One of the odd things about my preferred logic of vagueness, as set out in True, Truer, Truest is that disjunctions can be perfectly true even though no disjunct is perfectly true. Some people (including me some of the time) find this a little unintuitive. It also has a quirky logical consequence.
I hadn’t been thinking about this at the time, but one consequence of that is that the simplest way to do multiple conclusion logic in the theory will lead to some odd failures of validity. For instance, using ‘:’ as a turnstile, it is natural to think that (1) will fail.
(1) A ∨ B: A, B
If our understanding of validity is that on any interpretation, at least one of the conclusions is at least as true as the conjunction of the premises, then (1) is false. That’s because when B is, say, ¬A, then the premise will be perfectly true, but each disjunct will be less than perfectly true.
Objection: It’s a little odd to have this definition of validity. It requires, in effect, that we treat the premises as a long conjunction, but the conclusions separately. If the rule for validity was that the disjunction of the conclusions must be at least as true as the conjunction of the premises, then (1) would be valid.
Response: I think this objection ultimately doesn’t take the multiple conclusion form of (1) seriously enough. It’s true that you could turn around and say that the theory of validity in True, Truer, Truest doesn’t take the multiple premise form of arguments seriously enough. But to that I happily plead guilty; I’m happy to say that arguments really have one long premise, with the parts sometimes joined by ‘∧’, and sometimes by ‘.’.
I mentioned briefly in the earlier paper that you could do an intuitionist version of the semantics in True, Truer, Truest. The classical version that I developed there said that truer generated a Boolean lattice over (equivalence classes of) propositions (where the classes are of equally true propositions). If instead you said that truer generates a Heyting lattice over (equivalence classes of) propositions, you get an intuitionist version of my theory. (That is, all and only intuitionist theorems are logical truths, and and all and only intuitionistically valid sequents are really valid.)
Now some Heyting lattices have the property that the join of a and b can be the top element even though neither a nor b is the top element. But not all Heyting lattices have this property. In particular, the Lindenbaum-Tarski algebra has this property. That’s the algebra defined over sentences in a countable language ordered by intuitionist entailment. So if we supplement the theory of True, Truer, Truest with the requirement that A ∨ B is perfectly true iff A is perfectly true or B is perfectly true, we get a consistent intuitionist theory.
Having said that, we still don’t make (1) valid. It can still be the case that A ∨ B is strictly weaker than both A and B, and in that case, (1) will be invalid. The trick, I think, will be to change the definition of validity.
New Account of Validity: An argument is valid iff whenever the premises are perfectly true, at least one of the conclusions is perfectly true.
The New Account of Validity doesn’t change what sequents are valid in the classical case, but it does change some validities in the (multiple-conclusion) intuitionist version of the ‘truer’ theory. Note that the New Account doesn’t imply that p entails that p is perfectly true, because p can be perfectly true without it being perfectly true that p is perfectly true.
If we think the algebra of truth values is a Lindenbaum-Tarski algebra and we adopt the New Account of Validity, then (1) will be valid. I’m not sure whether this is too high a price to pay to keep (1) valid, but I think there is some motivation for each of these assumptions.
UPDATE: I fixed the broken link to True, Truer, Truest. Thanks to Seamus Bradley for alerting me to this.
Posted by Brian Weatherson in Uncategorized