Sunday, January 31, 2010
Kaplanian content is not compositional
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 = {x1, 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 α1,..., 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 ).
* [[πα1,..., 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?
Monday, November 09, 2009
Random randomly silent
With the further condition:
* Whether Random answers ‘da’ or ‘ja’ or whether Random answers at all should be thought of as depending on the toss of a fair three-sided dice hidden in his brain: if the dice comes down 1, he doesn't answer at all; if the dice comes down 2, he answers ‘da’; if 3, ‘ja’.
Thursday, October 15, 2009
`He' and `she' demonstrating me
Tuesday, July 21, 2009
A.N. Prior on double indexing
``Hans Kamp devised in 1967 a consistent semantic interpretation for `now' which can be presented, with slight modifications, as a new sort of UT-calculus, in which T ties each tense-logical proposition not to one instant but to two, i.e. our basic form is not Tap but Tabp. The proposition p is related to the instants a and b in different ways; the essential difference is that the elimination of complexities from what is put after Tab may take us to other instants than a, but never to other instants than b. And wherever we may have been taken from a by operators like G and H, the one place to which we are always immediately taken by J [`It is now the case that'] is the instant b, i.e. the instant represented by the second argument of T. We might read the form Tabp as `From b it is the case at a that p', and `From b it is that case at a that p--now' = `From b it is the case at b that p'." -- Prior, A. N.: 1968, `"Now"', Nous 2(2), 101–119.
The 1967 work that Prior refers to is Kamp, H.: 1967, "The treatment of `now' as a 1-place sentential operator", multilith circulated to a graduate seminar at the University of California in Los Angeles. The results were reported in A.N. Prior (1968) and full treatment was published in Kamp (1971),`The formal properties of ``Now"', Theoria 37, pp. 227-274.
Prior presents problematic sentences such as,
(A) It will be the case that it is now the case that I am sitting,
and attributes to Kamp the above ``the first solution offered" to such problems. Prior agrees that this works but goes on to present an alternative (singly-indexed) treatment by introducing an instant-constant n for a designated instant (and then doing some other tricks due to Prior & Meredith (1953) -- I don't think he just gives extensional treatment of the operators but it seems very similar).
