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Logical Constants: a Modalist Approach 11 Respect to the Producible Theorems, One System Is No Better Than Another, Save for Reasons of Efficiency Or Aesthetic Values NOUSˆ 47:1 (2013) 1–24 Logical Constants: A Modalist Approach1 OTAVIO´ BUENO University of Miami SCOTT A. SHALKOWSKI University of Leeds 1. Introduction Philosophers sometimes take refuge in logic in a way befitting a domain free of controversy. Metaphysical claims are thought to be dubious in ways that logical claims are not.2 Metaphysical matters cannot be settled in any straightforward way, whereas logical issues typically can be. There is more than a little self-deception contained in this contrast, however. In this paper, we begin with a theoretical dis- agreement in logical theory. This disagreement carries over to the characterization of logical constants. After presenting Tarski’s very general account of the nature of the constants, and Gila Sher’s more detailed development of the Tarskian approach, we return to the subject of logical disagreement and show the deficiencies with the basic Tarskian framework. We argue that a modalist alternative should supplant it. Our goal in the paper is to offer a modalist account of the status of logical constants. We are not developing a full-fledged modalist account of logical conse- quence. We take only the first step in that direction by examining the ineliminable role that modality plays in shaping our understanding of logical constants. The modalist treatment of logical consequence is left for another occasion. 2. The Model-Theoretic Approach to Logical Constants: Some Features Logical theory is a partial theory of good argumentation. It is a partial theory be- cause it concerns only the formal or structural component of good argumentation, and good arguments are about more than structure. Good arguments are also about truth, warranted belief, the transmission of warrant, and the like. Disagreements regarding any aspect of good argumentation may well generate disagreements re- garding which arguments are valid and which are not. Intuitionist logicians have maintained that the Law of Excluded Middle is not a logical truth and that re- ductio ad absurdum is not a valid argument form. Paraconsistent logicians have maintained that logics codifying well-managed inference should not be explosive, i.e., they should not treat as valid the inference of an arbitrary conclusion from inconsistent premises. It is usual and agreed among the advocates of divergent treatments of logic that expressions for first-order quantification, negation, conjunction, disjunction, C 2012 Wiley Periodicals, Inc. 1 2 NOUSˆ and the material conditional are permitted only their own respective invariant interpretations. All parties agree that permitting all expressions to have variable interpretations makes formal treatments of validity impossible, since no argument form would preserve truth under all interpretations of the logical “constants”, and all agree that permitting no expressions to have variable interpretations does not allow for multiple instances of a given logical form, thus precluding the study of formal logic. Hence, all agree on the need for logically relevant constants. There is also widespread, even if not universal, agreement that the Tarskian model-theoretic framework is the proper framework for characterizing the various theories of logical truth and logical consequence. In his original account of logical consequence, Tarski saw the need for the interpretation of some expressions to remain fixed, but he provided only an implicit account of what those fixed expres- sions were by employing the constants as he did, and he provided no account of what makes an expression appropriate for only an invariant interpretation (Tarski [1935]). Tarski filled this gap in a posthumous paper on the notion of a logical con- stant (Tarski [1966]). There he provided a very general model-theoretic framework that, in principle, could be accepted by all participants in the debate regarding the proper characterization of logical consequence. The key idea behind eligibility for invariant interpretation is that an expression has an invariant interpretation when its interpretation is unaffected by all permutations of the objects in the domain. Tarski’s own development of his approach was extremely general and abstract. Gila Sher provided a much more detailed presentation of a model-theoretic account of the logical constants (Sher [1991], Chapter 3, and Sher [2003]). Her account can be accepted entirely by classical logicians. For our purposes, the key features of her account are that logical constants are: (1) extensional in character, (2) defined over all models, and (3) defined by functions that are invariant over isomorphic structures. More specifically (see Sher [1991], pp. 54–56, and Sher [2003], pp. 189– 190): C is a logical constant iff C is a truth-functional connective or C satisfies the following conditions: (A) A logical constant C is syntactically an n-place predicate or functor (functional expression) of level 1 or 2, n being a positive integer. (B) A logical constant C is defined by a single extensional function and is identified with its extension.3 (C) A logical constant C is defined over models. In each model A over which it is defined, C is assigned a construct of elements of A corresponding to its syntactic category. Specifically, C should be defined by a function fC such that given a model A (with universe A) in its domain: n (a) If C is a first-level n-place predicate, then fC (A)isasubsetofA . n (b) If C is a first-level n-place functor, then fC (A) is a function from A into A. (c) If C is a second-level n-place predicate, then fC (A)isasubsetofB1 x ...x Bn, m where for n ≥ i ≥ 1, Bi = A if i(C) is an individual, and Bi = P(A )ifi(C) is an m-place predicate (i(C)beingtheith argument of C). Logical Constants: A Modalist Approach 3 (d) If C is a second-level n-place functor, then fC (A) is a function from B1 x ...x Bn into Bn +1,whereforn+1 ≥ i ≥ 1, Bi is as defined in (c). (D) A logical constant C is defined over all models (for the logic). (E) A logical constant C is defined by a function fC which is invariant over isomorphic structures. That is, the following conditions hold: (a) If C is a first-level n-place predicate, A and A are models with universes A and n n A respectively, b1, ...,bn∈A , b 1, ...,b n∈A , and the structures A, b1, ...,bn and A ,b 1, ...,b n are isomorphic, then b1, ...,bn∈fC (A)iff b 1, ...,b n∈fC (A ). (b) If C is a second-level n-place predicate, A and A are models with universes A and A respectively, D1, ...,Dn∈B1 x ...x Bn, D 1, ...,D n∈B 1 x ...x B n (where for n ≥ i ≥ 1, Bi and B i are as in (C.c)), and the structures A, D1, ...,Dn and A , D 1, ...,D n are isomorphic, then D1, ...,Dn∈ fC (A)iffD 1, ...,D n∈fC (A ). (c) Analogously for functors. Sher’s is a disjunctive account, according to which, given the first disjunct, any truth-functional connective is automatically a logical constant. To avoid exclud- ing non-truth-functional connectives by fiat, the second disjunct is offered. First, syntactically logical constants are functional expressions (condition (A)) associated with a single extensional function (condition (B)) defined over all models for the logic (conditions (C) and (D)). Finally, the crucial condition is that a logical con- stant is a function that is invariant over isomorphic structures (condition (E)), i.e., the extension of the function does not change across isomorphic structures. In other words, if we permute the objects of the domains of these structures, the extension of that function will remain invariant. This is the core of the Tarskian model-theoretic approach, and Sher develops it carefully. 3. The Model-Theoretic Approach to Logical Constants: Some Troubles The model-theoretic account faces several difficulties. The most significant lies in the model-theoretic framework itself. Models are useful and informative only to the extent that they model something and fail to model other things. In the context of logic, models serve primarily as invalidators of inferences. If there is a model in which the premises of an argument are jointly true and yet the conclusion is false, the argument is invalid. So far, we have said nothing about what models are, and not just any old models will do. Not all of the Lego models in the world will do what the logician requires. The reason for this, of course, is that all of the Lego models in the world fail to model all that there is. The world itself manages what the Lego models do not; it manages to model all that there is because it is all that there is. If the world and an argument conspire so that the premises are true and the conclusion false, that is sufficient for the invalidity of the argument. The logician, however, is still not satisfied with the world as the (domain of) models. Even if all inferences of “At no time is Mars inhabited” from “At some 4 NOUSˆ time Earth is inhabited” begin and end in truth, the connection between the truths in those inferences is insufficiently tight. Intuitively, we have contingency where we require necessity. The model-theoretician maintains that ‘the world’ was inter- preted too narrowly. The world is not merely the concrete world of plums, planets, princesses, and peas. It contains all the wonders of the abstract. While there is no concrete model in which the premise is true while the conclusion is false, there is an abstract model invalidating that inference. There are two ways of interpreting the model-theoretic account: one is platonist, the other is nominalist.
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