Logic of Conditional Statements

Conditional symbol

→

p in a Conditional

hypothesis; antecedent

q in a Conditional

Conclusion; consequent

Conditionals are also called

Implications

Regular Conditionals in English (p leading)

If p, then q; p implies q; p only if q;

Regular Conditionals in English (q leading)

q whenever p; q unless ~p; q if p

Necessary and Sufficient in English

p if, and only if, q

When is the only time a conditional is False

p is True, q is False (promise not kept)

Logical equivalence of conditional

p → q ≡ ¬p ∨ q

Negation of a Conditional

¬(p → q) ≡ p ∧ ¬q

Variations of Conditionals

Converse, Inverse, and Contrapositive

Converse

q → p; Flip the order

Inverse

¬p → ¬q; Flip the signs

Contrapositive

¬q → ¬p; Flip both

Biconditional

p ↔ q; p if and only if q

Logical Equivalence of Biconditional

(p → q) ∧ (q → p)

Where do conditionals and biconditionals appear in the order of operation?

Together, after AND/OR

Sufficient Condition

r → s; R guarantees S

Necessary Condition

¬r → ¬s; R is required for S to happen

Converse is equivalent to

Inverse

Contrapositive is equivalent to

Regular Conditional

Sufficient is to Regular Conditional as Necessary is to

Converse

Necessary AND sufficient

r ↔ s

p → q, in terms of sufficient and necessary

p is sufficient for q, q is is necessary for p