Abstract
Second-order logic is generally thought problematic by the philosophical populace. Philosophers of mathematics and logic may have sophisticated reasons for rejecting second-order logic, but ask the average philosopher-on-the-street what’s wrong with second-order logic and they will probably mumble something about Quine, ontological commitment, and set theory in sheep’s clothing. In this paper, I try to get more precise about exactly what might be behind these mumblings. I offer four potential arguments against second-order logic and consider several lines of response to each. Two arguments target the coherence of second-order quantification generally, and stem from concerns about ontological commitment. The other two target the expressive power of ‘full’ (as opposed to ‘Henkin’) second-order logic, and give content to the concern that second-order logic is in fact “set theory in sheep’s clothing”. My aim is to understand the dialectic, not take sides; still, second-order logic comes through looking more promising than we might have initially thought.
Thanks to Aaron Cotnoir, Daniel Elstein, Robbie Williams, Stephen Yablo, and an audience at the University of Leeds Centre for Metaphysics and Mind for helpful comments and discussion.
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Notes
- 1.
This can be generalized in various ways. For instance, in many-valued logics, models assign sentences one of several truth-values, one or more of those values is ‘designated’, and model-theoretic consequence is understood as preservation-of-desigation-in-all-models. Other generalizations are possible, of course. For our purposes, though, we can stick with our first-pass understanding.
- 2.
Gillian Russell [20] argues that there is a crucial ambiguity in what we take ‘arguments’ to consist in which leads to a logical pluralism of a different sort than that advocated in [2]. I have slid over this ambiguity, and am going to proceed by assuming (hoping!) it won’t affect any of what’s to come.
- 3.
These principles make free use of truth and falsity, and thus may be subject to worries of the sort Field [6] has levelled against what he calls ‘The Validity Argument’. I think the situation here can be finessed with use of conditionals and infinite conjunctions, but for our purposes won’t bother.
- 4.
Systems that allow the variables but not the binding are also possible—see e.g. [21, p. 62]—but we won’t consider them here.
- 5.
Since properties are generally taken to be intensional entities, a better characterization would use sets of possibilia. We will stick to extensional contexts here, though, so we can safely ignore the difference.
- 6.
[21, p. 105] Note that (NCH) is not simply the negation of (CH). On models with countable domains, both (CH) and (NCH) are true.
- 7.
This is failure of what is called ‘weak completeness’—not all model-theoretic truths are theorems—and thus is stronger than the failure of what is called ‘strong completeness,’ which happens when not all model-theoretic consequences are proof-theoretic ones.
- 8.
The argument of Sect. 20.3.2 may provide such an explanation; but then we can consider it in its own right, rather than as an adjunct to Quine’s.
- 9.
Another question: If this argument isn’t Quine’s, what is Quine’s, and is it any good? Unfortunately, the closest thing I can find to an argument in Quine is at p. 66 of his Philosophy of Logic, and it’s a howler. I’ve left it out in interests of space, but had I included it I would not have said anything more (or better) against it than was said by Boolos [3, pp. 510–511].
- 10.
The Fregean has other troubles, though: expressibility problems that relate to the so-called ‘concept horse problem’. I cannot hope to pursue that huge literature here. I’ll simply focus on the qualitative conception instead.
- 11.
Or, perhaps, by ostending their meanings; but I take this option to be unavailable for second-order quantifiers.
- 12.
I don’t know anywhere this argument is explicitly presented in this form; van Inwagen presents a similar argument against substitutional quantification in [23], and comes close to giving this one [25, p. 124]. In the latter he also ascribes something like the present argument to Quine in Philosophy of Logic, but I cannot quite find that argument there.
- 13.
‘Things somehow\(_{X^{n}}\) relate is to be interpreted (roughly) as ‘things are-or-aren’t somehow\(_{X^{n}}\) related; see [18, p. 84].
- 14.
I’m assuming here that the second-orderist is happy to reason instrumentally with models in this way, even if she insists that, in all seriousness, second-order variables aren’t in the ‘ranging over’ business.
- 15.
Likewise, it may be that however Scooby, Shaggy, and Velma aren’t related, there will be three other things also so unrelated; this takes care of the ‘don’t’ part of the ‘do-or-don’t’ clause mentioned in Footnote 13.
- 16.
See [19] for related worries.
- 17.
Slightly more carefully, we can prove that, if \(\vdash _{F}\) (CH), the continuum hypothesis is true, and vice versa; someone who follows the line of thought outlined in Sect. 20.4.2.3 will have room to resist concluding that if \(\vdash _{F}(CH)\), (CH) is genuinely valid.
- 18.
Thanks to Aaron Cotnoir for suggesting this example to me.
- 19.
At least, if the ‘we’ are philosophers; Shapiro [21] argues at length that actual mathematical practice, which presumably includes that of model-theory, is rife with second-order reasoning. I cannot evaluate that claim here.
- 20.
I assume that if it is an error in reasoning to A, then we have an epistemic obligation to not A. I will sometimes slide between error-talk and obligation-talk in the text.
- 21.
It may not be the right diagnosis, of course. Another plausible diagnosis [4] has it that the Rose Argument is sound and we should satisfy the obligations of (iv′) by getting ourselves into a position where it’s indeterminate whether we believe that the rose is red or believe instead that the rose is pink.
- 22.
Should it count as a full second-order system? It’s genuinely weaker than the (usual) full system, but stronger than Henkin systems (see the next note). I doubt usage is fixed enough to settle this question.
- 23.
The quick-and-dirty way to show this is to note that full second-order logic has sentences that characterize infinite models. So long as our notion of infinity isn’t itself indeterminate, this means that (determinately) we have a sentence that is entailed by an infinite set (‘there is at least one thing,’ ‘there is at least two things,’ …) but not any of its finite subsets. So, unlike Henkin systems, the system F D is not compact.
- 24.
This may not be an entirely fair characterization of Shapiro’s view, in large part because he is considering a normative constraint somewhat stronger than the one outlined in Sect. 20.1.2. Still, this captures the basic idea, and is a move available here to friends of second-order logic worried about our weaker Normativity constraint.
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Turner, J. (2015). What’s So Bad About Second-Order Logic?. In: Torza, A. (eds) Quantifiers, Quantifiers, and Quantifiers: Themes in Logic, Metaphysics, and Language. Synthese Library, vol 373. Springer, Cham. https://doi.org/10.1007/978-3-319-18362-6_20
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