbox itself contains sufficient tools to assess arguments. The two
threads of logic will permeate virtually all of our discussions.
Hopefully, the topics introduced here will have provoked some
critical questions. For example, it was somewhat cavalierly asserted that
an argument form was valid if and only if all arguments of that form were
valid. That, and similar statements, should raise your critical hackles.
After all, could one ever know that all arguments of a certain form are
valid? Certainly this could not be established by simply producing
examples. Examples, after all, are always of some not all. These, and
related questions, remain to be addressed.
You may also have had critical thoughts about the concept of logical
form. Logical form and substitution instances of forms are more complex
than might seem at first glance. In the argument ‘All Porsches are
automobiles; therefore, all Porsches are automobiles’, the form can be
represented by ‘P; therefore P’. Because the premise and the conclusion
are the same proposition, using ‘P’ to stand for an entire sentence provides
an acceptable representation of the form of the argument, even though the
propositions themselves can be analyzed at a finer level of detail.
But the situation changes with a simple change in the argument.
Suppose the argument is changed to ‘All Porsches are automobiles;
therefore, all non-automobiles are non-Porsches.’ This argument is valid,
and the premise and conclusion are not the same proposition. The form of
the argument is not adequately represented by ‘P; therefore Q’ because
that form is invalid, and therefore, does not provide an adequate analysis
of the argument. Clearly, the interior of each proposition must be analyzed
to account for the fact that the premise and conclusion although different
propositions, share critical content and have related structures.
Consider another example. ‘John and Bill are freshmen; therefore
John is a freshman.’ This is valid, but assigning one variable to the
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premise and another to the conclusion cannot represent its form. This is
easily resolved by restating the premise as ‘John is a freshman and Bill is
a freshman.’ Using single variables for each proposition will then yield the
desired result.
There are also issues that relate to form and proper substitution
instances. For example, ‘John and Bill are freshmen and Bill and Sam are
freshmen; therefore, John and Sam are freshmen.’ Again, this is evidently
valid, but it is easy to point to an invalid argument that appears to be of the
same form. If ‘friends’ replaces ‘freshmen’, the result is clearly invalid but,
if ‘siblings’ replaces ‘freshmen’ the result is valid. Accounting for this will
require a significant expansion of the few tools we have used to represent
propositional form.
Although there are interesting issues in characterizing logical form,
the good news is that for the most part it is easy to recognize when a
proposed analysis of structure fails for want of addressing sufficient detail.
More importantly, as you study different levels of logic, additional and more
powerful tools will be introduced to address increasingly complex cases of
logical form.
Different parts of modern logic will provide you with different tools to
analyze arguments and determine validity. Your study of logic has begun
with relatively simple foundational building blocks. As your studies expand
you will be progressively adding more complex elements, but the simpler
elements remain unchanged and become a part of the advanced systems.
This is a point of some consequence. It means that if an argument is
proven valid using the simpler tools, its validity will be retained in the
advanced forms of logic. It also means that if an argument is not proven
valid with simpler tools, the possibility remains (in some cases) that it is
valid and that more powerful tools are required to demonstrate its validity.
As you will see, logic builds upon itself in an organized, controlled,
and ordered manner yielding integrated and coherent structures that have
both utility and beauty of form. Logic has practical utility as well as
aesthetic value.
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§2.9
EXERCISES
1. Using the conventions regarding single quotation marks, correct the
following to make them (presumably) true.
a) John is a four-letter word.
b) Congress is a noun.
c) Bachelor means unmarried male.
d) ‘Athens’ is in Greece.
e) Agricola is a Latin word.
f) The second letter in cat is a.
g) Shakespeare’s real name, according to some people, is
Bacon.
h) William was named William after his maternal grandfather, but
his nickname is Bill.
2. Classify the following propositions as analytic or synthetic. For the
analytic propositions decide which are formally and which are
semantically analytic. Identify any instances presenting unusual
difficulties.
a) All roses are red.
b) All roses are flowers.
c) What will be will be.
d) The population of Cleveland is greater than five million.
e) Cows give milk.
f) If John is older than Mary, then Mary is younger than John.
g) No man can know another’s pain.
h) The lights are on or it is not the case that the lights are on.
i) All open windows are windows.
j) The lights are on or the electrical circuit is not complete.
3. Decide whether each of the following is an argument. In the case of
arguments, decide if they are deductive or inductive and isolate their
premises and conclusions.
a) The car overheated because the radiator hose ruptured.
b) There is no carbon in sulfuric acid; therefore, because all
organic compounds contain carbon, sulfuric acid cannot be an
organic substance.
c) The compound is most probably silver since all of the samples
tested conducted electricity and dissolved in nitric acid. And
silver has these properties.
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d) If Ptolemy is correct then there should be no stellar parallax.
On the other hand if Copernicus is correct there should be.
But Betel has shown that the phenomenon of stellar parallax is
observable. Therefore, Ptolemy is wrong and Copernicus is
correct.
e) If John’s argument for the existence of a divine being is
formally valid, then all arguments of that form must be valid
arguments. Mary’s argument for the contingency of all
existence is identical in form to John’s argument. But Mary’s
argument has true premises and a false conclusion.
Therefore, Mary’s argument must be invalid, and that implies
that John’s argument is invalid. Consequently, there must be
no divine being.
f) We know that a mixture of barley and oats was loaded into the
railroad car. The problem is to determine in what proportions
the grains were mixed. The first sample taken from the top of
the car contained 4 parts of barley to 2 parts of oats. The
second sample, taken from the bottom of the car, contained
2.1 part of barley to 1 part of oats. The third sample from the
middle of the car contained 5.9 parts barley to 3 parts of oats.
We conclude that the original mixture of barley and oats was 2
parts of barley to 1 part of oats.
g) There are six black balls and two red balls in the first urn. The
second contains five red balls and three black balls.
Therefore, the probability of picking a black ball from the first
and the second urn is nine thirty-seconds.
4. Answer and explain the following questions.
a) Is it possible to have a valid argument whose propositions are
all synthetic?
b) Is it possible to have a valid argument whose propositions are
all analytic?
c) Is it possible to have a valid argument whose premises are
synthetic and whose conclusion is analytic?
d) Is it possible to have a valid argument whose premises are
analytic and whose conclusion is synthetic?
e) Is it true that for every valid argument there is a corresponding
formally analytic conditional sentence?
f) Is it true that for every analytic conditional sentence there is a
corresponding valid argument?
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5. Decide whether each of the following is true or false.
In a valid argument…
a) the conclusion cannot be false if the premises are true.
b) if the conclusion is true, the argument is sound.
c) if the conclusion is false, the argument is unsound.
d) the conclusion cannot be true if the premises are false.
e) the conclusion cannot be false if the argument is sound.
f) that is unsound, the conclusion must be false.
g) if the conclusion is false, all of the premises must be false.
h) if the conclusion is false, at least one of the premises must be
false.
6. Suppose some makes the following observation and recommendation.
“The definition of ‘valid’ ensures that the conclusion of a valid
argument will be true if the premises are in fact true. But, this
has the unacceptable result that valid arguments, in general, do
not necessarily prove anything true. It would be much better to
have valid arguments result in proving something true. Hence,
we would be better served by defining a valid argument as one
which does have true premises and which does ensure the truth
of its conclusion. After all, there is no point in arguing from
premises that are not true.”
Carefully consider the pros and cons of this proposal with a view to
deciding whether it is reasonable and desirable.
7. Is it possible for a proposition to be a substitution instance of different
propositional forms? Is it possible for an argument to be a substitution
instance of different argument forms?
8. Is it possible for a valid argument to be a substitution instance of an
invalid argument form?
9. Suppose you are told that there are three different propositions and it is
not possible for all of these three propositions to be true. Does that
information alone allow you to construct any valid arguments using the
three propositions?
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10. The following table represents information regarding an argument with
two premises. Complete each row of the table on the basis of the
information given in that row. Note that in some instances the
appropriate answer might be ‘unknown’.
First
Premise
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
True
True
False
True
True
False
False
Second
Premise
True
True
True
False
True
True
False
True
Valid or
Conclusion
Invalid
Valid
Sound or
Unsound
Unsound
Unsound
True
Valid
Unsound
False
Valid
Invalid
Valid
Sound
True
True
False
False
Invalid
True
Invalid
Valid
Invalid
Unsound
False
11. Show that the following are not formally valid by constructing a
refutation by logical analogy or a counter example.
1. Some large animals are not mammals since all whales are
mammals and some large animals are not whales.
2. No X are Y, hence since all X are Z, no Z are Y.
3. Los Angeles is to the west of Cleveland and Boston. And
Chicago is to the west of Boston. Therefore, Chicago is to the
west of Cleveland.
4. All X are Y and no W are Z, therefore, no X are W since all Z
are Y.
5. If blue litmus paper turns red in the solution, then the solution
is acidic. The blue litmus paper did not turn red in the
solution. Therefore, the solution is not acidic.
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Symbolic Logic – A Primer
Part I: Introduction
§2: Basic Concepts
§3.0
QUICK REFERENCE:
SUMMARY OF BASIC CONCEPTS
PROPOSITIONS
• A proposition is sentence (or a fragment of a sentence able to
function as a sentence), which is understood to have a fixed
meaning and is either true or false; in addition, the truth or falsity of a
proposition holds at all times and is independent of the proposition’s
author.
• A propositional form is the form of a proposition.
• A proposition is a substitution instance of a propositional form if and
only if it can be obtained by uniform substitution from that form.
ANALYTIC PROPOSITIONS
• An analytic proposition is a proposition that is logically (necessarily)
true.
• A proposition is formally analytic if and only if every proposition of
the same form is analytic.
• A propositional form is analytic if and only if all substitution instances
of that form are analytic.
ARGUMENTS
• An argument is an organization of propositions claiming that the
truth of some of its propositions (the premise or premises)
constitutes evidence for the truth of another proposition in the
argument (the conclusion).
• A deductive argument claims that it is not logically possible for its
premises to be true and its conclusion false.
• An argument form is the form or structure of an argument.
• An argument is a substitution instance of a given form if and only if it
can be obtained by uniform substitution from that form.
VALID / INVALID ARGUMENTS
• A deductive argument is valid if and only if it is not logically possible
for the premises to be true and the conclusion false.
• A deductive argument is invalid if and only if it is logically possible for
the premises to be true and the conclusion false. (Or, more
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Symbolic Logic – A Primer
Part I: Introduction
§2: Basic Concepts
concisely: A deductive argument is invalid if and only if it is not
valid.)
• An argument is formally valid if and only if all arguments of that form
are valid.
VALID / INVALID ARGUMENT FORMS
• An argument form is valid if and only if all substitution instances of
that form are valid arguments.
• An argument form is invalid if and only if not all substitution
instances of that form are valid, i.e., there is at least one substitution
instance of the argument form with true premises and a false
conclusion.
SOUND /UNSOUND
• A deductive argument is sound if and only if it is (i) valid and (ii) all of
its premises are true.
• A deductive argument is unsound if and only if it is (i) invalid or (ii) it
has at least one false premise, or both. (Or, more concisely: A
deductive argument is unsound if and only if it is not sound.)
CORRESPONDING CONDITIONALS AND VALIDITY
• The corresponding conditional for any deductive argument, or
argument form, is the conditional whose antecedent is the
conjunction of all of the premises in the argument and whose
consequent is the conclusion of the argument.
• An argument is valid if and only if its corresponding conditional is
analytic.
• An argument is formally valid if and only if its corresponding
conditional is formally analytic.
• An argument form is valid if and only if its corresponding conditional
is an analytic propositional form.
Sherwin Iverson
University of Nevada, Reno
August 2010
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