Validity Now and Then

It is often said that an argument is valid if and only if it is impossible for its premises to be jointly true and its conclusion false. Usually there is little harm in saying this but it places the concept of truth at the very heart of logic and, given how complex and obscure that concept is, one might wonder if trouble arises from this.

It does — in at least two contexts. One of these was explored in the first half of the fourteenth century by Jean Buridan and by the mysterious figure known as the Pseudo-Scotus of the Questions on the Prior Analytics printed in the edition of Scotus's works edited by Luke Wadding. Buridan thought that the bearers of truth were particular sentence-tokens; he thought of truth as a property of those tokens and he thought that nothing had properties unless it existed. There were, he pointed out, cases in which things could be as a sentence claimed them to be but in which the sentence could not be true. Such were cases in which the existence of the sentence was incompatible with the claim made by the sentence. For example, the sentence

1. 1. 'No sentence is negative.'

cannot be true because any situation in which it exists contains at least one negative sentence — to wit that one. Still if sentence tokens are contingent objects it could be that there are no negative ones so things could be as the sentence claims them to be.

Consider the argument:

                    Every sentence is affirmative.

Therefore, no sentence is negative.

Intuitively this is a valid argument but its premise can be true and its conclusion cannot be true, so if validity is connected with truth in the way usually said then the argument is not valid. It is possible for the premise to be true and the conclusion not to be true.

Buridan and the nominalist tradition of which he was part were a little unusual in making it out that contingent objects — sequences of sounds in the air, collections of marks on surfaces and the like — were the bearers of truth and one might think that if one were a little more liberal ontologically, if, for example, one introduced eternal abstract objects such as the propositions or statements as the bearers of truth, these problems would disappear. But, as the Pseudo-Scotus, who does not seem to have been a tokenist, and as Buridan himself points out, the Liar Paradox and its relatives provide another class of cases in which intuitive arguments are problematic on the usual account of validity.

Here is a case that Buridan considers:

1. 1. A human is a donkey or (1) is false.

2. 2. A human is not a donkey.

3. Therefore, 3. (1) is false.

This is intuitively a good argument — a case of disjunctive syllogism. Suppose the usual account of validity on which a good argument preserves truth. (2) is clearly true so (1) is true if and only if its second disjunct is true. Suppose the second disjunct is true. Then (1) is true (since it is a disjunction with a true disjunct). Hence the conclusion (3) is true (since both premises are true and the argument is a good one). But then (1) is false after all (since that is what (3) claims) and so both disjuncts are false and so the conclusion (3) is after all false. We have thus reasoned by truth-preserving reasoning to a contradiction — that (3) is both true and false.

Buridan's solution to these problems was:

1. 1. to deny the necessity of the conditional, if p then Tp, and insist that further conditions beyond it being the case that p need be met for it to be the case that Tp.

And,

1. 2. to argue that validity and logical relations more generally did not involve truth but only what Hans Herzberger, reflecting on Buridan's proposal, termed correspondence.

(Herzberger 1973)

To deal with the Liar and its ilk, Buridan took a third step. The form of this step varied a bit over his career. In his early work he...