Logicality and Meaning

Abstract

In standard model-theoretic semantics, the meaning of logical terms is said to be fixed in the system while that of nonlogical terms remains variable. Much effort has been devoted to characterizing logical terms, those terms that should be fixed, but little has been said on their role in logical systems: on what fixing their meaning precisely amounts to. My proposal is that when a term is considered logical in model theory, what gets fixed is its intension rather than its extension. I provide a rigorous way of spelling out this idea, and show that it leads to a graded account of logicality: the less structure a term requires in order for its intension to be fixed, the more logical it is. Finally, I focus on the class of terms that are invariant under isomorphisms, as they render themselves more easily to mathematical treatment. I propose a mathematical measure for the logicality of such terms based on their associated Löwenheim numbers.

Logicality and Model Classes

“…In sum: in this paper we develop further this aspect of logicality identified by Sagi, namely that it is graded, albeit with a different philosophical agenda than is laid out in [46]. We especially point out problems in the Löwenheim-Skolem properties of L ∞∞ , along with other ways in which it diverges from first order logic.…”

“…Thus L(Q 0 ) is an absolute logic because it has a recursive syntax, just like first order logic, and its semantics is absolute in transitive models of (even weak) set theory. 46 On the other hand, L(Q 1 ) is not absolute, even though it has a recursive syntax. 47 Burgess's [14] forges a link between absoluteness and the idea of having a proof procedure.…”

“…We relate this to modeltheoretic characteristics of abstract logics in which the model class is definable. This results in a graded concept of logicality in the terminology of Sagi [46]. We investigate which characteristics of logics, such as variants of the Löwenheim-Skolem Theorem, Completeness Theorem, and absoluteness, are relevant from the logicality point of view, continuing earlier work by Bonnay, Feferman, and Sagi.…”

“…As G. Sagi notes in her [46], the Löwenheim number of a logic encodes the degree to which the logic is indifferent to cardinality. This is due to the fact that if a sentence satisfied by a model is also satisfied by a model of size less than or equal to a given cardinality κ, then all the validities 12 which are captured in the full range of models are captured already on an initial segment (of V ).…”

On Williamson's new Quinean argument against nonclassical logic

“…That's all I'm sketching here. For recent debate on both logical vocabulary and its relation to logical form see Gil Sagi's work[42]; Sagi's work challenges some of the assumptions of this paper but, for purposes of the current paper (but only for such purposes), I set the challenges aside.7 This is logically redundant -it is always treated as the null operator, taking a sentence and delivering a logically equivalent (and everywhere intersubstitutable) sentence -and, so, standardly left off the list. I mention it here only to complete the symmetric picture of duals.8 This is sometimes useful to express using the truth operator, just to distinguish from tense-intensive conjunctions (e.g., 'dynamic' conjunctions).…”