Definition: The implication is the relationship between the 2 statements of a conditional statement A, called the antecedent and B, called the consequent, when A implies B.
The statement before the arrow is called the antecedent (ante =before) and the statement following the arrow is the consequent (consequī =follow closely). In if A→B, A= the antecedent and B= the consequent.
Other words that indicate a cause-and-effect relationship are:
of cause: as, because, in order that, since, so that.
of condition: even if, if, in case, provided that, unless.
of Time (general): when, while.
In the (material) implication the 2 statements have a different type of relationship with each other than we have seen so far. In a conjunction or disjunction the 2 statements’ order can be reversed while their compound statement’s meaning stays the same. This relation is called commutative. In the implication however, the 2 statements have a cause-and-effect relationship. This means that the order of the 2 statements cannot be reversed without changing the compound statement’s meaning.
If A=T, then B follows from A. But if A=F, intuitively we might think the statement then is irrelevant, or that the resulting truth value is F. However, from A=F, nothing in particular can be inferred from A; B can follow or not.
The logical implication excludes only 1 meaning: cause without effect (A=T, B=F).
Included are: cause and effect (A=T, B=T), effect without cause (A=F, B=T), without both cause and effect (A=F, B=F).
What does it mean if when we say ‘If it rains, the roofs get wet’? Actually it means 4 things. All those 4 statements combined make up the entire meaning of this single statement.