Double Math

# Implication or conditional

A compound statement of the form if p then q , also written p implies q , is called a
conditional or an implication, p is called the antecedent or hypothesis and q is called the
consequent or the conclusion.
A conditional is regarded as false only when the antecedent is true and consequent is
false. In all other cases it is considered to be true. Its truth table is, therefore, of the adjoining
form.
Entries in the irst two rows are quite in consonance with
common sense but the entries of the last two rows seem to be
against common sense. According to the third row the conditional
If p then q
is true when p is false and q is true and the compound proposition
is true (according to the fourth row of the table) even when both its
components are false. We attempt to clear the position with the help Table (4)
of an example. Consider the conditional
If a person A lives at Lahore, then he lives in Pakistan.
If the antecedent is false i.e., A does not live in Lahore, all the same he may be living in
Pakistan. We have no reason to say that he does not live in Pakistan.
We cannot, therefore, say that the conditional is false. So we must regard it as true. It must be
remembered that we are discussing a problem of Aristotlian logic in which every proposition
must be either true or false and there is no third possibility. In the case under discussion there
being no reason to regard the proposition as false, it has to be regarded as true. Similarly,
when both the antecedent and consequent of the conditional under consideration are false,
there is no justiication for quarrelling with the proposition. Consider another example.