Is ‘determinately’ a logical constant?

I’ve had a very interesting discussion about the logical status of the determinately operator over at Theories n’ Things recently. One counter (due to Tim Williamson I think) to the claim that supervaluationism preserves classical logic is the fact that bad things happen when a (’determinately’) operator is added to the language. For example, if validity is thought of as preservation of supertruth, then the deduction theorem fails.

• If then

Any model in which p is supertrue is one in which is supertrue. However, take a model in which p is true on some but not all precisifications. Let v be a precisification on which p is true. Since there are precisifications on which which p is false, is false on all precisifications, in particular v. So the conditional is false on v, and is thus not supertrue.

I don’t particularly like the definition of validity used here. If you think that truth is fundamentally a property of propositions, and you think vague sentences express multiple propositions then the idea that validity preserves truth can be fleshed out as valid arguments preserve truth at a precisification (we’re starting off with a relation, , between propositions, and we’re raising it to a relation between sentences which are ambiguous.)

But of course, even if we go for validity-as-preservation-of-supertruth, the argument is still not straightforward, for it depends on whether we’re keeping the interpretation of fixed across models. If we’re not, then the deduction theorem holds again - interpret as negation and the antecedent fails. Robbie pointed me to a nice paper where he made essentially this point.

So why shouldn’t we treat as a logical constant? Well one test is to see if it is invariant under arbitrary permutation of the domain. It’s not clear how to apply this to intensional operators, but there are similar invariance tests you can apply to the accessibility relation. As far as I can tell fails (but I still need to check up on what the MacFarlane tests are here.)

Another reason to think that isn’t logical is as follows: the logical constants cannot be vague. Why is this? Well the supervaluationist will presumably say logical truths must be true at all precisifications (and all worlds etc…) But it is independently plausible (and also widely believed, see e.g. the Lewis/Sider argument for unrestricted composition.)

I want to argue that in a setting which countenances higher order vagueness, is vague, and thus cannot be a logical constant. Unfortunately the standard tests for vagueness are difficult to apply. If our term is of the type of a predicate, <e, t>, then we can just see if it is susceptible to a sorites paradox (this isn’t perfect since it doesn’t give a positive test for when a predicate isn’t vague, i.e. precise, but its good enough for most purposes.) Then we can recursively apply the test to other types as follows: assume we can test for vagueness for the lower types. Combine the higher type term we want to test with precise terms (we can test these for precision by inductive hypothesis.) Finally test the result (of a lower type) for vagueness.

But in our case, if is vague, then there is a good case for saying that p wasn’t precise in the first place. If a term is precise, this fact should itself be precise. Not only is it determinately true that everything is self identical, but it’s determinately determinately true, and so on. But in the context of higher order vagueness, this is exactly the principle we want to fail.

It’s not looking so good for the idea that is vague. But nonetheless the existence of higher order vagueness (in p, say - so ) strikes me as compelling evidence that there is vagueness located in rather than in p. I was hoping to run a more precise version of this argument by constructing a sorites for directly. Of course its not that simple since it’s an operator and not a predicate.

Assume the following

• The operation of lambda abstraction is a precise operation. (I also think its an interesting case of a logical operation - it does not fit into the type heirarchy.)
• Existential quantification is precise.

Sider has a cute argument for the second claim: suppose and are different precisifications of universal quantifiers. Then they must have different extensions. That is, there must be an object, x, which one ranges over but the other doesn’t. But then one of them would not be an admissible precisification of the universal quantifier since it missed an object out.

Going back to our test for vagueness: if composing with precise operations yields a vague term, then is vague. Thus, if

• 

is vague, then must be vague (in what follows, abbreviate to Vague(F).) Note that this predicate is true of a predicate F iff it has a borderline case. But it turns out this predicate is vague, by an argument due to Sorensen. Define:

• x is k-small iff x < k or x is small

Since being 0-small is equivalent to being small, 0-small satisfies Vague(F) since it has a borderline cases. Since n := Rayo’s number isn’t small, being n-small is equivalent to being less than n, which is precise, so n-small does not satisfy Vague(F). Since the predicates ‘k-small’ for k < n form a sorites of Vague(F), Vague(F) must be vague. And hence, by our assumption, must be vague.