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Infinitary Languages and the Barcan Formulae

January 2, 2008

I’ve recently been reading this paper by Yablo on abstract objects. The thing that really caught my interest was his proposal and defence of the following logicist thesis: “The real content of any arithmetical truth is a logical truth” (p39.) He goes on to make similar remarks for ZFC, but I feel that they stand or fall with the weaker arithmetical claim. His argument relies on the following principles

(i) Basic arithmetical statements are logical truths (e.g. 3+5 = 8, 9 > 4, etc…)
(ii) An existentially (universally) quantified sentence has the same logical content as an infinite disjunction (conjunction.)
(iii) An arbitrary conjunction (disjunction) where each (some) of its conjuncts (disjuncts) is logically true, is logically true.

Given these principles it is relatively easy to see that every arithmetical statement has the same logical content as a logical truth. The case for (i) is that a statement like 3+5 = 8 can be translated into first order logic as

((\exists !_3 xFx \wedge \exists !_5 xGx \wedge \neg \exists x (Fx \wedge Gx)) \rightarrow \exists !_8 x(Fx \vee Gx)).

Although controversial (for a start they have wildly different logical forms) I’m willing to grant this for now. I find (iii) very compelling. For finite conjunctions/disjunctions it should be uncontroversially true (for classical logicians anyway.) If you think infinite disjunction is a logical operation then presumably the infinite case holds for similar reasons. I am personally sympathetic with the view that the logical operators are those definable in L_{\infty, \infty} (see here), so this falls out fairly cleanly for those of similar persuasion.

My problem is with (ii). If two statements have the same logical content, they should be necessarily equivalent. But, in general, you can’t always find a disjunction that is necessarily equivalent to an existentially quantified statement. Suppose there are two people: Jim, and Susan. Jim is happy, Susan is sad. According to (ii) a) and b) should have the same logical content:

a) Someone is happy
b) Jim is happy or Sue is happy

But they don’t seem to have the same modal status. Consider another world containing Jim, Sue and Bob. In this world Jim and Sue are sad, and Bob is happy. In this world a) is true, and b) false – a) and b) are not necessarily equivalent.

There are still a few kinks to be worked out. If you believe that the converse Barcan formula is true, then presumably you won’t be moved by these counterexamples. In fact, (ii) plausibly entails the Barcan formulae, so you might even take this as a reason for accepting them. Even if you reject the Barcan formulae, you might argue that b) is the wrong candidate for a’s equivalent; it should be a disjunction of propositions including non-existent objects. However, on this account all universally quantified statements come out truthvalueless (or false), provided propositions containing non-existent entities are truthvalueless, and conjunctions including truthvalueless propositions are truthvalueless (this is not to mention the odd consequence that \exists x would range over non-existent things.) Lastly, numbers exist necessarily, so these problems won’t affect a restricted version of (ii) for quantification over numbers – but then it would be puzzling why, as a matter of logic, (ii) holds in this case but not unrestrictedly, when numberhood is not a logical property.

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