 ## Is ‘determinately’ a logical constant?

April 28, 2008

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 $\Delta$ (‘determinately’) operator is added to the language. For example, if validity is thought of as preservation of supertruth, then the deduction theorem fails.

• If $p\models \Delta p$ then $\models (p \rightarrow \Delta p)$

Any model in which p is supertrue is one in which $\Delta p$ 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, $\Delta p$ 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, $\models$, 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 $\Delta$ fixed across models. If we’re not, then the deduction theorem holds again – interpret $\Delta$ 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 $\Delta$ 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 $\Delta$ fails (but I still need to check up on what the MacFarlane tests are here.)

Another reason to think that $\Delta$ 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, $\Delta$ 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 $\Delta p$ 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 $\Delta$ is vague. But nonetheless the existence of higher order vagueness (in p, say – so $\nabla \Delta p$) strikes me as compelling evidence that there is vagueness located in $\Delta$ rather than in p. I was hoping to run a more precise version of this argument by constructing a sorites for $\Delta$ 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 $\forall_1$ and $\forall_2$ 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 $\Delta$ with precise operations yields a vague term, then $\Delta$ is vague. Thus, if

• $\lambda F \exists x \nabla Fx$

is vague, then $\Delta$ 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, $\Delta$ must be vague.

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### 5 comments

1. I’m not sure one can use the construction of a sorites series for some expression E as evidence that E is vague, when the sorities itself is construction by concatenating E with expressions that might themselves be vague.

Normally we idealize away these sorts of worries when constructing sorites. E.g. we presume that we’ve got a way of picking out a series of men in a way that has no relevant vagueness, and then the series of claims “man number n is bald” is supposed to shed light on the vagueness of “bald”.

But suppose we consider cases where the referential terms have relevant indeterminacy. E.g. suppose we have a paradigm electron, Sparky, and a paradigm non-electron, Robbie. Let “item number n” be introduced as the x satisfying the following reference-fixing description: either x is Sparky, and n is small; or x is Robbie, and n is not small. Now the first few terms (where x is determinately small) will determinately refer to Sparky, and the last few (where x is determinately not small) will determinately refer to Robbie. But “item number n” for borderline-small n will be indeterminate in reference between the two.

Now consider the collection of claims: (0) “item 0 is an electron”; (1) “item 1 is an electron”…. (n) “item n is an electron”. This’ll display the characteristics of a forced-march sorites, I guess, and we could turn it into a sorites paradox without too much trouble. Does it show “electron” is vague? Not at all—intuitively the reason we get a sorites is because of the vagueness in the referential terms.

And similarly at the case at hand, we can construct a soritical series: (0) “Vague(0-small)” (1) “Vague(1-small)” … etc. But since at least some of the k-small expressions are vague, I’m not sure what we are allowed to conclude about “Vague”.

By the way, what semantic treatment of lambda-expressions (and variables) do you intend? I’m a bit worried that if you do things in a standard way (where the assignments of values to variables is not relative to any precisification) then you won’t get even the first premise: that your higher order predicate Vague has 0-small as an instance. Williamson discusses the formal issues here in his survey paper “Vagueness in Reality”, cf. especially his discussion of the result that on supervaluationist settings (and supposing a standard treatment of variables) it’s provable that $\neg \exists F \exists x \nabla Fx$.

2. Hi Robbie. Thanks for your comment (sorry it got swallowed up originally – for some unfathomable reason WordPress thought it was spam!)

That’s an interesting example. But for the vague argmument to be analogous, it has to be that there are n for which $\exists x \nabla n-small(x)$ is true on some precisifications, and not on others (just as there are n for which ‘item number n’ is an electron on some but not all precisifications.) But it seems to me that the interpretation of n-small can’t depend on precisifications in this context because, as it were, the precisifications are all ‘bound out front’ by the $\nabla$ operator.

If vague is a perfectly precise concept, what can it mean to say that n-small is vague on a precisification, and not vague on another. Surely it’s impossible to completely precisify all predicates in your language and still have vague predicates.

Anyway, I guess the Sorensen argument is left wanting – I’ll have to think more about this! As for the lambda stuff, I assume that the inference from $\exists x \nabla small(x)$ to $\exists F\exists x \nabla F(x)$ is valid (or at least should be), so the theorem you mentioned would entail that there are no vague predicates in the language. Did you say that Williamson could prove this from principles in the object language, or did he give a semantics on which that came out true?

3. Hi—sorry it took so long to get back.

Just on the direct point you raise. I was thinking that “item x” is analogous to “x-small”. Just as “item x” is for at least some cases indeterminate in extension, “x-small” is indeterminate in extension, for at least some cases. So the cases seemed kinda parallel—at least when in versions that don’t explicitly utilize lambda-expressions.

The *general* moral of the electron case, I thought, was that the presence of a sorites series for F can’t in general be sufficient for F being vague. And I hoped that would then place the burden of proof on others to come up with the hard task of saying what conditions have to be met for a sorites series to imply vagueness of the predicate involved. Minimally, if the all terms involved in the sorites series other than the predicate F were precise, then it looks like the predicate must be vague. Where the terms are themselves indeterminate in extension, I just don’t know what to conclude—and that seemed to me the situation we faced with the Sorenson argument.

But I may well be missing the point here. I find this all quite head-spinning.

Maybe the use of lambda-expressions in your example could help out (cf. your “bound out front” thought). Is the idea that the relevant sorites series could be represented as formed of

(n) $\exists F((F\equiv\text{n-small})\wedge \exists x \nabla Fx)$

Compare this to the analogous move in the electron case:

(n) $\exists x(x=\text{item n}\wedge \text{electron}(x))$

Now in the latter case, we have a sorites just as before. What happens in the former case? It all depends on how you do the semantics for the higher-order quantifiers… if the variable assignment associates functions from precisifications to extensions to predicates, and we understand $\equiv$ properly, then we get a situation analogous to the one we had with the electron case, and (I think) similar comment go through. If assignments just give extensions to variables, independently of precisification, then we’re in the Williamson situation, and none of the quantified-out (n) come out true. So we don’t get a sorites series.

So I was thinking there’s a dilemma—with the Williamson reading of quantification, we don’t get a sorites at all, so there’s no argument that “Vague” is vague. But if we have a reading that allows the kind of relativity to precisifications required to generate a sorites, I don’t yet see the disanalogy between this and the parody case of “electron”.

But I’d be the first to admit there are lots of moving parts here, and I’m not sure I’m tracking them right.

(By the way, Williamson’s argument is entirely semantical, IIRC. He does talk a bit about strange claims that come out true if you don’t go for the way of handling quantification he favours, so it’s not like he just lays down the semantics w/o defense.)

4. Hi Robbie,

Thanks for your reply. I completely agree with the general moral – (sorry if I wasn’t clear that I agreed) – the electron thing does undermine the use of Sorites here.

Anyhow, I was trying to draw a disanalogy between the two Sorites in that post, but I’m afraid my thoughts where quite scattered then. Let’s try a different tact.

Let “item number k” refer (determinately) to the predicate “is k-small”.* Now we can run a Sorites as follows: (i) item number 0 is vague (ii) if item number k is vague, so is item number k+1, (iii) item number n isn’t vague, where n := Rayo’s number.

The difference here is that all of the terms used in the Sorites sequence are precise, unlike with the electron case. (Presumably it doesn’t matter if what is designated is vague – that’s just what we’re going to need anyway.) The claim now being that: if all the terms in the Sorites sequence are precise – then the Sorites sequence is sufficient to show vagueness. What do you think? Does this Sorites do any better?

* It depends here what kinds of things you think are vague. You might think it’s the predicate “k-small”, or the semantic value of “k-small” that’s vague. (So in the latter case the semantic values themselves are vague – functions from precisifications to extensions, in the former propositions etc… are precise, its just indeterminate which proposition is designated.) I don’t think it matters to the argument though.

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