# logical

```Traditional Logic:
Introduction to Formal Logic
Martin Cothran
Introduction
Logic: The Basics (1)
• Logic: The science of
right thinking.
• German philosopher
Immanuel Kant called
Aristotle “the father of
logic.”
• Formal logic has
changed little since
Aristotle.
Two Branches of Logic (2)
• 1.) Material “major”: concerned with the
content of argumentation.
– Deals with the truth of the terms and propositions
in an argument.
• 2.) Formal “minor”: interested in the form of
the structure of reasoning.
– Truth is a secondary consideration; concerned
with the method of deriving one truth from
another.
G. K. Chesterton (2)
“Logic and truth… have very little to do with each
other. Logic is concerned merely with fidelity and
accuracy with which a certain process is performed,
a process which can be performed with any
materials, with any assumption. You can be as
and pigs… Logic, then, is not necessarily an
instrument for finding out truth; on the contrary,
truth is a necessary instrument for using logic—for
using it, that is, for the discovery of further truth…
Briefly, you can only find truth with logic if you have
This means what? (2)
• We should refer to statements as true or false,
not logical or illogical.
• Likewise, arguments are not true or false, but
valid or invalid.
– Validity: helps describe if an argument is logical.
– Truth: the correspondence of a statement to
reality.
Argument Anatomy (3)
• Expect your arguments to take on the general
structure:
Argument
Term
All men are mortal
Premise
Socrates is a man
Premise
Therefore, Socrates is
mortal
Conclusion
Mental Act vs. Verbal Expression
Mental Act
Verbal Expression
Simple Apprehension
Term
Judgment
Proposition
Deductive Inference
Syllogism
Mental Acts (3-4)
• Simple Apprehension: occurs when we first form
in our mind a concept of something.
– EX: thinking of your logic book
• Judgment: to affirm or deny
– You think: “This book is boring.”
• Deductive Inference: when we make the logical
connections in our mind between the terms in
the argument in a way that shows us that the
conclusion either follows or does not follow from
the premises; i.e., when we make progress
Verbal Expression (4)
• Term: the verbal expression of a simple
apprehension
• Proposition: the verbal expression of a
statement
• Syllogism: the verbal expression of a
deductive inference
Drawing everything together… (5)
• Imagine moving from one room to another.
– Moving your foot -&gt; Simple Apprehension
– Taking steps -&gt; Performing judgment
– Everything together -&gt; Deductive inference
Chapter 1
What is Simple Apprehension?
What is Simple Apprehension? (9)
• The introduction said Simple Apprehension
occurred when we first form in our mind a
concept of something.
• Example: Looking at a chair
– 1. We perceive it with our senses
– 2. We form an image of it with our minds
– 3. We conceive of its meaning
1. Sense Perception (9)
• “Sense perception” is
common vocabulary in
all branches of
philosophy
• Definition: The act of
seeing or hearing or
smelling or tasting or
touching.
1. Sense Perception (9)
• We have sense perception while we are in
contact with objects
– Your sense perception of a chair ends when
you stop looking at the chair, etc.
– The sense perception of “chair” is different
than the chair itself because it is in your
mind
2. Mental Image (9-10)
• Definition: The image of an object formed in
the mind as a result of a sense perception of
that object.
• Occurs when an image continues after sense
perception ceases
• Different than both the chair itself, and the
sense perception that it creates
3. Concept (10-11)
• Understanding without a mental image or
sense perception.
• “When you grasp the concept of something,
like a chair, you understand what a chair is.”
• “Simple apprehension is an act by which the
mind grasps the concept of general meaning
of an object without affirming or denying
Other Terms (11)
• Essence: the meaning of a thing
• Abstraction: The process by which a simple
apprehension is derived from a sense
perception and a mental image
– Helps raise a chair from the senses to the intellect
• To affirm or deny a Simple Apprehension is to
engage in judgment
– However, thinking merely “chair” is Simple
Apprehension
Chapter 2
Comprehension and Extension
First things… (15)
• This chapter will discuss the properties of
Simple Apprehension.
– Definitions explain what something is.
– Properties distinguish objects from each other.
• The two properties of Simple Apprehension
– Comprehension: tells the essence of a thing
– Extension: tells us the things to which that
essence applies
Comprehension (15)
• Defined as “the completely articulated sum of
the intelligible aspects, or elements (or notes)
represented by a concept.”
• Note this is NOT the definition you grew up
with. Welcome to life.
• Not all concepts are simple.
– Plato’s definition of man: a “featherless biped”
– Plato later said man is a “rational animal”
What is an animal? (16)
• Animals break into four simple concepts:
– Substance: something rather than nothing
– Material: to have a non-spiritual body
– Living (self-explanatory)
– Sentient: to have senses
• These concepts are called “notes,” or
intelligible aspects represented by a concept.
– See explanation of “Comprehension.”
– A chair has four notes.
What is a man? (16)
• “Man” breaks into five
concepts or notes:
–
–
–
–
–
Substance
Material
Living
Sentient
Rational
• Comprehension of man,
then, equals the sum of
said five notes.
The Porphyrian Tree (17)
• Invented by thirdcentury logician
Porphyry
• Helps us break down
complex concepts into
simple concepts
• Comparable to our
in terms of specificity
Porphyrian Categories (17)
• Substance: material or nonmaterial
– Unicorns have no substance yet chairs do. Do you
know why?
•
•
•
•
Body: living or nonliving (mineral)
Organism: sentient or nonsentient (plant)
Animal: rational or nonrational (brute)
Logical species: man
Extension (19)
• Extensions tells us the things to which that
essence applies. Think “example!”
• What is the extension of man?
– All the men who have ever lived, who are now living,
and who will live in the future
• The greater number of notes a concept has, the
less extension it has.
• Man has five notes while animals have four.
– “Man” is more specific than “animals”
– There are more animals than man.
Important! (19)
• The greater the comprehension a concept has,
the less extension it has; and the more
extension it has, the less comprehension.
• Example: Man has five notes while animals
have four. Thus, man is more specific and
applies to less things.
• The higher on the tree, the more to which the
object applies. The lower, the less.
Simple Apprehension Wrap Up
• Processes of Simple Apprehension
– Sense Perception, Mental Image, Concept
• Two properties of Simple Apprehension
– Comprehension: a description plus categories
– Extension: describes the things to which the
concept applies
• Next week we will have our last real
vocabulary lesson!
Chapter 3: Signification and
Supposition
Overview
• Term: a word or group of words which verbally
expresses a concept (23).
– There are two properties of “terms”
• Signification: defined by if the term is
univocal, equivocal, or analogous (23)
• Supposition: refers to types of existence, such
as verbal, mental, or real (25)
Univocal Terms
• Definition: have exactly
the same meaning no
matter when or how
they are used (23)
• Latin: “unus” (one) +
“vox” (voice)
• EX: photosynthesis,
screwdriver, drill bit
Equivocal Terms (24)
• Definition: although spelled and pronounced
exactly alike, have entirely different and
unrelated meanings
– Latin: “aequus” (equal) + “vox” (voice)
• Example: plane, jar, hang
• “We must all hang together, or assuredly we
will all hang separately.” (Ben Franklin)
Analogous Terms (24)
• Definition: applied to different things but have
related terms
• Unlike equivocal terms, their differing
meanings are related
• Example: “set of wheels”
– Means both “car” and “new tires”
Why does this matter? (24)
• Logic requires an
accurate and consistent
use of the terms
• The English language
has many equivocal and
analogous terms
• In real life, language
confusion is the source
of many arguments
Example Argument (25)
• All NBA basketball players are men
• Dennis Rodman is a good NBA basketball
player
• Therefore, Dennis Rodman is a good man
– This argument is invalid because “good” is used
analogously
– This problem will be explained more in detail in
later lessons
Supposition (25)
• Verbal existence: refers to material
supposition
– EX: “Man” is a three-letter word
• Mental existence: logical supposition
– EX: “Man” has five notes
• Real existence: real supposition
– EX: censored
Summary of Chapters 1-3 (26)
• Three aspects of logic: simple apprehension,
judgment, deductive inference
• Verbally expressed by terms, propositions, and
syllogisms
• In future chapters we will discuss terms in
propositions, then syllogisms (arguments)
Chapter 4: What is Judgment?
Chapter 5: The Four Statements of Logic
Judgment (31)
• From the outset, Judgment (Mental Act) aligns
with Proposition (Verbal Expression)
• Judgment: the act by which the intellect
unites by affirming, or separates by denying
• EX: Man is an animal.
– We are joining “Man” and “animal”
Uniting Concepts in Judgments (31)
• Judgments are made of subjects and
predicates
• Subjects: that about which we are saying
something; the concept which we are
affirming or denying
• Predicates: what we are saying about the
subject; what we are affirming or denying
The Proposition (32)
• Definitions: (1) the verbal expression of a
judgment; (2) a sentence or statement which
expresses truth or falsity
• Not all sentences are propositions (such as
questions, commands, exclamations, etc.)
• Examples
– It is raining today.
– There is a fly in my soup.
Elements of Proposition (32)
• There are three elements to any proposition:
1. The subject-term (S), verbal expression of
subject of a judgment
2. The predicate-term (P), verbal expression of a
predicate of a judgment
3. The copula (C), the word that connects or relates
the subject to the predicate; a form of “to be”
such as “is” or “are”
Examples of Propositions (32)
• Man (S) is (c) an animal (P).
• The little brown-haired boy is very loud.
– Subject: little brown-haired boy
– Predicate: very loud
• Notice how this is similar to algebraic
statements, such as X = Y.
• Modern logic takes this to an extreme,
whereas Classical Logic does not.
Logical Sentence Form (33)
• Sentences must be placed into a proper form
to be handled logically.
• EX 1: “The little brown-haired boy screams
very loudly” is not in logical form.
– We need to rework the predicate portion
• EX 2: “The little brown-haired boy is a child
who screams very loudly.”
The Four Statements of Logic (39)
• Formal Logic has four basic categorical forms:
– A: All S is P.
– I: Some S is P.
– E: No S is P.
– O: Some S is not P.
• The letters come from the Latin “affirmo” and
“nego,” or “to affirm” and “to deny.”
• Note: Non-categorical propositions will not be
covered in this curriculum.
To Affirm or Deny? (40)
• Affirmo
– A: All S is P. (EX: All men are mortal)
– I: Some S is P. (EX: Some men are mortal.)
• Nego
– E: No S is P. (EX: No men are mortal.)
– O: Some S is not P. (EX: Some men are not mortal.)
• Notice the pattern?
The Quantifier (40-41)
• Quantifiers tell us quality and quantity
– Four kinds: All, Some, No, Some… not.
• Quality: affirmative or negative?
– EX: “All men are mortal” affirms about “All men.”
• Quantity: universal or particular?
– Universal: refers to all, not some
– Particular: refers to some, not all
Distinguishing Universals (41)
• When there is no quantifier, we must
determine whether they are universal or
particular.
– EX: “Frogs are ugly”  “All frogs are ugly”
• General rule: All is intended unless some is
clearly indicated.
• EX: Men have gone to the North Pole.
– Does not mean “all.”
– “Some men have gone to the North Pole.”
Closing Thoughts
• Universal/Particular cont.: In “Socrates is a
man,” statement is singular (41).
• We can summarize quality-quantity like this:
– A: Affirmative-Universal
– I: Affirmative-Particular
– E: Negative-Universal
– O: Negative-Particular (42)
Contrary Statements
Chapter 7: Subcontraries and
Subalterns
Categorical Relations (49)
• Categorical statements are related via the
relationship of opposition or equivalence.
– The former has four relationships, the latter three.
• Four ways of opposition.
– Contrary
– Subcontrary
– Subalternate
differ in both quality and quantity.
• Which statements differ in both quality and
quantity?
differ in both quality and quantity.
• Which statements differ in both quality and
quantity?
– A is contradictory to O.
– I is contradictory to E.
• What specific examples can we apply to these
categories?
The First Law of Opposition (53)
Contradictories cannot at the same time be
true nor at the same time false.
• Does this hold for Example 1?
– A: All men are mortal.
– O: Some men are not mortal.
• And for Example 2?
– E: No men are gods.
– I: Some men are gods.
Two More Rules (54)
The Rule of Contraries
– Two statements are contrary to one another if
they are both universals but differ in quality.
– Only one pair: A vs. E (All S is P and No S is P.)
The Second Law of Opposition
– Contraries cannot at the same time both be true,
but can at the same time both be false.
– What statements would this apply to?
The Rule of Subcontraries (61)
• Two statements are subcontrary if they are
both particular statements that differ in
quality.
• Which statements would these apply to?
The Rule of Subcontraries (61)
• Two statements are subcontrary if they are
both particular statements that differ in
quality.
• Which statements would these apply to?
– I and O (“Some S is P” vs. “Some S is not P.”)
– Some men are mortal. vs. Some men are not
mortal.
The Third Law of Opposition (63)
• Subcontraries may at the same time both be
true, but cannot at the same time both be
false.
• Explain the following example:
– I: Some S is P.
– O: Some S is not P.
• Explain something more specific…
The Rule of Subalterns (63)
Two statements are subalternate if they have
the same quality, but differ in quantity.
• They are not opposite, but related
nonetheless.
• Which statements would fall under this
category?
The Rule of Subalterns (63)
Two statements are subalternate if they have
the same quality, but differ in quantity.
• They are not opposite, but related
nonetheless.
• Which statements would fall under this
category?
– A (“All S is P”) and I (“Some S is P”)
– E (“No S is P”) and O (“Some S is not P”)
The Fourth Law of Opposition (63-65)
Subalterns may both be true or both be false. If
the particular is false, the universal is false; if the
universal is false, then the particular is true;
otherwise, their status is indeterminate.
• For A and I statements, if “Some S is P” is
false, then “All S is P” is false.
• For E and O statements, if “Some S is not P” is
false, then “No S is P” is false.
Square of Opposition
Chapter 8: Distribution of Terms
Distribution (71)
• Distribution is the status of a term in regard to
its extension.
– Reminder: Extension is a description of the things
to which a concept applies (19).
• Reminder: Subjects and predicates.
– Where is the subject in “All S is P”?
– Where is the predicate in “All S is P”?
• Distributed is when a term is used universally.
What, then, would undistributed mean? (72)
Distribution of Subject-Terms (73)
Rule: The subject-term is distributed in
statements whose quantity is universal and
undistributed in statements who quantity is
particular.
– Look to the quantifier: All, Some, No, Some… not
Type of Sentence
Subject-Term
A (“All S is P”)
Distributed
I (“Some S is P”)
Undistributed
E (“No S is P”)
Distributed
O (“Some S is not P”)
Undistributed
Distribution of Predicate-Terms (72)
Rule: In affirmative propositions the predicateterm is always taken particularly (and therefore
undistributed), and in negative propositions the
predicate is always taken universally (and
therefore distributed).
Type of Sentence
Subject-Term
Predicate-Term
A (“All S is P”)
Distributed
Undistributed
I (“Some S is P”)
Undistributed
Undistributed
E (“No S is P”)
Distributed
Distributed
O (“Some S is not P”) Undistributed
Distributed
Distribution of the
Predicate-Term in A Statements
Animal
All men are animals.
Man
Distribution of the
Predicate-Term in I Statements
Some dogs are vicious animals
Dogs
Vicious
Things
Distribution of the
Predicate-Term in E Statements
No man is a reptile.
Men
Reptiles
Distribution of the
Predicate-Term in O Statements
Some men are not blind.
Different ways
to Diagram I statements
Some men are carpenters.
Why are the lines dotted?
Different ways to
diagram O statements
Some men are not carpenters.
Chapter 9: Obversion,
Conversion, and Contraposition
Review: Categorical Relations
• Categorical statements are related via the
relationship of opposition or equivalence.
• Four modes of opposition: Contradictory,
Contrary, Subcontrary, and Subalternate (49)
• Three modes of equivalence (81)
– Obversion: works on all statements (82)
– Conversion: for E and I statements (85)
– Contraposition: for A and O statements (85)
Obversion, Part 1 (81)
• To obvert: (1) change the quality and (2)
negate the predicate.
– (1) If affirmative, negate; if negative, affirm.
– Warning: Do not change the quantity.
Statement
Step 1
A: All S is P
No S is P
E: No S is P
All S is P
I: Some S is P
Some S is not P
O: Some S is not P
Some S is P
Obversion, Part 1 (82)
• To obvert: (1) change the quality and (2)
negate the predicate.
– Place “not” in front of it.
• What would this look like for each statement?
Statement
Statement Obverted
A: All S is P
No S is not P
E: No S is P
All S is not P
I: Some S is P
Some S is not non-P
O: Some S is not P
Some S is not P
Double Negation of Predicate-I (82)
• How do we handle “Some S is not non-P”?
1.
2.
3.
4.
Switch the “non” and “not” (sounds better!)
Add prefix in the predicate: im, un, in, ir
Rule of double negation.
• EX 1: Some men are not non-mortal.
• EX 2: Some Pok&eacute;mon are not non-Fire-types.
Double Negation (83)
• The rule of double negation says that a term
which is not non negated is equivalent to a
term that is negated twice (and visa-versa).
– Example: “not not P” to “P”
• O: Some S is not P  Some S is not not not P
– Thus, and obverse O statement is equivalent
Conversion (84)
• Interchange the subject and predicate.
– E: No S is P  No P is S
– I: Some S is P  Some P is S
• Partial conversion of the A
– All dogs are animals  Some animals are dogs.
– Why does this make sense?
• Will this work for A and O statements?
Contraposition (84)
• Three steps: (1) obvert, (2) convert, (3) obvert
and the statement again.
• Example for A statement: All men are mortal.
– Obvert: No men are non-mortal
– Convert: No non-mortals are men.
– Obvert: All non-mortals are non-men.
• For O statements (steps condensed):
– Some S is not P  Some non-P is S.
To Be Updated…
• Review for Chapters 4-9 (85-87)
• Upcoming Chapters
– 10: What is Deductive Inference?
– 11: Terminological Rules for Categorical Syllogisms
– 12: Quantitative Rules for Categorical Syllogisms
– 13: Qualitative Rules for Categorical Syllogisms
Chapter 10:
What is Deductive Inference?
Introduction (95)
• Chapters 4-9 have discussed proposition.
Mental Act
Verbal Expression
Simple Apprehension
Judgment
Deductive Inference
Term
Proposition
Syllogism
• Deductive inference is one kind of reasoning.
• Reasoning: the act by which the mind acquires
new knowledge by means of what it already
knows.
Introduction (96)
• Deductive inference: the act by which the mind
establishes a connection between the antecedent
and the consequent.
• Syllogism: a group of propositions in orderly
sequences, one of which (consequent) is said to be
necessarily inferred form the others (antecedent)
Argument
Term
Definition
All men are mortal.
Premise/Antecedent
Goes before
Socrates is a man.
Premise/Antecedent
Goes before
Therefore, Socrates is mortal.
Conclusion/Consequent
Conclude
Validity (96-97)
• Essential Law of Argumentation: If the
antecedent is true, the consequent must also
be true.
Two corollaries
• 1.) If the syllogism is valid and the consequent
is false, then the antecedent must be false.
• 2.) In a valid syllogism with a true consequent,
the antecedent is not necessarily true.
Corollary 1 (97)
All men are sinners.
My dog Spot is a man.
Therefore, my dog Spot is a sinner.
• This syllogism is valid (premises are true,
therefore conclusion is true).
• But the conclusion is true. Corollary 1 says a
premise is false. Which one is false?
Corollary 2 (97)
All vegetables are philosophers.
Socrates is a vegetable.
Therefore, Socrates is a philosopher.
• The conclusion is true: Socrates is a
philosopher.
• Corollary 2 says although the consequent is
true, its antecedents are false.
Terms in a Syllogism (97-98)
• Major Term: the predicate of the conclusion
• Minor Term: the subject of the conclusion
• Middle Term: appears in both premises, but
not the conclusion.
All menM are mortalP.
SocratesS is a manM.
Therefore, SocratesS is mortalP.
Syllogism Simplified (98)
• Chapters 4-9 matter because of the following:
Argument
Simplified
All menM are mortalP.
All M is P.
SocratesS is a manM.
All S is M.
Therefore, SocratesS is mortalP Therefore, All S is P.
• Notice terms boil into subjects and predicates,
which makes syllogisms possible.
Proper Form (98-99)
• A syllogism is properly formed if the major
premise if first, the minor premise is second,
and the conclusion is third.
– Major premise: contains the major term.
– Minor premise: contains the minor term.
• However, not a syllogisms are properly
formed, meaning we may have to rearrange
them.
Syllogistic Principles (99)
• Principle of Reciprocal Identity: two terms
that are identical with a third term are
identical to each other.
• EX: mortal vs. man, man vs. Socrates
Argument
All menM are mortalP.
Term
SocratesS is a manM.
Therefore, SocratesS is mortalP
S=M
M=M
S = M, or Subject = Predicate
Syllogistic Principles (99)
• Principle of Reciprocal Non-Identity: two
terms, one of which is identical with a third
term and the other of which is non-identical
with that third term, are non-identical to each
other.
Syllogism
Term
No men are angels.
M≠A
Socrates is a man.
S=M
Therefore, Socrates is not an angel.
S ≠ A or S ≠ P
Syllogistic Principles (99)
• Dictum de Omni: what is affirmed universally
of a certain term is affirmed of every term
that comes under the term.
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
• Since every man is mortal, Socrates, an
extension of man, is therefore mortal.
Syllogistic Principles (100)
• Dictum de Nullo: what is denied universally of
a certain term is denied of every term that
comes under that term.
No man is God.
Socrates is man.
Therefore, Socrates is not God.
• This argument denies divinity universally of
men.
Chapter 11: Terminological Rules
for Categorical Syllogisms
The Seven Rules (107)
Terminological Rules
• I. There must be three and only three terms.
• II. The middle term must not occur in the
conclusion.
Quantitative Rules
• III. If a term is distributed in the conclusion,
then it must be distributed in the premises.
• IV. The middle term must be distributed at
least once.
The Seven Rules (107)
Qualitative Rules
• V. No conclusion can follow from two negative
premises.
• VI. If the two premises are affirmative, the
conclusion must also be affirmative.
• VII. If either premise is negative, the
conclusion must also be negative.
Terminological Rules (107-108)
Rule 1: There must be three and only three
terms.
• Can be violated in two ways.
• 1. Fallacy of Four Terms: when there are more
than three clearly distinguishable terms.
• 2. Fallacy of Equivocation: when we use a
term for both its meanings
Fallacy of Four Terms (108)
All mammals (S) have hair (P)
All horses (?) have manes (?)
Therefore, some mammals (S) have hair (P)
Note: None of these terms actually connect.
However, changing one term makes it valid:
All mammals (M) have hair (P)
All horses (S) are mammals (M)
Therefore, all horses (S) have hair (P)
Fallacy of Equivocation (108)
• This fallacy is less obvious. Example below.
– All planes are two-dimensional
– All 747s are planes
– Therefore, all 747s are two-dimensional
• Because the middle terms are use equivocally,
there are four terms.
• Therefore, the argument is not valid.
Terminological Rules (109)
• Rule II: The middle term must not occur in the
conclusion.
• The role of the middle is to connect the major
and minor terms. If it were in the conclusion,
it would stand in for the others.
– All plants are living things
– All animals are living things
– Therefore, all living things are plants or animals
Chapter 12: Quantitative Rules
for Categorical Syllogisms
Quantitative Rules (116)
• Rule III: If a term is distributed in the
conclusion, then it must be distributed in the
premises.
• Prevents us from trying to say more in the
conclusion than what is contained in the
premises.
• Distribution: the status of a term in regard to
its extension (what a term refers to).
Quick Reminders
Two Things To Remember…
• Quantity: Universal or Particular? (115-116)
• Distribution by Category (Chapter 8)
Quantitative Rules (116)
• All angels___ are spiritual beings___
• No men___ are angels___
• Therefore, no men___ are spiritual beings___
– This conclusion presumes all spiritual beings are
angels, but that is not stated in the premises.
– The conclusion says more than the premise allows.
– While the premises are both true, they are not valid.
*Use the Distribution Chart and fill this in using S, P,
and M, and “d” for Distributed, and “u” for…
Quantitative Rules (116-117)
• All angels (Md) are spiritual beings (Pu)
• No men (Sd) are angels (Md)
• Therefore, no men (Sd) are spiritual beings
(Pd)
– Notice the Predicate is distributed in the
Conclusion, but not the Premise. Thus, it violates
Rule III.
Rule III Fallacies (117-118)
Syllogisms that violate Rule III commit the Fallacy of
Illicit Process.
• 1. Fallacy of Illicit Major: when the major term is
distributed in the conclusion, but not the
premises
• 2. Fallacy of Illicit Minor: when the minor term is
distributed in the conclusion, but not the
premises
• Remember: Fallacies are so because they are easy to mix
up. Reach each term for what it is; not what it says in its
respective premise/conclusion.
Rule III Fallacies (118)
Which Rule III fallacy does this fit?
• All men___ are animals___
• All men___ are mortals___
• Therefore, all mortals___ are animals___
– Fill in the terms as they are, and then add the
distribution.
Rule III Fallacies (118)
Which Rule III fallacy does this fit?
• All men (Md) are animals (Pu)
• All men (Md) are mortals (Su)
• Therefore, all mortals (Sd) are animals (Pu)
– The Fallacy of the Illicit Minor, because the minor
term is not distributed in the minor premise.
Quantitative Rules (118)
• Rule IV: The middle term must be distributed
at least once.
• The Fallacy of the Undistributed Middle
• Fill in the following syllogism
– All angels___ are spiritual beings___
– All men___ are spiritual beings___
– Therefore, all men___ are angels___
Quantitative Rules (118)
• All angels (Pd) are spiritual beings (Mu)
• All men (Sd) are spiritual beings (Mu)
• Therefore, all men (Sd) are angels (Pu)
– Notice the middle, although used, was not
distributed. Thus, this syllogism is not valid.
Chapter 13: Qualitative Rules for
Categorical Syllogisms
Review of Rules I-IV (125)
Terminological Rules
• I. There must be three and only three terms.
• II. The middle term must not occur in the
conclusion.
Quantitative Rules
• III. If a term is distributed in the conclusion, then
it must be distributed in the premises.
• IV. The middle term must be distributed at least
once.
Making Predictions…
• If Quantitative Rules dealt with Quantity, that
is, distribution, what might today’s rules do?
• Quality is defined as what?
• How might that affect syllogisms?
Qualitative Rules (126)
• Rule V: No conclusion can follow from two
negative premises.
• Prevents us from saying more in the
conclusion than is stated in the premises.
• When broken, commits The Fallacy of
Exclusive Premises.
An Example (126)
• The following syllogism breaks Rule V
– No plants are animals
– Some minerals are not animals
– Therefore, some minerals are not plants
• While both premises are true, the conclusion
does not follow.
• Remember: No conclusion can follow from
two negative premises!
Qualitative Rules (126)
• Rule VI: If the two premises are affirmative,
the conclusion must also be affirmative.
• Syllogism: Fallacy of Drawing a Negative
Conclusion from Affirmative Premises.
– All men are mortals
– All mortals make mistakes
– Therefore, some things that make mistakes are
not men
Qualitative Rules (126)
• Rule VII: If either premise is negative, the
conclusion must also be negative.
• If breaks, Fallacy of Drawing an Affirmative
Conclusion from a Negative Premise.
– All cannibals are bloodthirsty
– Some accountants are not bloodthirsty
– Therefore, some accountants are cannibals
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