Propositions and Truth Values – Study Notes

Propositions and Truth Values (Overview)

• A proposition makes a claim (an assertion or a denial) that may be either true or false.
• It must have the structure of a complete sentence.
• Propositions can be represented by letters (e.g., p, q).
• Truth values: True (T) or False (F).
• A truth table collects the truth values for propositions across all possible combinations of truth values of the involved propositions.
• Examples use p and q as basic propositions; truth values are T or F.

Definitions

• Proposition: a declarative sentence that can be classified as true or false.
• Not every sentence that looks like a sentence is a proposition (e.g., imperatives are not propositions).
• Sentences may be represented by letters such as p, q to simplify discussion of logical forms.
• Truth values: T = true, F = false.
• Truth table: a table listing each possible combination of truth values for the propositions involved and the resulting value of a compound statement.

Example 1 – Propositions

• a) "Please, sit down over there." → Not a proposition (an imperative).
• b) "All cats dislike dogs." → A proposition; its truth value can be determined.
• Truth value given: the claim is false because some cats do like some dogs.

Example 2 – Truth Tables

• Given statements: p, q.
• Possible combinations (p, q):
• (T, T)
• (T, F)
• (F, T)
• (F, F)
• For the conjunction p ∧ q, the truth values are:
  • (T, T) → T
  • (T, F) → F
  • (F, T) → F
  • (F, F) → F
• Notation: the conjunction is written as p \land q with truth table showing that it is true only when both p and q are true.

Negation

• Negation of a proposition p is a new proposition ¬p (often written as ~p).
• Semantics: if p is true, ¬p is false; if p is false, ¬p is true.
• Truth table for negation:
• p: T → ¬p: F
• p: F → ¬p: T
• Notation: The symbol for negation is the tilde, ~ (or in some texts ¬).
• Expression form: not p is represented as ~p.

Example 3 – Negations

• a) "Tom is a cat." → Negation: "Tom is not a cat."
• b) "Jerry is not a mouse." → Negation: "Jerry is a mouse."

Double Negation

• Law: ¬¬p is logically equivalent to p (same truth value).
• Truth-table for p, ¬p, ¬¬p:
• p = T: ¬p = F, ¬¬p = T
• p = F: ¬p = T, ¬¬p = F
• Expressed succinctly: \neg(\neg p) \equiv p.

Example 4 – Radiation and Health (Natural-language to logic)

• Scenario: A health scientist is quoted about a potential association between low-level radiation and cancer among younger workers.
• Quoted claim: "My opinion is that it’s unlikely that there is no association." (i.e., not having an association is unlikely.)
• Question: Does the scientist think there is an association between low-level radiation and cancer among younger workers?
• Analysis: The phrase contains a double negation in ordinary language terms; the key idea is that the statement is interpreted as asserting an association, via a double negation form.
• Translation: The statement is interpreted as suggesting there is an association. The reasoning notes that the phrase the way it’s constructed leads to a form equivalent to ¬(¬p) where p means “there is an association.” Therefore, it is interpreted as supporting the proposition p (there is an association).
• Conclusion: The scientist’s statement effectively expresses that it is likely that there is an association between low-level radiation and cancer among younger workers.

Logical Connectors

• Propositions are often joined by logical connectives: and, or, if…then.
• Example setup: p = "I won the game."; q = "It was fun."
• Conjunction: p ∧ q means "p and q"; true only if both p and q are true.
• Disjunction: p ∨ q means "p or q"; truth table will show it is true if at least one of p, q is true.
• Conditional: p → q means "if p, then q"; truth table shows it is true except in the case p is true and q is false.
• Notation: the conditional is written as p \rightarrow q.
• Hypothesis and conclusion: In p → q, p is the hypothesis (or antecedent); q is the conclusion (or consequent).
• Alternative phrasings for conditionals:
• p is sufficient for q
• q is necessary for p
• p will lead to q
• q if p
• p implies q
• q whenever p

Conjunctions (∧)

• Definition: The statement p ∧ q is a conjunction; true only if both p and q are true.
• Truth table (repeated for clarity):
• p T, q T → T; p T, q F → F; p F, q T → F; p F, q F → F
• Symbol: ∧

Or (Disjunctions)

• Two interpretations of “or”:
• Inclusive or: means "either or both" (the default when not specified).
• Exclusive or: means "one or the other, but not both." (explicitly distinguished when needed).
• In logic, or is assumed inclusive unless stated otherwise.

Disjunctions (p ∨ q)

• Definition: The statement p ∨ q is a disjunction; true unless both p and q are false.
• Truth table:
• p T, q T → T
• p T, q F → T
• p F, q T → T
• p F, q F → F
• Symbol: ∨

Conditionals (If p then q)

• Definition: The statement p → q is a conditional; true unless p is true and q is false.
• Truth table:
• p T, q T → T
• p T, q F → F
• p F, q T → T
• p F, q F → T
• Interpretation: p is the hypothesis (or antecedent); q is the conclusion (or consequent).
• Notation: p \rightarrow q
• Alternative phrasings (listed again for emphasis):
• If p, then q
• p is sufficient for q
• q is necessary for p
• p will lead to q
• q whenever p

Variations on the Conditional

• Given concrete statements:
• p = It is raining.
• q = I will bring an umbrella.
• Conditional: If it is raining, then I will bring an umbrella.
• Converse: If I bring an umbrella, then it is raining.
• Inverse: If it is not raining, then I will not bring an umbrella.
• Contrapositive: If I do not bring an umbrella, then it must not be raining.
• Logical relationships:
• If p then q and If q then p are not logically equivalent (these are different directions).
• If not p, then not q is the inverse; If not q, then not p is the contrapositive of the converse, etc. (understanding of how these relate is part of the study).

Logical Equivalence

• Two statements are logically equivalent if they share the same truth values across all possible assignments.
• Common equivalences to remember:
• p → q ≡ ¬p ∨ q
• p → q ≡ ¬q → ¬p (contrapositive)
• q → p ≡ ¬p → ¬q (inverse of the converse)
• p ↔ q is logically equivalent to (p → q) ∧ (q → p)
• Summary idea: some pairs of statements always share the same truth values, making them interchangeable in logical reasoning.

Conditional vs Contrapositive; Converse vs Inverse (Logical Equivalence)

• Conditional statements and contrapositive statements are logically equivalent: p → q ≡ ¬q → ¬p.
• Converse statements and inverse statements are logically equivalent: q → p ≡ ¬p → ¬q.

Example 5 (Practice)

• Prompt: Let’s come up with a conditional statement and find the converse, inverse, and contrapositive.
• Note: The slide ends here with an exercise for you to apply the concepts above.

Quick reference recap (key symbols)

• Conjunction: p \land q
• Disjunction: p \lor q
• Conditional: p \rightarrow q
• Negation: \neg p or \sim p
• Equivalence/bi-conditional: p \leftrightarrow q
• Contrapositive: \neg q \rightarrow \neg p
• Conjunction truth: true only when both inputs are true; disjunction is true if at least one input is true.

Notation and conventions used in these notes

• Uppercase T/F denote truth values.
• Lowercase p, q denote simple propositions.
• Logical connectives are presented in standard symbolic form with their respective truth tables.
• LaTeX formatting for equations is enclosed in double dollar signs as requested:
• Examples: p \rightarrow q, p \land q, p \lor q, \neg p, \neg \neg p, p \leftrightarrow q