INTERMEDIATE LOGIC: Lesson 4

Some Types of Implication a Conditional Can Imply

1. Cause/Effect 2. Definition 3. Promises 4. Conditions (both sufficient and necessary)

The Conditional

Also called the hypothetical or material implication, this operator asserts that one component (the antecedent) IMPLIES the other (the consequent). It uses the symbol called a horseshoe (⊃) and is often an if/then-type proposition. It is FALSE IF AND ONLY IF the ANTECEDENT is TRUE and the CONSEQUENT is FALSE.

Rule of Transposition

If p then q IS EQUIVALENT TO If not q then not p. It is symbolized like this (p⊃q) ≡ (∼q⊃∼p)

Antecedent

The proposition that follows the "if."

Consequent

The proposition that follows the "then."

Defining Truth Table for the Conditional

p q p⊃q ________ T T T T F F F T T F F T Note: A conditional is always true if the antecedent is false.

If you have the proposition "IF p THEN q," its translation is...

p⊃q

If you have the proposition "WHEN p, q," its translation is...

p⊃q

If you have the proposition "p IMPLIES q," its translation is...

p⊃q

If you have the proposition "p IS SUFFICIENT FOR q," its translation is...

p⊃q

If you have the proposition "p ONLY IF q," its translation is...

p⊃q

If you have the proposition "p IF q," its translation is...

q⊃p

If you have the proposition "p IS NECESSARY FOR q," its translation is...

q⊃p

If you have the proposition "UNLESS p, q," its translation is...

∼p⊃q

If you have the proposition "p UNLESS q," its translation is...

∼q⊃p