Monday, March 21, 2011

Names, quantifiers, Aristotelian logic and one-sided relations

This is going to be a pretty technically involved post and it will be written very badly, as it's really just notes for self. Start with this objection to Aristotelian logic. A good logical system reveals the deep logical structure of sentences. But Aristotelian logic takes as fundamental sentences like:
  1. Everyone is mortal.
  2. Socrates is mortal.
In so doing, Aristotelian logic creates the impression that (1) and (2) have similar logical form, and it is normally taken to be that modern quantified logic has shown that (1) and (2) have different logical forms, namely:
  1. x(Mortal(x))
  2. Mortal(Socrates).
I shall show, however, that there is a way of thinking about (1) and (2), as well as about (3) and (4), that makes them have the same deep logical form, as Aristotelian logician makes it seem. (This is a very surprising result for me. Until I discovered these ideas this year, I had a strong antipathy to Aristotelian logic.) Moreover, this will give us some hope of understanding the medieval idea of one-sided relations. The medievals thought, very mysteriously, that creation is a one-sided relation: we are related to God by the created by relation, but God is not related to us by the creates relation.

Now to the technical stuff. Recall Tarski's definition of truth in terms of satisfaction. I think the best way to formulate the definition is by means of a substitution sequence. A substitution sequence s is a finite sequence of variable-object pairs, which I will write using a slash. E.g., "x1"/Socrates,"x2"/Francis,"x3"/Bucephalus is a substitution sequence. The first pair in my example consists of the variable letter "x1", a linguistic entity (actually in the best logic we might have slot identifiers instead of variable letters) and Socrates—not the name "Socrates" (which is why the quotation marks are as they are). We then inductively define the notion of a substitution sequence satisfying a well-formed formula (wff) under an interpretation I. An interpretation I is a function from names and predicates to objects and properties respectively. And then we have satisfaction simpliciter which is satisfaction under the intended interpretation, and that's what will interest me. So henceforth, I will be the intended interpretation. (I've left out models, because I am interested in truth simpliciter.) We proceed inductively. Thus, s satisfies a disjunction of wffs if and only if it satisfies at least one of the wffs, and so on, the negation of a wff if and only if it does not satisfy the wff, and so on.

Quantifiers are a little more tricky. The sequence s satisfies the wff ∀xF iff for every object u, the sequence "x"/u,s (i.e., the sequence obtained by prepending the pair "x"/u" at its head) satisfies F. The sequence s satisfies ∃xF iff for some object u, the sequence "x"/u,s satisfies F.

What remains is to define s's satisfaction of an atomic wff, i.e., one of the form P(a1,...,an) where a1,...,an are a sequence of names or variables. The standard way of doing this is as follows. Let u1,...,un be a sequence of objects defined as follows. If ai is a variable "x", then we let ui be the first object u occuring in s paired with the variable "x". If for some i there is none such pair in s, then we say s doesn't satisfies the formula. If ai is a name "n", then we let ui=I("n"). We then say that s satisfies P(a1,...,an) if and only if u1,...,un stand in I(P).

Now notice that while the definition of satisfaction for quantified sentences is pretty neat, the definition of satisfaction for atomics is really messy, because it needs to take into account the question of which slot of the predicate has a variable in it and which one has a name.

There is a different way of doing this. This starts with the Montague grammar way of thinking about things, on which words are taken to be functors from linguistic entities to linguistic entities. Let us ask, then, what kind of functors are represented by names. Here is the answer that I think is appealing. A name, say "Socrates", is a functor from wffs with an indicated blank to wffs. In English, the name takes a wff like "____ likes virtue" and returns the wff (in this case sentence) "Socrates likes virtue". (The competing way of thinking of names is as zero-ary functors. But if one does it this way, one also needs variables as another kind of zero-ary functor, which I think is unappealing since variables are really just a kind of slot, or else one has a mess in treating atomics differently depending on which slots are filled with names and which with variables.) We can re-formulate First Order Logic so that a name like "Socrates" is (or at least corresponds to) a functor from wff-variable pairs to new wffs. Thus, when we apply the functor "Socrates" to the wff "Mortal(x)" and the variable "x", we get the wff (sentence, actually) "Mortal(Socrates)". And the resulting wff no longer has the variable "x" freely occurring in it. But this is exactly what quantifiers do. For instance, the universal quantifier is a functor that takes a wff and a variable, and returns a new wff in which the variable does not freely occur.

If we wanted the grammar to indicate this with particular clarity, instead of writing "Rides(Alexander, Bucephalus)", we would write: "Alexanderx Bucephalusy Rides(x,y)". And this is syntactically very much like "∀xy Rides(x,y)".

And if we adopted this notation, the Tarski definition of satisfaction would change. We would add a new clause for the satisfaction of a name-quantified formula: s satisfies nxF, where "n" is a name, if and only if "x"/I("n"),s satisfies F. Now once we got to the satisfaction of an atomic, the predicate would only be applied to variables, never to names. And so we could more neatly say that s satisfies P(x1,...,xn) if and only if every variable occurs in the substitution sequence and u1,...,un stand in I(P) where ui is the first entity u occurring in s in a pair of the form "xi"/u.  Neater and simpler, I think.

Names, thus, can be seen as quantifiers. It might be thought that there is a crucial disanalogy between names and the universal/existential quantifiers, in that there are many names, and only one universal and only one existential quantifier. But the latter point is not clear. In a typed logic, there may be as many universal quantifiers as types, and as many existential ones as types, once again. And the number of types may be world-dependent, just as the number of objects.

If I am right, then if we wanted to display the logical structure of (1) and (2), or of (3) and (4) for that matter, we would respectively say:
  1. x Mortal(x)
  2. Socratesx Mortal(x).
And there is a deep similarity of logical structure—we simply have different quantifiers. And so the Aristotelian was right to see these two as similar.

Now, the final little bit of stuff. Obviously, if "m" and "n" are two names, then:
  1. "mnF(x,y)" is true iff "nmF(x,y)" is true,
just as:
  1. "∀xyF(x,y)" is true iff "∀yxF(x,y)" is true.
But the two sentences in (8), although they are logically equivalent, arguably express different propositions. And I submit that so do the two sentence in (7). And we even have a way of marking the difference in English, I think. Ordinarily, what the left hand side in (7) says is that u has the property PxnyF(x,y) while the right hand side in (8) says that v has the property PymxF(x,y), where u and v are what "m" and "n" respectively denote, and PxH(x) is the (abundant) property corresponding to the predicate H (the P-thingy is like the lambda functor, except it returns a property, not a predicate). These are distinct claims.

The medievals then claim that in the case of God we have this. They say that "Godx nF(x,y)" is true in virtue of "ny GodF(x,y)" being true. It is to the referent of "n" that the property Py GodF(x,y) is attributed, and the sentence that seems to attribute a property to God is to be analyzed in terms of the one that attributes a property to the referent of "n".


Jonathan D. Jacobs said...

We could, if we liked, represent predicates in the same way. (Cf: contemporary ways of being theorists.) Wouldn't that mitigate against the similarity between Aristotelian and modern logic?

Alexander R Pruss said...

I don't see how to represent all predicates in the same way. The best I can do is that a predicate is a functor from sequences of variables and/or names to wffs. And I think that's not so neat, because then the base clause of the Tarski truth definition is messy, as it has to take both variables and names into account.

I can, however, represent all but one predicates in the above way. Namely, have a single variable grade predicate "Instantiates" and then names for all the properties. So instead of "Mortal(Socrates)", we'd say:
Socratesx Mortalityy Instantiates(x,y)
I.e., x instantiates y, where x is Socrates and y is Mortality. But we still have a special predicate that is not handled in this way, Instantiates.

You might try something second-order like:
Socratesx MortalityF F(x),
where capital letters are predicate variables. But that's just Instantiates all over again, except that we use "F(x)" as shorthand for "Instantiates(x,F)".

Jonathan D. Jacobs said...

Socratesx Mortalityy x=y

Jonathan D. Jacobs said...

(At any rate, that's how the ways of being theorists do it.)

Alexander R Pruss said...

That's not an "=" of identity, is it?

But whatever it is, isn't the "=" a predicate, so you still have one predicate which isn't handled in the same way as all the others?

Jonathan D. Jacobs said...

Yes, identity. (It's not my view, for what little that's worth. Neither is it, by the way, the ways of being theorists' view. It's one way to put the ways of being stuff to use in this context. For the actual view, see Kris McDaniel and Jason Turner's work.)

So, yes, there will be one predicate handled unlike the others. (In this context, where we treating each predicate like a way of being, this will have the result that identity can be contingent and temporary.) I suppose your proposal will treat identity like any other relation. I don't really have a view on which proposal is more simple.