implication (logic)

In logic, an implication, also known as a material conditional or material consequence, is a binary, logical operator that defines a relationship between two statements, an antecedent and a consequent.  Implications are most often symbolized using a right arrow, →. Given two statements p and q, a conditional statement of the form pq is read as “if p then q.” In this statement, p is the antecedent and q is the consequent.

A material conditional statement, an implication, does not imply causality. p does not necessarily cause q, but whenever p is true, so is q. An implication tells us nothing of the truth value of p when we only know that q is true.  In this case p can be either true or false. The only way the state pq is false is if p is true and q is false.

The truth table for pq is:

p q pq
T  T T


As the truth table above shows pq is equivalent to ¬p ∨ q (not p or q).

pq is also equivalent to ¬q→¬p (not p implies not q).  This is called a contraposition.

Importantly, pq is not the same ¬p→¬q. This called an inversion, and it is not true. Not p does not imply not q.

¬p→¬q the same as saying qp (q implies p), called a conversion, which is not true because, as was stated above, knowing the truth value of q tells us noting about the truth value of p.

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