God – Bas van Fraassen's philosophy blog

Almost twenty years before I met Fitch, he had published what he described as an empirical, a posteriori, proof of the existence of God:

“On God and Immortality”. Philos. and Phenom. Research 8 (1948), 688-693.

In that proof God had turned out to be a theory.

At first blush the argument looks, besides having a conclusion entirely irrelevant to religion, not just simplistic, but dated, and most likely trivial. However, I became quite intrigued with the argument when I started asking myself what unstated assumptions Fitch was bringing to bear. Most especially, what must the language be like in which such an argument could be formulated?

I began to realize that it was a shot fired in the war Curry and Fitch were waging with a conventional wisdom derivative from Tarski and Carnap.

The crucial claim in this paper, argued to be well confirmed, indeed as well-confirmed as any empirical theory could be, is that every class of facts in the universe has an explanation.

Reading his preliminaries we are at once taken aback by how explanation and confirmation are defined. Explanation is equated with implication by a consistent theory, and confirmation with positive instances. But this was a time when the agenda for those notions was set by Hempel, and they were acccepted in the literature just as Fitch presents them.

His one departure from the orthodoxy of the time was this: a fact disconfirms a theory X iff it confirms some theory inconsistent with X. Thus confirmation and disconfirmation may come at the same time! Many facts confirmed both the Ptolemaic and Copernican theory, and thus disconfirmed both of them as well.

The crucial claim looked trivial. It is critique rather than applause for Freudian theory when people say it explains everything. Also, I’m used to reading “theory” in the way, introduced by Tarski in his ‘calculus of systems’, as “set of sentences closed under deduction, i.e. containing all its own consequences”. Then if Z is any set of true sentences the deductive closure of Z is a theory, which implies — and so, by the above definition, explains — all of Z. Trivial! But Fitch has a stronger, more demanding concept of theory, requiring a character familiar in philosophy of science:

“A theory is a proposition (that is, an hypothesis) which involves no mention of any particular thing, event, or moment of time, though propositions mentioning particular things, events, and moments of time may be deducible from it.”

So read the crucial claim is not trivial, but looks obviously false, as long as explanation equated to implication. For a theory of this sort will not imply the sort of singular facts that scientific theories take as data or boundary conditions. To be charitable, let’s just say that Fitch needs a considerably more sophisticated concept of explanation, to run his argument. A theory of that sort might unify a whole realm of particular facts in a way that could count as explanatory, for example. But the details in Fitch’s deductive steps in the overall argument do not hinge on this, anyway, so, charitably, let’s continue.

Now Fitch formulates a theory that will bear the brunt of his reasoning, the theory S:

“Let us designate by S the proposition that every fact or class of facts has at least one explanation. The proposition S is a theory because it does not mention any particular thing or any particular event or any particular moment of time. The proposition S is also clearly consistent; hence it is an explanation of any fact that it implies, and any such fact is a con- firmation of it.”

S is an empirical theory, and is very well confirmed, he argues. For it is confirmed every time any theory is confirmed, any time when any class of facts is explained. And even if some theory is disconfirmed, which is a disconfirmation of S as well, by definition some other theory was confirmed, and that is an additional confirmation of S. Hence S is as well confirmed as any theory is, was, or ever will be! This conclusion is going to have remarkable consequences.

But stop, for a moment: what is the language like in which this argument is conducted?

S is a theory about theories, for it says that fore every class of facts there is a theory that explains that. To introduce this is not just to assume that theories can be formulated in the language in use, but that they can be talked about in that language. And indeed, S itself purports to be one of those explaining theories, about the class of facts which involve relations between facts and theories, so it is also about itself. This is where we see the contra-Tarski theme of many of his writings: our language is one in which much of the semantics of that very language can be formulated.

Theories are sets of sentences in this language, but here it is presumed possible to talk about theories in general in this language, about implication in the language, about relation to facts, about truth and falsity. So to a significant extent, it is possible in this language to talk about that language itself, about a great deal of what we would classify as belonging to its semantics.

Thus Fitch is relying on something he argued for , notably in “Universal metalanguages for philosophy” and “Self-reference in philosophy”. (I’ll have to devote another post to this!) It is part of the controversy in which Curry and Fitch were on one side, and the only side now generally accepted or even known at all, was Tarski’s.

Well, let’s quickly finish up with the startling consequences of theory S, including the existence of God:

“A corollary of S is that the class of all facts has an explanation.”

Is it possible to talk about the class of all facts, in his language in use, given that he equates facts with propositions, and propositions with what his sentences express? Paradoxes threaten, but they are just the paradoxes that he takes himself to have solved or dissolved or otherwise avoided, in his conception of language.

Let us call an explanation of the class of all facts an ultimate explanation . That there is such an explanation, Fitch argues, “means that the universe as a whole is a system”.

He derives at once that an ultimate theory, by definition consistent, must be true, for if it implied the contradictory of any fact it would still have to imply the fact itself (since it implies all facts), and so would be inconsistent. Furthermore, it is unique:

“there can be only one true explanation of all the facts in the universe, for if there were two different explanations, the fact that one of the explanations was true would be a fact to be accounted for by the other explanation, and so the two explanations would imply each other and hence be equivalent.”

And this ultimate explanation, that we must all agree to call God. For it is the “guarantee that no event in the universe is altogether without connection and relevance to the over-all scheme of things” — what else, then, could it be, other than God?

Of course he sees the obvious objection. Whatever the conception of God might be, it is not a conception of a mere theory! But, Fitch says, this is not a mere theory! For it is true and it is the ultimate explanation of everything.

To his commentator Charles Baylis at the APA, who did not see anything worthy of worship in even such a theory, Fitch retorts:

“I grant that my conception of God as the ultimate system in the universe differs considerably from the anthropomorphic conception of deity held by many people in both primitive and modern times. God seems to me indeed worthy of worship, but my account of the nature of worship cannot easily be summarized here.” (same journal and issue, page 698)

I kept an eye out for the secret smile that I suspected behind much of what Fitch used to say, in his characteristically gentle, courteous voice. But sometimes it is hard to see.

In 1966 Fitch ended with the conclusion that God is a proposition, the one true proposition that implies all true propositions. In that case, why would it be so difficult to prove that God exists? Is it so difficult to establish that there is such a proposition?

I don’t know what Fitch’s proof was — his Nachlass is in the Yale University Library archives, and may contain it. (He published an earlier proof in 1948, to which I may devote another post.) But let’s just think about it ourselves for now.

If there were only finitely many true statements, or if all the true statements could be summed up in just a finite subclass of them, we could just take the conjunction: it would be true, and would imply all of them. At first blush it seems unlikely that this is so. Some physicists’ dreams of a final theory are of this, but it seems doubtful that they are more than dreams. Leibniz came close: he took as his two premises God’s goodness and the principle of sufficient reason. Given those, the idea that there are more than one possible world that God might, or could, have created, leads to a contradiction. God did create one, so it must be the only one, among all the possible ones, that could be actual. Hence (still assuming the premises) all true propositions are necessary. But Leibniz added that how they follow from the premises typically cannot be captured in a finite deduction, so is typically beyond human comprehension.

What if, as seems much more likely, the truths are an irreducibly infinite class, not finitely or recursively axiomatizable? Nothing could be, or play the role of, their conjunction — language does not have the resources.

Enter propositions! In 1950, in the Review of Metaphysics (3, 367-384) Fitch presented his view of propositions as abstract entities, universals. Although he discusses actuality and possibility as applied to propositions, treats the concept of logical consistency as applying to propositions without further ado, characterizes false propositions as contradictories of the actual ones, and emphasizes that propositions are timeless, his discussion does not broach the question of just what propositions there are, or how we could establish anything about which there are.

Informally we can just say that propositions are what sentences can express, but there may be many more of them than there are sentences. Some of them may indeed be inexpressible by any sentences, given the factual limitations of language. So perhaps one of the true ones which is inexpressible implies all the other true ones; and “this”, I suppose Fitch would say, echoing earlier proofs, “is what we all agree in calling God”.

But I wrote “perhaps“. For, what propositions are there? From my ametaphysical point of view, there is nothing to go on. We could postulate lots about them, for example that there is an operation o on propositions such that if sentences p and q respectively express propositions P and Q then (p & q) expresses o(P,Q). That could be step one in a semantic analysis of language which would be based on propositions as semantic content.

Let’s call that a P-semantic theory, for the language under discussion. Might be an attractive theory, indeed. But what could establish that this operation on propositions exists? Or more generally, that for any two propositions there exists a third which is their greatest lower bound in the implication ordering?

Metaphysicians, however, might have principles that just have to be true, about the logical structure of the universe, which entail all sorts of facts about propositions, the relations between them, the operations on them.

The P-semantic theory could function as an explanation of why the logic of our language is what it is. For example, this theory might imply that a sentence of form (p v ~p) is always true, because it expresses a proposition that can’t fail to be true, and hence explain why Excluded Middle is a correct logical principle.

On the other hand, if we could take for granted what our logic is, independently, and also embraced a bit of set-theory, then we could offer an ametaphysical theory of propositions. For the proposition expressed by sentence p could be identified with, or as, the set of all sentences logically equivalent to p. These propositions would form an algebra with its operations and relations induced by the logical relations among sentences: the Lindenbaum algebra.

Note well, though, that this, taken as a theory of propositions expressed by sentences, would neither establish why our logic is what it is (it would not have that explanatory function), nor would it yield any proposition that could play the role fo being true and implying all truths … there is no path from the logical construction of the Lindenbaum algebra to the God of the logicians.

There is another, familiar, identification of propositions in semantics, namely in the ‘standard’ semantic analysis of modal logic. There sentences have a truth-value in each of the items called ‘worlds’, and the proposition expressed by a given sentence A is identified as |A|: the set of worlds in which it is true. That is certainly a useful way of thinking, for it allows us to display the semantic value of any complex sentence as a definable function of the semantic values of its components. (Does anyone remember the phrase ‘the California hypothesis’?) For example, |A&B| is the intersection of |A|and |B|. And, in the version where relative possibility among worlds is a relation R, the proposition |Necessarily, A| is the set {w: A is true in each world u such that wRu}.

It may be natural in that case to just say: propositions are sets of possible worlds, any set of possible worlds is a proposition. In a context where possible world talk is acceptable, that would answer the question what propositions there are. But, well, I said “natural“. What does that mean?

If we were to take propositions seriously, in the way Fitch did, then we could not say that propositions are sets of worlds, but only that sets of worlds represent, or can be used to represent, propositions. In that case, of course, there is nothing natural about the assumption, or postulate, that all sets of worlds represent propositions: they can only represent propositions that there are! So once again, the question of what propositions are there, in the sense in which it should arise for Fitch, has no answer. At least, no answer short of answers he could derive from metaphysical principles, whatever they may have been.

Tag: God

A memory of Frederick Fitch (3) God 1948

A memory of Frederick Fitch (2) Propositions