Zimmermann basics

Zimmermann is interested in analyzing (unembedded) disjunctions as "lists of epistemic possibilities." Here, what "list" crucially involves is closure: a list of items is closed when nothing else can be added to the list: (intuitively, the list items union to and *cover* the contextually determined set over which we are quantifying--so nothing can be added to the list except a set which is a union of subsets the sets that are already in it). Various "closure operators" which may be added to "open" lists are discussed (and many of them involve Montague type-lifting the items in the list.)

What this involves for choice sentences, syntactically speaking, is that all choice sentences are analyzed as wide disjunctions (the contrast here with Simons couldn't be greater.) So the following inference does follow for Zimmermann:

Might(A)

Therefore, Might (A v B)

...it's just that the conclusion is not a parse that occurs in natural language. As far as I can tell, this gives [[or]] quite a complicated semantic entry. It is going to be a recursive, raised-type thing, but the basis for the recursion will be:

[[or]] = \lambda p . \lambda q . might p and might q.

In order to figure out how the recursion goes, I'll need to learn more about Montague lifts.

Post-Meeting 2/19/10

--Challenge: Consider two felt entailments:

(1) (Might A) v (Might B) => (Might A) & (Might B)

(2) Might (A v B) => (Might A) & (Might B)

Considering that we will, according to Simons, need recourse to pragmatic mechanisms to explain the first felt entailment, isn't a semantics that gives us a semantic entailment in (2) redundant?

--Answer: I'm not sure. It depends on:

1) whether the first felt entailment is really that strong. I guess I don't think it is--especially not on the assumption that epistemic modal operators work in the same way that deontic and other modals do (but perhaps this is not a good assumption to make...) Even without relying on the analogy with other forms of modality, it seems like the felt entailment is NOT CANCELLED by the rider "but I don't know which", but rather forced to a reading in which the entailment was never felt in the first place!

There is Zimmerman's argument that A v B => Might A & Might B. This is a good one in most circumstances. But this is definitely pragmatic.

2) the status and viability of my/Simons's claim that epistemic modal operators by default take type-lifted arguments--that is, sets of propositions rather than bare propositions. Reply: but the Hamblin Type-shift will make it the case that bare propositions are also (singleton) sets of propositions. Counter-reply: ok, well, I was considering an alternative semantics in which the type-lift doesn't occur until the derivation hits an operator that demands it: that means that, when [[Might]] hits a set {p1, p2} of propositions, it will try to compose with the whole unit before trying to compose with the individual disjuncts. Since it CAN compose with the whole unit, that's what it will do--it will never go to the fallback step. (Simons thinks that it does, sometimes, but I don't think so.)

Q) How devastating is it, for Simons, that on her semantics [[must]] and [[might]] aren't duals?

A) It's not so good, but note that in her favor they do come out duals in the single-proposition case: that's the case about which we have the strongest prima facie intuitions.

Counter-answer) Yeah, but, those negations of the second type (``You can't take French or Spanish, you HAVE to take Spanish") sound awfully metalinguistic!

[what is the status of metalinguistic negation...?]

Intriguing Quotes Dept.: Dummett

[why not rely on a "know it when you see it" characterization of proper names--as Frege apparently did?]

"It is not merely that there would be a great many borderline cases which some would be inclined to accept as proper names and others to reject, with no principle on which either could defend his judgment. It is, rather, that such an account would be unable to display the connection between the classification of some expressions as proper names and the use of those expressions. Being funny and being red are detachable qualities: someone with no sense of humor or someone totally color-blind misses a lot, but there is nothing else which we can infer that he misses in virtue of his missing these things. By contrast, someone who cannot recognize an expression as a proper name must either fail to understand the expression, or else must simply fail to grasp the concept `proper name' but yet be capable of coming to grasp it. Of someone totally color-blind we may well say that he is never capable of acquiring the concept red: but someone who understand language must already have that by means of which he could come to learn which expressions are proper names and which are not, if we could but find the correct means to explain this concept to him."

Dummett, Frege: phil. lang, 55

Things I would like to know:

1) What typed lambda-calculus has to do with avoiding set-theoretic paradoxes.

2) What assignments have to do with different ways of stating the recursive truth-conditions for the Universal and Existential quantifiers.

3)

Inconclusive Conclusions II

Thisness and Vagueness

Forbes, Synthese, Jstor not working, wtf?!?!

Inconclusive Conclusions I

Schiffer's wide-scoping objection and Heck's we-don't-even-need-wide-scoping objection.

Contraposing MP-restricted conditionals*

Consider the following MP-restricted conditionals:

(1) If p, then Def(p).

(2) If the miners are in shaft A, we should block shaft A.

They are "restricted" in the sense that they cannot be used in conditional proof, reductio ad absurdum, or proof by cases (disjunction elimination). Straightforward contraposition yields:

(1') If ~Def(p), then ~p.

(2') If we shouldn't block shaft A, the miners aren't in shaft A.

(2') leads to the following disaster:

1. We should block neither shaft.

2. We should not block A and we should not block B. (from (1))

3. If we shouldn't block A, the miners aren't in A.

4. If we shouldn't block B, the miners aren't in B.

5. The miners aren't in A and they aren't in B.

Note that there are no subordinate deductions here! Uh-oh!

Keefe proposes that if a MP-restricted conditional is "p -> q", one may contrapose with "~q -> ~D(p)", which gives

(1'') ~Def(p) -> ~Def(p)

(2'') If we shouldn't block A, the miners aren't definitely in A.

[(2''') If the miners are definitely in A, then we should block A.]

Let's see the horrible proof again with Keefe's contraposition instead of the standard contraposition:

1. We should block neither shaft.

2. We shouldn't block A and we shouldn't block B.

3. If we shouldn't block A, they aren't definitely in A.

4. If we shouldn't block B, they aren't definitely in B.

5. They aren't definitely in A and they aren't definitely in B.

That sounds excellent. What to make of this??

Prima facie, it looks bad for the MP-is-invalid-ers, because it supports the alternative hypothesis that the conditional in question is just plain false: it's a conditional for which MP fails and contraposition fails. But considering that the only way to prove that "~q -> ~p" follows from "p -> q" is to use reductio and conditional introduction, perhaps this is all to be expected. (Note that contraposition also fails to preserve the intuition or truth for McGee's infamous "Republican will win" conditional.)

What to make, though, of Keefe's rule of contraposition? Because by contraposing twice, you should get back to what you originally started with:

(2) If the miners are in shaft A, we should block shaft A.

(2') If we shouldn't block shaft A, it's not the case that the miners are definitely in A.

(2'') If the miners are definitely in A, we should block shaft A.

This seems to indicate that (2'') is the real meaning of (2). We have gone from three truth-values in the metalanguage to an operator, "Def", appearing in the object language.

What's the difference between it's being true that Def(p) and its being definitely true that p? Perhaps the sketchy remarks at the end of Heck give some indication. Heck writes that it is emphatically not the intuitionist's viewpoint that there are three truth-values, or any number of truth values n s.t. n > 2:

"Intuitionists reject the principle of bivalence and so deny that every statement is determinately either true or false. Yet, according to Intuitionism, no statement can be neither true nor false: the Intuitionist's view is emphatically not that, instead of two truth-values, there are many. The view is...rather that mathematics does not merit the kind of objectivity we are inclined to accord to it: in mathematics, on this view, we may speak of what is true only in terms of what is provable; of what is provable, only in terms of what we can, in principle, prove. Intuitionism rejects any notion of truth according to which the truth or falsity of a mathematical statement is independent of our epistemic capacities; hence, the notion of a statement which is objectively "neither true nor false"--one which could neither be proven nor refuted by any intuitive proof--is, at best, not one for which Intuitionists have any use and would be regarded, by many Intuitionists, as dubiously intelligible." (292)

(Hey fantastic disambiguation! Use "emphatically" rather than "definitely"! Just like "queasy" rather than "vague"!)

Perhaps a way of glossing Intuitionism would be this. According to Intuitionism, truth-values themselves are subjective. So to claim that there is an objectively existing third truth-value is to miss the point, because the whole idea behind the truth-value gap was that truth-values are subjective, rather than objective.

**

MacFarlane and Kolodny, "Ifs and Oughts," ms.

Rosanna Keefe, "Theories of Vagueness," CUP.

R. G. Heck, "That There Might Be Vague Objects (so Far As Concerns Logic)," Monist 1991.