Sunday, January 31, 2010

Kaplanian content is not compositional

I claim that Kaplanian content isn't compositional. The tension lies in Kaplan's treatment of variables and the compositional semantics for quantifiers -- there is an inconsistent tetrad between the direct reference of variables, the semantics of quantification and the compositionality principle.

To see this let's just focus on a fragment of Kaplan's LD that has to do with variables and quantification (if you know it you can skip down to after the semantic clauses).

In the syntax of LD we have an infinite set of individual variables, V = {x
1, x2, x3,...}, an infinite set of n-place predicates Π = {F1, F2, F3,...} and the two quantifiers ∀, ∃. For these we have the following (relevant) formation rules:

* If π is an n-place predicate and α
1,..., αn are variables, then πα1,..., αn is a formula.

* If φ is a formula and α ∊ V, then ∀αφ and ∃αφ are formulae.

For the semantics we have a structure
A = {C, W, T, U, I}, where C is the set of contexts, W is the set of worlds, T is the set of times, U is the set of individuals, and I is an interpretation function (which gives extensions to predicates at t,w).

A point of evaluation is a quadruple (c,f,t,w) where c ∊ C, t ∊ T, w ∊ W and f is an assignment function. An assignment function f is a function from variables to individuals, f: V → U. We write f[α/x] to denote the assignment that is just like f except it assigns x to α . And we write [[γ]]
c,f,t,w for "the extension of γ at (c,f,t,w)" (we omit the structure).

Given this we recursively define 1 ("truth") at a point of evaluation as follows.

* [[α ]]
c,f,t,w = f(α ).

* [[πα
1,..., αn]]c,f,t,w = 1 iff ([[α1]]c,f,t,w ,..., [[αn]]c,f,t,w ) ∊ Iπ(t,w).

* [[∀αφ]]
c,f,t,w = 1 iff for all i ∊ U, [[φ]]c,f[α/i],t,w = 1.

* [[∃αφ]]
c,f,t,w = 1 iff there is an i ∊ U, [[φ]]c,f[α/i],t,w = 1.


That is the basics. I wanted to make sure I didn't leave anything out. And that is all fine as far as it goes. The lexical entries for the quantifiers here are syncategorematic, so its left implicit what the exact compositional mechanisms are -- what is it that quantifiers are functions from function from? Yet its obvious what they must be. Back to this in a second.

The Kaplanian
content of an expression (at a context) is meant to be the propositional contribution that the expression makes. This generally means two things: (i) content is "what is said" or expressed by an expression, and (ii) content is compositional, i.e. the content of a compound expressions is a function of the contents of its parts. Ignore the first bit. What we are interested in here is refuting the contention that Kaplanian content is compositional.

Kaplan says "A variables first and only meaning is its value" and "Free variables under an assignment of values are paradigms of ... directly referential terms". Saying that a variable α is directly referential is to say that the propositional (semantic) contribution of α at an assignment f is simply the object f(α) (or in non-Russellian lingo a constant function from W x T to that object). And this is just what Kaplan tells us in the "Remarks..." on LD. He introduces the notation {γ}
c,f to mean "the Content of γ in the context c under the assignment f" and tells us that the content of a variable is as follows.

* If α is a variable, then {α}
c,f = that function which assigns to each t ∊ T, w ∊ W, [[α ]]c,f,t,w .

That is, the content of α at a context c and assignment f is λt,w.[[α ]]
c,f,t,w, where for any (t',w'), λt,w.[[α ]]c,f,t,w(t',w') = f(α), i.e. a constant function from circumstances to f(α).

Now there is a clash between the claim that the content of a quantified sentence, like `∀xGx', is compositionally determined by the contents of its parts and Kaplan's statement above about what the content of a variable is (i.e. the claim that variables are directly referential).

The content of the formula `Gx' at a context and assignment (c,f) is a function which assigns 1 to a circumstance (t,w) iff f(x) is in the extension of G at (t,w). In other words, the content of `Gx' is a "singular proposition" to the effect that f(x) is G. A function from circumstances to truth-values may be the right entity for modal and temporal operators to take as argument but it is not the right input to a quantifier over individuals. We can't get the right result by compositionally combining a function of this type with the semantic value of the quantifier ∀x!

What the quantifier needs to take as argument is a function over individuals (or assignments) -- a functon like this: λi.[[Fx]]
c,f[x/i],t,w. This is because the semantic value of ∀α at a context c is something like λp. for all i, p(f[x/i]) = 1 at t,w, i.e. a function from assignments to truth-values. Thus, the compositional semantic value of `Gx' at a context is different from the Kaplanian content. Kaplanian content is not compositional.

Someone might respond by insisting that I've simply taken Kaplan's claims about "free" variables and applied them to "bound" variables. But there is no such distinction between different kinds of variables in LD. There are simply the members of V which can occur both free and bound and there is no semantic/syntactic difference made between them. So my argument restricted to the language of LD is undeniable.

It is true, however, that Kaplan seems to want to make a distinction between the semantics of free and bound occurrences of a variable (note the similarity here between deictic and anaphoric uses of pronouns). But what is this distinction exactly? In the formula `∀xFx & Fx' is it something about binding one of the occurrences of `x' that "changes" the semantic value? How does that work compositionally? Is is ambiguity? Are there two
homographic expressions `x' and `x' in the formula - a variable and a schmariable? Then in LD we should actually have two syntactic categories of variables Vfree and Vbound. And we can give these different contents (semantic values) in contexts.

It seems to me the much better options are either (i) to adjust the commitments about the contents of variables, i.e. to give up the claim that free variables are directly referential or (ii) to give up the claim that Kaplanian contents do the compositional work.

So, Kaplan followers what do you do?

10 comments:

Tom said...

Hi Brian,

I'm not as familiar with the details of Kaplan's system as you are but what you say sounds like it's a real worry.

My first response would be something along the lines of the analogy with pronouns that you describe. Take the 'deictic' reading of
(1) Susan went to her office.
where the referent of 'her' is some contextually salient female. We can also get a bound 'anaphoric' reading, as you say. To get the analogy with your case we could put a quantifier like 'Every professor' in place of 'Susan'. This doesn't make us think that 'her' is ambiguous, it's just a variable. (We need data where we get co-variance without c-command for that, as I'm sure you know.) Now, this doesn't mean that we have a violation of compositionality because compositional systems can handle it. The variable always contributes its nature as variable, as it were.

Is the problem that Kaplan rules this sort of response out for his system because he says that variables always contribute objects, but ones bound by quantifiers don't? In that case couldn't the Kaplanian give up that claim i.e. the claim that all variables are directly referential but keep the claim that some, namely the free ones, are. That allows her to keep (i), as far as I can see.

Is there some problem with this strategy I'm not seeing? If not, it looks like your problem is a real problem but is also easily fixed by being a bit more careful in setting up LD.

Brian Rabern said...

hey Tom,

Right. The following situation doesn't present a challenge to compositionality:

(2) Susan is an x such that x went to the office of x.
(3) Susan is a y such that y went to the office of x.

We don't have to posit an ambiguity w.r.t. the last expression of (2) and (3).

The problem is adding in Kaplan's commitment on what the semantic value of a variable is. The problem is that if a variables semantic value in a context is not a function over assignments (this can be glossed in the Russellian object lingo as you do or in the parameters/functions lingo) then we will get the wrong results when we compose the semantic values of variables and the semantic values of quantifiers.

Now you suggest: "couldn't the Kaplanian give up...the claim that all variables are directly referential but keep the claim that some, namely the free ones, are."

Yes, thats right but that is just to make the syntactic/semantic distinction between free and bound variables that I said Kaplan probably wants to do -- I said "in LD we should actually have two syntactic categories of variables Vfree and Vbound. And we can give these different contents (semantic values) in contexts."

My modest claim was that LD is broken in this respect -- an so LD is not compositional w.r.t. Kaplanian content. But I would like to go further and suggest that positing this "ambiguity" or homonymy is not good methodology. What work does it do to hold on to the "direct reference of variables"?
But I didn't give an argument against the ambiguity view so your right that that is a way out. I just think its not well motivated.

Maybe there is a motivation I am not aware of.

Wolfgang said...

Hi Brian,

good catch! Kaplan clearly misplaced the assignment function when he put it into context rather than the index. But as you say, putting it in the index would seem to undermine some of his doctrines about content.

Perhaps the most conservative fix would be to go schmentencite, as in Lewis's "Index, Context and Content". One would then say that formulas with free variables aren't sentences and don't express propositions. (After all, they cannot be used to say anything.) They merely express functions from assignments to propositions. The truth-value of a real sentence never depends on the assignment function, so it can be dropped from the index.

Otherwise I suppose one would have to introduce an intermediate level of content: words in contexts express functions from assignments to proper contents. Proper contents are relative to a context and an assignment function, and the proper content of a variable is its referent. OTOH, compositional semantics happens on the intermediate level rather than the level of proper content.

Another alternative would be to drop assignment functions altogether and interpret quantifiers by "recursion on truth", as many newer logic textbooks do it (see e.g. 84-86 in Bostock's _Intermediate Logic_). The resulting semantics is then not strictly compositional though.

No matter what we do, the idea that the sole semantic contribution of a free variable is its referent has to go.

Brian Rabern said...

Hey wolfgang,

thanks for the suggestions.

I was actually thinking of this in terms of the schmentencite strategy. I was going to spin the post more that way and call it "Kaplan is a schmentencite" or something.

The way I understand the schmentencite strategy here is the same as the ambiguity strategy I mention in the post. We say that the formula `Gx' in a context expresses a singular proposition but we must distinguish this stand-alone expression from the `Gx' that occurs in `\forall x Gx'. This constituent `Gx' is a schmormula -- we must distinguish the constituent schmormula `Gx' from the homonymous formula, which is not a constituent of anything. Once we do this we can let the unembedded `Gx' express a proposition and have the embedded `Gx' express a function from assignments to propositions.

It seems to me that the essential schmentencite move is making this ambiguity/homonymy move. In this case we posit a distinction between variables and schmariables (free variables and bound variables).But, again, I don't know why we would do this.

Wolfgang said...

Oh, I see. My idea was to treat all open sentences as schmentences, whether embedded or not. On this account, you couldn't "say" anything by uttering 'Gx'. There would be no ambiguity because there are no schmariables, just variables.

But I guess it depends on how you want to use unembedded open sentences. If you want to allow assertions like 'Gx' (and don't want to give them the closure interpretation, reading 'Gx' as '(Ax)Gx'), you might have to postulate the kind of ambiguity you're talking about.

Brian Rabern said...

yeah. I take it that Kaplan wants `Gx' at a context to expresses a singular proposition, if free variables are directly referential.

So I don't think saying that formulas with free variables aren't sentences and don't express propositions saves Kaplan here. But ignoring Kaplan it is good option -- probably the one I prefer.

The analogy with the schmentecite is very tight. The semantic value of the sentence "Its raining." at a context is a truth-value. Whereas the semantic value of the schmentence `its raining' in "PAST its raining" is a function from times to truth-values.

DilipNinan said...

This is interesting. But wouldn't Kaplan be saved by (a) putting the assignment function in the index, and (b) retaining his definition of contents as functions from indices to truth values? Or does that conflict with the claim that variables are devices of direct reference?

Brian Rabern said...

hi Dilip,

right, that would make the system compositional at the level of content but since the content of variables are now "assignment-unspecific" they are not directly referential.

in other words, he can't hold onto the equivalence between "what is said" and compositional semantic value, while insisting that "what is said" by a variable (or an open formula) is insensitive to all parameters (including the assignment).

this is essentially an "operator argument" but of course "quantifier" arguments work just as well.

DilipNinan said...

Oh, I see. So to be directly referential an expression has to be a constant function from indices to an object.

Brian Rabern said...

right. i think it is a necessary condition on being directly referential that an expression's content be a constant function from indices to extensions.

however, sam cumming's "variablism" view on names has it that the semantic values of names are not constant functions of the index (since they vary with assignment and this is in the index) but he insists that this still "comports with our intuitions that names are nondescriptive and directly referential", p. 550.

yet the way Kaplan talked about it he clearly intended "content" to be what you got after both fixing the context and assigning values to variables.