Chapter 1b: Propositions and Truth Values – Study Notes

Propositions and Truth Values

  • Definitions

    • A proposition makes a claim (either an assertion or a denial) that may be either true or false.

    • It must have the structure of a complete sentence.

    • Every proposition has two possible truth values: T\;=\;true and F\;=\;false.

    • A truth table is a table with a row for each possible set of truth values for the propositions being considered.

  • Key terms

    • p, q, etc. represent propositions used in truth-functional logic.

Negation (Opposites)

  • The negation of a proposition p is a proposition that makes the opposite claim of p.

    • If p is true (T), \neg p is false (F).

    • If p is false (F), \neg p is true (T).

  • Symbol: \neg p (often written as ~p in plain text).

  • Example: Negation

    • Original proposition: "Amanda is the fastest runner on the team."

    • Negation: "Amanda is not the fastest runner on the team."

    • If the negation is false, the original statement must be true (Amanda is the fastest).

Double Negation

  • The double negation of a proposition p, \neg(\neg p), has the same truth value as p.

  • Symbolically: \neg(\neg p) \equiv p.

Example: Radiation and Health (1 of 3)

  • Context

    • Data show an association between low-level radiation and cancer among older workers.

    • A health scientist is asked about the possibility of a similar association among younger workers.

  • The scientist’s quoted statement is a double negation example.

  • Step-by-step translation

    • Let p be: "it's likely that there is an association (between low-level radiation and cancer)."

    • The phrase "it's unlikely that there is an association" is the negation: \neg p.

    • The phrase "no association" modifies the last statement to become "it's unlikely that there is no association," which is \neg\neg p \equiv p.

Example: Radiation and Health (2 of 3)

  • Conclusion about the sentence structure

    • The double negation means the original statement is equivalent to the negation of a negation, i.e., the same as p.

    • Therefore, the scientist’s remark effectively expresses that there is likely an association.

Example: Radiation and Health (3 of 3)

  • Takeaway

    • Because the double negation has the same truth value as the original proposition, we conclude that the scientist believes it likely that there is an association between low-level radiation and cancer among younger workers.

Propositions joined with logical connectors

  • Logical connectors covered: and, or, if…then.

  • Examples with p and q

    • p = "I won the game."

    • q = "It was fun."

    • New propositions:

    • Conjunction: p \land q ("I won the game and it was fun.")

    • Disjunction: p \lor q ("I won the game or it was fun.")

    • Conditional: p \rightarrow q ("If I won the game, then it was fun.")

Given two propositions p and q

  • The statement p \land q is true only if both p and q are true.

  • The statement p \lor q is true if at least one of p or q is true (true unless both are false).

Example: And Statements (1 of 2)

  • Statement set

    • a. "The capital of France is Paris and Antarctica is cold." -> Both propositions true; their conjunction is true.

  • Structure

    • This is an instance of a conjunction where the truth of the combined statement depends on the truth of both parts.

Example: And Statements (2 of 2)

  • Statement set

    • b. "The capital of France is Paris and the capital of America is Madrid." -> The first is true; the second is false; their conjunction is false.

n An inclusive or and an exclusive or

  • Inclusive or means “either or both.”

  • Exclusive or means “one or the other, but not both.”

  • In logic, or is typically interpreted as inclusive unless specified otherwise.

The Logic of Or (Disjunctions)

  • Given p and q, the disjunction p \lor q is true unless both p and q are false.

  • Example: Smart Cows? Airplanes can fly or cows can read.

    • Propositions: (1) airplanes can fly (true), (2) cows can read (false).

    • Since at least one is true, p \lor q is true.

A statement of the form if p, then q (Conditional propositions)

  • Definition

    • A conditional is true unless p is true and q is false.

  • Components

    • Hypothesis: p

    • Conclusion: q

  • Common alternative phrasings

    • p \rightarrow q is equivalent to saying: "q is necessary for p" and/exchangeable forms such as "q if p" and "whenever p, q".

Converse, Inverse, Contrapositive (Variations on the Conditional)

  • Given the base form: If it is raining, then I will bring an umbrella to work.

    • Converse: If I bring an umbrella to work, then it must be raining. q \rightarrow p

    • Inverse: If it is not raining, then I will not bring an umbrella to work. \neg p \rightarrow \neg q

    • Contrapositive: If I do not bring an umbrella to work, then it must not be raining. \neg q \rightarrow \neg p

  • Summary

    • p is the hypothesis; q is the conclusion.

    • Variations show different truth-value relationships between the pair (p, q).

Two statements are logically equivalent

  • Definition

    • Two statements are logically equivalent if they share the same truth values under all interpretations.

  • Term used: logical equivalence.

  • Key point in implications: a statement and its contrapositive are logically equivalent; the contrapositive has the same truth value as the original.

Example: Logical Equivalence (1 of 3)

  • Original statement: If a creature is a whale, then it is a mammal.

    • Let p: "a creature is a whale" and q: "a creature is a mammal."

  • Converse: q \rightarrow p → "If a creature is a mammal, then it is a whale." This is false because most mammals are not whales.

  • Determine logical equivalence between original and converse/inverse/contrapositive as discussed below.

Example: Logical Equivalence (2 of 3)

  • Inverse: \neg p \rightarrow \neg q → "If a creature is not a whale, then it is not a mammal." This is also false; e.g., dogs are not whales but are mammals.

  • Contrapositive: \neg q \rightarrow \neg p → "If a creature is not a mammal, then it is not a whale." This is true, because all whales are mammals.

Example: Logical Equivalence (3 of 3)

  • Note

    • The original proposition and its contrapositive have the same truth value and are logically equivalent.

    • Similarly, the converse and inverse have the same truth value and are logically equivalent.

Connections to the material

  • Propositions, truth values, and logical connectives form the basis of evaluating logical structure.

  • Understanding negation, double negation, and the effects of phrasing (e.g., words like 'unlikely' or 'no' in a sentence) is crucial for proper logical analysis and interpretation of statements.

  • Logical equivalence explains why certain forms (contrapositive, and converses/inverses in some pairs) share truth values, which is essential for logical reasoning and proving statements.

Quick reference of key formulas

  • Negation: \neg p

  • Double negation: \neg(\neg p) \equiv p

  • Conjunction: p \land q

  • Disjunction (inclusive): p \lor q

  • Conditional: p \rightarrow q

  • Converse: q \rightarrow p

  • Inverse: \neg p \rightarrow \neg q

  • Contrapositive: \neg q \rightarrow \neg p

  • Logical equivalence: two statements have the same truth values; original and contrapositive are equivalent; converse and inverse are equivalent.