The unqualified claim that second-order logic is incomplete cannot stand in any case; it needs to be made more precise. The deductive system is not semantically incomplete in and of itself; rather, it is incomplete with respect to some specified formal semantics. The deductive system of second-order logic, for example, is in- complete with respect to the standard model-theoretic semantics. As laid out in chapter 2, a model in the standard model theory consists of a domain of objects and an interpretation function. This function assigns objects in the domain to names in the languages, subsets of the domain to one-place predicates, subsets of the Carte- sian product of the domain with itself to two-place predicates (sometimes also called binary relation symbols), and so on. The first-order quantifiers range over the do- main, while the second-order quantifiers range over the subsets of the domain in case the quantifier binds a one-place predicate variable, over the subsets of the Cartesian product of the domain with itself in case the quantifier binds a two-place predicate variable, and so on.
It is well known that standard semantics is not the only semantics available. Henkin semantics, for example, specifies a second domain or the upper case constants and variables. The second-order quantifiers binding predicate variables, e.g., can be thought of as ranging over a subset of the full powerset of the first-order domain. What is relevant to the present discussion is that the deductive system of second-
order logic is sound and complete with respect to a Henkin semantics.2 (For details see chapter 2.) To pick up on a thought of Shapiro’s, one might think that standard (as opposed to Henkin) semantics does not provide enough models to invalidate all sentences of the language of second-order logic that are not theorems. Shapiro dismisses this view.3 He rather thinks that Henkin models are insufficient, in a sense, since the second-order variables are not guaranteed to range over the full powerset of the domain in a Henkin semantics. Shapiro argues in various places4 that second-order logic with standard model-theoretic semantics is the right logic, at least for mathematical practice. (The discussion of this view is the topic of my chapter 7.)
I will not take sides here in the debate regarding whether standard or Henkin semantics is the “right” semantics. Note, however, that with respect to a Henkin se- mantics we have a completeness theorem for second-order logic. It would be bizarre, though, to claim that the incompleteness complaint is thereby refuted: it cannot be enough to just provide some semantics for which a completeness proof is possible. Note that Henkin and standard semantics are not the only available options. As we have already seen in chapter 4, there is, for example, the plural interpretation of second-order logic that was suggested by George Boolos.5 Other options include (presumably) game-theoretical semantics, and (definitely) category-theoretical or topological semantics.6 Peter Simons suggested a Le´sniewskian semantics.7 Surely, those do not exhaust the alternatives. It seems not unlikely that the important
2(Henkin, 1950).
3(Shapiro, 1998), p. 141; but see also his fn. 10 on the same page.
4See e.g. (Shapiro, 1985), (Shapiro, 1991), (Shapiro, 1997), (Shapiro, 1998), (Shapiro, 1999).
5See (Boolos, 1984b), (Boolos, 1985a), (Boolos, 1994).
6(Awodey and Butz, 2000); see also (Awodey and Reck, 2002b),§3.
philosophical issues concerning completeness and logical consequence will be ob- scured rather than elucidated by bringing up too many different approaches to for- mal semantics and a discussing which one “gets it right”.
I will therefore in this chapter talk about formal or model-theoretic semantics in general, in the same way that I do not single out one specific deductive system. I will consider model-theoretic approaches to characterise logical consequence in general, irrespective of which model theory (if any) might be the right one to codify logical consequence, on the one hand. On the other hand, I will consider deductive systems in the abstract, again irrespective of which specific deductive system (if any) might properly capture logical consequence. Further, the discussion will turn out to be easier if a term is available that applies to both deductive systems and model-theoretic semantics. I will use ‘formal system’ to cover both.
So, we are looking for formal systems which capture, codify, axiomatise, charac- terise and/or formalise in some way the notion of logical consequence. One might hope that the meta-theoretic results of soundness and completeness deliver some- thing informative concerning the question whether we succeeded or failed in our en- deavour. One might, for example, be tempted to read the soundness result as: “We will not deduce a sentence from a class of premises that is not a logical consequence of them” (we will come back to this), and the completeness result accordingly as: “We can deduce every sentence from a class of premises that is a logical consequence of them”. This should give us pause, however.
Logical truths are (by definition) logical consequences of the empty class of premises and hence, by monotonicity, logical consequences of every class of premises. Certainly a logical truth of first-order logic cannot be deduced in the deductive sys- tem of propositional logic. Propositional logic is complete, and is a proper logic
if anything is. Yet a logical truth like ‘(∀x(F x ⊃ Gx)∧F a) ⊃ Ga’ escapes its consequence relation. The solution to this “puzzle” is of course simple. Propo- sitional logic lacks the expressive power that first-order sentences require. It can therefore not be accused of not capturing logical truths of predicate logic. Some- thing similar seems to hold for the relation between first- and second-order logic. Also note that, as stated above, completeness is a meta-theoretic relation that holds (or fails to hold) between a deductive system and a formal semantics. The two consequence relations we are dealing with are thus a deductive consequence relation and the model-theoretic consequence relation of a formal semantics. Completeness shows that every model-theoretic consequence of a set of sentences is also a deduc- tive consequence of those sentences. But how doeslogical consequence get into the picture?
We are dealing with two formal systems here, the deductive system and the formal semantics, each of them equipped with their own consequence relation. If there is a soundness and a completeness proof we know that this duplication does not matter: their respective consequence relations are co-extensive. If we now know for some independent reason that we can infer nothing but logical consequences of given premises according to one system, then this will be the case for the other system, too. If completeness fails, however, this guarantee is not available.
If in such a case one decides that it is the deductive system that faithfully cap- tures the pre-theoretic notion of logical consequence, one will presumably hold that the model theory is defective in the sense that it produces a surfeit of consequences of a set of sentences which are not actually logical consequences of it. One might then say that the model theory does not provide an appropriate model of the logical consequence relation that is specified by the deductive system, and consequently
reject the semantics. John Etchemendy seems to hold such a view with respect to second-order logic.8 (I discuss Etchmendy’s views on logical consequence in chapter 9.) The failure of completeness does not disqualify the deductive system from cap- turing logical consequence and therefore from being a proper logic (other features might still have this effect, of course). If, on the other hand, one has convinced oneself that model theory is the system that properly codifies logical consequence, one will presumably think that the right thing to do, when one wants to do logic, is just that, viz. model theory. It is at least prima faciehard to see why the lack of a complete deductive system should cast doubt on the model-theoretic system as a logic if one has independent reasons to believe the model theory to properly capture logical consequence.
Part of the importance of soundness and completeness results is thus that they inform us about important features of the two systems. For example, they show that results from the one system can be carried over to the other.9 If I prove a theorem of first-order logic then I know, by soundness, that this sentence is valid, i.e. it is true in all models. If I provide a model that makes a sentence false, again by soundness I know that I will not be able to prove it. Completeness allows us to make these transitions in the reverse direction. A model-theoretic argument can establish that a sentence is a consequence of some other sentences. If completeness holds one knows that there is also a derivation in the deductive system that establishes this result. The reason to prefer a proof to be carried out in one or the other system might have to do with the time it takes to carry out the proof, lemmata that are available that will aid one in the proof, or other matters of convenience. Essentially,
8(Etchemendy, 1990), esp. pp. 158–159.
however, we have two formal systems that in a sense do the same job, and deliver the same results, which cannot be a disadvantage.
Now consider the case of a second-order inference and imagine a logician who is convinced that the standard model theory properly captures the logical consequence relation. A derivation in the deductive system is, by soundness, as good as a model- theoretic argument as a means of showing that the conclusion of the derivation is a logical consequence of its premises. The mere fact that there are semantic consequences for which there is no derivation in the deductive system does not throw any doubt on it tracking the logical consequence relation (again, other considerations might well do so).
I said above that one might be tempted to think that a soundness proof shows us that one can only derive logical consequences in a deductive system. It seems quite obvious now that this can only be the case if one already has established independently that the model-theoretic consequence relation embraces only logical consequences.
Soundness proofs can, however, establish something like relative consistency. If we know, for whatever reason, that the model theory is consistent, then a sound- ness proof assures us that the deductive (or axiomatic) system that is sound with respect to this (consistent) model-theoretic semantics, is consistent, too. This latter point makes it clear that we need a proviso concerning all the mentioned benefits of the meta-theoretic proofs of soundness and completeness: the benefits are only available to the extent that the meta-theory can be trusted. For the extreme case, assume the meta-theory used to carry out the soundness and completeness proofs is inconsistent. This means thateverythingcan be proven in it.10 On this assumption,
the soundness and completeness proofs carried out in this inconsistent meta-theory do not show anything, of course. There are also less dramatic ways in which the meta-theory might not be beyond doubt. It might, for example, make some substan- tial assumptions that should not be taken for granted in a given context. For the purpose here, I will assume that everything is in good order with the meta-thoery.