Commenter Jan, the Polish physicist, gave me the idea for the following post.
An antilogism is an aporetic triad, an array of exactly three propositions which are individually plausible but collectively inconsistent. For every antilogism, there are three corresponding syllogisms, where a syllogism is a deductive argument with exactly two premises and one conclusion. Here is the antilogism I want to discuss:
1. Possibly, the number of objects is finite.
2. Necessarily, if sets exist, then the number of objects is not finite.
3. Sets exist.
The modality at issue is 'broadly logical' and sets are to be understood in the context of standard (ZFC) set theory. 'Object' here just means entity. An entity is anything that is. (Latin ens, after all, is the present participle of the infinitive esse, to be.)
Corresponding to the above antilogism, there are three syllogisms. The first, call it S1, argues from the conjunction of (1) and (2) to the negation of (3). The second, call it S2, argues from the conjunction of (2) and (3) to the negation of (1). The third, call it S3, argues from (1) and (3) to the negation of (2).
Note that each syllogism is valid, and that the validity of each reflects the logical inconsistency of the the antilogism. Note also that for every antilogism there are three corresponding syllogisms, and for every syllogism there is one corresponding antilogism. A third thing to note is that S3 is uninteresting inasmuch as it is surely unsound. It is unsound because (2) is unproblematically true.
This narrows the field to S1 which argues to the nonexistence of (mathematical) sets and S2 which argues to the impossibility of the number of objects (entities) being finite. Our question is which of these two syllogisms we should accept. Obviously, both are valid, but both cannot be sound. Do we have good reason to prefer one over the other?
Here are our choices. We can say that there is no good reason to prefer S1 over S2 and vice versa; that there is good reason to prefer S1 over S2; or that there is good reason to prefer S2 over S1.
Being an aporetician, I incline toward the first option. Peter Lupu, being less of an aporetician and more of dogmatist, favors the third option. Thus he thinks that the antilogism is best solved by rejecting (1). Peter writes:
(a) If there are infinitely many numbers, then (1) is false. Are there infinitely many numbers? Very few would deny this. How could they, for then they would have to reject most of mathematics. [. . .]
To keep it simple, let's confine ourselves to the natural numbers and the mathematics of natural numbers. (The naturals are the positive integers including 0.) If there are infinitely many naturals, then there are infinitely many objects. If so, then presumably this is necessarily so, whence it follows that (1) is false.
I fail to see, however, why there MUST be infinitely many naturals. I am of course not denying the obvious: for any n one can add 1 to arrive at n + 1. With a sidelong glance in the direction of Anselm of Canterbury: there is no n that fits the description 'that than which no greater can be computed.' In plain English: there is no greatest natural number. But this triviality does not require that all of the results of possible acts of +1 computation actually be 'out there' in Plato's heaven. When I drive along a road, I come upon milemarkers that are already out there before I come upon them. But why must we think of that natural number series like this? I don't bring the road and its milemarkers into being by driving. But what is to stop us from viewing the natural number series along Brouwerian (intuitionistic) lines? One can still maintain that the series is infinite, but the infinity is potential not actual or completed. Peter's first argument, as it stands, is not compelling. (Compare: Everyone will agree that every line segment is infinitely divisible. But it does not follow that every line segment is infinitely divided.)
(b) If propositions exist, then there are infinitely many propositions. Are there propositions? Kosher-nominalists obviously will have to deny that propositions exist. Sentences do not express propositions. But, then, what do they express?
I am on friendly terms with Fregean (not Russellian) propositions myself. And I grant that it is very plausible to say that if there is one proposition then there is an actual infinity of them. Consider for example the proposition *p* expressed by 'Peter has a passion for philosophy.' *P* entails *It is true that p* which entails *It is true that it is true that p,* and so on infinitely. But again, why can't this be a potential infinity?
The following three claims are consistent: (i) Declarative sentences express propositions; (ii)Propositions are abstract; (iii) Propositions are man-made. Karl Popper's World 3 is a world of abstracta. It is a bit like Frege's Third Reich (as I call it), except that the denizens of World 3 are man-made.
I am agreeing with Peter and against the illustrious William that there are (Fregean) propositions, understood as the senses of context-free declarative sentences. I simply do not understand how a declarative sentence-token could be a vehicle of a truth-value. But why can't I say that propositions are mental constructs? (This diverges from Frege, of course.)
(c) Are there sentence types? A nominalist will have to deny the existence of sentence types. But, then, it is difficult to see how any linguistic analysis can be done.
Peter may be conflating two separate questions. The first is whether there are any abstract objects, sentence types for example. The second is whether there is an actual infinitity of them. He neeeds the latter claim as a countrerexample of (1). So again I ask: why couldn't there be a finite number of abstract objects: a finite number of sets, propositions, numbers, sentence types, etc. This would make sense if items of this sort were Popperian World 3 items.
I conclude that, so far, there is no knock-down refutation of (1). But there is also no knock-down refutation of (3) either, as Peter will be eager to concede. So I suggest that the rational course is to view my (or my and Jan's) antilogism as a genuine intellectual knot that so far has not been definitively solved.