32. The Square of Opposition and Contradictory Statements

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Lecture Notes

Main Topics #

The Square of Opposition #

• Four types of categorical statements arranged in a square:
  • Universal Affirmative (A): Every B is A
  • Universal Negative (E): No B is A
  • Particular Affirmative (I): Some B is A
  • Particular Negative (O): Some B is not A

True Contradictories vs. Contraries #

• Contradictories (diagonals): One must be true and one must be false, without exception
  • A and O are contradictories
  • E and I are contradictories
• Contraries (top horizontal): Can both be false, but cannot both be true
  • Example: “Every man is wise” and “No man is wise” can both be false
  • “Every man is white” and “No man is white” are both false
• Particular statements (bottom horizontal): Can both be true
  • “Some man is wise” and “Some man is not wise” can both be true

The Problem with Universal Subjects #

• When the subject is universal rather than singular, ambiguity arises about what “some” means
• “Man is wise” is ambiguous—does it mean every man, some man, or potentially multiple interpretations?
• This ambiguity necessitates the square of opposition for clarity

Key Logical Relationships #

• If “Every B is A” is true, then “Some B is A” must be true (subalternation)
• If “Every B is A” is true, then both “No B is A” and “Some B is not A” must be false
• If “Some B is A” is true, then “No B is A” must be false (by contradiction with its diagonal)
• Particular statements cannot determine the truth/falsity of universal statements

Universals vs. Collections #

• Universal: Something one, set of many (not a collection)
• Collection/Class: An aggregation of individuals
• Example distinction: Man is not the same as mankind (a multitude of men)
• Porphyry’s two meanings of genus illustrate the difference
• Modern tendency to think of universals as classes is a mistake arising from imagination

Introduction to the Logic of the Third Act #

• Reasoning is distinct from understanding
• Reasoning is to understanding as motion is to rest
• The English word “understanding” comes from “to stand,” reflecting its static nature
• Reasoning involves movement from one thing to another

Definition and Nature of Reasoning #

• Reasoning = coming to know or to guess a statement from other statements already known or accepted, and because of them
• The phrase “because of them” is essential—mere temporal sequence is insufficient
• Example: One may know “a whole is larger than its parts” and “no odd numbers are even” without these statements being the reason for knowing “man is not a stone”
• Reasoning involves either knowledge (certainty) or guessing (probability)

Knowledge vs. Guessing #

• Knowledge: Certainty about a statement
• Guessing: Probable opinion (stronger form = opinion; weaker form = suspicion)
• Both can have true conclusions, but differ in degree of certitude
• Socrates uniquely claims to know the difference between knowledge and guessing while claiming not to know other things
• The principle of non-contradiction grounds certainty: something cannot both be and not be at the same time

The Principle Underlying Certainty #

• If everything were merely guessing, nothing could be known
• The very claim “I don’t know” presupposes knowledge of the difference between knowing and not knowing
• To deny that statements exist is self-refuting (it is itself a statement)
• To deny that some statements are true is self-refuting (that denial would need to be true)

Key Arguments #

Against Mr. Anti-Statement #

• Objection 1: “Statements don’t exist”
  • Response: This is itself a statement, so statements exist
• Objection 2: “All statements are false; no statement is true”
  • Response: If no statement is true, then this statement is true (contradiction); therefore, some statements are true
• Objection 3: “I hate statements”
  • Response: This is itself a statement; one cannot even express hatred of statements without making one
• General principle: The principle of non-contradiction (something cannot both be and not be) makes certain statements self-evidently true

The Universal Subject Problem #

• When we say “man is wise,” the universal subject creates ambiguity
• We must clarify whether we mean: all men, some men, or potentially both interpretations depending on context
• The square of opposition resolves this ambiguity by distinguishing particular and universal forms

Why Some Universals Cannot Be Both False #

• If “Every man is sitting” is false, it means at least one man is not sitting
• But this alone doesn’t make “No man is sitting” true—many might be sitting
• Therefore both universals can be false simultaneously
• Only the diagonals guarantee one true and one false

Important Definitions #

• Contradictory statements: Statements with the same subject and predicate, one affirmative and one negative, such that one must be true and one must be false, regardless of what the subject and predicate are
• Contrary statements: Universal statements (top of square) that cannot both be true but can both be false
• Subalternation: The relationship where a universal statement’s truth guarantees the corresponding particular’s truth, but not vice versa
• Universal: Something one, set of many (πολλά—polla), not a collection but a single nature common to many
• Reasoning (τὸ διανοεῖσθαι): Motion of the mind from one thing to another; going from known or accepted statements to another statement because of them
• Knowledge (ἐπιστήμη—epistēmē): Certainty and sureness about a statement
• Guessing (δόξα—doxa): Probable opinion about a statement

Examples & Illustrations #

Courtroom Truth #

• Legal proceedings seek simple statement truth: guilty or not guilty, did he or did he not commit the crime
• This illustrates concern with the truth-value of simple propositions

Women’s Beauty #

• Question posed to student: “If I say ‘women are beautiful,’ what do you mean?”
• Illustrates the ambiguity with universal subjects—do you mean all women or some women?

The Square Applied #

• “Every woman is beautiful” vs. “No woman is beautiful”—both can be false
• “Every man is wise” vs. “Some man is wise”—if the first is true, the second must be true (subalternation)
• “Some man is wise” vs. “Some man is not wise”—both can be true

Sitting Men #

• “Every man is sitting” (false) vs. “No man is sitting” (false)—both false simultaneously
• “Some man is sitting” (true) vs. “Some man is not sitting” (true)—both true simultaneously
• Demonstrates why only diagonals are true contradictories

Porphyry’s Meanings of Genus #

• Adam as a genus: one man from whom many descended
• The Adamites as a genus: the multitude descended from one
• Both are meaningful but distinct from the universal “man” used in logic

Self-Refuting Claims #

• “Statements don’t exist”—this very statement contradicts itself
• “No statement is true”—if this is true, then it is false
• “I hate statements”—this expresses hatred through a statement

Notable Quotes #

“Reasoning is to understanding as motion is to rest.” — Berquist, paraphrasing Aristotle

“We allow squares in logic that there are no circles in logic.” — Berquist, on why the square of opposition uses a square, not circular diagrams

“Something one, set of many.” — Berquist’s definition of a universal, explaining how universals relate to individuals

“You know if they had the same B in both of those and the same A, that one of those is true and the other is what, false?” — Berquist, emphasizing that contradictories require no knowledge of what B and A are

“Let us not guess at random about the greatest thing.” — Socrates, quoted by Berquist on the importance of reasoned rather than wild guessing

“You always say the same thing about the same things.” — Accusation against Socrates; Socrates’ response: “That’s better than always saying opposite things about the same thing.” — Berquist, defending consistency against the charge that consistency shows lack of thought

“If one half of the contradiction I see is only probable, then the other half I can’t see as necessarily false, can I?” — Berquist, on the logical relationship between probability and certainty

Questions Addressed #

What is the difference between contradictory and contrary statements? #

• Contradictories (diagonals in the square): Cannot both be true or both be false; one must be true and one must be false
• Contraries (horizontals at top): Cannot both be true but can both be false
• This distinction is crucial for understanding logical opposition with universal subjects

Why do we need the square of opposition? #

• With singular subjects (“Socrates is wise” vs. “Socrates is not wise”), contradiction is obvious
• With universal subjects (“man is wise”), ambiguity arises about meaning
• The square clarifies which pairs of statements are true contradictories

Can both universal statements be false? #

• Yes. “Every man is sitting” is false, and “No man is sitting” is false (some are, some aren’t)
• Therefore the universals are not true contradictories—this is why we must look to the diagonals

Why can particular statements both be true? #

• “Some man is wise” being true does not deny that every man is wise
• “Some man is not wise” being true does not deny that some man is wise
• Therefore both particulars can be true simultaneously

What makes something a self-refuting statement? #

• A statement that, in order to be true, requires its own denial
• “Statements don’t exist” requires the existence of a statement
• “No statement is true” requires itself to be true
• These demonstrate the principle of non-contradiction

How does Socrates claim not to know while knowing? #

• Socrates knows fundamental logical truths (like the difference between knowledge and guessing)
• But he claims not to know about virtue, justice, and other complex matters
• This is consistent because basic logical principles are self-evident while virtue is not

What is the relationship between understanding and reasoning? #

• Understanding is the first act of the intellect (simple apprehension of what something is)
• Reasoning is movement from one understanding to another
• Reasoning presupposes understanding just as motion presupposes rest

Historical and Philosophical Context #

Aristotle and Logic #

• The square of opposition is foundational to Aristotelian logic
• Aristotle’s principle of non-contradiction underlies the self-refuting arguments against Mr. Anti-Statement
• Berquist references Aristotle’s treatment in the Wisdom (likely Metaphysics) and the Sophistic Refutations

Porphyry’s Contribution #

• Porphyry’s Isagoge (Introduction) discusses meanings of genus that precede logic
• His work helps transition from linguistic meanings to logical universals

Thomas Aquinas #

• Thomistic philosophy underlies Berquist’s treatment of the acts of the intellect
• Aquinas’s ordering of the three acts (simple apprehension, judgment, reasoning) structures this lecture series

Medieval Logic #

• Medieval logicians used the term “discourse” (discursus) for reasoning
• The square of opposition was a standard medieval teaching tool

Pedagogical Approaches #

Use of Venn Diagrams and Criticism #

• Berquist warns against using Venn diagrams or circles to represent the square
• These visual aids mislead because they suggest “some” means “definitely some but not all”
• This can lead to false understanding of what particular statements mean
• Reference to Plato’s Parmenides: imagining universals as spread over individuals leads to confusion

The “Road from Senses to Reason” #

• The four kinds of arguments (example, induction, enthymeme, syllogism) form a progression
• This progression moves from singular to universal
• This pedagogical framework helps students understand different types of reasoning

Use of Counterexamples #

• When students hesitate about whether both universal statements can be false, Berquist provides clear examples
• “Every man is white, no man is white”—both false
• “Every man is black, no man is black”—both false
• This makes the point vivid and memorable

Anticipated Confusion #

• Students often mistake “some B is A” for meaning “some but not all”
• Berquist clarifies that “some B is A” is true whether all B are A or only some are A
• This is a critical point for understanding the square correctly