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.
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