The fundamental logical concepts – implication

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 AB, 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.

implication meaning

A

B

 implication If it rains, then the roofs get wet

T

T

T

If it rains, then the roofs get wet

T

F

F

It is not the case that if it rains, then the roofs don’t get wet

F

T

T

If it doesn’t rain, then the roofs get wet.[there could be another cause that gets the roofs wet]

F

T

F

If it doesn’t rain, then the roofs don’t get wet.
Advertenties

Thank you for your action!

Vul je gegevens in of klik op een icoon om in te loggen.

WordPress.com logo

Je reageert onder je WordPress.com account. Log uit / Bijwerken )

Twitter-afbeelding

Je reageert onder je Twitter account. Log uit / Bijwerken )

Facebook foto

Je reageert onder je Facebook account. Log uit / Bijwerken )

Google+ photo

Je reageert onder je Google+ account. Log uit / Bijwerken )

Verbinden met %s