 Most other proposals have tried to delineate in some other way the Aristotelian idea that the logical expressions have some kind of “insubstantial” meaning, so as to use it as a necessary and sufficient condition for logicality. One recent suggestion is that logical expressions are those that do not allow us to distinguish different individuals. One way in which this has been made precise is through the characterization of logical expressions as those whose extension or denotation over any particular domain of individuals is invariant under permutations of that domain. (See Tarski and Givant 1987, p. 57, and Tarski 1966; for related proposals see also McCarthy 1981, Sher 1991, ch. 3, McGee 1996, Feferman 1999, Bonnay 2008, Woods 2016 and Griffiths and Paseau 2022, among others.) A permutation of a domain is a one-to-one correspondence between the domain and itself. For example, if \(D\) is the domain {Aristotle, Caesar, Napoleon, Kripke}, one permutation is the correspondence that assigns each man to himself; another is the correspondence \(P\) that assigns Caesar to Aristotle (in mathematical notation, \(P(\text{Aristotle})=\text{Caesar}\)), Napoleon to Caesar, Kripke to Napoleon, and Aristotle to Kripke. That the extension of an expression over a domain is invariant under a permutation of that domain means that the induced image of that extension under the permutation is the extension itself (the “induced image” of an extension under a permutation \(Q\) is what the extension becomes when in place of each object \(o\) one puts the object \(Q(o)\)). The extension of “philosopher” over \(D\) is not invariant under the permutation \(P\) above, for that extension is \(\{\text{Aristotle}, \text{Kripke}\}\), whose induced image under \(P\) is \(\{\text{Caesar}, \text{Aristotle}\}\). This is favorable to the proposal, for “philosopher” is certainly not widely applicable, and so non-logical on most views. On the other hand, the predicate “are identical” has as its extension over \(D\) the set of pairs 