Wednesday, June 04, 2008

The insufficiency of permutation invariance

MacFarlane has two related criticisms of the permutation invariance criterion of logicality, which says that an operation is logical iff it is invariant under all permutations of the domain.

The first criticism is that not all permutations are considered. It is only permutations of the domain of objects and not of truth values that are considered. There are not many classical truth functions that are invariant under permutation of the domain of truth values. However, if no permutation of the domain of truth values is allowed, then all functions on just that domain are logical. This might not be so bad when considering classical logic since all classical truth functions are equivalent to some combination of conjunction and negation. It is not enough to say that we only consider permutations on the domain of objects. If we are working with a modal frame, we consider permutations of the domain of worlds just as readily as those of the domain(s) of objects. In fact, there are also not many operations that are invariant under all permutations of the domain of worlds. The objection takes an even more concrete form when considering larger domains of truth values than the classical one, {t,f}. MacFarlane's suggestion is to consider all permutations that respect what he calls intrinsic structure. This is a relation of some sort on a domain, considered as just a set. The intrinsic structure of classical logic is: null structure on the objects, f≤t and not t≤f.

The notion of intrinsic structure sets up the second criticism of permutation invariance. This criticism is that the sorts of intrinsic structure that the permutations must respect is in need of justification. Why is it, MacFarlane presses, that there should be no structure among the objects? This rules out the epsilon of set theory as a logical notion. Similarly, for modal frames, if there is no structure among the worlds then we get an S5 necessity as logical rather than the operations of any of the other modal logics. Null structure on the objects is structure, albeit of a degenerate sort, and stands in need of justification. MacFarlane seems to be doubtful that any non-question begging reason can be obtained without recourse to an antecedent view of logic. The objection underlying both of these can be summed up as: at best permutation invariance systematizes some antecedent intuitions about logicality; it does not explain the notion of logicality.

MacFarlane's own suggestion is that something more is needed, so that logicality can not be specified just in terms of permutation invariance. He suggests that the extra is an understanding of intrinsic structure that connects it to a Kantian notion of logic, i.e. as normative for thought as such. This gap is supposed to be bridged by noting that "intrinsic structure belongs to a type in virtue of the most general purpose of logical theory: the study of the semantic relationships that hold between sentences solely in virtue of their capacity for being asserted and used in inferences." He continues: "On this ground, one might say that notions that are sensitive only to intrinsic structure are applicable to thought as such..." Intrinsic structure ought to be the minimal structure demanded by the above Kantian view of logic. There is fleshed out using some distinctions from Belnap that I'm not going to go into in this post (because I need to read the relevant article, "Under Carnap's Lamp"), the distinctions between presemantics, semantics and postsemantics. Interestingly, this suggestion ends up ruling that the non-S5 modal operators are not logical because the accessibility relation on worlds is not minimal in his sense, whereas the structure in various logical lattices, like the standard one for {t,b,n,f} is minimal. Even more interestingly, it suggests that the structure needed for Prior-Thomason(-Belnap)-style indeterministic tense logics has some claim to minimality and so to putting tense operators in the logical box.

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