This paper presents a way of formalising definite descriptions with a binary quantifier ι, where ιx[F, G] is read as ‘The F is G’. Introduction and elimination rules for ι in a system of intuitionist negative free logic are formulated. Procedures for removing maximal formulas of the form ιx[F, G] are given, and it is shown that deductions in the system can be brought into normal form.

In this paper I am concerned with an analysis of negative existential sentences that contain proper names only by using negative or neutral free logic. I will compare different versions of neutral free logic with the standard system of negative free logic (Burge, Sainsbury) and aim to defend my version of neutral free logic that I have labeled non-standard neutral free logic.

Sentences containing definite descriptions, expressions of the form ‘The F’, can be formalised using a binary quantifier ι that forms a formula out of two predicates, where ιx[F, G] is read as ‘The F is G’. This is an innovation over the usual formalisation of definite descriptions with a term forming operator. The present paper compares the two approaches. After a brief overview of the system INFι of intuitionist negative free logic extended by such a quantifier, which (...) was presented in (Kürbis 2019), INFι is first compared to a system of Tennant’s and an axiomatic treatment of a term forming ι operator within intuitionist negative free logic. Both systems are shown to be equivalent to the subsystem of INFι in which the G of ιx[F, G] is restricted to identity. INFι is then compared to an intuitionist version of a system of Lambert’s which in addition to the term forming operator has an operator for predicate abstraction for indicating scope distinctions. The two systems will be shown to be equivalent through a translation between their respective languages. Advantages of the present approach over the alternatives are indicated in the discussion. (shrink)

This article presents a sequent calculus for a negative free logic with identity, called N . The main theorem (in part 1) is the admissibility of the Cut-rule. The second part of this essay is devoted to proofs of soundness, compactness and completeness of N relative to a standard semantics for negative free logic.

Free logics are a family of first-order logics which came about as a result of examining the existence assumptions of classical logic (Hintikka _The Journal of Philosophy_, _56_, 125–137 1959 ; Lambert _Notre Dame Journal of Formal Logic_, _8_, 133–144 1967, 1997, 2001 ). What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential (...) import. Free logics reject the claim that names need to denote in (ii). _Positive_ free logic concedes that some atomic formulas containing non-denoting names (including self-identity) are true, _negative_ free logic treats them as uniformly false, and _neutral_ free logic as taking a third value. There has been a renewed interest in analyzing proof theory of free logic in recent years, based on intuitionistic logic in Maffezioli and Orlandelli (_Bulletin of the Section of Logic_, _48_(2), 137–158 2019 ) as well as classical logic in Pavlović and Gratzl (_Journal of Philosophical Logic_, _50_, 117–148 2021 ), there for the positive and negative variants. While the latter streamlines the presentation of free logics and offers a more unified approach to the variants under consideration, it does not cover neutral free logic, since there is some lack of both clear formal intuitions on the semantic status of formulas with empty names, as well as a satisfying account of the conditional in this context. We discuss extending the results to this third major variant of free logics. We present a series of G3 sequent calculi adapted from Fjellstad (_Studia Logica_, _105_(1), 93–119 2017, _Journal of Applied Non-Classical Logics_, _30_(3), 272–289 2020 ), which possess all the desired structural properties of a good proof system, including admissibility of contraction and all versions of the cut rule. At the same time, we maintain the unified approach to free logics and moreover argue that greater clarity of intuitions is achieved once neutral free logic is conceptualized as consisting of two sub-varieties. (shrink)

Linnebo and Shapiro have recently given an analysis of potential infinity using modal logic. A key technical component of their account is to show that under a suitable translation ◊ of nonmodal language into modal language, nonmodal sentences ϕ 1, …, ϕ n entail ψ just in case ϕ 1 ◊, …, ϕ n ◊ entail ψ ◊ in the modal logic S4.2. Linnebo and Shapiro establish this result in nonfree logic. In this note I argue that (...) their analysis of potential infinity should be carried out in a free logic. I then extend their key theorems to the setting of negative free logic. (shrink)

In this paper I aim to defend a first‐order non‐discriminating property view concerning existence. The version of this view that I prefer is based on negative (or a specific neutral) free logic that treats the existence predicate as first‐order logical predicate. I will provide reasons why such a view is more plausible than a second‐order discriminating property view concerning existence and I will also discuss four challenges for the proposed view and provide solutions to them.

Semantic externalism in the style of McDowell and Evans faces a puzzle formulated by Pryor: to explain that a sentence such as 'Jack exists' is only a posteriori knowable, despite being logically entailed by the seemingly logical truth 'Jack is self-identical', and hence being itself a logical truth and therefore a priori knowable. Free logics can dissolve the puzzle. Moreover, Pryor has argued that the existentially hedged 'If Jack exists, then Jack is self-identical', when properly formalised, is a logical (...) truth in a system of neutral free logic and therefore a priori knowable, while it does not entail that Jack exists. The latter holds also for negative free logic. In response, Yeakel has argued that on any system of neutral free logic existence hedges will either entail some unwanted existence claims or they will not entail some wanted existence claims. The dilemma also holds for any non-positive free logic. It will be shown that the extension of one of the systems of neutral free logic with a truth operator escapes Yeakel's dilemma, whereas no other non-positive free logic when extended with the truth operator does the same (or it breaks quantifier exchangeability). (shrink)

Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic. What those assumptions are varies, but the central ones are that the domain of interpretation is not empty, every name denotes exactly one object in the domain and the quantifiers have existential import. Free logics usually reject the claim that names need to denote in, and of the systems considered in this paper, the positive (...) class='Hi'>free logic concedes that some atomic formulas containing non-denoting names are true, while negative free logic rejects even the latter claim. Inclusive logics, which reject, are likewise considered. These logics have complex and varied axiomatizations and semantics, and the goal of this paper is to present an orderly examination of the various systems and their mutual relations. This is done by first offering a formalization, using sequent calculi which possess all the desired structural properties of a good proof system, including admissibility of contraction and cut, while streamlining free logics in a way no other approach has. We then present a simple and unified system of abstract semantics, which allows for a straightforward demonstration of the meta-theoretical properties, and offers insights into the relationship between different logics. The final part of this paper is dedicated to extending the system with modalities by using a labeled sequent calculus, and here we are again able to map out the different approaches and their mutual relations using the same framework. (shrink)

Free logics aim at freeing logic from existence assumptions by making them explicit, e.g., by adding an existence premisse to the antecedence of the classical axiom-schema of Universal Instantiation. Their historical development was motivated by the problem of empty singular terms, and that one of simple statements containing at least one such singular term: what is the referential status of such singular terms and what truth-value, if any, do such statemants have? Free logics can be classified with (...) regard to their respective answers to these problems. Negative free logics assume that non-existent objects cannot have any properties at all; hence, in particular, they cannot be self-identical or rotate. Positive free logics believe that non-existents can be self-identical according to the Leibnizian concept of identity. Neutral free logics think that statements of self-identity are truth-valueless because of the Fregean principle of compositionality. Since only in negative free logic, but not in positive free logic, two statements of the forms "a = a" and "E!a" are logically equivalent, one also can define only for NFL, but not for PFL, existence by self-identity. (shrink)

This paper presents rules in sequent calculus for a binary quantifier I to formalise definite descriptions: Ix[F, G] means ‘The F is G’. The rules are suitable to be added to a system of positive free logic. The paper extends the proof of a cut elimination theorem for this system by Indrzejczak by proving the cases for the rules of I. There are also brief comparisons of the present approach to the more common one that formalises definite descriptions (...) with a term forming operator. In the final section rules for I for negative free and classical logic are also mentioned. (shrink)

This paper investigates the key drivers behind the origins of value-for-money (VFM) audit in Canada and the aims, intents, and logics ascribed by the original proponents. Drawing on insights from governmentality and New Public Management, the paper utilizes analysis methods adapted from case study research to review a wide range of primary documentation (e.g., Hansards from the Public Accounts Committee, House of Commons debates, the so-called Wilson report and the FMCS study) and secondary documentation (newspaper articles, Office of the Auditor (...) General internal publications, journal articles). Major findings indicate a rise of a management consulting culture within the Auditor General’s office following the appointment of James Johnson Macdonell. VFM legislation effectively operationalized practices drawn from management consulting expertise by invitation of the consultant. It was offered as an answer to the growing scope and size of government, which had become problematized by the Auditor General in terms of parliament losing control over the public purse. The Auditor General’s invocation of financial crisis led to a substantial broadening of mandate and increase in resources for that office. (shrink)

Thanks to the Inlaks Foundation in India, I was able to do my doctoral research on Our Talk About Nonexistents at Oxford in the early eighties. The two greatest philosophers of that heaven of analytical philosophy - Peter Strawson and Michael Dummett - supervised my work, reading and criticising all the fledgling philosophy that I wrote during those three years. At Sir Peter's request, Gareth Evans, shortly before his death, lent me an unpublished transcript of Kripke's John Locke Lectures. Work (...) on the Appendix about Indian Philosophy was supervised by the late Professor Bimal Krishna Matilal with whom informal but intense philosophical conversations used to spill over into dinner at his place almost every other day. It was Professor Matilal who sent me, over a summer, to study a tough Navya-Nyaya text under his own Nyaya teacher Pandit Visvabandhu Tarkatirtha at Calcutta. All four of these teachers were as kind to me as my life-long mentor in philosophy Professor Pranab Kumar Sen, whose clarity and depth remain the unreachable regu lative ideal of my intellect. When I came back to India, my life became blissfully free of the agonising anxiety to publish, until, after a conference at Jadavpur University where I gave an impromptu paper, ironically enough, on Non-doings, I met Derek Parfit. He had a six-hour conversation with me, explicitly planning my life. Five years had already elapsed since I had finished my D. Phil, but Derek read my thesis and liked it. (shrink)

A cut-free Gentzen sequent calculus for Ewald’s intuitionistic tense logic \ is established. By the proof-theoretic method, we prove that, for every set of strictly positive implications S, the classical tense logic \ is embedded into its intuitionistic analogue \ via Kolmogorov, Gödel–Genzten and Kuroda translations respectively. A sufficient and necessary condition for Glivenko type theorem in tense logics is established.

From Leibniz to Krauss philosophers and scientists have raised the question as to why there is something rather than nothing. Why-questions request a type of explanation and this is often thought to include a deductive component. With classical logic in the background only trivial answers are forthcoming. With free logics in the background, be they of the negative, positive or neutral variety, only question-begging answers are to be expected. The same conclusion is reached for the modal version (...) of the Question, namely ‘Why is there something contingent rather than nothing contingent?’. The categorial version of the Question, namely ‘Why is there something concrete rather than nothing concrete?’, is also discussed. The conclusion is reached that deductive explanations are question-begging, whether one works with classical logic or positive or negative free logic. I also look skeptically at the prospects of giving causal-counterfactual or probabilistic answers to the Question, although the discussion of the options is less comprehensive and the conclusions are more tentative. The meta-question, viz. ‘Should we not stop asking the Question’, is accordingly tentatively answered affirmatively. (shrink)

Free Semantics is based on normalized natural deduction for the weak relevant logic DW and its near neighbours. This is motivated by the fact that in the determination of validity in truth-functional semantics, natural deduction is normally used. Due to normalization, the logic is decidable and hence the semantics can also be used to construct counter-models for invalid formulae. The logic DW is motivated as an entailment logic just weaker than the logic MC of (...) meaning containment. DW is the logic focussed upon, but the results extend to MC. The semantics is called 'free semantics' since it is disjunctively and existentially free in that no disjunctive or existential witnesses are produced, unlike in truth-functional semantics. Such 'witnesses' are only assumed in generality and are not necessarily actual. The paper sets up the free semantics in a truth-functional style and gives a natural deduction interpetation of the meta-logical connectives. We then set out a familiar tableau-style system, but based on natural deduction proof rather than truth-functional semantics. A proof of soundness and completeness is given for a reductio system, which is a transform of the tableau system. The reductio system has positive and negative rules in place of the elimination and introduction rules of Brady's normalized natural deduction system for DW. The elimination-introduction turning points become closures of threads of proof, which are at the points of contradiction for the reductio system. (shrink)

I present the traditional debate about the so called explanation of Russell’s paradox and propose a new way to solve the contradiction that arises in Frege’s system. I briefly examine two alternative explanatory proposals—the Predicativist explanation and the Cantorian one—presupposed by almost all the proposed solutions of Russell’s Paradox. From the discussion about these proposals a controversial conclusion emerges. Then, I examine some particular zig zag solutions and I propose a third explanation, presupposed by them, in which I emphasise the (...) role of an implicit premise in the derivation of the paradox. In conclusion, I propose a different zig zag solution obtained by the adoption of a negative free logic. (shrink)

I present the traditional debate about the so called explanation of Russell’s paradox and propose a new way to solve the contradiction that arises in Frege’s system. I briefly examine two alternative explanatory proposals—the Predicativist explanation and the Cantorian one—presupposed by almost all the proposed solutions of Russell’s Paradox. From the discussion about these proposals a controversial conclusion emerges. Then, I examine some particular zig zag solutions and I propose a third explanation, presupposed by them, in which I emphasise the (...) role of an implicit premise in the derivation of the paradox. In conclusion, I propose a different zig zag solution obtained by the adoption of a negative free logic. (shrink)

Fix a finite ordinal \ and let \ be an arbitrary ordinal. Let \ denote the class of cylindric algebras of dimension \ and \ denote the class of relation algebras. Let \\) stand for the class of polyadic algebras of dimension \. We reprove that the class \ of completely representable \s, and the class \ of completely representable \s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety \ between polyadic algebras (...) of dimension \ and diagonal free \s. We show that that the class of completely and strongly representable algebras in \ is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class \ is not closed under \. In contrast, we show that given \, and an atomic \, then for any \, \ is a completely representable \. We show that for any \, the class of completely representable algebras in certain reducts of \s, that happen to be varieties, is elementary. We show that for \, the the class of polyadic-cylindric algebras dimension \, introduced by Ferenczi, the completely representable algebras coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension \ and \s for \ an infinite ordinal, proving negative results for the first and positive ones for the second. (shrink)

This paper contributes to the semantic typology of dependent indefinites, by accounting for the distribution and interpretation of the Romanian indefinite vreun. It is shown that its occurrences are restricted to negative polarity and a subset of modal contexts. More specifically, the study of its behavior in intensional environments reveals that vreun is systematically incompatible with non-epistemic operators, a restriction we capture by proposing a novel empirical generalization (‘the epistemic constraint’). To account for the observed pattern, we adopt the (...) unitary approach to polarity in Chierchia (Logic in Grammar: polarity, free choice and intervention, 2013a; in: Fălăuş (ed.) Alternatives in semantics, 2013b) and derive the properties of vreun from its obligatory association with alternatives. Its distributional restrictions are argued to follow from the interplay between the types of alternatives it activates (scalar and subdomain alternatives), the way these alternatives are factored into meaning (via an independently motivated mechanism of exhaustification) and the lexical semantics of the operators in the context of occurrence. We propose that the epistemic constraint arises from the lexicalization of an inference (anti-total variation) that non-epistemic operators can never satisfy. (shrink)

The author integrates, expands, and deepens his previous publications about irregular (or "metalinguistic") negations. A total of ten distinct negatives-several previously unclassified-are analyzed. The logically irregular negations deny different implicatures of their root. All are partially non-compositional but completely conventional. The author argues that two of the irregular negative meanings are implicatures. The others are semantically rather than pragmatically ambiguous. Since their ambiguity is neither lexical nor structural, direct irregular negatives satisfy the standard definition of idioms as syntactically complex (...) expressions whose meaning is non-compositional. Unlike stereotypical idioms, idiomatic negatives lack fixed syntactic forms and are highly compositional. The final chapter analyzes other "free form" idioms, including irregular interrogatives and comparatives, self-restricted verb phrases, numerical verb phrases, and transparent propositional attitude and speech act reports. (shrink)

Russell's introduction of negative facts to account for the truth of "negative" sentences or beliefs rests on his collaboration with Wittgenstein in such efforts as the characterization of formal necessity, the theory of logical atomism, and the use of the Ideal Language. In examining their views we arrive at two conclusions. First, that the issue of negative facts is distinct from questions of meaning or intentionality; what a sentence or belief means or is about rather than what (...) makes it true or false. Second, that the ontological use of the Ideal Language is incompatible with the requirements of its employment in the logical study of inferences. On this basis we conclude that despite elaborations by recent proponents, the doctrine of negative facts lacks adequate support, and perhaps more importantly, it is proper ontological method to free the Ideal Language from the exigencies of a symbolism constructed for logical investigation. (shrink)

Sentences such as Olivia can take Logic or Algebra (‘♢∨-sentences’) are typically interpreted as entailing that Olivia can take Logic and can take Algebra. Given a standard semantics for modals and disjunction, those ‘Free choice’ (FC) readings are not predicted from the surface form of ♢∨-sentences. Yet the standard semantics is appropriate for the ‘double prohibition’ reading typically assigned to ¬♢∨-sentences like Olivia can’t take Logic or Algebra. Several extant approaches to FC can account for those (...) two cases, but face challenges when ♢∨, ¬♢∨ and related sentences appear embedded in certain environments. In this paper, we present a novel account of FC that builds on a ‘grammatical’ theory of scalar implicatures — proposed by Bassi et al. (2021) and Del Pinal (2021) — according to which covert exhaustification is a presupposition trigger such that the prejacent forms the assertive content while any excludable or includable alternatives are incorporated at the non-at issue, presuppositional level. Applied to ♢∨, ¬♢∨, and similar sentences, ‘presuppositional exhaustification’ predicts that their default interpretations have an assertive component (roughly, the classical interpretation of the prejacent) and a homogeneity presupposition which projects in standard ways. Those predictions, we then show, support a uniform account of the puzzling behavior of ♢∨, ¬♢∨, and related sentences when embedded under (negative) factives (Marty & Romoli 2020), disjunctions (Romoli & Santorio 2019), and in the scope of universal, existential (Bar-Lev & Fox 2020) and non-monotonic quantifiers (Gotzner et al. 2020). (shrink)

The Logic of Partial Terms LPT is a strict negative free logic that provides an economical framework for developing many traditional mathematical theories having partial functions. In these traditional theories, all functions and predicates are strict. For example, if a unary function (predicate) is applied to an undefined argument, the result is undefined (respectively, false). On the other hand, every practical programming language incorporates at least one nonstrict or lazy construct, such as the if-then-else, but nonstrict (...) functions cannot be either primitive or introduced in definitional extensions in LPT. Consequently, lazy programming language constructs do not fit the traditional mathematical mold inherent in LPT. A nonstrict (positive free) logic is required to handle nonstrict functions and predicates. Previously developed nonstrict logics are not fully satisfactory because they are verbose in describing strict functions (which predominate in many programming languages), and some logicians find their semantics philosophically unpalatable. The newly developed Lazy Logic of Partial Terms LL is as concise as LPT in describing strict functions and predicates, and strict and nonstrict functions and predicates can be introduced in definitional extensions of traditional mathematical theories. LL is "built on top of" LPT, and, like LPT, admits only one domain in the semantics. In the semantics, for the case of a nonstrict unary function h in an LL theory T, we have $\models_T h(\perp) = y \longleftrightarrow \forall x(h(x) = y)$ , where $\perp$ is a canonical undefined term. Correspondingly, in the axiomatization, the "indifference" (to the value of the argument) axiom $h(\perp) = y \longleftrightarrow \forall x(h(x) = y)$ guarantees a proper fit with the semantics. The price paid for LL's naturalness is that it is tailored for describing a specific area of computer science, program specification and verification, possibly limiting its role in explicating classical mathematical and philosophical subjects. (shrink)

Libet considers “positive free voluntary acts” as mere illusions, admitting free will only as Veto. This essay shows seven ways by which we can gain evident knowledge about positive and negative free will, through: (1) the immediate evidence of free will in the cogito, (2) the light of the necessary essence of free will, (3) the experience of moral “oughts” in whose experience freedom is co-given, (4) any denial of human free will entails (...) its assertion or recognition, (5) the objects and subjects of certain acts disclose free will, (6) in a world without free agents there would be no explanation of the beginning of efficient causality, and (7) Veto-power of the will logically presupposes positive free will. Libet’s experiments confirm that the free decision to act at a certain time and the preceding and accompanying free acts make new energy to burst forth in the brain. (shrink)

In this extended critical discussion of 'Kant's Modal Metaphysics' by Nicholas Stang (OUP 2016), I focus on one central issue from the first chapter of the book: Stang’s account of Kant’s doctrine that existence is not a real predicate. In §2 I outline some background. In §§3-4 I present and then elaborate on Stang’s interpretation of Kant’s view that existence is not a real predicate. For Stang, the question of whether existence is a real predicate amounts to the question: ‘could (...) there be non-actual possibilia?’ (p.35). Kant’s view, according to Stang, is that there could not, and that the very notion of non-actual or ‘mere’ possibilia is incoherent. In §5 I take a close look at Stang’s master argument that Kant’s Leibnizian predecessors are committed to the claim that existence is a real predicate, and thus to mere possibilia. I argue that it involves substantial logical commitments that the Leibnizian could reject. I also suggest that it is danger of proving too much. In §6 I explore two closely related logical commitments that Stang’s reading implicitly imposes on Kant, namely a negative universal free logic and a quantified modal logic that invalidates the Converse Barcan Formula. I suggest that each can seem to involve Kant himself in commitment to mere possibilia. (shrink)

The logic of partial terms (LPT) is a variety of negative free logic in which functions, as well as predicates, are strict. A companion paper focused on nonconstructive LPTwith definite descriptions, called LPD, and laid the foundation for tableaux systems by defining the concept of an LPDmodel system and establishing Hintikka's Lemma, from which the strong completeness of the corresponding tableaux system readily follows. The present paper utilizes the tableaux system in establishing an Extended Joint Consistency (...) Theorem for LPDthat incorporates the Robinson Joint Consistency Theorem and the Craig-Lyndon Interpolation Lemma. The method of proof is similar to that originally used in establishing the Extended Joint Consistency Theorem for positive free logic. Proof of the Craig-Lyndon Interpolation Lemma for formulas possibly having free variables is readily had in LPTand its intuitionistic counterpart. The paper concludes with a brief discussion of the theory of definitions in LPD. (shrink)

A free logic is one in which a singular term can fail to refer to an existent object, for example, `Vulcan' or `5/0'. This essay demonstrates the fruitfulness of a version of this non-classical logic of terms (negative free logic) by showing (1) how it can be used not only to repair a looming inconsistency in Quine's theory of predication, the most influential semantical theory in contemporary philosophical logic, but also (2) how Beeson, (...) Farmer and Feferman, among others, use it to provide a natural foundation for partial functions in programming languages. Vis à vis (2), the question is raised whether the Beeson-Farmer-Feferman approach is adequate to the treatment of partial functions in all programming languages. Gumb and the author say No, and suggest a way of handling the refractory cases by means of positive free logic. Finally, Antonelli's solution of a problem associated with the Gumb-Lambert proposal is mentioned. (shrink)

Judaic Logic is an original inquiry into the forms of thought determining Jewish law and belief, from the impartial perspective of a logician. Judaic Logic attempts to honestly estimate the extent to which the logic employed within Judaism fits into the general norms, and whether it has any contributions to make to them. The author ranges far and wide in Jewish lore, finding clear evidence of both inductive and deductive reasoning in the Torah and other books of (...) the Bible, and analyzing the methodology of the Talmud and other Rabbinic literature by means of formal tools which make possible its objective evaluation with reference to scientific logic. The result is a highly innovative work – incisive and open, free of clichés or manipulation. Judaic Logic succeeds in translating vague and confusing interpretative principles and examples into formulas with the clarity and precision of Aristotelean syllogism. Among the positive outcomes, for logic in general, are a thorough listing, analysis and validation of the various forms of a-fortiori argument, as well as a clarification of dialectic logic. However, on the negative side, this demystification of Talmudic/Rabbinic modes of thought (hermeneutic and heuristic) reveals most of them to be, contrary to the boasts of orthodox commentators, far from deductive and certain. They are often, legitimately enough, inductive. But they are also often unnatural and arbitrary constructs, supported by unverifiable claims and fallacious techniques. Many other thought-processes, used but not noticed or discussed by the Rabbis, are identified in this treatise, and subjected to logical review. Various more or less explicit Rabbinic doctrines, which have logical significance, are also examined in it. In particular, this work includes a formal study of the ethical logic (deontology) found in Jewish law, to elicit both its universal aspects and its peculiarities. With regard to Biblical studies, one notable finding is an explicit formulation (which, however, the Rabbis failed to take note of and stress) of the principles of adduction in the Torah, written long before the acknowledgement of these principles in Western philosophy and their assimilation in a developed theory of knowledge. Another surprise is that, in contrast to Midrashic claims, the Tanakh (Jewish Bible) contains a lot more than ten instances of qal vachomer (a-fortiori) reasoning. In sum, Judaic Logic elucidates and evaluates the epistemological assumptions which have generated the Halakhah (Jewish religious jurisprudence) and allied doctrines. Traditional justifications, or rationalizations, concerning Judaic law and belief, are carefully dissected and weighed at the level of logical process and structure, without concern for content. This foundational approach, devoid of any critical or supportive bias, clears the way for a timely reassessment of orthodox Judaism (and incidentally, other religious systems, by means of analogies or contrasts). Judaic Logic ought, therefore, to be read by all Halakhists, as well as Bible and Talmud scholars and students; and also by everyone interested in the theory, practise and history of logic. (shrink)

Halbach has argued that Tarski biconditionals are not ontologically conservative over classical logic, but his argument is undermined by the fact that he cannot include a theory of arithmetic, which functions as a theory of syntax. This article is an improvement on Halbach's argument. By adding the Tarski biconditionals to inclusive negative free logic and the universal closure of minimal arithmetic, which is by itself an ontologically neutral combination, one can prove that at least one thing (...) exists. The result can then be strengthened to the conclusion that infinitely many things exist. Those things are not just all Gödel codes of sentences but rather all natural numbers. Against this background inclusive negative free logic collapses into noninclusive free logic, which collapses into classical logic. The consequences for ontological deflationism with respect to truth are discussed. (shrink)

Motivations for systems of free logics are reviewed and systems are divided according as they are positive (asserting atomic truths with non-denoting terms) negative (denying all such sentences) or neutral. A positive theory is developed and defended. One of the major considerations in favor of the theory is that it allows (via translation) representation of the other points of view. Finally, the relation between free logic and truth theories is elaborated.

Motivations for systems of free logics are reviewed and systems are divided according as they are positive (asserting atomic truths with non-denoting terms) negative (denying all such sentences) or neutral. A positive theory is developed and defended. One of the major considerations in favor of the theory is that it allows (via translation) representation of the other points of view. Finally, the relation between free logic and truth theories is elaborated.

Motivations for systems of free logics are reviewed and systems are divided according as they are positive (asserting atomic truths with non-denoting terms) negative (denying all such sentences) or neutral. A positive theory is developed and defended. One of the major considerations in favor of the theory is that it allows (via translation) representation of the other points of view. Finally, the relation between free logic and truth theories is elaborated.

We characterise the intermediate logics which admit a cut-free hypersequent calculus of the form \, where \ is the hypersequent counterpart of the sequent calculus \ for propositional intuitionistic logic, and \ is a set of so-called structural hypersequent rules, i.e., rules not involving any logical connectives. The characterisation of this class of intermediate logics is presented both in terms of the algebraic and the relational semantics for intermediate logics. We discuss various—positive as well as negative—consequences of (...) this characterisation. (shrink)

Arguments for Restrictivism – the position that we are rarely free– have been proposed by incompatibilists Peter van Inwagen and David Vander Laan among others. This article is concerned much more with these arguments than with quantifying the frequency of free actions. There are two general ways to argue for restrictivism. First, one may take a Negative Strategy, arguing that the situations in which one is not free are common and predominant. Second, one may focus on (...) situations in which one is apparently free, and argue directly that these situations are rare – the Inventory Strategy. I conclude that both types of arguments for restrictivism are unconvincing. (shrink)

Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate semantics and proof systems, whose philosophical interpretations and computational properties are found wanting. In this paper we investigate congruential non-classical negations that live inside very natural systems of normal modal logics over complete distributive lattices; these logics are further enriched by adjustment connectives that may be used (...) for handling reasoning under uncertainty caused by inconsistency or undeterminedness. Using such straightforward semantics, we study the classes of frames characterized by seriality, reflexivity, functionality, symmetry, transitivity, and some combinations thereof, and discuss what they reveal about sub-classical properties of negation. To the logics thereby characterized we apply a general mechanism that allows one to endow them with analytic ordinary sequent systems, most of which are even cut-free. We also investigate the exact circumstances that allow for classical negation to be explicitly defined inside our logics. (shrink)

The essays in philosophical logic collected in this volume are dedicated to Henry S. Leonard who was one of the first American philosophers to urge the application of modern logic to non-mathematical areas. Leonard also inspired the development of certain areas of contemporary philosophical logic discussed in some of the papers of this volume. This is especially clear in the case of free, or presupposition free, logics which Leonard's early work on a logic of (...) existence inspired. In one essay of this volume Bas C. Van Frassen further develops his work on the semantics of free logic. In another, Milton Fisk relates free logic to modal logic, suggesting a new semantics of strength to deal with various problems in the area and in a third essay, Karel Lambert relates free logic to certain logical puzzles about quantum theory. Leonard was also one of the first to entertain the idea of and discuss a logic of questions and an erotetic logic or logic of interests. The volume contains essays on these topics by Nuel D. Belnap and David Harrah. Leonard's interest in modal logic is reflected in Fisk's article as well as in an article by Richmond Thomason on modal logic and metaphysics. His interest in epistemic logic is seen in an essay on the logic of belief by Jon Vickers and an essay by Hintikka applying the logic of belief and knowledge to problems about the ontological argument. In addition to these essays, there are five others of high quality on diverse topics: R. M. Martin on intentions, Wilfrid Sellars on the metaphysics of the person, R. M. Chisholm on agency, Frederick Fitch on combinatory logic and negative numbers, and H. E. Hendry and G. J. Massey on Sheffer functions. This is a worthy memorial to a quietly influential American philosopher.--R. H. K. (shrink)

This article is devoted to the analysis of arguments from empirical science against free will. Its main purpose is to reveal their deep anti-naturalism. This anti-naturalism lies in the use of a concept of free will that cannot be the subject of naturalistic consideration, as well as in the various explanatory and ontological paradoxes that arguments from empirical science lead in case when someone is trying to generalize the explanatory principles underlying them. At the beginning of the article, (...) the author gives a general notion of the free will problem, a working definition of naturalism and the place of arguments from empirical science in discussions about free will. To achieve the main goal of the article, the author suggests a classification of arguments from empirical science, which includes five types: from prediction, from manipulation, from the brain, from illusion, from the substitution of concepts. In accordance with this classification, the structure of the article is defined, where each of the presented types is sequentially considered. The logic of considering each type of argument is approximately the same: explication of the essence of the argument of a particular type, its analysis, identification of basic principles and their generalization, demonstration of the negative consequences that it leads to, and answers to possible objections. In the course of the consideration, the author formulates an ontologically neutral concept of free will as a set of abilities associated with the agent’s control over his actions. At the end of the article, the main points is summed up, the idea of naturalistic compatibilism is proposed, the role of arguments from empirical science in discussions about free will is clarified, as is the problem of free will itself, the question of the sources of “chimerization” of naturalism is briefly highlighted, and the problem of completeness of the proposed reasoning is touched upon. (shrink)

Free logic, an alternative to traditional logic, has been seen as a useful avenue of approach to a number of philosophical issues of contemporary interest. In this collection, Karel Lambert, one of the pioneers in, and the most prominent exponent of, free logic, brings together a variety of published essays bearing on the application of free logic to philosophical topics ranging from set theory and logic to metaphysics and the philosophy of religion. (...) The work of such distinguished philosophers as Bas van Fraassen, Dana Scott, Tyler Burge, and Jaakko Hintikka is represented. Lambert provides an introductory essay placing free logic in the logical tradition beginning with Aristotle, developing it as the natural culmination of a trend begun in the Port Royal logic of the 1600s, and continuing through current predicate logic--the trend to rid logic of existence assumptions. His Introduction also provides a useful systematic overview of free logic, including both a standard syntax and some semantical options. (shrink)

Free logic is an important field of philosophical logic that first appeared in the 1950s. J. Karel Lambert was one of its founders and coined the term itself. The essays in this collection explore the philosophical foundations of free logic and its application to areas as diverse as the philosophy of religion and computer science. Amongst the applications on offer are those to the analysis of existence statements, to definite descriptions and to partial functions. The (...) volume contains a proof that free logics of any kind are non-extensional and then uses that proof to show that Quine's theory of predication and referential transparency must fail. The purpose of this collection is to bring an important body of work to the attention of a new generation of professional philosophers, computer scientists and mathematicians. (shrink)

What general conclusions can be drawn about the reception of zymase, its relation to the larger shift from a protoplasm to an enzyme theory of life, and its status as a social phenomenon?The most striking and to me unexpected pattern is the close correlation between attitude toward zymase and professional background. The disbelief of the fermentation technologists, Will, Delbrück, Wehmer, and even Stavenhagen, was as sharp and unanimous as the enthusiasm of the immunologists and enzymologists, Duclaux, Roux, Fernback, and Bertrand, (...) and of Pfeffer, the experimental plant physiologist. Other skeptics—Voit, Kupffer, and Fischer—were conservatives in traditional fields. In all these cases it seems clear that professional commitments and outlook profoundly influenced the reception of zymase.In some cases there is a correlation with a specific earlier statement favoring the protoplasm theory or predicting a fermentation enzyme. Reynolds Green is a striking case of both. In 1893 he stated his belief that life processes in higher plants and fungi were identical, both being mediated by protoplasm.109 Thus he claimed that digestion was not carried out by enzymes, as in higher animals, but by the whole protoplasm, and so too fermentation in yeast: “All the metabolic processes must be carried out in the unicellular organism in the same mass of protoplasm.”110 But Green also insisted that there was no essential difference between “organized” and “unorganized ferments”; enzymes were simply more highly differentiated and specialized forms of primitive protoplasm. He spoke of “the alcoholic ferment” (that is, enzyme), and of intracellular enzymes—very advanced beliefs for 1893. Thus it is understandable why Green saw no reason to doubt his first negative results with yeast juice, and rushed into print. But it is also understandable that he persisted and soon became an outspoken advocate of Buchner's view; like Wroblewski and Pfeffer, he had already accepted the idea of an intracellular fermentation enzyme.The physiological chemists also fit the pattern in a special way. Physiological chemistry is generally regarded as the source of biochemistry; indeed, many early biochemists were trained in physiological chemistry, and imbibed there a progressive interest in enzymes and metabolism. It is not surprising, therefore, that on the whole Neumeister, Madfadyen, Wroblewski, and Loew accepted cell-free fermentation. But what is surprising, and significant, is the diversity of opinion in this group, from Abeles' living protoplasm to Wroblewski's organized array of active proteins. The feelings of the physiological chemists were mixed, and there was enough of the old protoplasm view in Macfadyen's and even Wroblewski's views to allow Buchner to see them as attacks on zymase. The ideas of physiological chemistry were also changing; the idea of protoplasm was already adjusting to the new interest in enzymes in the early 1890's. But in contrast to immunochemistry, the change in physiological chemistry was gradual, uneven, undramatic, and relatively invisible. For the new biochemistry, the tradition of physiological chemistry was the source of traditional conservatism as well as of new ideas of enzymes.In general, then, I claim that it was not experimental facts that determined attitudes toward zymase so much as previous commitments, experience, and expectations. Initially, at least, zymase was less a determinant of opinion than a touchstone of pre-existing opinion. Like a prism, it revealed the spectrum of existing attitudes toward vital phenomena.Also relevant here is the fact that the experimental evidence was entirely ambiguous; the interpretation of Buchner's experiments with antiseptics or dried yeast depended entirely on how far one was willing to stretch the scope of the terms “protoplasm” and “enzyme”. Likewise, how important one judged the instability of zymase to be, or the low level of in vitro activity, depended largely on one's point of view. There was no crucial experiment, no certain proof. Logically zymase could be regarded as protoplasm. A. Fischer did so in 1900;111 Macfadyen was still wavering in 1907;112 and Beijerinck was holding out as late as 1916.113 There were undoubtedly others. Besides the special case of Reynolds Green and the technologists, there are no recorded cases of conversion.But if there was no evidence to convert a dedicated protoplasmist, there was plenty to encourage a dedicated belief in the enzyme theory, and that, I believe, is precisely what happened. The arguments of Buchner et al in support of zymase had the greatest effect not on the protoplasmists but on those who were already predisposed toward the biochemical view. The zymase debate enabled the emerging group of biochemists to recognizethemselves as a group with a community of out-look and objectives; it brought the issue of protoplasm vs. enzyme into the open and gave it a specific and concrete issue on which to turn. The debate sharpened the biochemists' awareness that their point of view was becoming the new mainstream of biochemical thought.All this of course had an important secondary effect on those who opposed the new view, or who were not immediately concerned with it. After the zymase debate, it would have been almost impossible to ignore the new ideas; the dramatic and wide publicity the debate enjoyed would have made it difficult not to take sides. Buchner's successful defense of zymase itself became an important influence on opinion; the new movement had to be looked at with respect. But the primary effect of the debate, I assert, was on those who were already inclined to the new view. The outcome was not a mass conversion of individuals, nor the ejection of one idea by another. The sides probably did not change much; the real change was in the way each side regarded itself and its place in history. The zymase debate may perhaps best be regarded as a process of selection. Buchner's party became aware of itself as the new progressive point of view, while those holding the traditional view could no longer expound it publicly without appearing obstinate or old-fashioned. Thus the way was clear for the gradually wider establishment of the new ideas, especially among the new generation of biochemists. In this specific sense, the acceptance of zymase was a social phenomenon.This way of looking at the zymase debate also illuminates its coincidence with the emergence of biochemistry as a profession. The new professional trappings and organizations were the social manifestation of the new self-awareness gained from the zymase controversy. One of the most important effects on those disposed toward the new view was the awareness of an intellectual community and of the need to give that community social forms. The almost mythical importance of Buchner's “victory” over the old protoplasm theory that later became current is readily understandable. It was a useful ideology for a group anxious to assert the novelty of their approach, a novelty which they in fact felt quite keenly.But if the primary effect of the zymase controversy was on those predisposed to accept zymase, then obviously the success of zymase depended on the fact that change had already begun to occur. Indeed, one of the most striking, and again unexpected, results of this study is the extent to which the importance of enzymes was already recognized in the early 1890's. The new immunology clearly pointed the way. Macfadyen and Hans Buchner were looking for active proteins inside the cell in 1893. Reynolds Green, Wroblewski, and Pfeffer had all anticipated the discovery of a fermentation enzyme. Other discoveries in enzymology in the 1890's pointed the same way.114 Moreover, it is clear from this study that the old protoplams theory was adapting to the new discoveries concerning the importance of enzymes. The sidechains of the old protoplasm molecule were simply reinterpreted as enzymes, as in Wroblewski's theory, for example, or in Paul Ehrlich's extremely influential “sidechain theory” of antibody formation (1897). Despite the later views of Hopkins et al., the old protoplasm and the new enzyme theories were not irreconcilable; a continuity of development had already begun when zymase came so dramatically on the scene. Buchner himself observed that the difference between the two views had become largely verbal; in retrospect we might say, not verbal, but social and historical. The gap between the continuous change in ideas and the contemporary feeling of sharply discontnuous change is a measure of the elusive but real social and historical implications of the zymase debate, of the demise of an old establishment, and the emergence of a new.In retrospect, then, it is clear that a trend was already under way in various quarters when zymase appeared. But it was still a collection of isolated and dispersed events. The zymase controversy made the trend visible and gave it a dramatic unity. The debate influenced those who had already begun to change. It was so effective because things had begun to change and because it coincided with the direction of that trend. (shrink)

The expression ‘free logic,’ coined by the author in 1960, is an abbreviation for ‘logic free of existence assumptions with respect to its terms, singular and general, but whose quantifiers are treated exactly as in standard quantifier logic.’ In more traditional language, such logics do not presume that either singular or general terms — the two distinct categories of terms emphasized in modern logical grammar — have existential import. A singular term ‘t’ has existential import (...) just in case t exists (or, equivalently, there exists an object the same as t) and a general term (or predicate) ‘G’ has existential import just in case G exist (or, equivalently, there exists an object that is G).1 Examples from colloquial English customarily taken to be singular terms are expressions such as ‘Socrates’, ‘the planet causing perturbations in the orbit of Mercury’, ‘5’, ‘5/0’, ‘the square of 3’ and ‘having a heart’. Some of these do not have existential import — in particular, ‘5/0’ and ‘the planet causing perturbations in the orbit of Mercury’. Examples from colloquial English customarily taken to be general terms are expressions such as ‘is a philosopher’, ‘is a planet causing perturbations in the orbit of Mercury’, ‘number’, ‘is divisible by 0’, and ‘has a heart’. Some of these general terms do not have existential import — in particular, ‘is a planet causing perturbations in the orbit of Mercury’ and ‘is divisible by 0’. To say that the quantifers are treated exactly as in standard quantifier logic is to say, roughly, that the operator symbol ‘∃’ (the existential quantifier) reads: ‘There exists an object’, and the operator symbol ‘∀’ (the universal quantifier) reads: ‘Every existent object’. (shrink)

This paper focuses on order-preserving logics defined from varieties of distributive lattices with negation, and in particular on the problem of whether these can be axiomatized by means Hilbert-style calculi that are finite. On the negative side, we provide a syntactic condition on the equational presentation of a variety that entails failure of finite axiomatizability for the corresponding logic. An application of this result is that the logic of all distributive lattices with negation is not finitely axiomatizable; (...) we likewise establish that the order-preserving logic of the variety of all Ockham algebras is also not finitely axiomatizable. On the positive side, we show that an arbitrary subvariety of semi-De Morgan algebras is axiomatized by a finite number of equations if and only if the corresponding order-preserving logic is axiomatized by a finite Hilbert-style calculus. This equivalence also holds for every subvariety of a Berman variety of Ockham algebras. We obtain, as a corollary, a new proof that the implication-free fragment of intuitionistic logic is finitely axiomatizable, as well as a new corresponding Hilbert-style calculus. Our proofs are constructive in that they allow us to effectively convert an equational presentation of a variety of algebras into a Hilbert-style calculus for the corresponding order-preserving logic, and vice versa. We also consider the assertional logics associated to the above-mentioned varieties, showing in particular that the assertional logics of finitely axiomatizable subvarieties of semi-De Morgan algebras are finitely axiomatizable as well. (shrink)

Completeness is shown for several versions of Girard's linear logic with respect to Petri nets as the class of models. One logic considered is the -free fragment of intuitionistic linear logic without the exponential !. For this fragment Petri nets form a sound and complete model. The strongest logic considered is intuitionistic linear logic, with ,&, and the exponential ! , and forms of quantification. This logic is shown sound and complete with respect (...) to atomic nets , though only once we add extra axioms specific to the Petri-net model. The logic is remarkably expressive, enabling descriptions of the kinds of properties one might wish to show of nets; in particular, negative properties, asserting the impossibility of an assertion, can also be expressed. Unfortunately, with respect to this logic, whether an assertion is true of a finite net becomes undecidable. (shrink)

A star-free relational semantics for relevant logic is presented together with a sound and complete sequent proof theory. It is an extension of the dualist approach to negation regarded as modality, according to which de Morgan negation in relevant logic is better understood as the confusion of two negative modalities. The present work shows a way to define them in terms of implication and a new connective, co-implication, which is modeled by respective ternary relations. The defined (...) negations are confused by a special constraint on ternary relation, called the generalized star postulate, which implies definability of the Routley star in the frame. The resultant logic is shown to be equivalent to the well-known relevant logic R. Thus it can be seen as a reconstruction of R in the dualist framework. (shrink)

Tanabe Hajime, another pole of the so-called Kyoto-School of Philosophy of modern Japan, attempts to construct a dialectical, triadic logic of genus, species and individual as a creative synthesis between Eastern and Western philosophy. Although the formal pattern of his method is influenced by the Hegelian dialectic, the way of his thinking is rather prevailed by Kantian dualism. This makes a sharp contrast to his mentor Nishida Kitaro, whose logic of Topos or Place qua Absolute Nothingness is criticized (...) as all-embracing and static in character by him. The difference between them might be parallel to that of Greek and Latin theology concerning the Trinity. Tanabe never presupposes any preexistent entity as the primordial One in the eternal dimension, but rather maintains the individuality as the free subjective agent in the field of history. The dichotomy between the universal and the individual is overcome in and through the mediation of the third term— the species — as the negatively self-realized, specific form of the genus. The species, however, turns out to be the self-estrangement, when it loses the perpetually negative mediation of the free subjective activity of the individual. (shrink)