Corresponding conditional


In logic, the corresponding conditional of an argument is a material conditional whose antecedent is the conjunction of the argument's premises and whose consequent is the argument's conclusion. An argument is valid if and only if its corresponding conditional is a logical truth. It follows that an argument is valid if and only if the negation of its corresponding conditional is a contradiction. Therefore, the construction of a corresponding conditional provides a useful technique for determining the validity of an argument.

Example

Consider the argument A:

Either it is hot or it is cold

It is not hot

Therefore it is cold


This argument is of the form:

Either P or Q

Not P

Therefore Q


or :


P Q

P

____________


The corresponding conditional C is:

IF THEN Q


or :


Q

and the argument A is valid just in case the corresponding conditional C is a necessary truth.
If C is a necessary truth then C entails Falsity.
Thus, any argument is valid if and only if the denial of its corresponding conditional leads to a contradiction.
If we construct a truth table for C we will find that it comes out T on every row in every row. These results confirm the validity of the argument A
Some arguments need first-order predicate logic to reveal their forms and they cannot be tested properly by truth tables forms.
Consider the argument A1:

Some mortals are not Greeks

Some Greeks are not men

Not every man is a logician

Therefore Some mortals are not logicians


To test this argument for validity, construct the corresponding conditional C1, negate it, and see if you can derive a contradiction from it. If you succeed, then the argument is valid.

Application

Instead of attempting to derive the conclusion from the premises proceed as follows.
To test the validity of an argument translate, as necessary, each premise and the conclusion into sentential or predicate logic sentences construct from these the negation of the corresponding conditional see if from it a contradiction can be derived Alternatively construct a truth tree and see if every branch is closed. Success proves the validity of the original argument.
In case of the difficulty in trying to derive a contradiction, one should proceed as follows. From the negation of the corresponding conditional derive a theorem in conjunctive normal form in the methodical fashions described in text books. If, and only if, the original argument was valid will the theorem in conjunctive normal form be a contradiction, and if it is, then that it is will be apparent.