Traditional Logic I by Martin Cothran

(Louisville: Memoria Press, 2000), 150

The kiln tests the potter's vessels; so the test of a man is in his reasoning.
-Sir-27-6

"Mom?"
"Yes?"
"What's lodge-ick?"
"Logic? Why, dear, logic is knowing what things are true and not true."
– Ray Bradbury, The Illustrated Man ("Zero Hour", 260)

Notes

Introduction: What is Logic?

Definitions

• Josiah Royce: "The science of order."
• Raymond McCall: "Logic in general is the science of right thinking."
• Jacques Maritain: "Logic is the art which enables us to proceed with order, ease, and correctness in the act of reason itself."
• Irving Copi: "The distinction between correct and incorrect reasoning is the central problem with which logic deals."

History of Logic

• Aristotle (384-322 BC) is the "father of logic"
• Chrysippus (279-206 BC) laid the foundation for symbolic logic, which was picked up by Leibniz (1646-1716)
• John Stuart Mill (1806-1873) pioneered induction

Two Branches of Logic

• Formal (minor) Logic: concerned with the *form or structure of reasoning, or the method of deriving one truth from another
• Material (major) Logic: concerned with the content of argumentation and the truth of the terms

You can only find truth with logic if you have already found truth without it.
–G.K. Chesterton

Truth, Validity, Soundness

• A statement is either true or false. A true statement corresponds to reality.
• The structure of an argument is valid if its conclusion follows logically from its premises, otherwise it is invalid.
• An argument as a whole is sound if all premises are true and the argument is valid.

Components of an Argument

• Term: the verbal expression of the mental act of simple apprehension
• Proposition: the verbal expression of the mental act of judgment, whereby we affirm that something is something, or deny that something is not something
• Syllogism: the verbal expression of the mental act of deductive inference, whereby we make the logical connections between the terms in the argument in a way that shows us that the conclusion either follows or does not follow from the premises

Simple Apprehension (Term)

Chapter 1: What Is Simple Apprehension?

• Sense Perception: the act whereby your senses present an object to your mind; you perceive an object by means of your senses
• Mental Image: the image of an object formed in the mind as a result of a sense perception of that object
• Simple Apprehension: is an act by which the mind grasps the general concept of an object without affirming or denying anything about it.
  • Simple apprehension allows us to grasp the essence of a thing.
• Abstraction: is the process by which a simple apprehension is derived from a sense perception and mental image.
• Sense perception → Mental image → Simple apprehension

Chapter 2: Comprehension and Extension

• The two properties of simple apprehension (concepts) are comprehension and extension
• Comprehension is the completely articulated sum of the intelligible aspects or elements (or notes) represented by a concept.
  • A comprehension answers the question, "What is a <blank>?"
  • A Note is the simple (atomic) concept that cannot be broken into simpler parts
  • The Porphyrian Tree (invented by third-century logician Porphyry) allows us to identify the notes associated with a concept

graph TD

a(<strong>Category:</strong> Substance)
b(Material)
c(Non-Material)
d(<strong>Remote Genus:</strong> Body)
e(Living)
f(Non-Living)
g(<strong>Remote Genus:</strong> Organsm)
h(Sentient)
i(Nonsentient)
j(<strong>Proximate Genus:</strong> Animal)
k(Rational)
l(Nonrational)
m(<strong>Logical Species:</strong> Man)

classDef r fill:#f96

a:::r --> b
a --> c
b --> d:::r 
d --> e
d --> f
e --> g:::r 
g --> h
g --> i
h --> j:::r 
j --> k
j --> l
k --> m

• Extension tells us to which things an essence applies.
  • An extensions answers the question, "To what does the concept <blank> refer?
• The greater number of notes a concept has, the less extension it has (something more specific is less applicable).

Chapter 3: Signification and Supposition

• A Term is the verbal expression of a concept.
• The two properties of terms are signification and supposition.
• Signification is a way of dividing terms:
  • Univocal terms are terms that have exactly the same meaning no matter when or how they are used.
  • Equivocal terms have different and unrelated meanings.
  • Analogous terms are applied to different things but have related meanings.
  • →Logic requires an accurate and consistent use of terms (which is an issue with equivocal and analogous terms).
• Supposition is a way of dividing terms:
  • Material supposition is when a term refers to something as it exists verbally
  • Logical supposition is when a term refers to something as it exists logically or mentally
  • Real supposition is when a term refers to something as it exists in the real world

Judgment (Proposition)

Chapter 4: What is Judgment?

• Judgment is the act by which the intellect unites the subject and predicate by affirming, or separates them by denying.
  • Judgment is a mental act whose verbal expression is a proposition.
• A Proposition is a statement which expresses truth or falsity. A proposition consists of three elements: the subject-term (S), the predicate-term (P), and the copula (c, usually "is" or "are"), which must be arranged in logical form.
  • $\text{(Man)}^S \text{ (is)}^c \text{ (an animal)}^P$
  • "The little brown-haired boy screams very loudly" becomes: $\text{(The little brown-haired boy)}^S \text{ (is)}^c \text{ (a child who screams very loudly)}^P$

Chapter 5: The Four Statements of Logic

• The Four Statements of Logic
  • A: All S is P (affirmo)
  • I: Some S is P
  • E: No S is P (nego)
  • O: Some S is not P
• Quantifiers include: all, some, no, some...not
• The Quality of a proposition is affirmative or negative
• The Quantity of a proposition has to do with whether it is universal or particular (or singular, treated as universal). Statements are assumed to be universal unless stated otherwise.
  • A: Affirmative-Universal
  • I: Affirmative-Particular
  • E: Negative-Universal
  • O: Negative-Particular

Chapter 6-7: Contradictory, Contrary, Subcontrary, and Subalternate Statements

• Categorical statements can be related in Opposition or in Equivalence
• Opposition: statements in opposition affirm and deny the same predicate of the same subject
  • Contradictory
  • Contrary
  • Subcontrary
  • Subaltern

• Equivalence

  • Obversion
  • Conversion
  • Contraposition

• Rule of Contradiction: Contradictory statements are statements that differ in both quality and quantity. Same colors (opposites) are contradictory:

• The Rule of Contraries: Two statements are contrary to one another if they are both universals but differ in quality. A (red) and E (purple) are contrary:

• The Rule of Subcontraries: Two statements are contrary to one another if they are both particular statements that differ in quality. I (red) and O (purple) are contrary:

• The Rule of Subalternates: Two statements are subalternate if they have the same quality, but differ in quantity. A (red) and I (red) are subalternate, and E (purple) and O (purple) are subalternate:

• First Law of Opposition: Contradictories cannot at the same time be true nor at the same time be false.

• The Second Law of Opposition: Contraries cannot at the same time both be true, but can at the same time both be false.

  • The Third Law of Opposition: Subcontraries may at the same time both be true, but cannot at the same time both be false.

  • The Fourth Law of Opposition: Subalterns may both be true or both be false. If the particular is false, the universal is false; if the universal is true, the particular is true; otherwise their status is indeterminate.

  • Logical Opposition Chart - Google Slides

Chapter 8: Distribution of Terms

• Distribution is the status of a term in regard to its extension.
  • A term is distributed if it refers to all the members of the class of things denoted by the term; a distributed term is universal.
  • A term is undistributed if it refers to only some members of the class denoted by the term.
• The subject-term is distributed in statements whose quantity is universal and undistributed in statements whose quantity is particular.

• The predicate-term is always undistributed for affirmative propositions and distributed for negative propositions.

Chapter 9: Obversion, Conversion, and Contraposition

• Two statements are logically the same if they are logically equivalent.
• Statements can be made equivalent through Obversion, Conversion, or Contraposition.
• Obversion: 1) Change the quality of the sentence, 2) Negate the predicate
  • 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 → All S is not non-P
  • O: Some S is not P → Some S is not P
• Conversion: 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 statement is done by interchanging the subject and predicate and changing the statement from universal to particular: All S are P → Some P are S
  • Conversion does not work with O statements and other A statements.
• Contraposition: 1) Obvert the statement, 2) Convert the statement, 3) Obvert the statement again
  • A: All S is P → All non-P is non-S
  • O: Some S is not P → Some non-P is S
  • Only works with A and O statements.
  • E statement can be partially converted.

Deductive Inference (Syllogism)

Chapter 10: What Is Deductive Inference?

• Reasoning is the act by which the mind acquires new knowledge by means of what it already knows.
• Deductive inference is a form of reasoning by which the mind establishes a connection between the antecedent and the consequent.
• A Syllogism is a group of propositions in orderly sequence, one of which (the consequent) is said to be necessarily inferred from the others (the antecedent).
  • A syllogism will always contain two premises and a conclusion.
• The Essential Law of Argumentation: If the antecedent is true, the consequent must also be true.
  • Corollary 1: If the syllogism is valid and the consequent is false, then the antecedent must be false.
  • Corollary 2: In a valid syllogism with a true consequent, the antecedent is not necessarily true.
• Terms in a Syllogism:
  • Major Term: predicate of the conclusion
  • Minor Term: subject of the conclusion
  • Middle Term: term that appears in both premises but not the conclusion
• Premises in a Syllogism:
  • Major Premise: premise which contains the major term
  • Minor Premise: premise which contains the minor term
• Principles of Syllogism:
  • The Principle of Reciprocal Identity: Two terms that are identical with a third term are identical to each other.
  • The Principle of Reciprocal Non-Identity: Two terms, one of which is identical with a third term and the other of which is nonidentical with that term, are nonidentical to each other.
  • The Dictum de Omni: What is affirmed universally of a certain term is affirmed of every term that comes under that term.
  • The Dictum de Nullo: What is denied universally of a certain term is denied of every term that comes under that term.

Chapter 11-13: Rules for Categorical Syllogisms

• Terminological Rules
  • There must be three and only three terms. (Violated if there are four unconnected terms, or by the Fallacy of Equivocation where a term is used in different senses.)
  • The middle term must not occur in the conclusion.
• Quantitative Rules
  • If a term is distributed in the conclusion, then it must be distributed in the premises. (Violated by the Fallacy of Illicit Process.)
  • The middle term must be distributed at least once. (Violated by the Fallacy of the Undistributed Middle.)
• Qualitative Rules
  • No conclusion can follow from two negative premises. (Violated by the Fallacy of Exclusive Premises.)
  • If the two premises are affirmative, the conclusion must also be affirmative. (Violated by the Fallacy of Drawing a Negative Conclusion from Affirmative Premises.)
  • If either premise is negative, the conclusion must be negative. (Violated by the Fallacy of Drawing an Affirmative Conclusion from a Negative Premise.)

Chapter 14: Review

Topic: Logic

Source

• Aquinas

Created: 2024-07-06-Sat
Updated: 2025-02-11-Tue