Saturday, February 19, 2011

Metaphysically Aristotelian quantification

There is a sense in Aristotelian metaphysics that "there are only substances". They are all there is a focal sense. Yet if we can talk about and quantify over accidents or modes, surely there are accidents or modes.

Here, then, is a simple quantified logic that preserves the Aristotelian intuition. This logic is developed only in the case of modes (or tropes) that are non-relational—that subsist in a single substance. The logic has the standard resources of first order sentential logic, together with the standard universal quantifier symbols ∀x and ∃x which quantify over substances x. But additionally there are two new quantifier symbols: ∀ax and ∃ax which quantify over a's modes x. Thus, "Some table has an accident" becomes:

  1. x(Table(x) and ∃xy(Accident(y,x))).

Then we can say that only the substances exist simpliciter—only they are quantified over by the standard quantifiers Ax and Ex. Modes "exist" only relative to the substance of which they are modes—they are grounded in that substance, as is indicated in the language by the subscripted quantifiers.

We can say that the mode-quantifier ∃ax yields existential quantification in an analogical sense. And we can spell out the analogy at least to some degree by giving rules of inference that are structurally analogous to those for the focal-sense quantifier ∃x.

Here's another application of the notion of relative existence. We might, for instance, hesitate to say that characters in novels really exist, but we might think (I am hesitant about that, too) that novels really exist. We might then think that for any novel N, there is a pair of quantifiers ∃Nx and ∃Nx over the entities-in-N. If S is some Star Trek novel, then when we say that ∃Sx(Klingon(x)), we are not really saying that there really are Klingons. We are saying that virtually, in-the-novel, relative-to-the-novel there are Klingons. This is not a fact about Klingons but about the novel, and our primary ontological commitment is to the novel. Of course then our logic then needs to be suitably designed so that we cannot infer from ∃Sx(Klingon(x)) that ∃x(Klingon(x)). This can all be done, and what I shall do below for modes can be done for characters in novels. Again, quantification over characters is quantification in an analogical sense.

The rest of this post is almost entirely technical and can be skipped.

We leave the truth-functional rules unchanged. We modify the quantificational rules as follows:

Universal elimination: From ∀xF(x) and Substance(a), you get to infer F(a). From ∀axF(x) and Mode(d,a) you get to infer F(d).

Universal introduction: If you have a subproof assuming Substance(c) and concluding with F(c), and the subproof cites nothing involving c from outside of itself, then you get to infer ∀xF(x). If you have a subproof assuming Mode(c,a) and concluding with F(c), and the subproof cites nothing involving c from outside of itself, then you get to infer ∀axF(x).

Existential elimination: If you have ∃xF(x) and a subproof from (F(c) and Substance(c)) to S, where the subproof cites nothing involving c from outside of itself and c does not appear in S, then you get to infer S. If you have ∃axF(x) and a subproof from (F(c) and Mode(c,a)) to S, where the subproof cites nothing involving c from outside of itself and c does not appear in S, then you get to infer S.

Existential introduction: From Substance(a) and F(a), you get to infer ∃xF(x), and from Mode(c,a) and F(c), you get to infer ∃axF(x).

And we add an additional equality introduction rule: If you have Mode(c,a) and Mode(c,b), then you get to infer a=b.

Models contain a substantial domain S and a function m that assigns to each member of S a set of objects, with the property m(x) and m(y) have no elements in common if x and y are distinct. We can define interpretations and satisfaction in a straightforward way, restricting the interpretations of the Substance and Mode predicates in such a way that I(Substance) is always equal to S and I(Mode) is the set of all pairs (x,y) such that x is in S and y is a member of m(x). (We don't put this rule in in the case of existence-in-a-novel.)

I haven't checked it, but I expect that we have soundness and completeness.

If, like Spinoza and unlike Aristotle, we want to allow for nested modes, this can be done, too.

6 comments:

Chris & Melinda said...

Alex,
#1 seems ill-formed to me. It is:

1.∃x(Table(x) and ∃_xy(Accident(y,x))).

Can the x in Accident(y,x) refer to something in a domain that ∃_xy cannot quantifier over?

Chris & Melinda said...

*quantify

Chris & Melinda said...

Silly mistake. Nevermind. :-)

Jonathan D. Jacobs said...

How should we translate "There are three quantifiers"?

Alexander R Pruss said...

Jonathan:

The relative-to-a-substance quantifier E_x is a quantifier analogically speaking. I do not know that Aristotelianism allows for quantifying over, say, "focal health and all its analogues". It could be that there is no limit to how far analogues of focal health might stretch. Likewise, there could be no limit to how far we could analogically stretch the notion of quantification. And if so, then it might not make sense to ask how many quantifiers there are.

A different, and theoretically more satisfying but also more risky, move is this. Introduce another analogical quantifier, this one over x's entia rationis. Let's call this E^x and A^x, using superscripts in place of the subscripts for the mode quantification. This makes no commitment to these entia being genuinely distinct beings. It could be that for finite x, the E^x and A^x quantifiers can be defined in terms of the mode-quantifiers, but not so in the case of God, since God is simple and has no modes distinct from him (in the system I offered in the post, I think this can only be expressed by saying that he has no modes).

Now, abstracta are identical with divine entia rationis. Functions are abstracta, and so are families of functions. A quantifier is a family of functions from n-ary schemata to (n-1)-ary schemata (a 0-ary schema is a truthbearer--a proposition or sentence (either type or token); a 1-ary schema is a property or unary predicate; a 2-ary schema is a binary relation or binary predicate). So, the God's-entia-rationis analogical quantifier lets us analogically quantify over quantifiers.

Alexander R Pruss said...

Anyway, the concrete answer to your question on my second proposal will be something like:

E^God x E^God y E^God z (x is distinct from y and y is distinct from z and x is distinct from z and Quantifier(x) and Quantifier(y) and Quantifier(z) and A^God w (if Quantifier(w) then w=x or w=y or w=z)).