a proposition logically implies the proposition if in every model in which is true, is also true. we write . the former is called the antecedent, and the latter is called the consequent.
a set of propositions logically implies the proposition if in every model in which all propositions of are correct is also correct. we write .
there is a slight difference between logical implication and material implication, as the latter is more "primitive".
let
| A | B | ||
|---|---|---|---|
| F | F | T | F |
| F | T | T | F |
| T | F | F | F |
| T | T | T | T |