Corresponding conditional

In logic, the corresponding conditional of an argument (or derivation) is a material conditional whose antecedent is the conjunction of the argument's (or derivation'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. The construction of a corresponding conditional therefore provides a useful technique for determining the validity of 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 (using standard symbols of the propositional calculus):

P $\lor$ Q
¬P
____________
Q

The corresponding conditional C is:

IF (P or Q) and not P) THEN Q

or (using standard symbols):

((P $\lor$ Q) & ¬P)→ 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 (The False).

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 (true) on every row (and of course if we construct a truth table for the negation of C it will come out F (false) 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 (you will need first-order predicate logic), 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 (a) translate, as necessary, each premise and the conclusion into sentential or predicate logic sentences (b) construct from these the negation of the corresponding conditional (c) see if from it a contradiction can be derived (or if feasible construct a truth table for it and see if it comes out false on every row.) Alternatively construct a truth tree and see if every branch is closed. Success proves the validity of the original argument.

In case of difficulty trying to derive a contradiction 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.

References

• First-order Logic: An Introduction

By Leigh S. Cauman Published by Walter de Gruyter, 1998 ISBN 3-11-015766-7, ISBN 978-3-11-015766-6, Page 19

• The Cambridge Companion to Mill

By John Skorupski Published by Cambridge University Press, 1998 ISBN 0-521-42211-6, ISBN 978-0-521-42211-6, PAge 40

• The Languages of Logic: An Introduction to Formal Logic

By Samuel D. Guttenplan Published by Blackwell Publishing, 1997 ISBN 1-55786-988-X, 9781557869883, page 90.

• The Value of Knowledge and the Pursuit of Understanding

By Jonathan L. Kvanvig Published by Cambridge University Press, 2003 ISBN 0-521-82713-2, ISBN 978-0-521-82713-3, page 175

• Logic

By Paul Tomassi Published by Routledge, 1999 ISBN 0-415-16696-9, ISBN 978-0-415-16696-6, page 153