There seems to be a fairly serious problem with the theory set out in Lewis’s “How to Define Theoretical Terms”, which I’ll set out here. I might be misinterpreting Lewis, and if so I’d like to know. And I’m probably just be repeating something that has been said elsewhere already, and if that’s so I’d really like to know. So comments from Lewis(iana) experts much appreciated.

The problem concerns what Lewis says about problems with unrealised theories, and what he says about problems with multiply realised theories. To a first approximation, Lewis says that terms t1, …, tn in theory T denote the unique x1, …, xn such that T[x1, …, xn] is true, where T[x1, …, xn] is the theory T with all of the terms t1, …, tn replaced with variables. That is, they denote x1, …, xn if there is a unique x1, …, xn, and nothing otherwise.

That seems fine in the case there is a unique realisation, but surprising in cases where there are either multiple realisations, or no realisations.

In cases where there are multiple realisations, we might think that the t1, …, tn are indeterminate in denotation between the various realisers. Lewis eventually (though I can’t right now relocate where, maybe _Naming the Colours_) granted that was true in some instances. But in HTDTT he said that any theory that was multiply realised was a failed theory, and failed theories don’t get to establish denotations. That seems rather strict to me, but let’s run with it. It is at least plausible that we should aim to have our theories T have unique realisations in Lewis’s sense.

In cases where there are no realisations, we might think it is natural to say the terms of the theory don’t denote. That’s, after all, exactly what we want to say about the phlogiston theory. But there are large variations in this class. Some theories are not even close to being true. Other thoeries just about make it. The latter type _do_ establish denotations for their theoretical terms.

Lewis’s solution to this is to look not at the explicitly endorsed theory T, but at a related theory T*. (He uses primes not stars, but I don’t know how to do them on the blog.) T* is the theory that says, roughly, T is close enough to being true. We can think of T* as a long disjunctions of enough conjuncts of T that if one of those disjuncts obtained, we’d regard T as close enough to the truth.

So now Lewis is committed to the following. Lewis says that terms t1, …, tn in theory T denote the unique x1, …, xn such that T*[x1, …, xn] is true, where T*[x1, …, xn] is the theory T* with all of the terms t1, …, tn replaced with variables. That is, they denote x1, …, xn if there is a unique x1, …, xn, and nothing otherwise.

But the problem with this should be obvious. Imagine a case where T _is_ perfectly satisfied, but there is a sequence y1, …, yn such that T*[y1, …, yn] is true, that for at least a few i, xi and yi are very different denotata (so this isn’t a regular case of vagueness), but that T*[y1, …, yn] is only true because one of its disjuncts representing a case where T is barely close enough to truth to count as term-defining obtains. So T has a perfect realiser, and a quite different imperfect realiser, but one that would count as close enough were T not independently perfectly realised.

According to the formalism in HTDTT, the theoretical terms in T are non-denoting. But that’s a crazy view – the theoretical terms in T denote x1, …, xn. According to the view Lewis later toyed with, the terms in T might be indeterminate between x1, …, xn and y1, …, yn. But that’s crazy do – these terms have determinate denotations.

I can sort of see what we’d want to say here. We don’t want to list the disjunctions in T*, but we want to rank them by order of closeness to T. If T* is multiply satisfied, the sequence that satisfies the disjunct closest to T determines the denotation. (Given Lewis’s later views, we’d want to look at the balance between closeness to T and naturalness of the denotata, but it would be anachronistic to include that here!) What I can’t see is how to express this in anything like the formal language that Lewis uses in HTDTT, where the meaning theory is set out in something like first-order logic.