6/27/2006

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 , where
- 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?

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7 Comments:

At 27/6/06, Anonymous P.D. Magnus said...

One might say: Models only specify meanings for the non-logical vocabulary like constants and predicates. Connectives are part of the logical vocabulary, and so their meaning is not part of the model. In the same way, identity is treated as part of the logical vocabulary and not merely as another predicate. Formally, the meaning of the the connectives and identity are built in to the definition of satisfaction.

This gives a straightforward class of analytic truths: the sentences that are true in all models.

Of course it is possible to describe formal structures with a non-standard interpretation of (eg) the conjunction symbol, but it is wrong to think that these are models of the sentences we started with. Such a non-standard interpretation changes the meaning of the conjunction symbol. So we should not be surprised if some of the analytic truths come out 'false' in such a structure. (The scare quotes indicate that it isn't really falsity, but merely some formal property that has some structural similarity to falsity.)

 
At 29/6/06, Blogger Andreas said...

You wrote:
"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."

My initial reaction to this is that what makes most sentences like 'Fb' non-analytic is that it is not a matter of the meaning of 'Fb' that f(b) is an element of the set f(F). What is a matter of the meaning of 'Fb' is merely that 'Fb' is true just when that is the case.

Whether f(b) is really included in f(F) depends on the nature of the world, to which the domain D - and more significantly it structure (that is the details about which elements of D are included in which subsets of D) - is supposed to be isomorphic.

The characteristics of the traditionally analytic sentences should be something like an inclusion relation between two subsets of D, as in forinstance 'All batchelors are unmarried'. For all elements x of D, either x is nor included in f(batchelors), or x is included in f(unmarried).

 
At 29/6/06, Blogger Aidan said...

I'm confused. Take again the sentence Andreas quotes:

"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)).'

Why does this license us to conclude that such languages determine the truth-values of every sentence? Aren't we just getting out truth-conditions, just as we'd hope and expect? It's left up to the world to decide whether the conditions stated on the RHS of such biconditionals is met, as Andreas says. The Carnap-analytic sentences might then be the ones where the RHS condition is met simply in virtue of the specification of the language in question. Where am I going wrong?

 
At 29/6/06, Blogger GF-A said...

P.D. --

I agree with everything you say, but I don't see how it goes against the claims of the original post. Is the idea that the "language" we are specifying is just the 'logical' language (connectives, quantifiers, identity, whatever else), and the other stuff (which is specified by the model) should not count as part of the language?

Andreas and Aidan --

Perhaps you are right and I'm wrong, but I don't see it yet.

I agree with Andreas that what makes a sentence like "Obscure and Confused Ideas is worthless" true is not the meanings of "OaCI" and "is worthless" alone -- however, if we have a model-theoretic formalization of this sentence, we can compute the truth-value of that sentence without further information. Why? The model assigns some element in D to "OaCI", and some subset of D to "is worthless". This determines the truth-value of the original sentence, since these two things determine whether the element is in the set or not. It's not really "left up to the world" whether the element is in the set, at least if by "left up to the world" you mean "not already determined by the language" -- because to specify which model-theoretic language you are speaking, you need to specify which set each of your predicates designate etc. There is nothing 'left open' by the language.

Maybe it would help to contrast this with a language that treats predicates intensionally (roughly: as properties, not as sets). One way to do this is Montague's (via Carnap's) notion of the intension of a predicate: f(F) then picks out not a set, but rather a function from possible worlds/ states-of-affairs to sets. It is a matter for empirical inquiry to determine (as much as is possible) which of those possible worlds we inhabit. If we don't know which possible state of affairs is the actual one, we can't say whether an arbitrary declarative sentence is true -- we can only state its truth-conditions, as Aidan (and I) want. ...Have I missed the point(s)?

(Note: This is a fumbling rehearsal of what Carnap says at the beginning of his Introduction to Semantics.)

 
At 29/6/06, Blogger Aidan said...

No, no point missing. At least not on your part - and I think I'm now clearer on what you were getting at. Thanks.

 
At 30/6/06, Anonymous P.D. said...

As I understood the original post, the point was that model-theoretic semantics makes all truth relative to the vagaries of a model. Since it leaves no way of distinguishing between different truths, it allows for no distinction between analytic and non-analytic truths.

My point was simply that this total collapse only happens if we make no distinction between logical and non-logical vocabulary. If the meaning of the logical vocabulary is included in the definition of truth rather than in the model itself, then there is a distinction between logical truths (tautologies) and contingent truths. This allows for at least some distinction between truths due to meaning and other truths.

 
At 1/7/06, Blogger Andreas said...

Yes, I think it's clearer now what point you were making.

Of course, I do not have to mention that the fact (if it is one, and I still have some doubts) that analyticity cannot be captures by a set-theoretically constructed model of a language L does not imply that there are no analytic sentences in L. This is especially so, I guess, if L is a natural language, since it seems obvious that what characterises a natural language is precisely that we cannot specify the meaning of its terms with the kind of precision required for model-theory (intensional or extensional).

Nevertheless, good point!

 

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