(L(p -> q) & Lp) -> Lq

are valid and that validity is closed under prefixing L's:

if p is valid, then so is Lp.

For example, L could be the modal operator 'necessarily', or it could mean the same as ''. If it means the same as '', then

is invalid, since it translates as

.

So the principle of Universal Instantiation

provided y is free for x in p

must be restricted (or we have to redefine 'free'). The following version, however, would be sound:

provided y is free for x in p.

Is there anything L could mean that would invalidate this version? Or do the principles for L guarantee that it holds?

(LL) x=y -> A(x) <-> A(y).

'x=y' is true iff 'x' and 'y' co-refer, and 'A(x) <-> A(y)' is true iff 'A(x)' and 'A(y)' have the same truth-value. So to say that all instances of (LL) are true is to say that

(E1) for any terms x and y, A(x) and A(y) have the same truth-value whenever x and y co-refer.

And wasn't this just the definition of an extensional context?

Well, this is one reading of the definition. But there is another, arguably more useful, reading. On this alternative reading, t occupies an extensional context in A(t) iff

(E2) for any terms x and y, A(x) and A(y) have the same truth-value whenever x in A(x) co-refers with y in A(y).

This second definition does not support (LL).

Two definitions -- so two notions of extensionality. They come apart if different occurrences of a term can differ in reference.

For example, let A(t) be the formula (x)(x=t) in a predicate logic in which free variables denote, just like constants. A(x) is true: it says that all things are self-identical. But A(y) may be false, even if x and y corefer: A(y) says that all things are identical to y. So by the standards of (E1), this is not an extensional context. The relevant form of (LL) is invalid:

x=y -> (x)(x=x) <-> (x)(x=y).

On the other hand, according to (E2) the context is extensional: In A(x), x has been captured by the quantifier; since bound variables don't refer, there is no co-refering term y that one could substitute for x in A(x).

This may sound like a technical trick, but the verdict delivered by (E2) seems to me correct: there is an important sense in which ordinary predicate logic has only extensional contexts; so (x)(x=t) should count as extensional.

For another example, let A(t) be 'Philip believes that t denounced Catiline'. A(Cicero) is true and A(Tully) false, although Cicero=Tully. So (LL) fails and the context is non-extensional according to (E1).

However, assume Frege was right about attitude reports. Then 'Philip believes that' opens a scope in which every expression denotes what is ordinarily its sense. So in A(Cicero), 'Cicero' denotes not Cicero, but the sense of 'Cicero'. Since the sense of 'Cicero' differs from the sense of 'Tully', the two names do not co-refer in A(Cicero) and A(Tully). An expression x that co-refers with 'Cicero' in A(Cicero) would have to be an expression whose (ordinary) sense is identical to the (ordinary) sense of 'Cicero'. Presumably, if there is such an expression x, then A(x) is true. So the context is extensional according to (E2).

Again, there is an important sense in which this really should count as extensional. The main point of Frege's theory is to ensure that the truth-value of every sentence is determined by the referents (i.e., extensions) of its parts -- that is, to render all contexts extensional.

So one shouldn't say that Leibniz' Law is valid whenever the relevant context A(t) is extensional.

(Incidentally, (E1) and (E2) can't be quite right, as they wrongly entail that 'Cicero' occupies an extensional position in 'Wo believes that either Cicero denounced Catiline or London has eight railway stations'. I'm not sure how to fix this.)

Many social norms are not constitutive of any particular activity: tipping in cafes or helping your neighbours are not things you have to do in order to partake in a particular activity -- except perhaps in the activity being a good citizen, i.e. being someone who follows the social norms. More importantly, norms that are constitutive of this or that activity need not be social norms at all. In many circumstances, there is no social norm to jump from roof-top to roof-top, although this is constitutive of a certain game popular among adolescents in France. Of course, if you are socially committed to partake in activity X, then the constitutive norms of X become social norms. In extension, social and constitutive norms may overlap, but the two concepts are still very different.

How do we check if something is a social norm? We see whether people are criticised for deviations, whether those who deviate tend to accept criticism and apologise or offer excuses, etc. This is exactly the kind of data philosophers often point at when they discuss norms of assertion. It is not clear why such facts should be relevant if the topic are constitutive norms of assertion. Perhaps the argument is that by uttering certain sentences, speakers commit themselves to partake in the activity of assertion, and hence the constitutive norms of assertion can be read off from the social norms then in force.

Both steps in this argument are questionable. By uttering "the door is locked", I certainly commit myself to various things. For instance, I commit myself to the door being locked. Do I also commit myself to partake in a special activity of assertion? Compare the social norms that come in force when I walk into the office with a big cake. I am obliged to offer my co-workers some of the cake; but is this because by entering the office with cake I have committed myself to partake in a special activity of, uhm, appropriately entering an office with cake, with its own constitutive norms? This is at best a rather stretched description of what is going on.

For any activity X, there is also an activity X+ of doing X while obeying the social norms. The social norms in force when doing X are then constitutive norms of X+. But talk of constitutive norms of X+ing just adds a superfluous level of descriptive complexity to what are really social norms of Xing.

The problem with the second step in the above argument is that "assertion" isn't quite a term like X+: one can make assertions that violate social norms. Some people do that all the time. But then it isn't obvious how one could read off the constitutive norms of assertion by looking at the social norms in place when someone makes an assertion.

It is unclear to me whether there are any constitutive norms of assertion, and if so, what use uncovering these norms would be in a theory of linguistic communication. Perhaps the only constitutive norm of assertion is to utter grammatical sentences, although we normally expect more from a speaker? Or perhaps the constitutive norm of assertion is to utter sentences that could not possibly be false, and we normally expect less?

The more interesting and important question is the second one, the one about social norms concerning assertions, or better: about social norms concerning utterances of declarative sentences. When I say "the door is locked", I first and foremost utter a declarative sentence. Whether I also made an assertion depends in part on the social norms that come in force.

(I recently tried to put this point to Williamson, but I think I didn't express myself clear enough: his reply was that there are no social norms for uttering declarative sentences.)

The first comes from intuitions about attitudes: when you think about your situation, couldn't you wonder where you will wake up? Couldn't you hope that it will be on Mars? Couldn't you imagine having the future on Venus? If the content of your wondering, hoping and imagining are centered possibilities, then these possibilities must have a unique, determinate future: since your hope to wake up on Mars differs in content from your fear to wake up on Venus, there must be one centered possibility verifying "I will wake up on Mars" and another verifying "I will wake up on Venus".

Dilip Ninan offers this motivation in his recent paper "Persistence and the First-Person Perspective", and he cites several authors who apparently put forward similar intuitions. I have to admit that I don't share these intuitions. In fact, they seem clearly false to me. If you wonder where you will wake up, you have misunderstood your situation. Perhaps you think you are a non-physical soul that won't fission at all. But that's not your actual situation.

If there were two pre-fission possibilities, it should make sense to ask which of them obtains. But the only sensible anwer is 'both'. And then you can't reasonably wonder which of them obtains. Moreover, if the two possibilities are genuine alternatives, it should be possible (in principle) to learn which of them obtains. But if you understand your situation, you know that anyone who comes along and tells you that you'll wake up on Mars and not on Venus is a liar. Even if it is God.

So the first reason doesn't work.

The second reason for postulating distinct possibilities comes from certain views about the metaphysics of people and the nature of centered worlds. On the popular "worm" view, there are two people in the fission scenario, one who wakes up on Mars (call her M) and one who wakes up on Venus (V). Both M and V exist today, although they somehow coincide. If centered worlds are something like world-time-individual triples, then we naturally have two possibilities at times t before the fission: <w,t,M> and <w,t,V>.

Something like this reasoning can be found in recent papers by David Wallace and Simon Saunders. I find both steps unconvincing. I don't believe in the worm view of persons, and I don't think centered worlds can be modeled as triples of a world, a time and an individual.

But the issue is subtle. Let's say that a centered world is a maximally specific way things might be, or a maximally specific property. Such a property must include historical information: "x is presently F and G and H" is less specific than "x is presently F and G and H and will be J and used to be K". So a maximally specific property can't be silent on whether and where x will wake up tomorrow. However, among the maximally specific futures a thing can have is to wake up both on Mars and on Venus. For instance, consider your present time-slice S. A maximal specification of S's properties would include having successors on both Mars and Venus. Or consider the fusion F of V and M. A maximal specification of F's properties would include having future parts on both Mars and Venus.

Lewis once suggested that although fissioning involves two persons M and V, any pre-fission thinking and wondering is done by a stage S that is common to M and V. This seems to imply that if you believe that you are temporally extended, then what you believe is false: the belief content is a class of extended objects, but the believer is unextended; so the content is false. Better say that what does the thinking and wondering is the fusion F; then your belief that you exist tomorrow on Mars is true, and so is your belief that you exist tomorrow on Venus.

Apart from S and F, of course the two worms M and V also exist, as do their maximally specific properties. But I don't think you can reasonably believe or hope to have one of these properties rather than the other. Compare the gerrymandered fusion of your present stage S together with some past stages of the Eiffel tower and some future stages of Barack Obama. You can't reasonably wonder whether you are this object rather than some other, equally gerrymandered fusion (unless, of course, you misunderstand your situation e.g. by thinking that only one of those fusions exists).

So the second reason also doesn't work.

Not only that, in both cases the relevant considerations seem to support exactly the opposite conclusion: that your distinct future possibilities do not correspond to distinct present possibilities.

This is unfortunate, because distinct pre-fission possibilities would be very convenient for the epistemology of branching subjects. They would help a lot to simplify my centered form of conditionalisation, and they might be useful in the confirmation theory of quantum mechanics (this is what Wallace and Saunders use them for). That's the third reason.