Invalid arguments with logically true conclusions

Emily asked:

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

Answer by Craig Skinner

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

Answer by Helier Robinson

In modern symbolic, no. In normal reasoning, yes. In symbolic logic validity is determined by truth-tables: a conditional (If p then q) is invalid only if q, the consequent, is false; if p, the antecedent, is false or if p and q are both true then the conditional is valid. But normal reasoning does not agree with this, so the truth-functional conditional is called material implication, to distinguish it from genuine implication. For example, ‘If roses are red then violets are blue’ is invalid in the sense that the colour of violets has nothing to do with the colour of roses. The definition of validity, other than in material implication, is that the truth of the antecedent necessitates the truth of the conclusion.

The peculiarity of material implication means that a false antecedent validly implies anything you please; or, to emphasise the absurdity of this, it means that from a contradiction you can deduce anything you please.

For example, ‘If circles are square then I am the king of France’ is valid. This flaw carries over into quantificational logic, in which anything said of something that does not exist is true. For example, if no mermaids exist then that validly means that all mermaids are butterflies, and also that no mermaids are butterflies. Most logicians believe that this peculiarity of logic is a genuine feature of logic which was hidden until George Boole invented truth-functional logic and revealed it; and it is something that must be simply avoided when using this logic. Others, myself included, think that this is a fatal flaw in the truth functional approach to logic: validity is not dependent simply on the truth of falsity of the antecedent and consequent, but on a relation of necessity between them. This necessity is grasped by the mind but not present between the words or symbols.


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