The paper suggests a definition of the idea of topic-neutrality, and indicates some of the consequences of identifying logicality with topic-neutrality so defined.

We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of (...) structural rules. These structured variables are interpreted semantically by means of a dependence relation. This relation is an analogue of the accessibility relation in modal logic. We then isolate a class of axioms for generalized quantifiers which correspond to first-order conditions on the dependence relation. (shrink)

One of the oldest topics in foundational metaphysics is the issue how particulars are to be individuated. To individuate a particular, x, means to find criteria that are necessary and sufficient to ensure the assertibility of x ≠ y, for all and only y that are distinct from x. One can distinguish two separate issues that are run under the heading of individuation. One is the question: what is it about a particular that makes it distinct from all other particulars? (...) The other is the question: how do we single out a particular qua distinct from all other particulars? The distinction closely resembles the one put forward by Hector- Neri Castañeda (1975), except that Castañeda talks about the first question in terms of the issue of the ‘internal constitution’ of objects. This way of speaking is misleading, as it insinuates that it is only intrinsic facts about particulars that could serve as candidates for individuation. Instead, in this essay, I will put forward a theory of individuation in which both questions are treated on equal footing, with the implication that both intrinsic and extrinsic, as well as some special properties that I will introduce play a role in individuating particulars. I call the theory ‘derivational contextualism’. It is an elaboration of some ideas I have put forward in an earlier work (Author forthcoming), regarding some basic conditions for individuation, with a structuralist component. I will.. (shrink)

Leibniz notoriously insisted that no two individuals differ solo numero, that is, by being primitively distinct, without differing in some property. The details of Leibniz’s own way of understanding and defending the principle –known as the principle of identity of indiscernibles (henceforth ‘the Principle’)—is a matter of much debate. However, in contemporary metaphysics an equally notorious and discussed issue relates to a case put forward by Max Black (1952) as a counter-example to any necessary and non-trivial version of the principle. (...) Black asks us to imagine, via one of the fictional characters of his dialogue, a world consisting solely of two completely resembling spheres, in a relational space. The supporter of the principle is then forced to admit that although there are ex hypothesi two objects in that universe, there is no property (except trivial ones), not even relational ones, to distinguish them, and hence the necessary version of the principle is falsified. In this essay I will argue that Black’s possible world, together with the dialectic between the potential friends and foes of the Principle as expounded by Black himself.. (shrink)

We investigate two large families of logics, differing from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant ff for “the false”) by adding various standard Gentzen-type rules for negation. The logics in the other family are similarly obtained from LJ+, the positive fragment of intuitionistic logic (again, with or without ff). For all the systems, we provide simple semantics which is (...) based on non-deterministic four-valued or three-valued structures, and prove soundness and completeness for all of them. We show that the role of each rule is to reduce the degree of non-determinism in the corresponding systems. We also show that all the systems considered are decidable, and our semantics can be used for the corresponding decision procedures. Most of the extensions of LJ+ (with or without ff) are shown to be conservative over the underlying logic, and it is determined which of them are not. (shrink)

A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true (we shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as her favourite truth-bearer: this would make little difference in the context). Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and false.

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 logical constants. (shrink)

Knowledge of the basic rules of logic is often thought to be distinctive, for it seems to be a case of non-inferential a priori knowledge. Many philosophers take its source to be different from those of other types of knowledge, such as knowledge of empirical facts. The most prominent account of knowledge of the basic rules of logic takes this source to be the understanding of logical expressions or concepts. On this account, what explains why such knowledge is distinctive is (...) that it is grounded in semantic or conceptual understanding. However, I show that this cannot be the correct account of knowledge of the basic rules of logic, because it is open to Gettier-style counter-examples. (shrink)

What is a logical constant? The question is addressed in the tradition of Tarski's definition of logical operations as operations which are invariant under permutation. The paper introduces a general setting in which invariance criteria for logical operations can be compared and argues for invariance under potential isomorphism as the most natural characterization of logical operations.

This paper contains five observations concerning the intended meaning of the intuitionistic logical constants: (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)

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 logical constants. 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 logical constants as a means to define the scope and nature of logic rests on highly problematic assumptions. (shrink)

Donald Dvaidson has claimed that a theory of meaning identifies the logical constants 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)

I shall argue in this paper that it should. To begin with, I shall defend (CP) against several criticisms that have been launched against it. These criticisms are of two kinds, which I shall call internal and external respectively. Internal objections are that a theory that includes (CP) fails to give an account of what it is rational to believe that is satisfactory by its own standards. In particular, since almost everyone agrees that belief in a contradiction is not rational, (...) (CP) is criticized on the grounds that it would imply that beliefs in some contradictions are rational. External objections maintain that the degree of idealization in a theory including (CP) is so great as to make it irrelevant to various real-world phenomena to which a theory of rational belief should be relevant.2. (shrink)

We can think of functional completeness in systems of propositional logic as a form of expressive completeness: while every logical constant in such system expresses a truth-function of finitely many arguments, functional completeness garantees that every truth-function of finitely many arguments can be expressed with the constants in the system. From this point of view, a functionnaly complete system of propositionnal logic can thus be seen as one where no logical constant is missing. Can a similar question be formulated for (...) quantified first-order logics ? How to make sense of the question whether, e.g., ordinary first-order logic is "functionaly" complete or have no logical constant missing ? In this note, we build on a suggestive proposal made by Bonnay(2006) and shows that it is equivalent to the criterion that a first-order logic L be functionaly complete if and only if every class of structures closed under L-elementary equivalence is L-elementary. Ordinary first-order logic is not complete in this sense. We raise the question whether any logic can be. -/- . (shrink)

There have been several different and even opposed conceptions of the problem of logical constants, i.e. of the requirements that a good theory of logical constants 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 logical constants involve requirements of a philosophically demanding nature which are probably not satisfiable by any minimally adequate theory. (shrink)

The model-theoretic analysis of the concept of logical consequence has come under heavy criticism in the last couple of decades. The present work looks at an alternative approach to logical consequence where the notion of inference takes center stage. Formally, the model-theoretic framework is exchanged for a proof-theoretic framework. It is argued that contrary to the traditional view, proof-theoretic semantics is not revisionary, and should rather be seen as a formal semantics that can supplement model-theory. Specifically, there are formal resources (...) to provide a proof-theoretic semantics for both intuitionistic and classical logic. We develop a new perspective on proof-theoretic harmony for logical constants which incorporates elements from the substructural era of proof-theory. We show that there is a semantic lacuna in the traditional accounts of harmony. A new theory of how inference rules determine the semantic content of logical constants is developed. The theory weds proof-theoretic and model-theoretic semantics by showing how proof-theoretic rules can induce truth-conditional clauses in Boolean and many-valued settings. It is argued that such a new approach to how rules determine meaning will ultimately assist our understanding of the apriori nature of logic. (shrink)

It may be that all that matters for the modalities, possibility and necessity, is the object named by the proper name, not which proper name names it. An influential defender of this view is Saul Kripke. Kripke’s defense is criticized in the paper.

In my dissertation (which I wrote under Bob’s supervision), I argued that in order to understand the confused state of contemporary debates about the demarcation of logic, one has to go back to Kant.1 Following tradition, Kant thought of logic as a normative discipline, with the job of identifying norms for thought. On this broad construal, it makes sense to talk of (say) the logic of jurisprudence, or of geometrical thinking, or of biological thinking. Kant called these “special logics.” But (...) in addition to all the special logics, which provide “rules of correct thinking as regards a certain kind of object,” Kant recognized a “general” logic, which provides rules for thought as such, regardless of its objects (and even regardless of whether it has an object). (shrink)

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 “logical constants.”[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 logical constants “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 logical constants. 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 “logical constants” of a language from its nonlogical expressions. (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 logical constants cannot all be given by their introduction and/or elimination rules; negation, in particular, comes under close scrutiny. (shrink)

It is sometimes held that rules of inference determine the meaning of the logical constants: the meaning of, say, conjunction is fully determined by either its introduction or its elimination rules, or both; similarly for the other connectives. In a recent paper, Panu Raatikainen (2008) argues that this view - call it logical inferentialism - is undermined by some "very little known" considerations by Carnap (1943) to the effect that "in a definite sense, it is not true that the standard (...) rules of inference" themselves suffice to "determine the meanings of [the] logical constants" (p. 2). In a nutshell, Carnap showed that the rules allow for non-normal interpretations of negation and disjunction. Raatikainen concludes that "no ordinary formalization of logic ... is sufficient to `fully formalize' all the essential properties of the logical constants" (ibid.). We suggest that this is a mistake. Pace Raatikainen, intuitionists like Dummett and Prawitz need not worry about Carnap's problem. And although bilateral solutions for classical inferentialists - as proposed by Timothy Smiley and Ian Rumfitt - seem inadequate, it is not excluded that classical inferentialists may be in a position to address the problem too. (shrink)

I discuss paradoxes of implication in the setting of a proof-conditional theory of meaning for logical constants. 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 logical constants. 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)

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)

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 logical constants aims at achieving.

In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.

The core language can be extended by defining additional logical constants. E.g., we can add ‘→’ (implication), ‘∨’ (disjunction), and ‘∀x’ (universal quantifiers). The choice of logical primitives is not as optional in DPL as it is in standard predicate logic.

Anti-realistic conceptions of truth and falsity are usually epistemic or inferentialist. Truth is regarded as knowability, or provability, or warranted assertability, and the falsity of a statement or formula is identified with the truth of its negation. In this paper, a non-inferentialist but nevertheless anti-realistic conception of logical truth and falsity is developed. According to this conception, a formula (or a declarative sentence) A is logically true if and only if no matter what is told about what is told about (...) the truth or falsity of atomic sentences, A always receives the top-element of a certain partial order on non-ontic semantic values as its value. The ordering in question is a told-true order. Analogously, a formula A is logically false just in case no matter what is told about what is told about the truth or falsity of atomic sentences, A always receives the top-element of a certain told-false order as its value. Here, truth and falsity are pari passu , and it is the treatment of truth and falsity as independent of each other that leads to an informational interpretation of these notions in terms of a certain kind of higher-level information. (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 logical constants. 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 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 logical constants. 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)

Extending the language of the intuitionistic propositional logic Int with additional logical constants, 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.

A formal system for , based on a game-theoretic analysis of the ukasiewicz prepositional connectives, is defined and proved to be complete. An Herbrand theorem for the predicate calculus (a variant of some work of Mostowski) and some corollaries relating to its axiomatizability are proved. The predicate calculus with equality is also considered.

Following the pioneer work of Bruno De Finetti [12], conditional probability spaces (allowing for conditioning with events of measure zero) have been studied since (at least) the 1950's. Perhaps the most salient axiomatizations are Karl Popper's in [31], and Alfred Renyi's in [33]. Nonstandard probability spaces [34] are a well know alternative to this approach. Vann McGee proposed in [30] a result relating both approaches by showing that the standard values of infinitesimal probability functions are representable as Popper functions, and (...) that every Popper function is representable in terms of the standard real values of some infinitesimal measure.Our main goal in this article is to study the constraints on (qualitative and probabilistic) change imposed by an extended version of McGee's result. We focus on an extension capable of allowing for iterated changes of view. Such extension, we argue, seems to be needed in almost all considered applications. Since most of the available axiomatizations stipulate (definitionally) important constraints on iterated change, we propose a non-question-begging framework, Iterative Probability Systems (IPS) and we show that every Popper function can be regarded as a Bayesian IPS. A generalized version of McGee's result is then proved and several of its consequences considered. In particular we note that our proof requires the imposition of Cumulativity, i.e. the principle that a proposition that is accepted at any stage of an iterative process of acceptance will continue to be accepted at any later stage. The plausibility and range of applicability of Cumulativity is then studied. In particular we appeal to a method for defining belief from conditional probability (first proposed in [42] and then slightly modified in [6] and [3]) in order to characterize the notion of qualitative change induced by Cumulative models of probability kinematics. The resulting cumulative notion is then compared with existing axiomatizations of belief change and probabilistic supposition. We also consider applications in the probabilistic accounts of conditionals [1] and [30]. (shrink)

In this paper I present a range of substructural logics for a conditional connective ↦. This connective was original introduced semantically via restriction on the ternary accessibility relation R for a relevant conditional. I give sound and complete proof systems for a number of variations of this semantic definition. The completeness result in this paper proceeds by step-by-step improvements of models, rather than by the one-step canonical model method. This gradual technique allows for the additional control, lacking in the canonical (...) model method, that is required. (shrink)

We briefly overview some of the historical landmarks on the path leading to the reduction of the number of logical connectives in classical logic. Relying on the duality inherent in Boolean algebras, we introduce a new operator ( Nallor ) that is the dual of Schönfinkel’s operator. We outline the proof that this operator by itself is sufficient to define all the connectives and operators of classical first-order logic ( Fol ). Having scrutinized the proof, we pinpoint the theorems of (...) Fol that are needed in the proof. Using the insights gained from the proof, we show that there are four binary operators that each can serve as the only undefined logical constant for Fol . Finally, we show that from every n -ary connective that yields a functionally complete singleton set of connectives two Schönfinkel-type operators are definable, and all the latter are so definable. (shrink)

ABSTRACT: The 3rd BCE Stoic logician "Chrysippus says that the number of conjunctions constructible from ten propositions exceeds one million. Hipparchus refuted this, demonstrating that the affirmative encompasses 103,049 conjunctions and the negative 310,952." After laying dormant for over 2000 years, the numbers in this Plutarch passage were recently identified as the 10th (and a derivative of the 11th) Schröder number, and F. Acerbi showed how the 2nd BCE astronomer Hipparchus could have calculated them. What remained unexplained is why Hipparchus’ (...) logic differed from Stoic logic, and consequently, whether Hipparchus actually refuted Chrysippus. This paper closes these explanatory gaps. (1) I reconstruct Hipparchus’ notions of conjunction and negation, and show how they differ from Stoic logic. (2) Based on evidence from Stoic logic, I reconstruct Chrysippus’ calculations, thereby (a) showing that Chrysippus’ claim of over a million conjunctions was correct; and (b) shedding new light on Stoic logic and – possibly – on 3rd century BCE combinatorics. (3) Using evidence about the developments in logic from the 3rd to the 2nd centuries, including the amalgamation of Peripatetic and Stoic theories, I explain why Hipparchus, in his calculations, used the logical notions he did, and why he may have thought they were Stoic. (shrink)

There are certain logical abilities that any rational creature must have. I call this thesis the Universality of Logic (UL). Something like UL is presupposed in Quinean and Davidsonian uses of the Principle of Charity. Their arguments for the Principle of Charity might be thought of as top−down arguments, establishing UL on the basis of very general considerations about meaning and belief. In this paper, I intend to argue for UL constructively, from the bottom up, as it were, by showing (...) just why and how rationality demands certain specific logical abilities. I shall begin by looking at two sources of opposition to UL, in order to locate it better in the current philosophical landscape and underline the interest and importance of the thesis. Next, I shall deal with important methodological issues surrounding my arguments for UL. This will involve addressing such questions as: what is a rational creature?; and, what are logical abilities? As a result, we will be led to a certain weakening of UL. Then, I shall argue for the weakened version of UL constructively, by demonstrating the universality of certain particular logical abilities. Finally, I shall examine some of the consequences of UL for other areas of philosophy. (shrink)

Two quite simple identities for exclusive disjunction and the biconditional are proven by mathematical induction. This proof is independently reprised in R.E. Jennings' /The Genealogy of Disjunction/ (OUP, 1994) pp. 6-7, esp. p. 7 which points out the consequences for the biconditional of the proof that runs from pages 6-7.

Natural deduction systems were motivated by the desire to define the meaning of each connective by specifying how it is introduced and eliminated from inference. In one sense, this attempt fails, for it is well known that propositional logic rules (however formulated) underdetermine the classical truth tables. Natural deduction rules are too weak to enforce the intended readings of the connectives; they allow non-standard models. Two reactions to this phenomenon appear in the literature. One is to try to restore the (...) standard readings, for example by adopting sequent rules with multiple conclusions. Another is to explore what readings the natural deduction rules do enforce. When the notion of a model of a rule is generalized, it is found that natural deduction rules express “intuitionistic” readings of their connectives. A third approach is presented here. The intuitionistic readings emerge when models of rules are defined globally, but the notion of a local model of a rule is also natural. Using this benchmark, natural deduction rules enforce exactly the classical readings of the connectives, while this is not true of axiomatic systems. This vindicates the historical motivation for natural deduction rules. One odd consequence of using the local model benchmark is that some systems of propositional logic are not complete for the semantics that their rules express. Parallels are drawn with incompleteness results in modal logic to help make sense of this. (shrink)

We illustrate, with three examples, the interaction between boolean and modal connectives by looking at the role of truth-functional reasoning in the provision of completeness proofs for normal modal logics. The first example (§ 1) is of a logic (more accurately: range of logics) which is incomplete in the sense of being determined by no class of Kripke frames, where the incompleteness is entirely due to the lack of boolean negation amongst the underlying non-modal connectives. The second example (§ 2) (...) focusses on the breakdown, in the absence of boolean disjunction, of the usual canonical model argument for the logic of dense Kripke frames, though a proof of incompleteness with respect to the Kripke semantics is not offered. An alternative semantic account is developed, in terms of which a completeness proof can be given, and this is used (§ 3) in the discussion of the third example, a bimodal logic which is, as with the first example, provably incomplete in terms of the Kripke semantics, the incompleteness being due to the lack of disjunction (as a primitive or defined boolean connective). (shrink)

Some formal properties of enriched systems of Lambek calculus with analogues of conjunction and disjunction are investigated. In particular, it is proved that the class of languages recognizable by the Lambek calculus with added intersective conjunction properly includes the class of finite intersections of context-free languages.

Tarski and Mautner proposed to characterize the logical operations on a given domain as those invariant under arbitrary permutations. These operations are the ones that can be obtained as combinations of the operations on the following list: identity; substitution of variables; negation; finite or infinite disjunction; and existential quantification with respect to a finite or infinite block of variables. Inasmuch as every operation on this list is intuitively logical, this lends support to the Tarski-Mautner proposal.

It is sometimes held that rules of inference determine the meaning of the logical constants: the meaning of, say, conjunction is fully determined by either its introduction or its elimination rules, or both; similarly for the other connectives. In a recent paper, Panu Raatikainen (2008) argues that this view - call it logical inferentialism - is undermined by some "very little known" considerations by Carnap (1943) to the effect that "in a definite sense, it is not true that the standard (...) rules of inference" themselves suffice to "determine the meanings of [the] logical constants" (p. 2). In a nutshell, Carnap showed that the rules allow for non-normal interpretations of negation and disjunction. Raatikainen concludes that "no ordinary formalization of logic ... is sufficient to `fully formalize' all the essential properties of the logical constants" (ibid.). We suggest that this is a mistake. Pace Raatikainen, intuitionists like Dummett and Prawitz need not worry about Carnap's problem. And although bilateral solutions for classical inferentialists - as proposed by Timothy Smiley and Ian Rumfitt - seem inadequate, it is not excluded that classical inferentialists may be in a position to address the problem too. (shrink)

In introductory logic courses the authors often limit their considerations to the truth-value operators. Then they write that conditionals and biconditionals of natural language ("if" and "if and only if") may be represented as material implications and equivalences ("⊃" and "≡"), respectively. Yet material implications are not suitable for conditionals. Lewis' strict implications are much better for this purpose. Similarly, strict equivalences are better for representing biconditionals (than material equivalences). In this paper we prove that the methods from standard first (...) courses in logic can be used for testing arguments with strict implications, strict equivalences and other operators which may represent connectives from natural language. (shrink)

We attempt to define the classical propositional logic by use of appropriate derivability conditions called Cn-definitions. The conditions characterize basic properties of propositional connectives.

interesting. In this paper, we combine nonclassical logics of negation and possibility in the presence of conjunction and disjunction, and then we combine the resulting systems with intuitionistic logic. We will nd that Kracht's results on the undecidability of classical modal logics generalise to a non-classical setting. We will also see conditions under which intuitionistic logic can be combined with a non-intuitionistic negation without corrupting the intuitionistic fragment of the logic.

As we’ve seen in the last chapter, there is good linguistic reason to categorize negations (and negative operators in general) by which De Morgan laws they support. The weakest negative operators (merely downward monotonic) support only two De Morgan laws;1 medium-strength negative operators support a third;2 and strong negative operators support all four. As we’ve also seen, techniques familiar from modal logic are of great use in giving unifying theories of negative operators. In particular, Dunn’s (1990) distributoid theory allows us (...) to generate relational semantics for many negations. However, the requirements of distributoid theory are a bit too strict for use in modeling the weakest negations. For a relational semantics to work, an operator must either distribute or antidistribute over either conjunction or disjunction; but the merely downward monotonic operators do not. Thus, a unifying semantics cannot be had in distributoid theory. In the (more familiar) study of positive modalities, there is a parallel result. Normal necessities distribute over conjunction, and normal possibilities over disjunction. When these distributions break down, a relational semantics is no longer appropriate. Here, there is a somewhat familiar solution: neighborhood semantics. In this chapter, I’ll adapt neighborhood semantics to the less familiar case of negative modalities, showing how it can be used to give a single semantic framework appropriate to all the pertinent sorts of negative operators. (shrink)

Presents a version of dynamic semantics for a language with presuppositions that predicts basic facts about presupposition projection in a non-stipulative way.

Imperatives cannot be true or false, so they are shunned by logicians. And yet imperatives can be combined by logical connectives: "kiss me and hug me" is the conjunction of "kiss me" with "hug me". This example may suggest that declarative and imperative logic are isomorphic: just as the conjunction of two declaratives is true exactly if both conjuncts are true, the conjunction of two imperatives is satisfied exactly if both conjuncts are satisfied—what more is there to say? Much more, (...) I argue. "If you love me, kiss me", a conditional imperative, mixes a declarative antecedent ("you love me") with an imperative consequent ("kiss me"); it is satisfied if you love and kiss me, violated if you love but don't kiss me, and avoided if you don't love me. So we need a logic of three -valued imperatives which mixes declaratives with imperatives. I develop such a logic. (shrink)

Model-theoretic proofs of functional completenes along the lines of [McCullough 1971, Journal of Symbolic Logic 36, 15–20] are given for various constructive modal propositional logics with strong negation.

One of the oldest topics in foundational metaphysics is the issue how particulars are to be individuated. To individuate a particular, x, means to find criteria that are necessary and sufficient to ensure the assertibility of x ≠ y, for all and only y that are distinct from x. One can distinguish two separate issues that are run under the heading of individuation. One is the question: what is it about a particular that makes it distinct from all other particulars? (...) The other is the question: how do we single out a particular qua distinct from all other particulars? The distinction closely resembles the one put forward by Hector- Neri Castañeda (1975), except that Castañeda talks about the first question in terms of the issue of the ‘internal constitution’ of objects. This way of speaking is misleading, as it insinuates that it is only intrinsic facts about particulars that could serve as candidates for individuation. Instead, in this essay, I will put forward a theory of individuation in which both questions are treated on equal footing, with the implication that both intrinsic and extrinsic, as well as some special properties that I will introduce play a role in individuating particulars. I call the theory ‘derivational contextualism’. It is an elaboration of some ideas I have put forward in an earlier work (Author forthcoming), regarding some basic conditions for individuation, with a structuralist component. I will.. (shrink)

Leibniz notoriously insisted that no two individuals differ solo numero, that is, by being primitively distinct, without differing in some property. The details of Leibniz’s own way of understanding and defending the principle –known as the principle of identity of indiscernibles (henceforth ‘the Principle’)—is a matter of much debate. However, in contemporary metaphysics an equally notorious and discussed issue relates to a case put forward by Max Black (1952) as a counter-example to any necessary and non-trivial version of the principle. (...) Black asks us to imagine, via one of the fictional characters of his dialogue, a world consisting solely of two completely resembling spheres, in a relational space. The supporter of the principle is then forced to admit that although there are ex hypothesi two objects in that universe, there is no property (except trivial ones), not even relational ones, to distinguish them, and hence the necessary version of the principle is falsified. In this essay I will argue that Black’s possible world, together with the dialectic between the potential friends and foes of the Principle as expounded by Black himself.. (shrink)

In The Principles of Mathematics, Russell writes: Whatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one, I call a term. This, then, is the widest word in the philosophical vocabulary. I shall use as synonymous with it the words unit, individual and entity. The first two emphasize the fact that every term is one, while the third is derived from the fact that every term has being, i.e. (...) is in some sense. A man, a moment, a number, a class, a relation, a chimera, or anything else that can be mentioned, is sure to be a term'. Now in this remark, a certain extremely general, topic-neutral use of ‘object’ is singled out, a use in which the expression is treated as equivalent to (equally neutral) uses of ‘term’, ‘entity’, ‘unit’, ‘individual’, and ‘thing’. In addition, a claim is made to the effect that the content of the expression, with its emphasis on one, is such as to be adequate to comprehend the sum-total of existence, to include whatever there may be. The paper addresses the question of what this claim means, and whether it might be true. (shrink)

Safe first-order formulas generalize the concept of a safe rule, which plays an important role in the design of answer set solvers. We show that any safe sentence is equivalent, in a certain sense, to the result of its grounding—to the variable-free sentence obtained from it by replacing all quantifiers with multiple conjunctions and disjunctions. It follows that a safe sentence and the result of its grounding have the same stable models, and that stable models of a safe sentence can (...) be characterized by a formula of a simple syntactic form. (shrink)

Kaplan (1989) insists that natural languages do not contain displacing devices which operate on character---such displacing devices are called monsters. This thesis has recently faced various empirical challenges (e.g. Schlenker 2003 and Anand & Nevins 2004). In this note, the thesis is challenged on grounds of a more theoretical nature. It is argued that the standard compositional semantics of variable binding employs monstrous operations. As a dramatic first example, Kaplan's formal language LD ("Logic of Demonstratives") is shown to contain monsters. (...) For similar reasons, the orthodox lambda-calculus-based semantics for variable binding is argued to be monstrous. This technical point promises to provide some far-reaching implications for our understanding of semantic theory and content. The theoretical upshot of the discussion is at least threefold: (i) the Kaplanian thesis that "directly referential" terms are not shiftable/bindable is unmotivated, (ii) since monsters operate on something distinct from the assertoric content of their operands, we must distinguish ingredient sense from assertoric content (cf. Dummett 1973, Evans 1979 and Stanley 2002), and (iii) since the case of variable binding provides a paradigm of semantic shift that differs from the other types, it is plausible to think that indexicals---which are standardly treated by means of the assignment function---might undergo the same kind of shift. (shrink)

We investigate a variant of the variable convention proposed at Tractatus 5.53ff for the purpose of eliminating the identity sign from logical notation. The variant in question is what Hintikka has called the strongly exclusive interpretation of the variables, and turns out to be what Ramsey initially (and erroneously) took to be Wittgenstein's intended method. We provide a tableau calculus for this identity-free logic, together with soundness and completeness proofs, as well as a proof of mutual interpretability with first-order logic (...) with identity. (shrink)

We investigate a rst-order predicate logic based on Wittgenstein's suggestion to express identity of object by identity of sign, and difference of objects by difference of signs. Hintikka has shown that predicate logic can indeed be set up in such a way; we show that it can be done nicely. More specically, we provide a perspicuous cut-free sequent calculus, as well as a Hilbert-type calculus, for Wittgensteinian predicate logic and prove soundness and completeness theorems.

Wittgensteinian predicate logic (W-logic) is characterized by the requirement that the objects mentioned within the scope of a quantifier be excluded from the range of the associated bound variable. I present a sound and complete tableaux calculus for this logic and discuss issues of translatability between Wittgensteinian and standard predicate logic in languages with and without individual constants. A metalinguistic co-denotation predicate, akin to Frege’s triple bar of the Begriffsschrift, is introduced and used to bestow the full expressive power of (...) first-order logic with identity on W-logic in the presence of constants. (shrink)