# Invalid arguments with logically true conclusions

Can a deductively invalid argument have a conclusion that is logically true?

Yes. A logical truth is a tautology, and so, being true full stop, needs no support from argument. Hence it can be the conclusion of any premises at all, or none, and therefore of an invalid argument, of a valid but unsound argument, of a valid and sound argument, or of a silly argument.

An illustration:

1. Invalid argument.

P1 All prime numbers are less than 20
P2 19 is less than 20
C 19 is a prime number

Here the C is logically true, but doesn’t follow from the Ps — it doesn’t follow from P1 that all numbers less than 20 are prime.

2. Valid but unsound argument.

P1 All numbers less than 20 are prime
P2 19 is less than 20
C 19 is a prime number

Now C does follow from the Ps. Argument therefore valid. But unsound because P1 is false

3. Valid, sound argument.

P1 A number divisible only by itself and 1 is a prime number
P2 19 is divisible only by itself and 1
C 19 is a prime number

4. Silly argument.

P1 Grass is blue
P2 Poodles are cats
C 19 is a prime number

Barely an argument, Ps completely irrelevant to C, but so what, C still true

That a logical truth follows from anything is one of the two ‘paradoxes of implication’ in classical logic. The other is that anything follows from a contradiction.

To tidy up the first, some logicians say that the Ps must be relevant to the C (relevantist logic) although it is hard to spell this out rigorously.

To tidy up the second, some logicians accept that there are some true contradictions (paraconsistency) but that this needn’t entail the ‘explosion’ whereby anything and everything follows (dialetheic logic).

These other logics are interesting, but for everyday living and for philosophical argument, classical logic is just fine.

Craig Skinner