What is the problem we are discussing?

We get into trouble right away, since there is no problem about whether there is such a thing as everything unless there is more than one possible answer. The two obvious possible answers seem to be,

Yes, there is such a thing as everything,

and

No, there is not such a thing as everything.

However, the second answer seems to say that "absolutely every single thing is such that that thing is not everything," and that answer seems to be committed to the coherence of "absolutely every single thing," and it is therefore self-defeating.

That is, to some extent, like the familiar problem Quine calls Plato's beard: When you claim, "Pegasus does not exist," the subject is Pegasus, and the claim is therefore self-defeating since it presupposes the existence of what it tries to deny the existence of.

Quine's proposal is to eliminate names in favor of predicates: not "Pegasus does not exist," but "Nothing Pegasizes."

For everything, that yields, "No thing among everything everythingizes," which doesn't really help.

Thus, we have a dilemma: there can't be such a thing as everything because of Russell's paradox. On the other hand, we can't coherently deny that there is such a thing as everything.

These are problems every theory about whether there is such a thing as everything must confront.

The introduction, being an introduction, just talks of accepting or denying the existence of such a thing as everything, and thus, in the choice of vocabulary, biases the discussion onto the positive side.

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Ian says that a natural response is just to make the claim that the concept of everything is ill defined, or (I'll add) that there is no such concept or something like that.

That is absolutely correct, but it is not the original claim. The introduction notes that there are two related problems:

• The metaphysical question (M): Is there an all-inclusive domain of discourse?
• The availability question (A): Could an all-inclusive domain be available to us as a domain of inquiry?

I, like Ian, think that a negative answer to the availability question entails that there is no metaphysical question. (%$ \lnot A\rightarrow \lnot (\exists x) x=M$%) Thus, we think that the availability question is more fundamental.

Many philosophers, including the authors of the introduction and the majority of authors of articles in the book think the situation is reversed: The really significant question is the first one, and it is only if it has a positive answer that the second question is of any interest.

One kind of argument against M is that M leads to a logical contradiction. "Indefinite extensibility is of that sort.

One kind of argument for M is exhibiting a way of thinking about the issue in accordance with which M does not lead to a logical contradiction. The "only" way of doing that is to exhibit a formal logical system of that sort. Most such arguments start with the basic idea that a property is not an object, that some property (say, the property of being self-identical) holds of absolutely everything, that a domain of quantification can be given by a property, and hence that we can quantify over absolutely everything without being committed to such an object. We want, not only to say things about every object, but about every property.

The nontechnical version of that idea is Cartwright's "all-in-one principle." The all-in-one principle states that if there are some things we can quantify over, then there is a collection of all such things. Denying that the property of being self-identical has an object that is the collection of all self-identical things is just a special case of denying the all-in-one principle.

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It may be that there is no general agreement on what we mean by "object." If that is right, then there can be distinct "everythings" for distinct meanings of "object."

Semantic indeterminacy: It turns out to be the case that if a set of sentences is true in model with an infinite domain, then it is true in many different models with domains of different sizes. But then, everything we can say still leaves even the size of a domain of absolutely everything undetermined. That suggests that we cannot communicate a concept of absolutely everything. From that some (including me) want to conclude that we cannot have a linguistic concept of absolutely everything. It is then natural, following out Ian's idea, to deny the coherence of the metaphysical question.

Homework:

Let %$P(x,y)$% be "%$x$% is a parent of %$y$%." Let %$F(x)$% be "%$x$% is female." We can then define "%$x$% is the mother of %$y$%" as %$P(x,y)\land F(x)$%. Ignore adoption, surrogacy, and miracles, and define grandparent, aunt, whatever. Can you define some of these in terms of others? For a real challenge, formalize IAmMyOwnGrandpa.

-- ShaughanLavine - 22 Jan 2009