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. Both emphasize the importance of models in modeling logi- cal consequence; they differ, however, on the status they assign to these models. The platonist reading insists that models exist and are abstract objects; the nominalist either denies their existence or their abstract character. We argue that on what- ever interpretation that is offered, there are troubles for the model-theoretician. In this respect, our point is perfectly general. We start by considering the platonist interpretation. It is perfectly fair for the platonist to pursue the model-theoretic cause in this way only if the platonist is entitled to claim that there is a sufficiently rich domain of abstract models to serve as invalidators for all and only the invalid inferences. Too many abstract models treated as invalidators and the model-theoretician would be forced to declare some valid arguments invalid; too few and the model-theoretician would be compelled to declare some invalid arguments valid. Platonist model theory treats ‘model’ as a term of art. It designates sets of a specific kind and these sets are alleged to be ontically significant. Platonists do not construe their talk of models as merely a fac¸on de parler that permits them to exchange the vocabulary of the logical or the modal for that of the model. On this account, logical notions have ontological underpinnings. Since the platonist’s is a substantive claim, this concern over warrant cannot be dealt with brusquely as though the proposed explanans is obviously adequate to the explanandum, as it could be were the discourse involving models merely a manner of speaking of no ontological consequence. The platonist must make plausible the claim that the model-theoretic account neither overgenerates nor undergenerates invalidators of arguments.4 When we consider the role of logic in both the choice of axioms and the derivation of theorems of a set theory, warrant for the model-theoretic account of the logical constants evaporates. The claim before us is that the logical constants are characterized model- theoretically. Wedded to a model-theoretic account of logical consequence, the result is that logical facts are model-theoretic facts. Since this claim is not intended to be trivial, epistemically speaking it might be incorrect. Its correctness would consist in the proper match of logical facts on the one hand with model-theoretic facts on the other, i.e., at a minimum the model-theoretic domain must be exten- sionally adequate. For the moment, let us take the basic, object level logical facts to be given and let us inquire, first, into the model-theoretic facts and our warrant for maintaining that the model-theoretic facts are this way rather than that. Even if the genesis of sophisticated set theories is misrepresented, let us begin with axioms of model theory; that is, general principles—typically formulated in a given set theory—that govern the construction and selection of models. On the Logical Constants: A Modalist Approach 5 platonistic framework we are granting to the model-theoretician, the axioms that concern us do not come from nowhere; they are not stipulative. They are defeasible claims that are the most perspicuous and most central claims about models.5 Since they are not stipulative, they—again, epistemically speaking—could be wrong. Once a set of axioms has been proposed, here is what we cannot do: inspect the domain of models to determine whether all and only the models are so characterized. No one thought that we could do such an inspection, but since the proposal to regard some claims as true axioms might be mistaken there must be some means of determining their fitness for use as axioms of a specific set theory. Waiving considerations of elegance that might weigh in favor of one set theory rather than another, the only test we have for a proposal like this is to see where it leads.6 Where it leads, quite obviously, is to theorems. Sometimes theorems are pedes- trian; sometimes surprising. Sometimes unexceptional; sometimes incredible. Some- times the incredible is just so far beyond the intellectual pale that an apparent theorem is treated not as a theorem at all, but as the basis for a reductio of the proposed axioms. Making our lives easy, consider the most obvious case in which a contradiction is derived, as Russell did from Frege’s axioms. Frege’s reconstruction of arithmetic was judged to be false, in light of Russell’s paradox. Why, exactly, was it so judged? Though the story is familiar, it is worth rehearsing the reasoning. Frege’s proposed reconstruction is inconsistent. There is no epistemic value in recognizing this inconsistency, unless one makes some assumption about the prospects of modeling inconsistencies. The natural thought is that inconsistencies claim things to be in ways they cannot be. If things cannot be that way, then they are not that way. Without this assumption and the obvious inference, there could be no grounds for doing some reductio exercise to conclude that Frege’s axioms are not jointly true. Though our immediate concern is not Frege’s reconstruction of arithmetic but instead axioms for a model theory appropriate for an account of the logical constants, the point remains: standard model theory makes substantive as- sumptions about how things could not be. In our example, mathematicians assume, i.e., they use a logic that encodes, that things cannot be in inconsistent ways.7 The semantics for the classical logic used in the derivation of theorems is, quite typically, model-theoretic. That semantics is supposed by its proponents to do the duty of the modal assumption. In particular, by using classical logic a platonist assumes that no inconsistent situation is modeled. Hence, no argument with incon- sistent premises is thought by such a platonist to have an invalidator, and thus, any such argument is taken by such a platonist to be valid. In particular, arguments with inconsistent premises are taken to entail any sentence of the language. Platonists, however, need not infer classically. If the platonist infers paracon- sistently, explosion is rejected and rejected precisely because some inconsistent situations are modeled and some of those modeled inconsistent situations invali- date the inference from contradictory premises to an arbitrary conclusion (see, e.g., da Costa, Krause, and Bueno [2007], and Priest [2006]). Similarly for logics that regiment inferences over incompletely specified situations and do not contain ex- cluded middle as a logical truth and do not treat disjunctive syllogism as valid. The difficulty for the platonist model-theoretician is, then, to find a principled way of 6 NOUSˆ deciding which logic should be adopted. If classical logic is assumed, all classically valid inferences are privileged; if some non-classical logic is adopted, not all of them are. In evaluating axioms we must reason and reasoning one way rather than another involves a prior assumption, encoded in the logic used, about key features of structures appropriate to invalidators of inferences. When the characterization of logical constants is at stake, this vitiates the entire approach: the logic that the model-theoretician adopts will privilege the characterization of certain constants as logical at the expense of others, but one is not entitled to privilege one particular characterization of the underlying models until after the set-theoretic axioms are fully evaluated. That is the general form of the problem for any invariance account of the logical constants that treats the invariance claims as substantive claims over a robust ontology. Of course, if we hold all and only the interpretations of these particular expressions invariant when we determine the range of invalidators for our inferences we will “discover” no invalidators for our officially sanctioned inferences. This produces no warrant for the substantive invariance claims. What we implicitly decide to hold fixed and the details of how we decide to hold it fixed fully determines what will turn out to be invariant because we thereby countenance nothing as a model within which those fixed expressions receive deviant interpretations. Thus, the model-theoretic account provides no deep insight into what the logical constants are in contexts where it is disputed exactly which those constants are. In contexts in which the enumeration of the constants is not in dispute, it provides no insight into the nature of those constants. Any account of the constants must provide at least one of these insights. Not everyone who adopts the model-theoretic account is a platonist, however. Tarski himself claimed to be a nominalist (despite all of his work which crucially uses set theory), and Sher seems sympathetic to nominalism as well. But it is not enough simply to assert that one is a nominalist: nominalists need to show that they are entitled to use talk of abstract objects without being committed to their existence. This requires a nominalization strategy for mathematics, including, in particular, the set-theoretic models that are used to give the model-theoretic semantics. It is controversial, however, whether nominalization strategies ultimately work (for a critical assessment, see Burgess and Rosen [1997]). Thus, the burden is now with nominalists to establish that it is viable to combine the model-theoretic conception with their view about the ontology of mathematics. Nominalization strategies for mathematics are of two types: one provides recon- structions of mathematical language in order to show that no ontological com- mitment to mathematical objects is forthcoming (this is the reconstructive type of nominalism); the other takes mathematical language literally and shows that there are features in that language that prevent the relevant ontological commitments (this is the non-reconstructive type of nominalism). Reconstructive nominalists include, e.g., Field [1980] and Hellman [1989], and a typical non-reconstructive nominalist is Azzouni [2004]. Let us now consider a model-theoretician who is a reconstructive nominalist. Since models are abstract objects, they need to be re- placed by suitable (i.e., nominalistically acceptable) counterparts. Whatever these Logical Constants: A Modalist Approach 7 counterparts turn out to be, however, the question arises as to whether these models, suitably reinterpreted according to reconstructive guidelines, represent the possibil- ities regarding what follows from what. If they don’t, the framework is ultimately inadequate; if they do, an underlying modal notion is doing the work to ground the representational adequacy of these models. In either case, as with the platonist interpretation, the model theoretician is in trouble. Exactly the same point applies to non-reconstructive nominalism. This type of nominalist will insist that it is per- fectly acceptable to quantify over models. All we need to realize is that quantifiers do not always carry ontological weight. The issue, however, arises as to whether such models suitably represent the possibilities regarding what follows from what, and the same difficulty faced by the reconstructive nominalist emerges. A precise form of the problem is faced by Sher’s account of the logical con- stants. As we saw, Sher’s account requires quantification over models. Thus, some framework is required to formulate the latter. Typically, this framework is offered by a particular set theory, which in turn presupposes some logic, which serves to associate some set of theorems with the chosen axioms. The logic is then invoked in the formulation of the background theory of models (see condition (D)). Let’s call this logic the meta-level logic. The logic in play at the meta-level, then, partially determines which structures count as models relevant to the characterization of the logical constants of the object level logic. The model-theoretic account favors the meta-level logic as specifying, in part, what counts as an object-level logical constant. Different choices of meta-level logics will have different consequences for the range of available models. Some choices will yield a richer domain of mod- els than others, and the richness of the domain will affect the resulting constants. For instance, it may affect the precise characterization of any individual constant. As an illustration, consider the case of a non-classical meta-level logic (such as a paraconsistent logic), and note the models in this instance yield a non-classical characterization of negation. Paraconsistent negation differs in significant ways from classical negation, after all. It is the richness of the available models that allows for the exploration of these non-classical logical constants.8 The failure to attend to the role of the meta-level logic explains, in part, the attraction of the model-theoretic approach to the logical constants. If we ignore the role of that logic, then the image of the model-theoretician’s project is the following faulty image. There is the domain of models and there is the (object-level) logic. We do independent investigations of the domain of models on the one hand and the behavior of the logic on the other. When the two investigations are sufficiently complete, investigators compare notes. The result, the model-theoretician maintains, is that the logical constants are all and only the expressions that behave in the appropriate manner over the domain of models.9 This image satisfies an important constraint on warranted reductive theories: the grounds for warranted beliefs about the reductive base are independent of both the grounds for warranted beliefs about the phenomena to be reduced and for specific reductive claims. Paradigmatic scientific reductions are warranted, in part, by the investigators’ ability to locate and assess the states of the reducing and the reduced phenomena independently of each other. Whether a beaker contains a liquid that 8 NOUSˆ is composed of two parts hydrogen for every part of oxygen is determinable in- dependently of prior knowledge that the beaker contains water. More obviously, warranted belief that it is water and not alcohol in the beaker can be obtained in the absence of any judgment at all about the chemical constitution of the liquid, since for centuries it was done without this knowledge. Suppose a community of researchers does not know that water is H2O. When these conditions of indepen- dent access hold, it is easy to see how they might obtain warrant for the hypothesis that water is H2O. Independent teams of investigators can determine the contents of the beaker and many other samples of liquids, and when they come together to compare results it may be discovered that all and only liquids composed of H2O are water. In the absence of any basis in physical theory to withhold the judgment that water is H2O, the claim is warranted. In reality, we do not require indepen- dent teams, since it is obvious that we have independent standards for determining whether we have water and for determining the chemical composition of liquids. A diagnosis of why this failure of independence is easily unnoticed is in order. Model theory is a branch of mathematics and grasping mathematical truth has tra- ditionally been thought to be grasping necessary truths of rather little controversy. While the grasp of necessary truths of little controversy might apply to truths of elementary arithmetic, the specifics of the axioms of model theory can hardly have the same hallowed status. In model theory the axiomatic cart was created prior to the horse of the grasp and use of the elementary theory of models, the reverse of what explains the status of number theory. For number theory, there was first a long and practical history of working with quantities. An objectual language in which numbers could be treated as objects was then developed. There was building and surveying, the development of a language and practice that facilitated building and surveying. Only then did Plato and others use philosophical arguments that were independent of the postulation of axioms and the inferences to mathematical theorems to convince many that numbers are indeed objects. Only later still was the axiomatic theory formulated. The axioms inherited their plausibility from all that went before. Things went somewhat differently with the “axioms” of model theory. The en- tirety of the theory is a relatively recent development. Without a long history of thinking that we have good reason for thinking that there are models about which we know a fair amount, “axioms” were assumed and the theories developed. Which “axioms”? The axioms that entail that there are enough models and that they are the things that can be all that we want them to be, mathematically speaking. While there may be no reason to think that the axioms were taken to be axioms so that a model-theoretic account of the logical constants could be formulated, background, logical considerations cannot be ignored. The function of the logical constants served as a constraint on the axioms. Suppose that a choice of axioms entailed that, surprisingly, there were no counter-models for an obviously invalid inference form. What would have the verdict been? Shocked wonderment that so many fright- fully intelligent people could have managed not to see that this argument form really is valid? The slightly less shocked wonderment that on this score the realm of the abstract differs from either the actual or the possible structure of the concrete? Or, Logical Constants: A Modalist Approach 9 the knowing sigh that the axiom set was inadequate? The final option seems to be the only one available, really. Of course, there is nothing peculiar to the classical case that is especially devas- tating. Exactly parallel things should be noted for any axiomatic model theory and any logic relied upon to derive the theorems of that theory. That there is a conve- nient match between how the domain of models is said by the theory to be and what an account of both logical consequence and the logical constants requires is just too convenient to produce any warrant for the philosophical claims. Begin with a constructive logic and the resultant domain of models recognized by the attend- ing model theory will contain members not contained in the domain of classical models. Constructive logics treat as invalid some classically valid inference forms, so constructive model theory must contain models to serve as invalidators of the relevant classical inferences. As it turns out, some of these models are incomplete structures (such as situations in which not all the properties of a given object have been specified) whereas all classical models are complete. Similarly, paraconsistent model theory recognizes inconsistent invalidators of some classical inferences.10 The moral of this part of the story is this. The thesis that the logical constants are those that receive invariant interpretations across the entire range of models is plausible to the extent that it is also plausible that two epistemologically in- dependent factors are in play: warranted belief about the existence and character of the relevant models and the interpretation of expressions within those models. The problem is that model theory depends on certain axioms and rules of infer- ence. There is no question of a predominantly empirically developed model theory, since not all models are empirically accessible. The concrete empirical world is too limited to be any real basis for reliable inductive inferences about other concrete models and it is an even poorer basis for inferences about abstract models. There is no question that users of any given logic fix interpretations of certain expressions. The question is whether that fixity is invariance across isomorphic mathematical structures. The case is rigged in favor of that thesis. Fault would be found with any candidate structure in which, despite being isomorphic with another, the rel- evant expression did not behave as the thesis requires. The determination of that fault would be solely on the basis of the deductive apparatus in place, not due to an appreciation of the space of models. The model-theoretic account, then, is not so much a substantive philosophical claim, as it is a translation manual from the language of ordinary logical operations to the language of isomorphic structures.

4. Logical Constants: A Modalist Approach Since the model-theoretic account of the logical constants is caught in an episte- mological bind, recall its provenance and motivation. Traditionally, valid inference was distinguished from merely truth-preserving inference because while both pre- served truth, valid inference did so necessarily. The formal study of reasoning isolated structural features of valid arguments that contribute to their necessarily truth preserving character. These are distinguished from any matters of the con- tent of particular arguments that contribute to their necessarily truth preserving 10 NOUSˆ features. This isolation of the structural sufficient conditions for necessary truth- preservation permitted the systematic study of formally valid reasoning and not merely the case-by-case study. For reasons that need not be detailed here, philosophers, logicians, and mathe- maticians at various historical points became suspicious that at least alethic modal- ities hid confusion and they held that clarity could be secured only by accounting for those modalities in less confused terms. Thus, philosophers were attracted to the model-theoretic account of logical consequence. Quantification over a suffi- ciently large and diverse range of models was to account for the necessarily truth preserving character of logic. Quite naturally, then, that very same framework was used to account for the logical constants. The latter portions of our discussion of this account are suffi- ciently general to put some pressure on both components of the model-theoretic approach to the general study of logic. In light of the troubles facing the model- theoretic approach—in particular, the use of models as representational devices in the characterization of logical constants—a return to a more traditional accounting of the logical constants is warranted. Consequently, the alternative offered here re- lies on no framework of abstracta. Rather, we offer a combination of two doctrines: modalism and logical pluralism. Modalism is the position according which modality is primitive (see Forbes [1985], Shalkowski [1994], and Shalkowski [2004]). Strictly speaking, the modal notion need not be primitive in one respect: when it is contrasted with certain essentialist notions such as what it is to be a (kind of) thing. We treat the modal notion as primitive here, but no part of our case depends on the rejection of serious essentialism (nor does it assume the latter), which takes matters of essence—the identity of an object—to be even more fundamental than possibility and necessity (Fine [1994]). Since the model-theoretic accounts of logical consequence and the logical constants were intended as substitutes for any explicitly modal account, our critique of that framework motivates reverting to an account inspired by the tradition that framework was to replace. Since we treat some modal notion as more primitive than distinctively logical notions, we adopt no meta-theory that quantifies over categorical structures of any kind. We rely on no inferences to distinctively mathematical objects such as sets nor do we rely on any distinctively philosophical referents of predicates such as universals or tropes. Consequently, we rely on distinctively ontological accounts of the nature of neither logical consequence nor the logical constants. Since our account is in no way referential, there is no need even to extend the notion of ‘reference’ to the recherche´ idea that logically constant expressions refer to a mathematical function from truth values to truth values. That ascent into a meta-theory, while convenient, obscures the entire point of developing formal logics, i.e., the general treatment of the structural features of inference that make for the most general or most minimal sufficient conditions for validity. To this end, we approach the constants via completely non-referential, object linguistic schemas, such as those used by the introduction and elimination rules of natural deduction. Since there are different systems of natural deduction that are equivalent with 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. Hacking [1979] has presented a version of Gentzen’s sequent calculus as the key to the nature of logic (see Gentzen [1935] and Prawitz [1965]). The modalist can do likewise. Rather than formulating the proposal in an ontologically loaded meta-theory with an attending meta-logic, the modalist takes the introduction and elimination rules as rules of the object language, expressed via suitable schemas. The modalist maintains that B logically follows from A only if the conjunction of A and the negation of B is impossible.11 Thus, the modalist understands the deduction re- lation (represented by ‘ٛ’) found at the top and the bottom lines of the rules of Gentzen’s sequent calculus in terms of this object language modal notion. As a result, both Gentzen’s structural and operational rules (Gentzen [1935]) can be for- mulated in modalist terms. The former embody basic features of deducibility (such as reflexivity, transitivity etc.), whereas the latter are devised in order to characterize operationally particular logical constants. For instance, the rules for conjunction can be formulated as follows (where ‘A’ and ‘B’ are schematic letters for particular formulas, and ‘’ and ‘’ for groups of such formulas, and ‘ٛ’ is understood in the modalist terms just mentioned):

,A ٛ , B ٛ ٛ A, ٛ B , ,A ∧ B ٛ , A ∧ B ٛ ٛ A ∧ B ,

There is no need, of course, to repeat Gentzen’s calculus here. Our point is simply to note that the calculus offers the modalist a framework for characterizing logical constants without invoking model-theoretical resources. The calculus also allows the modalist to highlight the fact that the primitive modal notion is central to the concept of logical consequence, given that deducibility is understood in modalist terms. Finally, the modalist reading of Gentzen’s calculus also offers a deflationary approach to the logical constants. Whereas the model-theoretic account attempts to specify the kind of object with which a logically constant expression is associated, namely, a mathematical function, the modalist finds the specification suspicious. Not any old set of introduction and elimination rules yield a logical constant. When formulating such rules, the modalist constraint on logical consequence is a requirement for the adequacy of such rules: B logically follows from A only if the conjunction of A and the negation of B is impossible. Any pair of introduction and elimination rules that violates this constraint is unacceptable. In particular, this is how we rule out Prior’s tonk (Prior [1960]). Recall that this is an expression whose introduction rule is like that of or (from A to A tonk B) and whose elimination rule is like that of and (from A tonk B to B). Even if we grant that the tonk rules (formulated Gentzen-style) provide us with inference rules, it does not follow that such rules are valid. Clearly, not all inference rules are. Trot out any number of “rules” from logic textbooks that are exposed as invalid. That we claim an inference pattern invalid is to acknowledge that there are (groups of) rules—even meaning- conferring rules, if one likes—that are invalid. This is not news and not something peculiar to Gentzen or to us. Since tonk permits the derivation of an arbitrary 12 NOUSˆ sentence from a set of premises, it violates the modalist constraint. The inference is then blocked. The tonk case illustrates the form of response to those challenges that try to undermine the extensional adequacy of the modalist account by providing alleged logical constants that have clear introduction and elimination rules. A different challenge emerges when one presses on the comprehensiveness of the modalist account. Once a set of introduction and elimination rules is proposed, how can the modalist establish that all logical constants are thereby formulated? In response we offer two remarks. First, we note that there is no disagreement among logicians about the availability of introduction and elimination rules for all the usual logical constants (conjunction, negation, disjunction, conditional, and so on). That part of the work has already been done in all standard systemizations of deductive inference. Second, suppose that an alleged logical constant is then offered for which introduction or elimination rules were unavailable. All would then, quite naturally, reject the idea that the expression is a logical constant at all. Logicians would think that we are dealing with nothing of that sort. There are no grounds for thinking that our modalist approach is not comprehensive. Something for which no introduction and elimination rules are available rightly would not be considered a logical constant. Thus the modalist account of logical constants has two key features: the require- ment that such constants be introduced via introduction and elimination rules, and that such rules satisfy the modalist constraint on logical consequence. As we saw, the modalist constraint is crucial to avoid overgeneration (as tonk case illustrates), and the introduction and elimination rules are central to prevent undergeneration (as the discussion of comprehensiveness above indicates). It should now be clear that the modalist account starts with a primitive notion of modality (possibility), from which the notion of logical consequence is formulated via the modalist constraint, which insists that B follows from A as long as the conjunction of A and the negation of B is impossible. Using this constraint and suitable introduction and elimination rules, an account of logical constants is then offered. It may be objected that, according to the modalist account, the concept of logical consequence presupposes logical constants, in particular conjunction and negation, since they are used to formulate the modalist constraint. In turn, the concept of logical consequence is used to characterize the logical constants. We seem to face a circle. The modalist account emphasizes the significance of a primitive notion of modal- ity in the formulation of the concept of logical consequence: in fact, that is the crucial feature of the account. The modalist, moreover, does not deny that informal notions have played a fundamental role in our thinking about the world—including informal notions of the logical constants themselves. In fact, we have invoked such informal notions when we used inferential machinery well before the development of formal logic. It is not surprising then that logical constants are crucial to the for- mulation of the concept of logical consequence. (This is true of the model-theoretic account as well: B follows from A if only if in every interpretation in which A is true so is B.) But using logical constants in the formulation of logical consequence is different from formally characterizing the logical constants. In order to do that Logical Constants: A Modalist Approach 13 we need, first, to have already identified the constants in question, and then provide suitable introduction and elimination rules that satisfy the modalist constraint. That is a technical and philosophical exercise, and as such it comes later in the game. Our account possesses the following virtues. First, it is neutral regarding the status of second-order logic, in the sense that it does not preclude that second-order quantification be taken as a logical notion. Of course, this virtue is also shared by those model-theoretic views that have second-order logic as their meta-level logic.12 Second, our account is neutral regarding the hoary dispute between platonists and nominalists. The neutrality emerges in two ways: one negative, one positive. On the negative side, nothing we have said here establishes that there are no set-theoretic models; nothing we have said here establishes their existence either. Perhaps they exist, perhaps they don’t. We simply insist that these models cannot be put to the sort of use for which model-theoreticians need them. We also emphasized that this point carries over even to nominalized versions of the model-theoretic approach, given that whatever replaces the abstract models on the nominalist view will have to play suitable representational role. Our challenge, then, is not to the particular ontological version adopted by the model-theoretician (platonist or nominalist), but to the overall model-theoretic framework. We remain neutral on the ontological front. On the positive side, both platonists and nominalists can adopt the Gentzen- style approach to logical constants we favor. They are likely to offer different interpretations to the proposal, but proponents of either view can invoke its central features. This is neutrality enough. Third, our account permits our meta-theory to introduce no new ontology, which would become, effectively, the substance of logic. The meta-theory within which the introduction and elimination rules are formulated is merely our original theory in which we reason, supplemented with the resources to schematize the patterns of inference we legitimize. The second and third virtues deserve some comment. We begin with models. That ‘model’ is a technical term when used in the philosophy of logic obscures the fact that models are models. They are not the genuine article; they are not the subject matter. They are the illustrations, the exhibits that illuminate the mind regarding the phenomenon of interest. They do so by making salient poorly understood features of that phenomenon. In science we are quite familiar with how models function. Bohr’s model of the atom, the twisted ladder of DNA, and—much more mundane— a scale model of a factory, and a cross-sectional model of an internal combustion engine each brings to the foreground important features that aid in understanding the basics—and perhaps even the essentials—of the item in question. Especially in the latter two cases, there is little temptation to mistake the model for the genuine article and there are well-known ways in which physical models must introduce distortions of some features to remain faithful to other features (see van Fraassen [2008]). From the modalist point of view, model-theoretic mathematical models are mod- els in the way the above models are. Granting the existence of one or more languages, sets, relations and functions defined over the languages and the sets (perhaps per impossibile, if need be), mathematical models can be used to model interesting 14 NOUSˆ features of many different things, logical consequence and the logical constants among them. Those models illustrate the difference between logical consequence narrowly construed and analytic consequence, for instance, with the latter depen- dent upon fixing the interpretation(s) of one or more expressions not usually thought to be logical. At most, thinking in quantificational terms over abstract objects is merely a way for our minds to think in objectual ways about logical mat- ters. Noting the pragmatic advantages of thinking in objectual terms does nothing, however, to show that what is modeled is model-theoretic in character, any more than an electron is small colored ball held in place by a colored stick around some tightly-grouped larger colored ball(s). The modalist may—and should—resist the temptation to ascend into the meta- language when treating logical consequence. Consider:

(1) All philosophers are mortal. (2) Socrates is a philosopher.

(3) Socrates is mortal.

What entails what? When introducing students to validity, we may well say that (1) and (2) entail (3). What are designated by the numerals, though? One might say that propositions are designated, but ‘propositions’ is ambiguous. It may extend over abstract objects or it might simply cover what is expressed by declarative sentences when they are used but not mentioned (assuming that there is a reasonable way of characterizing the content of declarative sentences without invoking abstract ob- jects). Having eschewed the model-theoretic account of the constants and returned to modalist tradition, we treat ‘necessarily’ as object-linguistic in its primary usage. Doing this permits us to treat entailment claims as object language claims, thus making arguments primarily object-linguistic in nature, as they indeed are. As contrary as this is to typical 20th and 21st century treatments of these matters it allows the focus of logic to be where it should be. When we are not giving reasoned evidence for some disputed claim, the numbered items would, in isolation, be used to report, but not about abstract objects or about metalinguistic facts about actual or ideal languages. Making sure that ‘fact’ is no philosophical term of art but simply a term that highlights that we speak of what are typically extra-linguistic worldly affairs, the inset argument above reports that some facts entail others. Some facts about philosophers and Socrates entail another fact about Socrates. When our interests are clearly formal and not about the particulars of that inference we produce:

(4) All P are M. (5) s is a P.

(6) s is M.

The formal version is no more about propositions or sentences than was the original. In fact, it is not about anything at all; that is precisely the point of the formal analysis of validity—structure and not content. It is, however, a useful Logical Constants: A Modalist Approach 15 representation of a phenomenon interesting to logicians: any argument with the form of the first is valid. It just does not follow from that universal claim about arguments, though, that validity is a metalinguistic phenomenon any more than the original argument is about metalinguistic phenomena simply because we must use language to provide the reasons for (3) contained in (1) and (2). Though mathematical models are useful in illustrating the validity of the form, validity is no more a model-theoretic phenomenon than it is a graphic phenomenon because we can use Venn diagrams to illustrate the validity of the form. That mathematical models are more versatile than Venn diagrams shows only that the former are better tools for modeling validity than are the latter. Confusing the models with the phenomenon to be modeled leads to forgetting the typical point of providing arguments when they are not themselves the object of study: giving reasons. Say that a particular claim is disputed. Grounds for the claim are requested. Reasons are given. Typically, parties to the dispute differ over the state of the world. One providing reasons for the disputed claim offers other claims about the world in evidence for the disputed claim. If the reasoner is fortunate enough to produce a valid argument for the conclusion that all parties see to be valid, then so long as none of them puts on a philosophical logician’s hat, each recognizes that it cannot be that the states of the world reported in the premises obtain without the state reported in the conclusion also obtaining. The concern is worldly from start to finish. How odd, then, that when academicians turn their attention to validity, the concern becomes metalinguistic. The mistake is the failure to recognize that the talk of ‘form’, ‘sentence’, ‘proposition’, ‘argument’ and the like are merely schematic mechanisms that permit us to turn our attention from Socrates and mortality to what can be recognized when there is similar concern and inference about Aristotle being the teacher of Alexander. Metalinguistic formulations are merely the means of expressing generality without the impossible examination of infinitely many instances. That means does not change the fact that in any particular instance it is the impossibility of some worldly affairs without another that makes that instance valid, which in turn provides part of the reasoned grounding for the conclusion of that instance. That the modalist introduces no new ontology gives the appearance of explana- tory advantage to the model-theoretic approach for both logical consequence and the logical constants. For consequence, the modal component is explained on the model-theoretic account as generality over a categorical domain; for constants, their uniqueness is explained as invariant interpretation over the same domain. In each case, a puzzling phenomenon or vocabulary is explained in apparently non-circular terms. In contrast, on the modalist alternative, the Gentzen approach is quite natural. There is no logically relevant ontology in terms of which to account for the nature of logical connectives and quantifiers and those expressions that play roles in logical consequence. We treat this notion as being irreducibly modal. If some modality is primitive, then it is no surprise that the most appropriate alternative to the model-theoretic characterization of the constants is a characterization in terms of introduction and elimination rules. Regardless of one’s primitives, whether they 16 NOUSˆ are modal or not, this is the way to deal with primitives, not further explanation. Anyone who fails to understand how ‘&’ functions in the logical symbolism, fails to know when ‘&’ statements can be inferred or what can be inferred from them. Linking ‘&’ with ordinary ‘and’ merely links the formal expression with precisely the knowledge of when and what one can infer validly from ordinary ‘and’ statements. Ending our explanation with natural deduction rules is, then, perfectly natural and no deficiency when compared to the model-theoretic account. Section 3 showed why the apparent explanatory advantage of the model-theoretic account was merely apparent. That the two components of the modalist account fit together naturally and in ways that one should expect when we are dealing with primitives shows that the alternative suffers no deficiency regarding appropriate explanation. Our advocacy of introduction and elimination rules as the means of character- izing logical constants for a given logic commits us to no formalist or syntactic understanding of the nature of consequence or constants. To think so is to import ontological explanations into the meta-theory in a way that clearly misrepresents the nature of inference and its formal study. We use the syntax of our metalanguage to show only how the logical symbols are used, normatively speaking, in inference. Valid inference is not thereby a syntactic phenomenon. Languages are constructed for communication. Both syntax and semantics play roles in making languages bet- ter or worse tools for communication. If syntax and valid inference are correlated in interesting ways, the syntax is interesting only insofar as it tracks inference, not the other way around. The other way around is what a distinctively syntactic theory of logical consequence or constants would require. Ours is no such theory.13 Why, according to the modalist, are the standard constants the genuinely logical constants? Since the context is the formal study of reasoning, the question reduces to what makes for form and what makes for content. If everything varies, then there are no constant forms to investigate; if nothing varies then there are constant forms, but there is no variation, so distinct arguments cannot share form. The modalist need not find a metatheoretic ontology and then see which expressions have invariant interpretations over that ontology. The groundwork is determining under what conditions we are inclined to judge that we speak the same language. The model-theoretic approach must, in the end, treat it as mysterious that ‘∀’ and ‘&’ have invariant interpretations and ‘is a philosopher’ and ‘is the teacher of Alexander’ do not. The modalist does not. Typical predicates and names “have variable interpretations” because typically the extensions of those predicates are not essential to them and names could name things other than what they do and named things could have different names. We are at a loss, though, to think that we “mean the same thing” if we used the standard logical vocabulary differently. ‘All’ just means all, and ‘and’ must mean and. No more than this is required to make sense of the formal study of first-order predicate logic. Extensions of that logic into modal, deontic or temporal logics (among others) are extensions in which we recognize that we are not speaking the same language if ‘possibly’ fails to mean possibly and ‘obligatory’ fails to mean obligatory. When studying the unextended logic we are concerned about “the logic of” declaration and/or predication. How much validity will declaration and predication secure on their Logical Constants: A Modalist Approach 17 own? Interesting question, but there is no reason to think that only they secure validity. One countenances the extensions of first-order predicate logic only to the extent that one thinks that modalizing, moralizing, or temporalizing also play roles in securing validity. Combining what modalizing secures with what both declaring and predicating secures yields quantified modal logic. Obviously and analogously for the others.14 An important doctrine shaping our view concerns the nature of logic. Logical pluralism is the doctrine according to which logics are domain-dependent, and that there is more than one adequate logic (see da Costa [1997], Bueno [2002], and Bueno and Shalkowski [2009]).15 In other words, there is no One True Logic, but a plurality of logics, each adequate for certain domains, and inadequate for others. Classical logic is, for the most part, perfectly adequate to deal with complete, consistent domains—that is, domains in which the objects under consideration are fully specified and in which there are no inconsistencies. However, if we are interested in reasoning about incomplete objects, whose properties are not fully specified, such as certain fictional characters or some mathematical constructions, a constructive logic is much more adequate. For example, suppose that mathematical objects do not exist independently of us, but are the result of certain mathematical constructions that we perform. Accordingly, a mathematical object is constructed in stages, and at each stage, the object will have only those properties that follow from those that have been explicitly specified up to that point. It is then perfectly possible that at a certain stage it is not determinate whether a mathematical object has or lacks a given property, since the relevant fact of the matter that settles the issue— namely, the explicit specification of the property in question or of a property that requires the one in question—is not yet in place. In this case, excluded middle fails. For those unfamiliar with constructive mathematics, the development of works of fiction serves the same purpose. Only as the novel is written do some facts regarding the characters and the course of events become established. Before Doyle specified that Holmes lived at 221B Baker Street, there was no fact of the matter—not even a fictional fact—about the precise location of his residence. Intuitively, there was, yet, no correct answer to the question “What is Holmes’ address?” It was not a truth that he either did or did not live at 221B Baker Street. The incompleteness of novels is reason not to reason classically about them. Sometimes, though, we must deal not with incompleteness but with inconsis- tency. Anything we can stipulate, we can stipulate inconsistently; anything we can legislate, we can legislate inconsistently. A particular legal framework might permit a legislative body to make legally mandatory the performance of an action, A. When the mists of time have obscured legislators’ memories of decisions by their predecessors, later legislation might make some action, B, mandatory and it may not be possible to perform both A and B. The law requires citizens to engage in inconsistent behavior. The propositional content of the whole body of law now entails that A be both done and not done. It would be very perverse legislators who would note this legal state of affairs, recall their elementary logical trainings and conclude—because everything follows from a contradiction—that anything and everything is now legal (and also illegal).16 18 NOUSˆ

Alternatively, suppose we are dealing with potentially inconsistent claims about certain objects, such as those found in huge databases. If we reason about these claims using classical logic, and the database is indeed inconsistent, then given the explosive nature of classical logic, every sentence in our language will follow logically. Even if no one actually derives an arbitrary statement from such an inconsistent database, the fact remains that everything validly follows from it. As a result, the reliability of any inference in this context is now doubtful. For arbitrary P, P follows from the information in the database, but we know that, in reality, for some P validly derivable from that information, P is false. If we are, however, interested in reasoning about such potentially inconsistent databases, e.g., in order to find out which bits of them should be excluded, adopting a paraconsistent logic is a good alternative. We can then reason about inconsistent claims without triviality; that is, without every claim logically following. It turns out that there are infinitely many paraconsistent logics (see da Costa, Krause, and Bueno [2007]), and virtually any one of them will be perfectly suited to the task. This illustrates how different domains may require different logics, and how more than one logic may be adequate for the same domain. We combine here modalism and logical pluralism. The modalist approach treats logical consequence as an essentially modal notion. Given the considerations above, it is not difficult to see why. The model-theoretic framework must characterize all and only valid arguments as valid. This presupposes that the models used in the characterization represent properly all the possibilities, so that no invalid arguments are characterized as valid, and all valid arguments are characterized as such. But, the adequacy of the model-theoretic framework depends on the logic that is adopted in our model theory. If we change the underlying logic, we change the nature of the models that serve the standard meta-logical functions ascribed to models. Hence, we change the extension of the logical consequence relation. This is part of the logical pluralist picture. So, we start with modality and get logical consequence in modal terms. Depend- ing on what is possible or impossible for a given domain, different specifications of the relation of logical consequence emerge. If all possibilities are complete and consistent, we obtain classical logic. If some possibilities are consistent and in- complete, we obtain constructive logic. If some possibilities are inconsistent and complete, we obtain paraconsistent logic. On the modalist picture, modality has priority over logic. In particular, our use of ‘consistent’ here does not assume a log- ical characterization of this notion. It is formulated via possibility and perhaps our understanding of conjunction as given by the introduction and elimination rules. Given the pluralist component, different logics are appropriate for reasoning about different domains. Once the relevant character of the objects reasoned about is known, an appropriate (family of) logic(s) is available and we can then use variations of Sher’s model-theoretic proposal for heuristic purposes. The form of her proposal is now restricted to some domain under consideration and only the models for the meta-logic in question are considered. Thus, depending on the meta-logic adopted, we obtain different constants as being logical or constants characterized in different ways. For example, suppose we have a domain codified Logical Constants: A Modalist Approach 19 by (first-order) classically valid reasoning. Thus, (first-order) classical logic is the meta-level logic. If we then invoke the model-theoretic account of logical constants, we will have as the logical constants—as expected—those from classical (first-order) logic, among others perhaps. Suppose, however, that we have a domain codified by constructive or by paracon- sistent reasoning. This means that the general features of what is possible for objects in that domain is now different from the classical scenario, since our concern would include situations that are consistent and incomplete, or inconsistent and complete, respectively. As a result, different characterizations of logical constants will emerge. In the case of constructive logics, we need to introduce “models” in which incom- plete situations are accounted for, and thus a distinctively constructive negation is characterized as logical. In the case of paraconsistent logics, the “models” for the logic will now include suitable infinite matrices, so that a non-truth-functional negation is characterized as a logical constant. Our favored approach, however, is to use the Gentzen rules, which also mesh very well with our logical pluralism. The introduction and elimination rules are ways of formalizing certain inferences and the corresponding constants, and they can be used to characterize a variety of logical constants. Once we move to a logical pluralist setting, the request to provide a unified account of all (and only) the logical constants is somewhat misguided. There is a plurality of logics and logical constants, and they need not be characterized in exactly the same way. The Gentzen framework provides all of the generality we need. More significant differences between the modalist account of logical constants and the model-theoretic account are now apparent. First, as opposed to the clearly monistic tendency of the model-theoretic view, we offer a pluralist alternative that makes better sense of the pluralism of logics and, hence, of the plurality of logical constants. The nature of the appropriate logical constants depends on the do- main under consideration. For different domains, different logics are appropriate, and hence, the distinctively logical vocabulary behaves differently. For the model- theoretician to obtain pluralism, they must, strictly speaking, give up the invariance constraint. To obtain different logics different groups of models must be counte- nanced. To restrict invariant interpretations of a constant to only some and not all models is to admit tacitly that the interpretation is not really invariant over all models, but over only some. If the specific restriction is not made on the basis of what is or is not possible for a given domain of ordinary objects, such as fictions or databases, it is hard to see why particular restrictions are made and not others. To wed pluralism with the model-theoretic account is to give up on the alleged advantage of trading modality for quantification over a given domain. Second, we do not reduce logical constants to a fixed class of (set-theoretic) models. Rather, we emphasize the role played by what is possible for different kinds of objects for the domains in question, and how different ranges of possibilities are reflected in the constants of an appropriate logic. Third, given that the modalist view does not take (set-theoretic) models as ulti- mate representational devices for what is possible, the challenge raised above to the model-theoretic framework does not apply to the modalist alternative. Possibility 20 NOUSˆ is more fundamental than models, and when appropriate (e.g., when we are deal- ing with consistent and complete domains) the usual talk of models used by the model-theoretic account can be employed, not as philosophically informative but as illustrative. Ultimately, there is no harm in using models as part of soundness and completeness proofs, since the models model the modality appropriate for the respective domains for which a given logic is appropriate. There is nothing logically revisionary in the modalist component of our project. The revision is only that the logical trousers are modal trousers, not model-theoretic. Finally, logic is understood as a local, domain-oriented matter. This fact pre- vents logical pluralism from collapsing into logical nihilism, the consequence of combining the model-theoretic account of logic with logical pluralism. If quantifi- cation over models is to secure the effect of necessary truth preservation for valid inference, the domain of models appropriate for, say, intuitionistic logics contain models that invalidate some classical inferences. Similarly, models appropriate for paraconsistent logics invalidate some intuitionistic inferences. Once the floodgates of models is opened, there are so many models, given progressively permissive logics such as quantum and non-adjunctive logics, there can hardly be any recognizable systematization of inference left. Hence, the abandonment of the quantificational approach eliminates the philosophical difficulties for the Tarskian tradition regard- ing the logical constants. Moreover, avoiding the quantificational approach and making sense of logical pluralism enables us in any given context to use the logic that permits us to extend our knowledge via inference as far as possible, with- out forgoing inferences about consistent and complete contexts simply because there are incomplete contexts within which some of those inferences would be unwarranted. The pluralist strand of our position forces a rejection of one common theme in discussions of the constants, i.e., that there is a single privileged class of expressions that are given constant interpretations and the search is on for the special character of some connectives and quantifiers. If logics are tools for reasoning and if there are distinct logics some of which sanction a given form of argument and some of which do not, then, as the case of negation shows, there is no single interpretation of negation. That component of the common theme falls most obviously. Non- adjunctive logics show that not even the stock of expressions treated as logical within various logics remains stable. The other component falls as well. If the appropriate logic to use is a context-sensitive matter, then it is unsurprising that the number and nature of the constants is likewise context-sensitive.

5. Conclusion There are, of course, several important issues that a full-fledged account of logical consequence needs to address: What role does formality play in logical inference? What is the source of normativity of logical consequence? How should both these features (formality and normativity) be explained? Model-theoretic accounts of logical consequence have addressed these issues (see, e.g., Sher [1991], [1999] and [2008]). The modalist also has an account of these issues (for considerations on Logical Constants: A Modalist Approach 21 some of them, see Bueno and Shalkowski [2009]). This is not the place, however, to provide such an account. We are here concerned with the particular issue of the status of logical constants. Our goal here is simply to motivate modalism as an account of these constants, thus paving the way for the development of a full- fledged modalist account of logical consequence. The latter task will be pursued in due course. Our defense of the modal characterization of the logical constants may strike some as incomplete in another respect. A comprehensive theory of argumentation must also address our epistemic considerations in inference. A major point of regimenting inference is to systematize safe, i.e., valid, inference. In this sense, logic permits us to extend our knowledge. An accounting of such safe inferences forces us into the domain of the epistemic, on which we have been silent. The separation between the modal and the epistemic is not great, though. Since the logical constants encode a modal character, grasping that character is what reassures one that some particular argument is either valid or invalid. Failing to grasp that modal character—either because of ignorance of the nature of valid inference and, hence, the modal character of the constants, or because of ignorance of the validity of this particular inference—warrants the lack of assurance of the safety of the inference in question. On the modalist picture, the modal character of inference (including, of course, the logical constants) grounds the latter’s validity. In turn, grasping the modal character of the constants is the basis for the assurance that the inferences in question are valid. Thus, satisfying any epistemic constraints on inference is part of an account of the modal character of the logical constants. However, spelling out the details of these epistemic constraints is the task for another occasion.

Notes 1 We would like to thank an anonymous referee for the extremely detailed and helpful comments. The paper has improved substantially as a result. 2 It is duly noted—and set aside—that the view of Wittgenstein as articulated in the Tractatus is that, strictly speaking, there are no purely logical claims. 3 Note that there is a shift between (A) and (B). In (A), a logical constant is an expression, an item of language. In (B), however, we have the constant identified with its extension, which typically is not an expression. 4 If coerced to admit that abstract reality contains too many invalidators, the model-theoretic account would require qualification. The standard clauses would apply only when quantifying over models of the right sort. See Bueno and Shalkowski [2009] for an exploration of this issue and some of the implications for orthodox philosophers of logic. Being coerced to admit that there are too few invalidators would make the model-theoretic project untenable, regardless of qualification. 5 For those who, following Feferman [1999], distinguish between structural and foundational axioms, our concern is with foundational axioms. 6 It is proper to waive concerns of elegance in this context, since the issue is whether the claims in set theory state truths of platonistic significance. For the record, we think that pragmatic concerns are not indicators of truth, but this is not the place to pursue this point. 7 That the assumption is not merely that things are not in inconsistent ways is shown by the universally-agreed incorporation of accepted rules of inference to counterfactual reasoning. The weaker assumption renders the rules undefined over counterfactual cases. 22 NOUSˆ

8 To clarify our point, note that we are not claiming that there is circularity in using at the meta-level logic precisely the logical constants that the invariance condition of the model-theoretic framework will eventually sanction as logical at the object level. To make such a point would require establishing that for every meta-level logic, the only logical constants that are sanctioned are those of the meta-level logic itself. We haven’t established, nor even attempted to establish, such a claim. Our point, rather, is to raise a suspicion about the choice of the meta-level logic, and how that choice impacts on which logical constants are sanctioned by the invariance condition. It seems to us that this is troublesome enough. (Some model-theoreticians, such as Sher [2008], explicitly reject vicious circularity, but allow some form of circularity as part of their holistic framework. Nothing that we say in this paper bears on that issue.) It may be objected that the invariance condition does not yield only those logical constants that are sanctioned by the meta-level logic. After all, quantifiers such as “few”, “most”, “finitely many”, and “uncountably many” are all yielded by that condition (Sher [1999], p. 222). We do not deny that these quantifiers emerge from the invariance account. Our concern, however, is whether these quantifiers should be considered to be logical constants. “Uncountably many” seems to rely fundamentally on the mathematical notion of the uncountable. And despite much progress on logicist reconstructions of analysis, it is still a contentious issue whether the uncountable can be formulated via logic alone (or even logic plus definitions). Similarly, the notion of finiteness seems to be fundamentally a mathematical rather than a logical notion. And a related point can be made about “few” and “most”. Sher can, of course, insist that since all of these quantifiers are yielded by the invariance condition, nothing more is needed to characterize them as logical constants. We wonder, though, whether this is the proper assessment of their status. 9 An additional difficulty surfaces at this point. As we saw above, on Sher’s formulation, a logical constant is defined over all models for the logic under consideration (as stated in condition (D)). From our perspective, this is a form of weaseling. A logical constant is defined over these models because they are models for the logic(s) in which the logical constant figures. But, why is it defined over those models? Two options emerge here: either those are all the models there are, or the other models are inappropriate for assessing this logic. However, neither option is correct. The first conflicts with pluralism about logic (a point to which we return below); the second conflicts with the idea of invariance over all models. In fact, this second option is a form of a “turn a blind eye” strategy: a logical constant is defined over all models—except when it isn’t. 10 It may be argued that our criticism of the model-theoretic approach does not recognize that model theorists can justify their axioms by means other than by examining their consequences. For example, set theorists often choose certain axioms because they provide a more unified account of the set-theoretic universe. We do not deny that some choices of axioms rely on theoretical virtues, such as unification. But note that to establish that such a virtue applies we need to examine the consequences of the relevant axioms (in order to determine that the system in question is indeed suitably unified or more unified than its rivals). Hence, our emphasis on the consequences. 11 We assume natural constraints on logical consequence such as formality (that B bears some structural relations to A) and normativity (that reasoning fallaciously violates some canon of seemly rational behavior), since they have no bearing on the matters that separate the modalist from the model-theoretician. 12 Some may reject second-order logic because its quantifiers range over properties rather than objects. But it is perfectly possible to interpret second-order quantifiers as devices of plural quantification without taking them to range over properties (see Boolos [1998]). Quine famously maintained that second-order logic is not really logic, but “set theory in sheep’s clothing” (Quine [1970], p. 66). Since set theory is stronger than second-order logic, Quineans who are already committed to set theory might reject second-order quantification in favor of set membership. Instead of writing ‘∃XXb’, the logician should write ‘∃α b∈α’. As Boolos has stressed ([1975], p. 40), however, this is neither validity-preserving nor implication-preserving. Although ‘∃X ∀xXx’isvalid,‘∃α ∀xx∈α’ is not. Moreover, although ‘x = z’ follows from ‘∀Y (Yx ↔ Yz)’ by logic alone, it does not follow from ‘∀α (x∈α ↔ z∈α)’ by logic alone. Some set theory is required. Quine’s recommendation turns a valid second-order claim into an invalid set-theoretic claim, and it fails to preserve the logical consequences from some second-order statements. Thus, second-order quantification and set-theoretical membership are, indeed, quite distinct. Logical Constants: A Modalist Approach 23

The Quineans’ rejection of second-order logic as the meta-level logic is unsupported by any reasonable interpretation and application of Quine’s slogan about second-order logic. 13 These remarks should make it clear that, unlike the model-theoretic account, ours does not reify logical constants, identifying the kind of object such constants are. In particular, we do not reify logical constants as syntactic constructions exemplified in introduction and elimination rules. Were we to do that, our account would just be a syntactic view in the end. Thus, Tarski’s [1935] complaint that syntactic views of logical consequence cannot accommodate logics that are incomplete does not apply to the modalist view we favor. 14 At the moment we are interested in how this phenomenon or that secures validity. We are interested in what follows from what. The necessities of the situation are revealed in a developed logic. Those necessities are implicit in the rules of the respective games. Without the implicit modal significance, the rules would cease to be interesting as rules of inference. 15 We adopt here a formulation of logical pluralism in terms of domains rather than cases as Beall and Restall do (Beall and Restall [2006]). On the latter version of pluralism, an argument is valid if, and only if, in every case in which the premises are true, the conclusion is true as well. However, given that cases can range over things as diverse as consistent and complete structures, inconsistent and complete situations, and consistent and incomplete situations, it is unclear that anything could satisfy the concept of validity. Note the quantification over all cases in the formulation of validity. This problem is not faced by the formulation of logical pluralism in terms of domains. After all, each domain will have different logics adequate for it. For instance, an inconsistent domain will have infinitely many paraconsistent logics adequate for it, such as the family of paraconsistent logics C (see da Costa, Krause, and Bueno [2007]). A consistent domain will also have infinitely many logics adequate for it: different formulations of classical logic (predicate and propositional, first- and second-order) as well as the family of paraconsistent logics C, which yield exactly the same valid inferences in a consistent domain as classical logics. 16 At first glance, it appears as though this is not first-class inconsistency. A is to be done and A is not to be done. This is a problem of “ought” and not “is”. Fair dinkum inconsistency is rooted in an inconsistent “is”. Since we are concerned with the formal study of logic here, this distinction is irrelevant. A is to be done and so is B. Doing B prevents doing A. For any action, if doing it prevents the doing of another, then that action and the other are inconsistent with each other. For any action, if it is to be done, then all actions inconsistent with are not to be done. Thus, A is not to be done. For any action, if it is not to be done, then it is not the case that it is to be done. Thus, it is not the case that A is to be done. Thus, A is to be done and it is not the case that A is to be done. Proper inconsistency.

References Azzouni, J. [2004]: Deflating Existential Consequence. New York: Oxford University Press. Beall, JC, and Restall, G. [2006]: Logical Pluralism. Oxford: Clarendon Press. Boolos, G. [1975]: “On Second-Order Logic”, Journal of Philosophy 72, pp. 509–527. (Reprinted in Boolos [1998], pp. 37–53.) Boolos, G. [1998]: Logic, Logic, and Logic. Cambridge, MA: Harvard University Press. Bueno, O. [2002]: “Can a Paraconsistent Theorist be a Logical Monist?”, in Carnielli, Coniglio, and D’Ottaviano (eds.) [2002], pp. 535–552. Bueno, O., and Shalkowski [2009]: “Modalism and Logical Pluralism”, Mind 118, pp. 295–321. Burgess, J., and Rosen, G. [1997]: A Subject With No Object: Strategies for Nominalistic Interpretation of Mathematics. Oxford: Clarendon Press. Carnielli, W, Coniglio, M., and D’Ottaviano, I. (eds.) [2002]: Paraconsistency: The Logical Way to the Inconsistent. New York: Marcel Dekker. da Costa, N.C.A. [1997]: Logiques classiques et non classiques. Paris: Masson. da Costa, N.C.A., Krause, D., and Bueno, O. [2007]: “Paraconsistent Logics and Paraconsistency”, in Jacquette (ed.) [2007], pp. 791–911. 24 NOUSˆ

Feferman, S. [1999]: “Does Mathematics Need New Axioms?”, American Mathematical Monthly 106, pp. 99–111. Field, H. [1980]: Science without Numbers. Princeton, NJ: Princeton University Press. Fine, K. [1994]: “Essence and Modality”, Philosophical Perspectives 8, pp. 1–16. Forbes, G. [1985]: The Metaphysics of Modality. Oxford: Clarendon Press. Gentzen, G. [1935]: “Investigations into Logical Deduction” in The Collected Papers of Gerhard Gentzen, Amsterdam: North Holland, 1969, pp. 68–131. Hacking, I. [1979]: “What is Logic?”, Journal of Philosophy 76, pp. 285–319. Hellman, G. [1989]: Mathematics without Numbers. Oxford: Clarendon Press. Jacquette, D. (ed.) [2007]: Philosophy of Logic. Amsterdam: North—Holland. Prawitz, D. [1965]: Natural Deduction. Stockolm: Almqvist and Wiksell. Prior, A. [1960]: “The Runabout Inference-Ticket”, Analysis 21, pp. 38–39. Quine, W.V. [1970]: Philosophy of Logic. Englewood Cliffs: Prentice Hall. Shalkowski, S. [1994]: “The Ontological Ground of the Alethic Modality”, Philosophical Review 103, pp. 669–688. Shalkowski, S. [2004]: “Logic and Absolute Necessity”, Journal of Philosophy 101, pp. 1–28. Sher, G. [1991]: The Bounds of Logic: A Generalized Viewpoint. Cambridge, MA: MIT Press. Sher, G. [1999]: “Is Logic a Theory of the Obvious?”, European Review of Philosophy 4, pp. 207–238. Sher, G. [2003]: “A Characterization of Logical Constants Is Possible”, Theoria 18, pp. 189–198. Sher, G. [2008]: “Tarski’s Thesis”, in D. Patterson (ed.), New Essays on Tarski and Philosophy. Oxford: Oxford University Press. Priest, G. [2006]: In Contradiction. (Second edition.) Oxford: Oxford University Press. Tarski, A. [1935]: “On the Concept of Logical Consequence”, in Logic, Semantics and Metamathematics, Indianapolis, Hackett, 1983, pp. 409–420. Tarski, A. [1966]: “What Are Logical Notions?”, History and Philosophy of Logic 7, pp. 143–154. van Fraassen, B. [2008]: Scientific Representation. Oxford: Clarendon Press.