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 = {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,..., α_n are variables, then πα₁,..., α_n 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,..., α_n]]^c,f,t,w = 1 iff ([[α₁]]^c,f,t,w ,..., [[α_n]]^c,f,t,w ) ∊ I_π(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 V_free and V_bound. 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?

Random randomly silent

In a new paper by Gabriel Uzquiano called ``How to solve the hardest logic puzzle ever in two questions" he shows how to solve the puzzle in two questions -- one way uses the ignorance of the gods with respect to Random's future answers (if such there be) and the other way uses potentially unanswerable self-referential questions. In either case, the trick is to get information from a god's inability to answer certain questions. Uzquiano suggests a further amendment to the puzzle to avoid such two question solutions: have Random randomly say 'ja', 'da' or remain silent (instead of silence one could have the god randomly suffer a head explosion -- doesn't matter as the logic is the same).

Puzzle. Three gods A, B and C are called, in some order, True, False and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely or whether Random speaks at all is a completely random matter. Your task is to determine the identities of A, B and C by asking yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for ‘yes’ and ‘no’ are ‘da’ and ‘ja’, in some order. You don't know which word means which.

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’.

It is left as an open question, whether this puzzle can be solved in two questions. Landon proves that it cannot be solved in two questions here.

`He' and `she' demonstrating me

Say I point to myself and utter,

(1) He is a student.

It was a bit awkward of me to do that but it seems to me that my utterance was true. But what if I point to myself and utter,

(2) She is a student.

It seems that here something much worse has happened. On some treatments of deictic pronouns both (1) and (2) would be undefined, i.e. they get no truth-value. The pronouns have certain "dominating features" and if the purported value doesn't satisfy one of the features it gets rejected as a value and the pronoun goes unassigned. For `he' the features are masculine, 3rd person, and singular -- where 3rd person is understood as being neither the speaker nor the audience of an utterance.

Pointing at yourself in the mirror without realizing that it is you is enough to remove the awkwardness of uttering (1). Is pointing at yourself in the mirror without realizing that it is you and without realizing that you are dressed as a female enough to remove the awkwardness of uttering (2)? Not sure.

Doesn't seem like `I' can refer to someone who fails to satisfy the 1st person feature. But maybe there is a strange case that removes the awkwardness.

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).

necessity, rigidity and co-extensionality

I have heard people say things like the following.

(K1) A true identity statement is necessary just in case the identity sign is flanked by two rigid designators. 

Such a principle seems to work alright for singular terms but it gets tricky when applying it to general terms -- it is notoriously tricky just how to extend the notion of `rigidity' to general terms. 

I prefer to extend the notion of rigidity to all expressions in the most straightforward manner: An expression is rigid (w.r.t. worlds) iff it has the same extension in all worlds. (This generalizes to temporal-rigidity, spacial-rigidity, agential-rigidity, etc.) 

With this conception expressions like `water' and `H2O' are not rigid. This result is thought to be a reason against generalizing the notion of rigidity in this way. The statement `Water is identical to H2O' is necessary so according to (K1) we should expect that the expressions that flank the identity sign be rigid designators. But with my preferred understanding of rigidity this is not the case, so such an explanation of the necessity of identity statements cannot be given.

I think (K1) is a bad principle. It is somewhat close, however, to the true principle, which is the following.

(B1) A true identity statement is necessary just in case the identity sign is flanked by two necessarily co-extensional expressions.

It is easy to see why one might mistakenly think that (K1) was the correct principle since all rigid designators that flank a true identity statement will be necessarily co-extensional. But it is not the case that all necessarily co-extensional expressions that flank a true identity statement are rigid designators. Consider,

(1) The inventor of bifocals is identical to the extension of `the inventor of bifocals'.*

This is a necessary truth in which the identity sign is not flanked by two rigid designators. Instead the identity sign is merely flanked by two necessarily co-extensional expressions. What is important for the necessity of identity statements is that the pattern of extension across worlds of their flanking expressions is the same not that they are rigid expressions. It just so happens that in cases like 

(2) The successor of one is identical to the smallest prime number,

the fact that the flanked expressions are rigid guarantees that their the patterns of trans-world extension agree. But it is the pattern not the rigidity which is of primary importance to the necessity of identity statements.

The same is true of the following identity statement.

(3) Water is identical to H2O.

The expression `water' and the expression `H2O' have the same pattern of counterfactual extensions. If `water' designates a certain set at world w, then `H2O' designates a certain set at world w.  This is true even though on my preferred understanding of rigidity `water' and `H2O' are not rigid. This seems like a nice simple way to make sense of these issues. What more do we want?

Homework: Think of necessary identity statements in which the expressions that flank the identity sign are non-rigid (but, of course, necessarily co-extensional). 

[*Ignore scope and assume this is an identity statement not a Russellian existentially quantified statement. And note that the expression "the extension of `the inventor of bifocals'" concerns our expression `the inventor of bifocals' not some other homophonic expression.]

The quick argument for double-indexing

Consider the following logical truths (or indexical validities) of English:

(1) It rains if and only if it rains now.

(2) It rains now if and only if it always rains now.

Assume (to reach a contradiction) a singly-indexed semantics. Given that (1) is a logical truth, then the semantic clause for the indexical sentential operator `Now' (or `It is now the case that') must be as follows:

[[Now(φ)]]^t =1  iff  [[φ]]^t =1

But then (2) is not valid (as long as there are times in structures where it rains and fails to rain). But (2) is valid. Contradiction.

[This is basically the argument given in Kamp (1971), "The formal properties of `now'"]

defense of two question solution

I have put a new paper online called "In defense of the two question solution to the hardest logic puzzle ever". It is mostly a reaction to a few blog post, e.g. this one (and some others that I can't find right now) and a manuscript. And we briefly discuss the generalization from this one.