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.

metalinguistic predicates vs. object language operators

I am interested in the distinction between metalinguistic predicates (sentential predicates) and object language operators (or sentential functors). This distinction seems to be very important for topics like (i) the Kripkean necessary aposteriori & contingent apriori, (ii) various implementations of two-dimensional semantics (especially metalinguistic versions?), (iii) issues surrounding context-dependence and monstrous operators, and probably many more things. 

But I don't have much to say about it because I don't fully understand the distinction and how exactly metalinguistic predicates work (how are they treated in the formal system?) So this is just a plea for help. Where is this distinction discussed? Where are metalinguistic predicates discussed? I can't seem to find anything but perhaps I am searching for the wrong thing.

The distinction I have in mind is illustrated below:

(1) It is F that S.

(2) `S' is F.

(1a) It is necessary that two is prime.

(2a) `Two is prime' is necessary.

(1b) It is true that I am hungry.

(2b) `I am hungry' is true.

(1c) It is believed by John that Cicero is Tully.

(2c) `Cicero is Tully' is believed by John.

(1d) It is a priori that bachelors are unmarried.

(2d) `Bachelors are unmarried' is a priori.

(1e) It is true at 2:00am that its raining now.

(2e) `Its raining now' is true at 2:00am.

etc.

Any help would be greatly appreciated.

inconstants?

Some expression are rigid in the sense that they have a constant (secondary) intension, while others are nonrigid. And some expressions are stable, in the sense that they have a constant character, while others are unstable. Moreover, some expressions have combinations of the rigidity and stability properties. 

Constants (e.g. logically proper names?) are both rigid and stable. Indexicals (e.g. `I', `tall') are rigid but unstable. Vacillators (e.g. `kangaroo', `the inventor of bifocals') are stable but nonrigid. 

Are there any expressions that are both unstable and nonrigid?

anaphoric pronouns with indexical antecedents

This probably has a simple and not very interesting explanation having to do with syntactic constraints or morphological constraints or something. But why is it that the starred sentences below go wrong, given that they are obtained from the relevant non-stars by substitution of co-extensive expressions (i.e. `I' for `Brian')? The source of the problem here is obviously not intensionality!

(1) Brian left his wallet on the table.
(1*) I left his wallet on the table.

(2) Brian is such that John believes that he is a spy.
(2*) I am such that John believes that he is a spy.

(3) Brian went to the store and he bought tofu.
(3*) I went to the store and he bought tofu.

Can you think of an example where the referent of `I' can be picked up by an anaphoric pronoun? Is `I' itself sometimes an anaphoric pronoun? How does this relate to the following Heim example?

(4) Only I got a question I understood.

philpapers

Here is a great new tool: philpapers. Its a massive directory of online philosophy articles with many useful features. Read more about it here.