Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logicalconstants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share (...) a common syntactic and semantic structure, because they differ in the placement of the logicalconstants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logicalconstants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logicalconstants” of a language from its nonlogical expressions. (shrink)
There is as yet no settled consensus as to what makes a term a logical constant or even as to which terms should be recognized as having this status. This essay sets out and defends a rationale for identifying logicalconstants. I argue for a two-tiered approach to logical theory. First, a secure, core logical theory recognizes only a minimal set of constants needed for deductively systematizing scientific theories. Second, there are extended logical (...) theories whose objectives are to systematize various pre-theoretic, modal intuitions. The latter theories may recognize a variety of additional constants as needed in order to formalize a given set of intuitions. (shrink)
A possibility of defining logicalconstants within abstract logical frameworks is discussed, in relation to abstract definition of logical consequence. We propose using duals as a general method of applying the idea of invariance under replacement as a criterion for logicality.
The problem of logicalconstants consists in finding a principled way to draw the line between those expressions of a language that are logical and those that are not. The criterion of invariance under permutation, attributed to Tarski, is probably the most common answer to this problem, at least within the semantic tradition. However, as the received view on the matter, it has recently come under heavy attack. Does this mean that the criterion should be amended, or (...) maybe even that it should be abandoned? I shall review the different types of objections that have been made against invariance as a logicality criterion and distinguish between three kinds of objections, skeptical worries against the very relevance of such a demarcation, intensional warnings against the level at which the criterion operates, and extensional quarrels against the results that are obtained. I shall argue that the first two kinds of objections are at least partly misguided and that the third kind of objection calls for amendment rather than abandonment. (shrink)
There is as yet no settled consensus as to what makes a term a logical constant or even as to which terms should be recognized as having this status. This essay sets out and defends a rationale for identifying logicalconstants. I argue for a two-tiered approach to logical theory. First, a secure, core logical theory recognizes only a minimal set of constants needed for deductively systematizing scientific theories. Second, there are extended logical (...) theories whose objectives are to systematize various pre-theoretic, modal intuitions. The latter theories may recognize a variety of additional constants as needed in order to formalize a given set of intuitions. (shrink)
There have been several different and even opposed conceptions of the problem of logicalconstants, i.e. of the requirements that a good theory of logicalconstants ought to satisfy. This paper is in the first place a survey of these conceptions and a critique of the theories they have given rise to. A second aim of the paper is to sketch some ideas about what a good theory would look like. A third aim is to draw (...) from these ideas and from the preceding survey the conclusion that most conceptions of the problem of logicalconstants involve requirements of a philosophically demanding nature which are probably not satisfiable by any minimally adequate theory. (shrink)
Work on the nature and scope of formal logic has focused unduly on the distinction between logical and extra-logical vocabulary; which argument forms a logical theory countenances depends not only on its stock of logical terms, but also on its range of grammatical categories and modes of composition. Furthermore, there is a sense in which logical terms are unnecessary. Alexandra Zinke has recently pointed out that propositional logic can be done without logical terms. By (...) defining a logical-term-free language with the full expressive power of first-order logic with identity, I show that this is true of logic more generally. Furthermore, having, in a logical theory, non-trivial valid forms that do not involve logical terms is not merely a technical possibility. As the case of adverbs shows, issues about the range of argument forms logic should countenance can quite naturally arise in such a way that they do not turn on whether we countenance certain terms as logical. (shrink)
The paper argues that a philosophically informative and mathematically precise characterization is possible by (i) describing a particular proposal for such a characterization, (ii) showing that certain criticisms of this proposal are incorrect, and (iii) discussing the general issue of what a characterization of logicalconstants aims at achieving.
Many philosophers claim that understanding a logical constant (e.g. ‘if, then’) fundamentally consists in having dispositions to infer according to the logical rules (e.g. Modus Ponens) that fix its meaning. This paper argues that such dispositionalist accounts give us the wrong picture of what understanding a logical constant consists in. The objection here is that they give an account of understanding a logical constant which is inconsistent with what seem to be adequate manifestations of such understanding. (...) I then outline an alternative account according to which understanding a logical constant is not to be understood dispositionally, but propositionally. I argue that this account is not inconsistent with intuitively correct manifestations of understanding the logicalconstants. (shrink)
There is as yet no settled consensus as to what makes a term a logical constant or even as to which terms should be recognized as having this status. This essay sets out and defends a rationale for identifying logicalconstants. I argue for a two-tiered approach to logical theory. First, a secure, core logical theory recognizes only a minimal set of constants needed for deductively systematizing scientific theories. Second, there are extended logical (...) theories whose objectives are to systematize various pre-theoretic, modal intuitions. The latter theories may recognize a variety of additional constants as needed in order to formalize a given set of intuitions. (shrink)
Fully adequate answers to these questions are best provided in a comprehensive philosophy of logic. Within shorter compass, it is nevertheless possible to be guided by some conditions that are necessary to adequate answers. These will be results of the analysis of propositions and statements. They are necessary, since no answers to the questions about the constants will be acceptable if, for example, it follows from the answers that propositions or statements cannot be unities, or that propositions or statements (...) cannot be true or false. (shrink)
The paper argues that a philosophically informative and mathematically precise characterization is possible by describing a particular proposal for such a characterization, showing that certain criticisms of this proposal are incorrect, and discussing the general issue of what a characterization of logicalconstants aims at achieving.
The paper investigates the propriety of applying the form versus matter distinction to arguments and to logic in general. Its main point is that many of the currently pervasive views on form and matter with respect to logic rest on several substantive and even contentious assumptions which are nevertheless uncritically accepted. Indeed, many of the issues raised by the application of this distinction to arguments seem to be related to a questionable combination of different presuppositions and expectations; this holds in (...) particular of the vexed issue of demarcating the class of logicalconstants. I begin with a characterization of currently widespread views on form and matter in logic, which I refer to as 'logical hylomorphism as we know it'—LHAWKI, for short—and argue that the hylomorphism underlying LHAWKI is mereological. Next, I sketch an overview of the historical developments leading from Aristotelian, non-mereological metaphysical hylomorphism to mereological logical hylomorphism (LHAWKI). I conclude with a reassessment of the prospects for the combination of hylomorphism and logic, arguing in particular that LHAWKI is not the only and certainly not the most suitable version of logical hylomorphism. In particular, this implies that the project of demarcating the class of logicalconstants as a means to define the scope and nature of logic rests on highly problematic assumptions. (shrink)
I offer new criticisms of invariantist views of logicality, objecting especially to Gila Sher’s arguments for invariantism’s ability to explain the formality, necessity, apriority and normative force of logic. I argue that the semantic conception of logic can do perfectly well without a model-theoretic notion of logicality, and that the descriptive and explanatory theoretical roles sometimes ascribed to invariantism can be played by a non-model-theoretic account of logicality, specifically by one in which some pragmatic properties of expressions play an important (...) role. I also develop the pragmatic account beyond the mere hints offered in its defense in my earlier work. (shrink)
The paper provides a new argument against the classical invariance criterion for logical terms: if all terms with a permutation invariant extension qualify as logical, then for any arbitrary true contingent sentence K of the meta-language, there would be a logically true object-language sentence 'φ' such that K follows from the sentence 'φ is true'. Thus, many logically true sentences would be a posteriori. To prevent this fatal consequence, we propose to alter the invariance criterion: not only the (...) term's extension, but also its semantic clause must satisfy certain invariance conditions. The paper ends with the observation that the new criterion makes explicit the dependency of the classification of terms into logical and non-logical ones at the different levels of the Tarskian hierarchy of languages. (shrink)
What is the relationship between logic and reasoning? How do logical norms guide inferential performance? This paper agrees with Gilbert Harman and most of the psychologists that logic is not directly relevant to reasoning. It argues, however, that the mental model theory of logical reasoning allows us to harmonise the basic principles of deductive reasoning and inferential perfomances, and that there is a strong connexion between our inferential norms and actual reasoning, along the lines of Peacocke’s conception of (...) inferential role. (shrink)
Extending the language of the intuitionistic propositional logic Int with additional logicalconstants, we construct a wide family of extensions of Int with the following properties: (a) every member of this family is a maximal conservative extension of Int; (b) additional constants are independent in each of them.
Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In particular, logical expressions are autonomous if given meaning by their introduction-rules, rules specifying the grounds for assertion of propositions containing them. If the elimination-rules do no more, and no less, than is justified by the introduction-rules, the rules satisfy what Prawitz, following Lorenzen, called an inversion principle. This connection between rules leads to a general form of elimination-rule, and when the rules have this form, they (...) may be said to exhibit “general-elimination” harmony. Ge-harmony ensures that the meaning of a logical expression is clearly visible in its I-rule, and that the I- and E-rules are coherent, in encapsulating the same meaning. However, it does not ensure that the resulting logical system is normalizable, nor that it satisfies the conservative extension property, nor that it is consistent. Thus harmony should not be identified with any of these notions. (shrink)
I discuss paradoxes of implication in the setting of a proof-conditional theory of meaning for logicalconstants. I argue that a proper logic of implication should be not only relevant, but also constructive and nonmonotonic. This leads me to select as a plausible candidate LL, a fragment of linear logic that differs from R in that it rejects both contraction and distribution.
In the theory of meaning, it is common to contrast truth-conditional theories of meaning with theories which identify the meaning of an expression with its use. One rather exact version of the somewhat vague use-theoretic picture is the view that the standard rules of inference determine the meanings of logicalconstants. Often this idea also functions as a paradigm for more general use-theoretic approaches to meaning. In particular, the idea plays a key role in the anti-realist program of (...) Dummett and his followers. In the theory of truth, a key distinction now is made between substantial theories and minimalist or deflationist views. According to the former, truth is a genuine substantial property of the truth-bearers, whereas according to the latter, truth does not have any deeper essence, but all that can be said about truth is contained in T-sentences (sentences having the form: ‘P’ is true if and only if P). There is no necessary analytic connection between the above theories of meaning and truth, but they have nevertheless some connections. Realists often favour some kind of truth-conditional theory of meaning and a substantial theory of truth (in particular, the correspondence theory). Minimalists and deflationists on truth characteristically advocate the use theory of meaning (e.g. Horwich). Semantical anti-realism (e.g. Dummett, Prawitz) forms an interesting middle case: its starting point is the use theory of meaning, but it usually accepts a substantial view on truth, namely that truth is to be equated with verifiability or warranted assertability. When truth is so understood, it is also possible to accept the idea that meaning is closely related to truth-conditions, and hence the conflict between use theories and truth-conditional theories in a sense disappears in this view. (shrink)
The logicalconstants are technical terms, invented and precisely defined by logicians for the purpose of producing rigorous formal proofs. Mathematics virtually exhausts the domain of deductive reasoning of any complexity, and it is there that the benefits of this refined form of language are felt. Pragmatic issues may arise — issues concerning the point of making a certain statement — for there will be more or less perspicuous and illuminating ways of presenting proofs in this language, and (...) we may be puzzled or misled when we wonder why the mathematician is taking some particular step. But this is hardly a compulsory topic in the philosophy of language. (shrink)
Donald Dvaidson has claimed that a theory of meaning identifies the logicalconstants of the object language by treating them in the phrasal axioms of the theory, and that the theory entails a relation of logical consequence among the sentences of the object language. Section 1 offers a preliminary investigation of these claims. In Section 2 the claims are rebutted by appealing to Evans's paradigm of a theory of meaning. Evans's theory is deliberately blind to any relation (...) of logical consequence among the sentences of the object language, and entails only what Evans takes to be a distinct and deeper relation of structural validity among the sentences of the object language. In Section 3 we turn to Evans's motivation in order to compare the two paradigms of a theory of meaning. Evans laid down criteria under which a theory of meaning gives what he called a ‘transcendent’ semantic classification of the lexicon of the object language, in contrast to a mere ‘immanent’ classification. However, when these criteria are applied we find that, pace Evans, they favour Davidson's paradigm over Evans's. In the final section we show that Evans's conception of structural consequence turns out to be a deeper formulation of logical consequence. (shrink)
Logicism is, roughly speaking, the doctrine that mathematics is fancy logic. So getting clear about the nature of logic is a necessary step in an assessment of logicism. Logic is the study of logical concepts, how they are expressed in languages, their semantic values, and the relationships between these things and the rest of our concepts, linguistic expressions, and their semantic values. A logical concept is what can be expressed by a logical constant in a language. So (...) the question “What is logic?” drives us to the question “What is a logical constant?” Though what follows contains some argument, limitations of space constrain me in large part to express my Credo on this topic with the broad brush of bold assertion and some promissory gestures. (shrink)
May the theory of radical interpretation developed by Donald Davidson on the basis of Quine's arguments for the indeterminacy of translation help fix the meaning of the logicalconstants? In particular, may the theory exclude ways of conferring meaning on the constants which, although developed within the Davidsonian framework, would lead to unexpected results? Could an interpreter fix the meaning of the constants in a non classical way, although still in accordance with the guiding principles of (...) the interpretative strategy? Or, on the contrary, does the theory incorporate constraints on interpretation which are stronger than those imposed by the possibility of non classical, or deviant logics? I examine the particular case of negation within the framework of the natural deduction system developed by Gentzen and Prawitz, and conclude that Davidson's theory, understood as an empirical theory of truth, leaves the problem of knowing which meaning should be assigned to the constant for negation without a satisfactory solution. Although interpetation has to be carried out according to the principle of charity, disagreements may come up regarding which meaning should be assigned to "not" in a properly regimented natural language. (shrink)
The thesis that, in a system of natural deduction, the meaning of a logical constant is given by some or all of its introduction and elimination rules has been developed recently in the work of Dummett, Prawitz, Tennant, and others, by the addition of harmony constraints. Introduction and elimination rules for a logical constant must be in harmony. By deploying harmony constraints, these authors have arrived at logics no stronger than intuitionist propositional logic. Classical logic, they maintain, cannot (...) be justified from this proof-theoretic perspective. This paper argues that, while classical logic can be formulated so as to satisfy a number of harmony constraints, the meanings of the standard logicalconstants cannot all be given by their introduction and/or elimination rules; negation, in particular, comes under close scrutiny. (shrink)
This paper tries to integrate a psychological account of propositional reasoning and a realistic conception of logicalconstants. I argue that the harmony between the norms of reasoning given by the semantics of ordinary logic and the psychology of reasoning holds good.
This paper contains five observations concerning the intended meaning of the intuitionistic logicalconstants: (1) if the explanations of this meaning are to be based on a non-decidable concept, that concept should not be that of 'proof'; (2) Kreisel's explanations using extra clauses can be significantly simplified; (3) the impredicativity of the definition of → can be easily and safely ameliorated; (4) the definition of → in terms of 'proofs from premises' results in a loss of the inductive (...) character of the definitions of ∨ and ∃; and (5) the same occurs with the definition of ∀ in terms of 'proofs with free variables'. (shrink)
This paper deals with Popper's little-known work on deductive logic, published between 1947 and 1949. According to his theory of deductive inference, the meaning of logical signs is determined by certain rules derived from ?inferential definitions? of those signs. Although strong arguments have been presented against Popper's claims (e.g. by Curry, Kleene, Lejewski and McKinsey), his theory can be reconstructed when it is viewed primarily as an attempt to demarcate logical from non-logicalconstants rather than as (...) a semantic foundation for logic. A criterion of logicality is obtained which is based on conjunction, implication and universal quantification as fundamental logical operations. (shrink)
ABSTRACT My aim in this paper is to present a pluralist thesis about the inferential role of logicalconstants, which embraces classical, relevant, linear and ordered logic. That is, I defend that a logical constant c has more than one correct inferential role. The thesis depends on a particular interpretation of substructural logics' vocabulary, according to which classical logic captures the literal meaning of logicalconstants and substructural logics encode a pragmatically enriched sense of those (...) connectives. The paper is divided into four parts: first, I introduce the motivation for the pluralist thesis of the paper; second I introduce the calculus for the different logics endorsed in the pluralist thesis; third, I motivate how the different behaviors of the logical vocabulary of these logics can be pragmatically interpreted, and fourth, I motivate how the different inferential roles that each substructural logic attribute to logicalconstants can coexist, and how we should reason with a logical constant c given this plurality. (shrink)
