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 p→q 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 p→q is false is if p is true and q is false.
The truth table for p→q is:
| p | q | p→q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
As the truth table above shows p→q is equivalent to ¬p ∨ q (not p or q).
p→q is also equivalent to ¬q→¬p (not p implies not q). This is called a contraposition.
Importantly, p→q 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 q→p (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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