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.