Two logical frameworks

A detailed discussion on which logic should be preferred for foundational purposes in mathematics goes beyond the purposes of this work. It suffices to say that this debate has become an important stream of research only recently.161 This issue is not a new one, though. In the history of logic, mathematicians have favoured different positions regard- ing this point: fathers of modern logic such as Frege and Dedekind dealt with a second order language while from G ¨odel’s first important results onward, first order languages took progressively the place that second order logic had in foundational studies. Indeed, G ¨odel was one of the “strongest (and probably the most influential) proponents of first- order languages”162. The issue, which has forcefully returned to the ongoing philosophi- cal debate, is crucial for our purposes. In fact, we have stressed that the expressive power of the formal language does have consequences for the occurrence of non-standard mod- els.

What we want to claim here is that the choice of which logic to employ should not be based on the different meta-properties that each logic hasper se, but rather on the foun- dational purpose in mathematics that we want to pursue.

161_{Commonly, [shapiro91] represents the landmark in this debate.} 162_{[shapiro85, p.715]}

3.7. DESCRIPTIVE VS DEDUCTIVE USE: 1ST VS 2ND ORDER LOGIC

Meta-properties of first and second order logic are recapitulated in the table 3.1. 1st order logic 2nd order logic

Sound x x

Complete (or Exhaustive) x –

Compact x –

Downward L ¨owenheim-Skolem x –

Upward L ¨owenheim-Skolem x –

Table 3.1: Meta-properties: 1st order versus 2nd order logic

Except for soundness (i.e. the basic requirement that inference rules are chosen to preserve truth), second-order logic does not present any of the beautiful meta-properties that first order-logic does. Nonetheless, what we are interested in here is not the logic as such, but rather the formal theories, formulated in a certain logic that are capable of capturing the features of a given mathematical structure.

As soon as we compare the first-order theory Peano Arithmetic with the formal ver- sion of Dedekind Theory, say DT*, formulated in second-order language, we move on to an account between arithmetical theories:

PA DT* Syntactically complete – – Semantically complete – x

Categorical – x

Decidable – –

Table 3.2: Meta-properties: PA versus DT*

By recalling what we said in section 3.3, by G ¨odel’s First Incompleteness Theorem we know that PA cannot be decidable and thus syntactically complete. By completeness of first-order logic, we also know that PA is not semantically complete and by corol- lary 2.5.0.10 it is not categorical either. Conversely, DT* has been proved categorical by Dedekind and therefore is also semantically complete. Still, since the incompleteness theorem applies to PA and its extensions, DT* as well as PA are neither decidable nor syntactically complete.

Now, if we recall Hintikka’s distinction between uses of logic, we can easily see that the choice of arithmetical theory (and accordingly of the underlying logic) may be based on the aim we want to achieve, namely either deductive or descriptive:

3.7. DESCRIPTIVE VS DEDUCTIVE USE: 1ST VS 2ND ORDER LOGIC

Task of logic Descriptive Deductive Logical system 2nd order logic 1st order logic

Aim Categoricity Syntactic completeness Table 3.3: Logical tasks in mathematics

a theory which is capable, for every formula in its language, to say whether the formula is provable or refutable (according to a given notion of deductive consequence). Although first-order logic is semi-decidable, i.e. if a statement is true, then the logic will eventually provide a proof for it, neither PA nor DT* are syntactically complete as a consequence of the First Incompleteness Theorem.

On the other hand, the Descriptive task of logic aims at categoricity, viz. to state a theory which fully captures the mathematical structure at hand. This is accomplished by ruling out the possibility for that theory to be satisfied by any non-standard model. To do so, the logic underlying the theory should not be chosen to be the first-order one. Let us discuss this point in detail.

In contrast to the deductive use, the descriptive goal can indeed be established notwith- standing the First Incompleteness Theorem. In fact, G ¨odel’s Incompleteness result shows the impossibility, under certain conditions, to formulate a syntactically complete theory for arithmetic. Even though the theorem applies to both PA and DT*, it does not follow that it is not possible to achieve any completeness whatsoever.

The syntactic completeness of a theory rests on two basic assumptions: the “exhaus- tiveness” of the fragment of the underlying logic, and the specification of the model of the theory for which we effectively check the logical theorems. Now, if we put aside the first ingredient and we adopt a second-order logic, thesyntacticincompleteness of DT* does not imply thesemantic incompleteness of DT*. Indeed, DT* is categorical as Dedekind showed, and therefore it issemantically complete.

To sum up, G ¨odel’s result did imply that the deductive task of logic in mathematics as defined earlier cannot be carried out. However, the same does not hold for the descriptive task, but in the end we can reach descriptive completeness, namely categoricity, only by sacrificing deductive completeness. That is, we can have either kind of completeness, but not both.