Abstract
Ignacio Jane has argued that second-order logic presupposes some amount of set theory and hence cannot legitimately be used in axiomatizing set theory. I focus here on his claim that the second-order formulation of the Axiom of Separation presupposes the character of the power set operation, thereby preventing a thorough study of the power set of infinite sets, a central part of set theory. In reply I argue that substantive issues often cannot be separated from a logic, but rather must be presupposed. I call this the logic-metalogic link. There are two facets to the logic-metalogic link. First, when a logic is entangled with a substantive issue, the same position on that issue should be taken at the meta- level as at the object level; and second, if an expression has a clear meaning in natural language, then the corresponding concept can equally well be deployed in a formal language. The determinate nature of the power set operation is one such substantive issue in set theory. Whether there is a determinate power set of an infinite set can only be presupposed in set theory, not proved, so the use of second-order logic cannot be ruled out by virtue of presupposing one answer to this question. Moreover, the legitimacy of presupposing in the background logic that the power set of an infinite set is determinate is guaranteed by the clarity and definiteness of the notions of all and of subset. This is also exactly what is required for the same presupposition to be legitimately made in an axiomatic set theory, so the use of second-order logic in set theory rather than first-order logic does not require any new metatheoretic commitments.
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Notes
Second-order logic typically also includes function variables. For the sake of simplicity I omit these, and also assume that there are no function symbols in the first-order fragment. Since we are concerned with the applications of second-order logic to the language of set theory, which has no function symbols, this does not affect the following discussion.
For a more detailed account of second-order logic, its semantics, and metatheoretical results, see [16].
For a logic that includes function variables a Henkin model will be a quadruple M H = 〈d, D, F, I〉 where F is a function specifying the range of function variables in a manner parallel to D as described in the main text.
For a system that also includes a pairing function, we can restrict the schema to the case n = 1.
For more detail or background on set theory, see e.g. [10].
Formally, these axioms are as follows: ∃x¬∃y(y ∈ x) and \(\exists {x}(\emptyset \in x \wedge \forall {y} (y \in x \rightarrow \bigcup \lbrace y, \lbrace y \rbrace \rbrace \in x)\). Somewhat more precisely, only the Axioms of the Empty Set and Infinity are \({{\Sigma }^{0}_{n}}\) sentences, the rest being \({{\Pi }^{0}_{n}}\) in the first-order formulation.
Notably, Shapiro links the notion of epistemic pedigree with formal deduction rather than model-theoretic proof: “the deductive notion of consequence sheds light on the mathematical practice of justifying one’s conclusions and the desire to keep track of what is presupposed or involved in a justification. In a formal deduction, all premises must be made explicit, so that the reader can read off exactly what is presupposed in the justification of the conclusion” [17, p. 50]. This suggests that the emphasis on epistemic pedigree may not carry the weight in the case of a choice of logic and semantics that Jané wants to put on it, because what has to hold for our proof of a theorem to go through might be different from what has to hold for the theorem to be true, as when proofs about a certain structure are carried out by embedding it in a richer structure and thus depend on facts about the embedding structure. The result is known to hold in the simpler structure, but the proof depends on facts about the richer structure. In such a case, what has to hold for the proof to go through is more than is needed for the theorem to hold. The axioms of the simpler structure thus do include all its substantive assumptions but do not suffice to make the epistemic pedigree explicit. So we must not conflate axioms containing all the content of a theory with their manifesting the epistemic pedigree of all the theorems of the theory. A foundational theory such as set theory cannot be embedded in a richer theory, but the semantic consequences can still outstrip the deductive consequences—indeed they will in second-order logic with standard semantics, since it is incomplete. Whether the epistemic pedigree of theorems can in fact be made explicit depends in part on whether it is possible to purge logic of substantive assumptions, an issue I take up in Section 4.
I thank an anonymous referee for suggesting the comparison with PA.
I say ‘central truths’ because Gödelian incompleteness precludes a (recursive) axiomatization from sufficing to prove all the truths of a given theory unless the theory itself is fairly weak. Nevertheless, PA suffices for a large amount of number theory and is generally taken as the canonical axiomatization of arithmetic.
Proof theoretically, ω-logic is the result of adding to first-order logic an infinitary rule of proof, according to which from ⊢ ϕ(1), ⊢ ϕ(2), ⊢ ϕ(3),... one can infer ⊢ ∀x ϕ(x). On the model theoretic side, an ω-model is simply a model whose domain is {1, 2, 3, ...}. See [4, 81ff.] or [16, §9.1.2].
There is another way mathematics can be presupposed in logic, which is if some amount of mathematics is needed to understand the notions of well-formed formula, deduction, etc. While it is an interesting question how much math is needed to even get a logic off the ground, this is clearly not the issue Jané is interested in. Thanks here to Neil Tennant.
The sections on intuitionism and fuzzy logic below largely follow a similar discussion in [25].
I have no precise account of what is a substantive issue or of what is separable from a logic, and use these notions in an intuitive sense. The gloss I gave above was that a substantive issue is the sort of philosophical thesis that makes for significant disagreement. We can say that an issue is separable from a logic when use of the logic presupposes no position on the issue. The examples that follow in the main text give a flavor for what these two notions amount to in practice.
This is because it entails an intuitionistically questionable principle, viz. Markov’s Principle [8, pp. 172-4]. In fact, the status of the completeness question for intuitionistic logic is remarkably unclear. See [12] for the view that Markov’s Principle may not be as intuitionistically dubious as is sometimes thought. See [19] and [22] for completeness results of intuitionistic logic with an intuitionistic metalogic but a slightly modified notion of model; but see [8, pp. 193-200] for reservations about whether these alternate models faithfully capture the intuitionist’s intended interpretation of the logical vocabulary. The fundamental point in the main text is not about the technical status of intuitionstic logic as complete or not but is rather about the dissonance of using a classical metatheory for intuitionistic logic. This point holds however the completeness question is to be settled.
Priest also argues that an adequate account of proof will necessitate admitting paradoxical sentences. I leave this motivation for LP to one side.
Although there are multiple ways of defining the truth-functions for propositional connectives in fuzzy logic, the following are a natural and common way of doing so. Let v be a valuation function. Then v(¬ϕ) = 1 − v(ϕ); v(ϕ∨ψ) = m a x(v(ϕ), v(ψ)); v(ϕ∧ψ) = m i n(v(ϕ), v(ψ)); v(ϕ → ψ) = min (1, 1 + v(ψ) − v(ϕ)). For a derivation of these functions from intuitive considerations, see [24, pp. 114–120]
In a similar vein, Thomas Weston writes: “For [formal theories], the intended interpretations of the constants, function symbols, etc. are invariably explained in the same way as the natural language technical terms of any part of mathematics or natural science. One explains the intended sense or ‘points to’ the intended reference in some natural language the speaker and his audience both understand. If these explanations and supplementary examples and audience guesses are inadequate to determine meaning or reference in the formal language case—as, no doubt, they sometimes are—they will certainly fail for natural languages as well” [23, p. 292]. My claim in the main text is the contrapositive: if there is an adequate determination of meaning in the natural language then there will be for the formal language as well. There is a similarity between the position Weston takes against Kreisel and the one I am taking against Jané. The core idea is that when a formal theory has an intended interpretation, as with set theory or number theory, that fact and the natural language expression of it can suffice to determine the meaning of terms in the formal theory. (Which of course does not rule out the existence of non-standard models—their existence is a technical fact. When there is an intended interpretation, though, non-standard models will require a reinterpretation of the relevant terms; this is what makes such models non-standard).
Cf. [16, §§ 5.3.1, 8.1]
Thanks to Stewart Shapiro, Kevin Scharp, Eileen Nutting, Neil Tennant, and an anonymous referee for helpful comments on earlier drafts.
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Brauer, E. Second-order Logic and the Power Set. J Philos Logic 47, 123–142 (2018). https://doi.org/10.1007/s10992-016-9422-x
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DOI: https://doi.org/10.1007/s10992-016-9422-x