Traditional Logic: Introduction to Formal Logic

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 logical about griffins and basilisks as about sheep 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 already found truth without it.”

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 -> Simple Apprehension – Taking steps -> Performing judgment – Everything together -> Deductive inference

Chapter 1What 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 anything about it.”

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 2Comprehension 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 third-century logician Porphyry

• Helps us break down complex concepts into simple concepts

• Comparable to our “Ladder of Abstraction” 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, table saw, Phillips head 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 about

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)

Chapter 6: Contradictory and 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. – Contradictory– Contrary– Subcontrary – Subalternate

The Rule of Contradiction (49-50)

Contradictory statements are statements that differ in both quality and quantity. • Which statements differ in both quality and

quantity?

The Rule of Contradiction (49-50)

Contradictory statements are statements that 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 predicate-term 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

Man

All men are animals.

Distribution of the Predicate-Term in I Statements

Dogs Vicious Things

Some dogs are vicious animals

Distribution of the Predicate-Term in E Statements

Men Reptiles

No man is a reptile.

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 1A: All S is P No S is PE: No S is P All S is PI: Some S is P Some S is not PO: 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 PE: No S is P All S is not PI: Some S is P Some S is not non-PO: 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. Have two adjacent “not’s”2. Switch the “non” and “not” (sounds better!) 3. Add prefix in the predicate: im, un, in, ir4. Rule of double negation.

• EX 1: Some men are not non-mortal. • EX 2: Some Pokémon 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)– Located in your book. Very helpful!

• 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.

• Deductive inference is one kind of reasoning. • Reasoning: the act by which the mind acquires new

knowledge by means of what it already knows.

Mental Act Verbal ExpressionSimple Apprehension Term

Judgment PropositionDeductive Inference Syllogism

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:

• Notice terms boil into subjects and predicates, which makes syllogisms possible.

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.

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. SocratesArgument TermAll menM are mortalP. M = MSocratesS is a manM. S = MTherefore, SocratesS is mortalP S = M, or Subject = Predicate


• 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 ≠ ASocrates is a man. S = MTherefore, Socrates is not an angel. S ≠ A or S ≠ P


• 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.


• 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)


• 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.


• 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___


The Correct Answer • 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!


• 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


• 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

Traditional Logic: Introduction to Formal Logic

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