Corresponding conditional (logic)

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In logic a corresponding conditional is a statement whose principal connective is the material implication symbol, and whose antecedent is the conjunction of the premises or an argument and whose consequent is the conclusion of that argument.

An argument is valid if and only of its corresponding conditional is a necessary truth.

Thus an if an argument A were of the form

Either P or Q

Not P

Therefore Q

Then its corresponding conditional C would be:

((PvQ) & ~P)-> Q

A is valid if and only if C is a necessary truth.

If C is a necessary truth then ~C entails The False. It follows that any argument A is valid if and only if the denial of its corresponding conditional leads to a contradiction.

In the above example, starting with denial of the corresponding conditional:

1. ~(((PvQ) & ~P)-> Q) premis
2. ((PvQ) & ~P)& ~Q) definition of ->
3. ((P&~P) v (Q&~P)) & ~Q de morgan
4. ((P&~P)&~Q) v ((Q&~P)&~Q)) de morgan
5. (P&~P) v (Q&~Q) weakening
6. The False
7. ~(((PvQ) & ~P)-> Q)-> The False
8. ~~(((PvQ) & ~P)-> Q) mod ton
9. ((PvQ) & ~P)-> Q) QED

We could of course have, instead, created a truth table for((PvQ) & ~P)-> Q) to show that it is a tautology.

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corresponding conditional