Propositions, Truth Values, and Logical Connectors in Logic

What is a proposition?

A proposition makes a claim that may be either TRUE or FALSE and must have the structure of a complete sentence.

What are the two possible truth values of a proposition?

T (TRUE) and F (FALSE).

Give an example of a proposition.

Amanda is the fastest runner.

Give an example of a non-proposition.

Close the door!

What is a truth table?

A structure used for tracking possible sets of truth values for the propositions being considered.

What is the negation of a proposition p?

The negation of p is another proposition that makes the opposite claim of p, symbolized as ∼p.

What does the double negation rule state?

The double negation of a proposition, not not p, has the same truth value as p.

What is a conjunction?

The statement 'p and q' is called a conjunction, symbolized as p ∧ q, and is true only if both p and q are true.

What is the truth value of the conjunction 'The capital of France is Paris and Antarctica is cold'?

True (T).

What is a disjunction?

The statement 'p or q' is called a disjunction, symbolized as p ∨ q, and is true unless both p and q are false.

What is the truth value of the disjunction 'Airplanes can fly or cows can read'?

True (T).

What is a conditional proposition?

A statement of the form 'if p, then q', where p is the hypothesis and q is the conclusion.

What is the truth value of a conditional proposition if the hypothesis is true and the conclusion is false?

False (F).

What is the contrapositive of a conditional statement?

The contrapositive is formed by negating both the hypothesis and conclusion and reversing them, symbolized as ∼q → ∼p.

What does it mean for two statements to be logically equivalent?

Two statements are logically equivalent if they share the same truth value.

What is the converse of a conditional statement?

The converse is formed by reversing the hypothesis and conclusion, symbolized as q → p.

What is the inverse of a conditional statement?

The inverse is formed by negating both the hypothesis and conclusion, symbolized as ∼p → ∼q.

How can logical equivalence simplify complex claims?

It allows you to switch the original proposition around with the same truth value to simplify or rephrase wordy claims.

What is the symbol for conjunction?

∧ (and).

What is the symbol for disjunction?

∨ (or).

What is the symbol for negation?

∼ (not).