Time Travel, Life, and Death

In this post, I'd like to describe a rather typical time travel case in order to elicit views concerning whether a certain claim is true in the case. Having presented the case and asked the question, I will explain some of the reasons for my interest in whether that claim is true.

Time Travelling Tim: Tim Travell was born at t1 in a rural area in Connecticut. He lived a normal (non-time travelling) life until t20, when he came into possession of a time machine. Using this time machine, Tim travelled forward in time to t500. However, soon becoming bored of the futuristic lifestyle, at t502, Tim travelled back in time to t21 and lived out the rest of his life without ever again time travelling. He died at t90.

Assuming that no (non-time travelling) funny business (for instance, resurrection) happened in this case, the claim concerning which I'd like to elicit views is:
(T) At t501, Tim is dead.
So, what do you guys think? Is (T) true or false?

Now let me explain some of the reasons why I am interested in whether (T) is true or false. First, if (T) is true, then being alive and being dead are not (contrary to common opinion) contraries. For it is clearly true that at t501, Tim is alive. So, if (T) is true, then there is a time at which something is both alive and dead.

Second (but relatedly), if (T) is true, then certain analyses of being dead are false. For instance, consider the following proposed analysis:
Proposed Analysis of Being Dead (PABD): Necessarily, for all x, x is dead (at t) iff (i) x was alive (at some time earlier than t) and (ii) x is not alive (at t).
If (T) is true, then (PABD) is false. For if (PABD) is true, then (T) is not true (since Tim is alive at t501 and hence, according to (PABD), isn't dead then).

Anyway, what do you guys think?

Possible Worlds and "Cases"

Some philosophers would like to (reductively*) analyze possibility and necessity in terms of possible worlds. As I understand their position, it is something like this:

Analysis of Possibility and Necessity (APN): There is a property P ("the property of being a possible world") and a relation R ("the relation of being true at") such that (i) being possible = being an x such that there is a y such that Py and Rxy and (ii) being necessary = being an x such that for all y, if Py, then Rxy.**

One argument some have offered in favor of (APN) is that treating possibility and necessity as existential and universal quantification over worlds, respectively, allows us to explain the validity of various inferences containing 'necessarily' and 'possibly'.*** The idea, I take it, is that sentences of the form:
(N) Necessarily, S.
are in fact ascriptions of the property of being necessary to the proposition that S:****
(N') That S is necessary.
which, given (APN), is equivalent to:
(N'') That S is such that for all possible worlds w, it is true at w.
Similarly for sentences of the form:
(P) Possibly, S.

Now it is interesting to note that to explain certain inferences containing 'necessarily' and 'possibly', proponents of (APN) are going to have to attribute hidden structure to sentences of English. For instance, consider the following inference:
(1) Necessarily, S.
(2) Therefore, S.
Given what has been said before, proponents of (APN) will claim that (1) is equivalent to:
(1') That S is such that for all possible worlds w, it is true at w.
But then whence the inference from (1) to (2)? Presumably, the proponent of (APN) will have to say that (2) is equivalent to something like this:
(2) Therefore, that S is true at a ("the actual world").

Now I want to draw an analogy between the argument for (APN) that I have been discussing and an argument for a different thesis. Call the (relevant) relation that obtains between the propositions that P and that Q iff if P, then Q being conditioned on. Now consider the following thesis:

Analysis of Being Conditioned On (ABC): There is a property P ("the property of being a case") and a relation R ("the relation of being true in") such that being conditioned on = being an x and a y such that for all z such that Pz and Rxz, Ryz.

The idea behind (ABC) is that a (material) conditional is true if and only if (the proposition expressed by) its consequent is true in all cases in which (the proposition expressed by) its consequent is true. Just as 'is a possible world' and 'is true at' are technical terms used to express whichever property and relation appear in the correct analysis of being necessary and being possible, so too 'is a case' and 'is true in' are technical terms used to express whichever property and relation appear in the correct analysis of being conditioned on.

Notice that just as (APN) allowed us to "explain" the validity of inferences containing 'necessarily' and 'possibly', so too (ABC) allows us to "explain" the validity of inferences containing the material conditional. For instance, hypothetical syllogism:
(1) If P, then Q.
(2) If Q, then R.
(3) Therefore, if P, then R.
HS can be explained by the proponent of (ABC) by treating (1), (2), and (3) as, respectively:
(1) For all cases x such that that P is true in x, that Q is true in x.
(2) For all cases x such that that Q is true in x, that R is true in x.
(3) Therefore, for all cases x such that that P is true in x, that R is true in x.

Also notice that a proponent of (ABC) will also need to attribute hidden structure in sentences of English to explain the validity of certain inferences, such as modus ponens:
(1) If P, then Q.
(2) P
(3) Therefore, Q.
In particular, they will need to postulate that (2) and (3) contain a case constant referring to the "real case".

It seems, then that the argument from the explanation of the validity of inferences for (APN) has an analogue: an argument from the explanation of the validity of inferences for (ABC). However, I take it that many of those who find the first argument plausible will not find the second argument plausible. But why? The cases seem relevantly similar. Via translation, we can explain the validity of inferences involving 'possibly' and 'necessarily' using possible worlds, and we can do the same for the validity of inferences involving the material conditional using cases. In addition, both views seem to involve positing hidden structure to English sentences. So what gives?

[One potential answer is that restricted quantification of the sort used in the statement of (ABC) is usually explained in terms of unresticted quantification and the material conditional. Hence, some may take (ABC) to be circular. However, restricted quantification needn't be construed in this way. Inference rules for restricted quantification can be introduced. So can a semantics. Thus, we are not forced to explain restricted quantification partly in terms of unrestricted quantification.]

* I include 'reductively' here because some philosophers make a distinction between reductive and non-reductive analyses. However, I must admit that I don't know what a non-reductive analysis is supposed to be.

** We could introduce an accessibility relation into the analysis as well without affecting the point I am trying to make.

*** I take it that only a "reductive" analysis would explain the validity of these inferences.

I got the idea for this post from teaching "baby logic" to undergraduates. They seem to grasp why certain inferences using the material conditional are valid/invalid best when those conditionals are translated into talk of cases and Venn diagrams are used.

Vagueness and 'part'

Let's restrict our attention to possible worlds where the meanings of words in the English language the same as they actually are. Given, this assumption, it is generally accepted that:

(V) Necessarily, if the English phrase 'is a part of' is vague, then so is the English phrase 'is identical to'.

It turns out that this is not a logical truth. There are models on which 'is a part of' is vague yet 'is identical to' is not. Moreover, some of these models obey Classical Extensional Mereology. But, there are some decent metaphysical reasons to believe that the claim is true. But, this is very troublesome since most philosophers want to link the content of 'is identical to' to the content of 'exists'. This all seems to lead to vague existence, which very few people want to accept.

The lesson is supposed to be that any vague phrases of the form 'a is part of b' are vague in virtue of the vagueness of the singular terms not in virtue of the vagueness of the word 'part'. I would like to challenge this lesson.

Consider the computer on by desk and call it 'Computer'. Now find an arbitrary atom near the surface of the computer that is such that 'that atom is a part of the computer' is vague when the complex demonstrative 'that atom' refers to the arbitrary atom. Now name the atom 'Fred'. Here is the vague phrase that I want to consider:

'Fred is a part of Computer'.

According to the lesson above, this phrase is vague in virtue of the vagueness of 'Fred' or the vagueness of 'Computer' but it is definitely not vague in virtue of the vagueness of 'part' (lest we fall prey to the vague identity and existence).

Now for the challenge. If 'Computer' is vague, then it is vague in virtue of the fact that there are multiple candidate referents of 'Computer'. Similarly, if 'Fred' is vague, then it is vague in virtue of the fact that there are multiple candidate referents of 'Fred'. Let's make the terms more precise. Let 'Computer*' name an arbitrary candidate referent of 'Computer' and let 'Fred*' name an arbitrary candidate referent of 'Fred'. Now, given that the names 'Computer*' and 'Fred*' are precise and that 'part' is precise as well, it is clear that the following sentence is not vague:

'Fred* is a part of Computer*'

This sentence is either definitely true or it is definitely false. Each case is relevantly just like the other. So, let's just suppose that it is definitely true. Now consider a continuous series of worlds each one just like the actual world except that the referent of 'Fred*' has been moved some small distance, n, away from it's actual position. There is a corresponding series of counterfactuals of the form:

(CS1) if Fred* had been n units from its actual position, then the English sentence 'Fred* is a part of Computer*' would be vague.

It seems that one of the counterfactuals in this series expresses a truth. But, since the names 'Fred*' and 'Computer*' are actually precise and since we restricting our attention to worlds where the meanings of English are held fixed, it is clear that:

(CS2) If Fred* had been n units from its actual position, then it would not be a fact that 'Fred*' is vague and it would not be a fact that 'Computer*' is vague.

But given the plausible necessary truth:

(N) Necessarily, if the English phrase 'Fred* is a part of Computer*' is vague, then either 'Fred*' is vague or 'Computer*' is vague or 'is a part of' is vague.

It follows from (CS1), (CS2), (N) and the obvious claim that it is possible that Fred* is n units from its actual position that:

(PV) Possibly, the English phrase 'is a part of' would be vague.

Of course, since we have been restricting our attention to worlds where the meanings of English are the same as they actually are, it follows that

(AV) it is actually the case that the English phrase 'is a part of' is vague.

But, this is contrary to the lesson that we were supposed to learn. (AV) in combination with (V) gets us the unwanted consequence that 'is identical to' is vague and ultimately gets us vagueness of existence.

Right now, I think that the best response is to reject that any instance of (CS1) is true. One might claim that some instance of (CS1) seems true because the following is true:

(VCS2) the sentence 'if Fred* had been n units from its actual position, then Fred* would be a part of Computer*' is vague.

We, the defender of the lesson might say, just think that (CS1) is true because (VCS1) is true. Moreover, the vagueness of the sentence talked about in (VCS1) is vague in virtue of the vagueness of subjunctive conditionals.

I am not happy with this response. But, I am having a hard time thinking of an alternative.