Abstract
Permutation invariance is often presented as the correct criterion for logicality. The basic idea is that one can demarcate the realm of logic by isolating specific entities—logical notions or constants—and that permutation invariance would provide a philosophically motivated and technically sophisticated criterion for what counts as a logical notion. The thesis of permutation invariance as a criterion for logicality has received considerable attention in the literature in recent decades, and much of the debate is developed against the background of ideas put forth by Tarski in a 1966 lecture (Tarski 1966/1986). But as noted by Tarski himself in the lecture, the permutation invariance criterion yields a class of putative ‘logical constants’ that are essentially only sensitive to the number of elements in classes of individuals. Thus, to hold the permutation invariance thesis essentially amounts to limiting the scope of logic to quantificational phenomena, which is controversial at best and possibly simply wrong. In this paper, I argue that permutation invariance is a misguided approach to the nature of logic because it is not an adequate formal explanans for the informal notion of the generality of logic. In particular, I discuss some cases of undergeneration of the criterion, i.e. the fact that it excludes from the realm of logic operators that we have good reason to regard as logical, especially some modal operators.
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Notes
Notice that Tarski’s concept of a ‘logical notion’ does not concern linguistic entities; so he is not after the partition of a language into logical and non-logical terms mentioned in his earlier paper. Rather, for Tarski (logical) notions are mathematical objects: functions, relations, operations etc.
See (Feferman 1999, section 7) for Feferman’s puzzlement over Tarski’s ‘answer’ to the problem.
That the criterion is not ontologically neutral is also a point made by Feferman (1999, 45), but he is then referring to the fact that it does not exclude higher-order quantification. We shall also see that it assumes a ‘null-structure’ for the base domain of objects.
It is well known that the finite cardinality quantifiers can be defined in first-order logic with identity, so anyone wishing to maintain the privileged status of first-order logic but to exclude the cardinality quantifiers from the class of logical constants must deny the status of logical constant to identity. This is precisely what Feferman does.
“First, my criterion for logical terms is based on an analysis of the Tarskian framework, which is insufficient for modals. Second, we cannot take for granted that the task of modal logic is the same as that of symbolic logic proper. To determine the scope of modal logic and characterize its operators, we would have to set upon an independent inquiry into its underlying goals and principles.” (Sher 1991, 54, my emphasis).
Feferman (1999, 33/4) develops Tarski’s suggestion in more precise terms. The key idea is that a permutation of the base domain of objects induces in a natural way a permutation in the higher domains.
I owe the point about the two latter kinds of frames to Andrew Bacon. MacFarlane (2000, 217) only recognizes frames with the universal accessibility relation as allowing for the modal operators to satisfy the permutation invariance criterion.
Indeed, it is often noted that the account of modalities in terms of possible-world semantics is just ‘disguised quantification’ (Harman 1972, 75), and thus not truly about modalities. For reasons of space, I cannot deal in detail with this charge here. Suffice it to say that I am sensitive to some of the arguments, but that my concern here is not so much with the philosophical significance of this approach to modal logic but rather with the fact that it is a well-entrenched subfield of current research in logic.
Because validity in S5 frames is provably equivalent to validity in universal frames, we may want to conclude that S5 modal operators are ‘logical’ according to the permutation invariance criterion. But technically, S5 modalities do not satisfy the criterion either, as this example shows. MacFarlane (2000, 217) seems to hesitate as to whether S5 modal operators do or do not satisfy the invariance criterion.
Interestingly, MacFarlane (2000, 6.7.5) thinks that the operators of tense logic do satisfy the criterion, even though the frames on which they are typically interpreted are special cases of Kripke frames.
I owe this example to an anonymous referee.
Harman (1972) suggests that we need not count modal logic as ‘logic’, and therefore should not count it as ‘logic’. I am suggesting here that, if we do this, as philosophers we will be excluding a vibrant portion of logical practices from the realm of analysis, which I take not to be recommended.
As can be gathered by this admittedly impressionistic description, on this conception logic is essentially an activity, a particular kind of practice that people engage in. As such, my approach is not incompatible with a realist view of logic, in the sense that logical principles may be constrained by the nature of reality. But to me, more important than establishing the nature of this underlying reality is to focus on what agents do against the background of this reality.
Feferman (1999, 33) briefly discusses how different portions of mathematics can be classified in terms of the automorphisms of structures in a given similarity class and the notions that are invariant under all such automorphisms.
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Dutilh Novaes, C. The Undergeneration of Permutation Invariance as a Criterion for Logicality. Erkenn 79, 81–97 (2014). https://doi.org/10.1007/s10670-013-9469-9
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DOI: https://doi.org/10.1007/s10670-013-9469-9