31. Truth and Falsity in Compound Statements

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Lecture Notes

Main Topics #

The Nature of Contradiction (Review) #

• Contradictory statements require: (1) same subject and predicate, (2) one affirmative, one negative
• “Three is an odd number” and “Three is an even number” are NOT contradictories—both affirmative, different predicates
• True contradictory: “Three is an odd number” / “Three is not an odd number”
• The term “some” (particular quantifier) does not imply its own negation
  • “Some women are beautiful” does NOT assert “Some women are not beautiful”
  • These are subcontraries; both can be true
  • Inference that some are not beautiful is a consequence, not what is said

Conditional Statements (If-Then) #

• Structure: Two simple statements joined by “if” and “then”
  • Antecedent: The “if” part (what comes before)
  • Consequent: The “then” part (what comes after)
• Example: “If this number is two, then this number is even”

Truth Conditions for Conditionals #

• Truth: The consequent follows necessarily from the antecedent
• Falsity: The consequent does not follow necessarily from the antecedent
• Crucially: Truth/falsity of a conditional is independent of the truth/falsity of its component statements

Key Examples #

1. True conditional with false components: “If Socrates is a mother, then Socrates is a woman”

   • Both parts are false
   • Yet the conditional is true because motherhood entails womanhood
   • The consequent follows from the antecedent

2. False conditional with true components: “If Mary is a woman, then Mary is a mother”

   • Both parts are true
   • Yet the conditional is false because womanhood does not entail motherhood
   • The consequent does not follow from the antecedent

3. True conditional (both true): “If Mary is a mother, then Mary is a woman”

   • Both parts are true and the consequent follows

Reversibility and Logical Relationships #

• Reversing a conditional does not preserve truth value unless special relationships hold
• Convertible: “If this is two, then this is half of four” is true both ways (half of four is a property of two in the strictest sense—belongs only to two, to every two, and always)
• Non-convertible: “If this is two, then this is less than ten” is true, but reversed is false (less than ten is more universal than the property two)
• When definition and property coincide, reversal is valid
• When a term is more universal than another, reversal fails

Disjunctive Statements (Either-Or) #

• Structure: Two or more simple statements joined by “either…or”
• Examples: “A number is either odd or even” / “A triangle is either equilateral, isosceles, or scalene”
• Contraction: Often contracted to appear as single assertions (e.g., “A triangle is equilateral or isosceles”) when logically expressing multiple alternatives

Truth Conditions for Disjunctives #

• Truth: The statement exhausts all possibilities in the domain
• Falsity: The statement fails to exhaust all possibilities
• Depends on the logic of division: a proper division must empty out the whole without remainder

Examples #

1. True: “A number is either odd or even” (exhaustive of all numbers)
2. False: “A triangle is either equilateral or isosceles” (omits scalene)
3. True: “A triangle is either equilateral, isosceles, or scalene” (exhaustive)

The Problem of Completeness in Division #

• Mathematical divisions are easier to verify as exhaustive (e.g., triangle types by angle and side)
• More abstract divisions require careful analysis (e.g., Porphyry’s five predicables: genus, species, difference, property, accident)
• Even in mathematics, one must sometimes reason to exclude apparent possibilities (e.g., no right-angled equilateral triangle exists because equilateral requires all angles equal, and if one is 90°, the others cannot also be 90°)
• Divisions can sometimes be probable rather than certain

Equivocity of Truth Across Statement Types #

• In simple statements: Truth = conformity of mind with things (what is the case)
• In conditionals: Truth = necessary consequence of antecedent to consequent (not what is the case, but what would follow if the antecedent were true)
• In disjunctives: Truth = exhaustion of all possibilities (completeness of division)
• These meanings are equivocal by reason—related but not identical; the word “true” does not mean the same thing in all contexts

The Role of Conditionals in Reasoning #

• A conditional alone does not tell us about reality; it expresses a logical connection
• To draw conclusions about reality, one must combine:
  1. An if-then statement (expresses necessity)
  2. A simple statement (asserts what is or is not the case)
  3. Conclusion: a simple statement (about reality)
• Example (from Euclid’s sixth theorem): If these sides were unequal, then you could cut off from the greater a line equal to the lesser, creating a contradiction. But this is impossible. Therefore, these sides must be equal.
• We cannot combine two conditionals and derive another conditional as a conclusion when we seek to know how things actually are

Key Arguments #

The Independence Argument (for Conditionals) #

• Truth in conditionals is independent of the truth-values of components
• One can have: (false antecedent + false consequent = true conditional), (true antecedent + true consequent = false conditional)
• Therefore, the truth of a conditional must be grounded in something other than the truth of its parts: namely, in the necessity of consequence

The Exhaustion Argument (for Disjunctives) #

• An either-or statement is true iff it completely divides the subject matter
• A proper division empties the whole; nothing remains outside
• This requires understanding the nature of the subject and recognizing all possible categories
• Mathematical examples make this clearer, but even abstract divisions (e.g., kinds of opposition in God) require careful reasoning

The Non-Equivocity of Truth Across Contexts #

• While truth means something different in simple, conditional, and disjunctive statements, these meanings are not purely equivocal
• There is a connection: every true conditional reflects how things are necessarily related; every true disjunctive reflects an actual division in reality
• But the connection is indirect and requires understanding the context

Important Definitions #

• Conditional statement (if-then statement): A compound statement joining two simple statements via “if” (antecedent) and “then” (consequent), such that truth depends on whether the consequent necessarily follows from the antecedent
• Antecedent: The conditional clause (“if” part); the condition
• Consequent: The resultant clause (“then” part); what is claimed to follow
• Disjunctive statement (either-or statement): A compound statement expressing alternatives, true only if all possibilities are exhausted
• Exhaustive division: A division that accounts for all members or possibilities within a domain, leaving nothing outside
• Equivocal by reason: A term whose meaning differs across contexts but whose uses are rationally connected rather than purely arbitrary

Examples & Illustrations #

Conditionals with Component Truth-Values #

1. False antecedent, false consequent → true conditional: “If Socrates is a mother, then Socrates is a woman”
2. True antecedent, true consequent → false conditional: “If Mary is a woman, then Mary is a mother”
3. True antecedent, true consequent → true conditional: “If Mary is a mother, then Mary is a woman”

Reversible vs. Non-Reversible Conditionals #

• Reversible (strict property): “If this is two, then this is half of four” ↔ “If this is half of four, then this is two”
• Non-reversible (looser predicate): “If this is two, then this is less than ten” (true) but “If this is less than ten, then this is two” (false)

Classroom Example #

• Teacher says, “Some of you have passed,” after grading half the papers
• This does NOT assert “Some of you have failed”; it only asserts those who passed
• Inference that some failed is a consequence, not what is said

Disjunctive Examples #

• True: “A number is either odd or even”
• False: “A triangle is either equilateral or isosceles” (missing scalene)
• True: “A triangle is either equilateral, isosceles, or scalene”
• Requires reasoning: No right-angled equilateral triangle exists (requires knowledge that equilateral = all angles equal = 60° each)
• Can be probable: “A man is either white, or black, or brown, or red, or yellow” (probably exhaustive but not provably so)

From Euclid’s Sixth Theorem #

• If these sides (of a triangle) are unequal, then you can cut off from the greater a line equal to the lesser
• This creates a sub-triangle with two equal sides and angle equal to the original
• By the fifth theorem, the bases must be equal
• But the base of the smaller triangle is part of the original base
• Thus part would equal whole—contradiction
• Therefore, the sides cannot be unequal; they must be equal
• The if-then statement is true; the consequent follows from the antecedent

Questions Addressed #

Q: Can a conditional statement be true if both its components are false? A: Yes, if the consequent follows necessarily from the antecedent. Example: “If Socrates is a mother, then Socrates is a woman” is true even though both parts are false.

Q: Can a conditional statement be false if both its components are true? A: Yes, if the consequent does not follow from the antecedent. Example: “If Mary is a woman, then Mary is a mother” is false even though both parts are true.

Q: What makes an either-or statement true? A: It exhausts all possibilities in the domain. The division must be complete, accounting for everything without remainder.

Q: Does reversing a conditional preserve its truth value? A: Not necessarily. Reversal preserves truth only when the terms are convertible (as in strict properties). Otherwise, the less universal term may not entail the more universal.

Q: Is truth the same concept in simple statements and compound statements? A: No. Truth is equivocal by reason: it means conformity with things in simple statements, necessary consequence in conditionals, and exhaustion of possibilities in disjunctives. These are related but distinct meanings.