Analyticity in model-theoretic languages
Part of why I am drawn to philosophy of science and logic is that I like to operate with clean and neat formulations of apparently messy concepts -- and these two sub-disciplines of philosophy embrace such tastes more than other sub-fields. Of course, I am not claiming that ethicists and metaphysicians are muddle-headed; most think about their sub-discipline's topics with far more clarity and rigor than I can. I am merely expressing a personal preference for studying deontic logic instead of the most recent form of consequentialism.
Enough autobiography -- I mention it only to explain my motivation for this post. And my point is this: if we adopt the usual formalization of an interpreted language (viz., the model-theoretic one), then we apparently cannot capture the notion of analyticity -- at least in the way Carnap, who is widely recognized as the champion of analyticity, conceives of it.
Conceiving of a language in model-theoretic terms is one widely-used way of introducing precision into a philosophical endeavor. Most readers probably can recite the definition of a model-theoretically understood langauge by heart, but for the innocent:
A language L consists of a ordered triple
- L carries grammatical information: which symbols belong to the language, which strings of symbols count as sentences, which grammatical category each symbol belongs to, etc.;
- M is a model =< D, f >, where the domain of discourse D is a set of individuals, and f is an interepretation function, which assigns an individual in D to each proper name in L, sets in D to one-place predicates, sets of ordered pairs drawn from D to two-place predicates, and so on; and
- r specifies the truth-values of certain compound sentences, given the truth-values of their components -- in other words, r basically specifies the truth-tables.
So much for the model-theoretic conception of language; what about analyticity? Carnap, throughout his career, identifies the analytic truths as those sentences that are true merely in virtue of the language one speaks. That is, if we specify that I am speaking a particular language, in the course of that specification, I might present enough information that the truth-values of certain sentences within that language are fixed. (For an obvious example: if I specify what 'and' and 'not' mean in my language via the usual truth tables for those words, any sentence of the form 'p and not-p' comes out false merely in virtue of the rules governing the language I am using.)
Now, after all that rehearsal of material most readers probably know well, I can get to my point. In a model-theoretically characterized language, the truth-values of ALL sentences are determined by the specification of that language. For example, the truth-value of atomic sentences such as 'Fb' are true iff the individual named by 'b' is in the extension of the set associated with 'F' (i.e. 'Fb' is true iff f(b) is an element of the set f(F)). And Carnap certainly never wanted every sentence of a (non-contradictory) language to be analytic.
The problem then is: one of my favorite tools for 'precisification' in philosophy -- model-theoretic languages -- apparently affords no way to characterize one of the concepts I'm most interested in: analyticity. What to make of this? The first, obvious thing to say is: "Of course there couldn't be any explication of analyticity in such languages, because such languages are extensional, and Carnap and Quine (who represent opposing positions in debates over analyticity) both basically agree that analyticity is an intensional notion."
This is right, but I think there is something further to note: in a straightforward sense, every sentence in a (classical) model-theoretic language has its truth-value determined by the specification of the language. That is, by specifying the language, we fix the truth-values of all the sentences in such a language. That seems odd -- the model-theoretic way of specifying a language that has proved very useful in certain situations, but it likely cannot be a fundamental and/or universally applicable one.
One further point: Carnap, Quine, and the other primary antagonists in battles over analyticity all agree that if there is any such thing as analytic truth, then the (so-called) logical truths are paradigm instances of analytic truths, i.e., truth in virtue of meaning (if you are thinking of "Two Dogmas" and don't believe me, look at Word & Object, sec. 14, fn.3, p.65). But the model-theoretic conception of language characterizes the logical truths as a class of sentences that are true across a set of related langauges. That is, to know whether a sentence is a logical truth in one model-theoretic language, you have to check whether that sentence is true in a bunch of other model-theoretic languages that share certain features with the first one.
So, one might think that the way to cash out analyticity in the idiom of the philosophical logician is to use something like Kripkean possible world semantics (which are used, with some variations, in modal, deontic, epistemic, and temporal logics). But these are usually not given linguistic interpretations, and it's not clear to me that it's possible to give a decent one... though I'd love to be wrong. Any thoughts?