Abstract
Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (Putnam, J Symbol Logic 45:464–482, 1980). Therefore, it is not clear how this logic should be interpreted. (3) Second-order logic also has strong ontological commitments: (a) it is ontologically committed to classes (Resnik, J Phil 85:75–87, 1988), and (b) according to Quine (Philosophy of logic, Prentice-Hall: Englewood Cliffs, 1970), it is nothing more than “set theory in sheep’s clothing”. (4) It is also not better than its first-order counterpart, in the following sense: if first-order logic does not characterize adequately mathematical systems, given the existence of non-isomorphic first-order interpretations, second-order logic does not characterize them either, given the existence of different interpretations of second-order theories (Melia, Analysis 55:127–134, 1995). (5) Finally, as opposed to what is claimed by defenders of second-order logic [such as Shapiro (J Symbol Logic 50:714–742, 1985)], this logic does not solve the problem of referential access to mathematical objects (Azzouni, Metaphysical myths, mathematical practice: the logic and epistemology of the exact sciences, Cambridge University Press, Cambridge, 1994). In this paper, I argue that the second-order theorist can solve each of these difficulties. As a result, second-order logic provides the benefits of a rich framework without the associated costs.
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Notes
The second-order existential quantifier is defined in the usual way: ∃X φ =def. ¬∀X ¬φ, and ∃f φ =def. ¬∀f ¬φ.
Shapiro provides further examples of mathematical concepts that can only be adequately expressed in second-order languages (1991, pp. 97–109).
As a further argument for second-order logic—at least for those with nominalist inclinations!—note that several nominalist proposals have adopted versions of this logic as part of their strategies to reduce ontological commitment to mathematical objects. This is the case of Field’s fictionalism [at least as presented in Field (1980), see Field (1989) for a first-order version] and Hellman’s modal-structural interpretation (Hellman 1989).
Monadic first-order logic has predicates with only one place, while monadic second-order logic has one-place predicates and second-order quantifiers binding one-place predicate variables.
Here is Putnam’s argument (1976, pp. 125–126). Let us assume that T is an ideal theory. It is consistent, complete, empirically adequate, simple, plausible etc. The only property of T that is left open (for the sake of argument) is its truth. (T may well be false.) Now, let us assume, with Putnam, that the world can be broken into infinitely many pieces, and that T states that there are infinitely many things. Since T is consistent, by the completeness theorem of first-order logic, it has a model. And since T has an infinite model, by the Löwenheim-Skolem theorem, it has a model of every infinite cardinality. Let us now select a model M that has the same cardinality as the world, and let us establish a one-to-one mapping from individuals of M into pieces of the world. We can use this mapping to define relations over M’s domain directly in the world, and as a result, we can define a satisfaction relation, sat, which establishes a ‘correspondence’ between T’s language and sets of pieces of the world. Now, provided that we interpret ‘true’ as true(sat)—that is, as the truth predicate that is defined by the relation sat, just as ‘true’ is defined in terms of ‘satisfies’ in Tarski’s account—the ideal theory T comes out true: indeed, it is true of the world. But then T cannot be false!
Kaplan’s proof of this fact is presented in Boolos (1984, pp. 432–433).
Supposing that the domain of discourse consists of critics, and Axy means ‘x admires y’, the Geach-Kaplan sentence becomes: ∃X (∃x Xx ∧ ∀x ∀y ((Xx ∧ Axy) → x ≠ y ∧ Xy)).
Boolos’s idea is that the informal metalanguage in which we give the semantics for (monadic) second-order logic contains the plural quantifier ‘there are objects’, which is then used to interpret the second-order monadic quantifiers. [For further discussion, see Shapiro (1991, pp. 222–226), and Simons (1997)].
As opposed to classical logic, paraconsistent logic distinguishes the notions of inconsistency and triviality. A theory T is inconsistent if it has a theorem of the form (A ∧ ¬A), and T is trivial if all the formulas of the language are theorems of T. The threat posed by inconsistencies in a classical logic setting is the triviality they bring. In a paraconsistent framework, however, there are inconsistent but non-trivial theories (see da Costa 1974; da Costa et al. 2007).
I am making here a distinction between naive set theory and an intuitive notion of set on the grounds that the former is a mathematical theory, whereas the latter incorporates intuitive, pre-theoretic assumptions that we may have about sets, collections etc.
As Shapiro remarks: “the major shortcoming of ω-languages is that they assume or presuppose the natural numbers. Therefore, such a language cannot be used to show, illustrate, or characterize how the natural number structure is itself understood, grasped, or communicated” (Shapiro 1985, p. 733).
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My thanks go to Philip Catton, Steven French, Sarah Kattau, Bernard Linsky, Judy Pelham, Jaroslav Peregrin, Peter Simons, Göran Sundholm, and Peter Woodruff for helpful discussions.
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Bueno, O. A Defense of Second-Order Logic. Axiomathes 20, 365–383 (2010). https://doi.org/10.1007/s10516-010-9101-4
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DOI: https://doi.org/10.1007/s10516-010-9101-4