Abstract
Tarski characterized logical notions as invariant under permutations of the domain. The outcome, according to Tarski, is that our logic, which is commonly said to be a logic of extension rather than intension, is not even a logic of extension—it is a logic of cardinality (or, more accurately, of “isomorphism type”). In this paper, I make this idea precise. We look at a scale inspired by Ruth Barcan Marcus of various levels of meaning: extensions, intensions and hyperintensions. On this scale, the lower the level of meaning, the more coarse-grained and less “intensional” it is. I propose to extend this scale to accommodate a level of meaning appropriate for logic. Thus, below the level of extension, we will have a more coarse-grained level of form. I employ a semantic conception of form, adopted from Sher, where forms are features of things “in the world”. Each expression in the language embodies a form, and by the definition we give, forms will be invariant under permutations and thus Tarskian logical notions. I then define the logical terms of a language as those terms whose extension can be determined by their form. Logicality will be shown to be a lower level analogue of rigidity. Using Barcan Marcus’s principles of explicit and implicit extensionality, we are able to characterize purely logical languages as “sub-extensional”, namely, as concerned only with form, and we thus obtain a wider perspective on both logicality and extensionality.
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Notes
We extend a bijection f between two domains to a function between the set-theoretic hierarchies built over those domains by recursion in the natural way. By “domains” we mean sets of ur-elements. The restrictions of domains to ur-elements enables the recursion by which f is extended to work, by blocking a set from being both a member and a subset of the domain.
Note that on the left we have an identity statement, and that the condition on the right (there is a bijection \(f:D\rightarrow D'\) such that \(f(ext_D(t))=ext_{D'}(t')\)) is an equivalence relation on pairs of domains and expressions, and thereby the bi-conditional has the structure of an abstraction principle.
We use “transformation”, as Tarski, to mean a one-one function from the space onto itself.
The group’s associated operation is composition of functions. Reflexivity is obtained from the identity element which is the identity function, symmetry is obtained from the existence of an inverse, and transitivity is obtained by closure under composition.
Though, through the influence of Klein, the mere idea of characterizing a discipline by a space and a group of transformations is thought of as “geometric,” referring to the method of classification rather than the distinctive content of geometric disciplines.
Especially as we may note that the idea of permutation invariance is prominent in Lindenbaum and Tarski (1983) and so was salient for Tarski at the time of writing the paper on logical consequence (Tarski 1936)—and yet Tarski does not appeal to it. Tarski notes that the results in Lindenbaum and Tarski (1983) were presented as early as 1932–1933. In a letter to Morton White, as late as 1944 (White and Tarski 1987), Tarski remains skeptical regarding a principled criterion for logicality, and again, does not even mention the results from the paper with Lindenbaum published in 1935.
Another interpretive option is to view Tarski as attending to the 1936 lacuna in his 1966 lecture, but not seeing himself as providing an absolute criterion. The characterization of logical notions as invariant under permutations still leaves room for relativity: what comes out as logical will depend on the background set theory (see Gómez-Torrente 2015). This interpretation sits well with the fact that Tarski does not declare that he had filled the gap of the 1936 paper, but it does not explain why Tarski, in the later lecture, avoids drawing any connection to that paper or to the concept of logical consequence.
But see the later Tarski and Givant (1987) where an explicit connection between logical terms in a given formal language and logical notions is made. Note, however, that when the link is finally made, it is in the context of a mathematical text on languages for set theory. The definition there of logical symbols (p. 57) serves for proving a mathematical theorem, and is not used elsewhere in the book. The definition is not backed by any philosophical analysis, and no connection is made to Tarski’s work on logical consequence (though other relevant works: Tarski’s lecture and Tarski’s earlier paper with Lindenbaum are mentioned).
Tarski refers here in a footnote to Whitehead and Russell (1910, III(2)).
Extensions for connectives can be defined as operators from sets of assignments satisfying subformulas to a set of assignments satisfying the complex formula. See Sect. 7, and in particular fn. 18, for more details. I shall not be concerned with modal operators and other expressions that belong solely to intensional languages.
There are reasons to keep models mathematically pure—consisting of only mathematical objects—in which case the extension of Dog in a domain will be a set of mathematical objects used as a surrogate for a set of dogs. I will leave these issues aside.
Recall that we extend a bijection f between two domains to a function between the set-theoretic hierarchies built over those domains by recursion in the natural way.
We shall set aside the issue of whether this is a welcome consequence. I do not rule out modifications to the adequacy condition that relieve this dependency on the cardinality of the domain (along the lines of proposed modifications to the Tarski–Sher criterion proposed in the literature (Feferman 1999; Bonnay 2008)).
Technically, we could opt for defining extensions, intensions and hyperintensions through invariance under transformations, but the result would be, it seems to me, unnatural and unenlightening.
The independence of logicality and rigidity would of course depend on our construal of possible world models and their relation to the domains alluded to here. To account for rigidity, we need to re-inflate the semantic apparatus. Recall that we have suggested considering extensions as derived from intensions applied to a distinguished actual world in each possible world model. Now note that rigidity will thus be relative to a possible world model, while logicality takes into account all models. However we re-inflate the semantics, if there are at least two distinct domains from which we draw extensions, we can provide examples for logical non-rigid expressions: a logical term defined over those domains whose extension is not empty will not be rigid. If, for instance, we treat the universal quantifier as a second-level predicate and take its extension in a domain D to be \(\{D\}\), its extension will differ in possible worlds with different domains, and so despite being logical, it will not be rigid. An example for a non-logical rigid expression would be an individual constant that denotes the same object in each possible world—no individual constant is logical, even if it is rigid.
Another example is that of i and \(-i\). i and \(-i\) share many properties, and in particular they have the same form, but they shouldn’t be equated. This issue touches on the so-called identity problem in structuralism in mathematics, see e.g. Button (2006).
One exception is the degenerate case where two expressions are of different categories both have an empty extension and therefore the same form.
We define L\(_{\infty \infty }\) a bit differently from McGee, and instead follow (Feferman 2010).
We are now able to give the forms of connectives, as promised in f.n. 8. In this setting, the extension of a connective is an operation on sequences. For example, the extension of conjunction is the intersection operation on sets of sequences over the domain: \(ext_D(\wedge )=\Big \{\langle \Sigma _1,\Sigma _2,\Sigma _3\rangle :\Sigma _1,\Sigma _2,\Sigma _3\in \mathcal {P}(D^\omega ), \Sigma _3=\Sigma _1\cap \Sigma _2\Big \}\).
This is a variant of what is known in model theory as the isomorphism property that holds of a logic if for every sentence \(\varphi \) and any models M and \(M'\) that are isomorphic vis-à-vis the nonlogical vocabulary in \(\varphi \), \(M\models \varphi \) if and only if \(M'\models \varphi \). This property holds in every logic in which the terms fixed as logical are invariant under isomorphisms (see Shapiro 1998; Sagi 2014), so we may also consider going beyond standard first-order languages.
So, for instance, if the language is of first-order and includes the truth-functional connectives and identity, every logical formula is equivalent over equinumerous domains to a formula containing purely logical vocabulary.
I appeal to the now accepted notion of logical truth which is derived from, but not identical to the notion of analytic sentence defined in Tarski (1936).
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Early versions of this paper were presented at the Logica Symposium in Hejnice and at the Reasoning Club conference at the University of Kent in 2014, and at the GroLog Seminar at the University of Groningen in 2015. I thank the audiences there for discussion. In addition, I thank the following people for very helpful comments on earlier drafts: Douglas Blue, Tim Button, Salvatore Florio, Carlos César Jiménez, Ran Lanzet, Pia Schneider, Stewart Shapiro, Jack Woods and two anonymous referees for this journal.
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Sagi, G. Extensionality and logicality. Synthese 198 (Suppl 5), 1095–1119 (2021). https://doi.org/10.1007/s11229-017-1447-3
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DOI: https://doi.org/10.1007/s11229-017-1447-3