Saturday, 14 June 2014

Relativization of quantifiers and relativizing existence

One can relativize a claim by inserting a qualifying predicate for each quantifier. For example,
(1) For any number $n$, there is a number $p$ larger than $n$.
(2) For any prime number $n$, there is a prime number $p$ larger than $n$.
This is called relativization of quantifiers. Whereas (1) is kind of obvious, (2) is not. Formally, (1) is called a $\Pi_2$-sentence, as it has the form, roughly,
(3) $\forall x \exists y \phi(x,y)$
Suppose (1) is true. When we relativize to the subdomain of prime numbers, it expresses a different proposition, and we can consider whether it remains true in the subdomain which is the extension of the relativizing predicate. I.e.,
(4) $\forall x (P(x) \to \exists y (P(y) \wedge \phi(x,y))$.
ln fact (4) is true, but it expresses something stronger than (3) does. We might write the relativized (4) more perspicuously as,
$(\forall x \in P)(\exists y \in P) \phi(x,y)$.
$(\forall x : P)(\exists y : P) \phi(x,y)$.
$(\forall x)_P(\exists y)_P \phi(x,y)$.
Nothing hinges much on this: it is pretty clear what is meant either way.

Suppose we relativize to a finite set. Let $D(x)$ mean "$x$ is either 0, 1, 2 or 3". Then
(5) $(\forall x \in D)(\exists y \in D) \phi(x,y)$
is now false.

If $\Theta$ is the original claim, then we sometime denote the claim relativized to $P$ as $\Theta^P$. The fact that $\Theta$ is true does not in general imply that $\Theta^P$ is true. In general, if $\Theta$ is a true $\Pi_1$-sentence, then its relativization $\Theta^P$ is true as well. (In model-theoretic lingo, we say that "$\Pi_1$-sentences are preserved in substructures".) On the other hand, if $\Theta$ is a true $\Pi_2$-sentence, then its relativization need not be true, as we saw above.

Here is a vivid example. Imagine a society which contains Yoko, who happens not to be married to herself, and in which the following $\Pi_2$-sentence is true:
(6) Everyone is married to someone.
Now restrict this claim to the unit set, $\{Yoko\}$. Clearly,
(7) Everyone who is Yoko is married to someone who is Yoko,
is false.

This tells us a bit about how to relativize the quantifiers in a sentence to a predicate.

It may be annoying to keep relativizing univocal quantifiers, and one might prefer a many-sorted notation, in which distinct styles of variables are used to range over separate "sorts". So, for example, in textbooks and articles, we generally know that
the letter "$n$" (and probably "$m$") is going to denote a natural number.
the letter "$r$" (and probably "$s$") is going to denote a real number.
the letter "$z$" is likely to denote a complex number.
the letter "$t$" is likely to denote a time instant.
the letter "$f$" is likely to denote a function.
the Greek letter "$\phi$" is likely to denote either a mapping or a formula.
the Greek letter "$\omega$" is likely to denote either the set of finite ordinals or an angular frequency.
the upper-case Latin letter "$G$" is likely to denote either a graph or a group, and "$g$" will denote an element of the graph or group.
With capital Latin letters, "$A$", "$B$", "$C$", $\dots$, all bets are off! But "$X$" or "$Y$" are likely to denote sets. So, if you see, e.g., the equation,
(8) $f(t) = r$
then intuitively, the intention is that the value of the function $f$ at time $t$ is some real $r$.

While these issues seem fairly clear, can sense be made of relativizing existence itself? That is, can we make sense of a claim like:
(9) $x$ and $y$ "exist in different senses"

For example,
(10) The Eiffel Tower and $\aleph_0$ exist in different senses.
(11) Dame Kelly Holmes and Sherlock Holmes exist in different senses.
We usually think such claims are meaningful -- surely they are. But what exactly do they mean? Probably, something like this,
(12) $x$ and $y$ are (from or members of) different kinds of things.
And this seems to mean,
(13) there are kinds (types, ontological categories, ...) $A,B$ such that $\square[A \cap B = \varnothing]$, and $x \in A$ and $y \in B$.
There are two necessarily disjoint categories and $x$ is in one, and $y$ is in the other.

Quine wrote a famous paper, "On what there is" (1948). Normally, following Quine, we treat "what there is" and "what exists" as synonyms. But it is not very interesting to inquire as to what "exists", if one insists that "exists" be a predicate. If one insists that "exists" be a predicate, then what then becomes interesting is what this predicate "$x$ exists" means. Everyone agrees that ordinary usage counts as grammatical both:
(14) There exists a lion in the zoo.
(15) Sherlock does not exist.
The first is normally, and uncontroversially, formalized using the quantifier "$\exists$" and the second seems, on its surface, to involve a predicate.

[I have a mini-theory of what "$a$ exists" means. I think a claim of the form "$a$ exists" means "$\exists x H_a(x)$", where $H_a$ is, loosely speaking, the property of being $a$.]

Quine stressed that the meaning of the symbol "$\exists$" is explained as follows:
(16) $\exists x \phi$ is true if and only if there is some $o$ such that $\phi$ is true of $o$.
In other words, we explain the meaning of "$\exists$" using "there is". I can't quite see how it might work otherwise, except: by a proof-theoretic "implicit definition", via introduction and elimination rules.

Consider the following idea: the idea that the following two claims
(17) $\exists x \phi$ is true
(18) there is nothing that is $\phi$ 
are compatible.

One finds something like this being advocated as a solution to some problems in the foundations of mathematics. I think - but I am not sure - that Jody Azzouni's view is that (17) is compatible with (18). This would imply that there being no numbers (say) is compatible with the truth of mathematics. I cannot make good sense of this, mainly because the technical symbol "$\exists$'' is introduced precisely so that (17) and (18) are incompatible. Similarly, claim like,
(19) The sentence "There are numbers" is ontologically committed to there being numbers
is simply analytic, since it is part of the definition of the phrase "ontological commitment".

Suppose someone says there are things that don't exist (e.g., fictional objects or perhaps mathematical ones). I assume that, in their idiolect, "exists" means "has some property", but what this is has been left unspecified. If so, it means
(20) There are things which lack property $\dots$.
And what this $\dots$ is, is somehow left unspecified. A crucial ambiguity can arise. For example, the claim
(21) Numbers don't exist.
can be taken to mean,
(22) If there are numbers, they don't "exist"
(23) There are no numbers.
With a charitable interpretation, the first claim (22) is true, but not very interesting, because "exists" probably just means (in the speaker's idiolect) "is a concrete thing". No one in the world asserts that numbers are concrete things! The second claim, (23), is exciting: it denies that there are numbers.

Returning to relativized existence claims, like a claim of the form
(10) The Eiffel Tower and $\aleph_0$ exist in different senses,
I don't really see how making sense of such a claim requires anything other than working with many-sorted logic, where the sorts are thought of as having some deep metaphysical significance. For example, the assumed significance might involve a Platonic theory of Being vs. Becoming, and then we might take (10) to be based on an assumption like
(24) The Eiffel Tower belongs to the world of Becoming, while $\aleph_0$ belongs to the world of Being.
One would need to be careful about trying to make this kind of approach work with a 1-sorted logic, for example using a pair of quantifiers $\exists_1$ and $\exists_2$, as a famous argument shows that an assertion of existence-in-sense 1 is logically equivalent to an assertion of existence-in-sense 2:
$\vdash \exists_1 x \phi(x) \leftrightarrow \exists_2 x \phi(x)$.
Proof. Suppose $\exists_1 x \phi(x)$. Skolemize, to give $\phi(t)$, where $t$ is a skolem constant. By Existential generalization, $\exists_2 x \phi(x)$. So, $\exists_1 x \phi(x) \to \exists_2 x \phi(x)$. Similarly in the other direction.

I believe that Kurt Gödel says somewhere that no sense can be made of relativizing existence itself, and Quine also makes a similar point in various writings.

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