This book aims to develop certain aspects of Gottlob Frege’s theory of meaning, especially those relevant to intensional logic. It offers a new interpretation of the nature of senses, and attempts to devise a logical calculus for the theory of sense and reference that captures as closely as possible the views of the historical Frege. (The approach is contrasted with the less historically-minded Logic of Sense and Denotation of Alonzo Church.) Comparisons of Frege’s theory with those (...) of Russell and others are given. It is in the end shown that developing Frege’s theory in these ways reveals serious problems hitherto largely unnoticed, including those possibly rendering a Fregean intensional logic inconsistent even if his naïve class theory is excluded. (shrink)

My aim in this paper is to explore how we might understand the relation of Deleuze’s early works to ethics, and to develop the connections between this way of understanding Deleuze and the work of Søren Kierkegaard. I will claim that we can view both figures as arguing that the sense or meaning we take from the world, and the metaphysical structure we ascribe to it, is secondary to an ethical stance we take in the face of a world (...) of becoming. As such, the central preoccupation of both Kierkegaard and Deleuze is how we make sense of an existence which is necessarily temporal. As we shall see, recognizing the importance of temporality involves a move away from the traditional resolution of the problem of sense that operates by making temporality an accidental aspect of a world. In the first half of this paper, I want to explore this claim in relation to Deleuze’s reading of Plato, tying his claim that Plato is essentially developing an ethical rather than metaphysical doctrine with the claim that “the task of modern philosophy has been defined: to overturn [renversement] Platonism” (DR 59). My emphasis in this section is how Deleuze, rather than “overturning” Platonism by simply rejecting the priority it gives to Forms or essences over appearances, the renversement of Deleuze’s reading inverts the relation to show how the search for essence is in fact grounded in an ethical decision, or “selection,” in the world of appearances/simulacra itself. As such, both Plato and Deleuze can be seen as presenting different trajectories for making sense of a world of becoming, either by grounding it in a transcendent, atemporal realm, or by seeing it as an intensive field of processes. In the second half, I will argue that Kierkegaard’s philosophy rests on a claim, similar to the one we find in Deleuze’s work, that we need to develop a logic of sense in response to a world of becoming. I will conclude that Deleuze’s reading of Kierkegaard is limited by a focus on the theological aspects of his thought, and that Kierkegaard provides a sophisticated and complementary taxonomy of the ways in which we try to make sense of our existence in a world of becoming by developing diverse ethical postures. (shrink)

What is the concept of sense developed by Deleuze in his 1969 Logic of Sense? This paper attempts to answer this question analysing the three dimensions of language that Deleuze isolates: the primary order of noises and intensities ; the secondary order of sense ; and the tertiary organisation of propositions. What renders language possible is that which separates sounds from bodies and organises them into propositions, freeing them for the expressive function. Deleuze argues that it (...) is the dimension of sense that brings about this genesis of language, and he analyses in detail the three syntheses that bring about the production of this surface of sense. Yet Deleuze also distinguishes between two types of non-sense: the nonsense of Lewis Carroll's portmanteau words, which remain ensconced in the dimension of sense, and the more profound nonsense of Antonin Artaud's psychotic scream-breaths, which penetrate the almost unbearable world of the primary order of noise and intensities. In the end, the focus of Logic of Sense is less the surface domain of sense than the primary depth of corporeal intensities. What Deleuze calls a ‘minor’ use of language is nothing other than an intensive use of language that constitutes a principle of metamorphosis. (shrink)

Deleuze’s The Logic of Sense- Irfan Ajvazi.

This paper discusses certain problems arising within the treatment of the senses of functions in Alonzo Church's Logic of Sense and Denotation. Church understands such senses themselves to be "sense-functions," functions from sense to sense. However, the conditions he lays out under which a sense-function is to be regarded as a sense presenting another function as denotation allow for certain undesirable results given certain unusual or "deviant" sense-functions. Certain absurdities result, e.g., an (...) argument can be found for equating any two senses of the same type. An alternative treatment of the senses of functions is discussed, and is thought to do better justice to Frege's original theory. (shrink)

This article presents formal correspondences between the ontological and logical structures of Deleuze’s theory of sense-events in the Logic of Sense as a “post-Cantorian orientation of thought” (Livingston 2012), grappling with an essential incompleteness or inconsistency at the heart of both Being and thought, one which Deleuze champions positively under the equation Ungrounding = Becoming. Through it, Deleuze’s sometimes slippery use of the concept of singularity (and its relation to the virtual) is elaborated, elucidating a post-Cantorian metaphysics (...) of events, distinct from and preceding Badiou’s, that concretely defines the role of the singular in Deleuze’s early major works. (shrink)

Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen (...) as the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic. (shrink)

In the paper, original formal-logical conception of syntactic and semantic: intensional and extensional senses of expressions of any language L is outlined. Syntax and bi-level intensional and extensional semantics of language L are characterized categorically: in the spirit of some Husserl’s ideas of pure grammar, Leśniewski-Ajukiewicz’s theory syntactic/semantic categories and in accordance with Frege’s ontological canons, Bocheński’s famous motto—syntax mirrors ontology and some ideas of Suszko: language should be a linguistic scheme of ontological reality and simultaneously a tool of its (...) cognition. In the logical conception of language L, its expressions should satisfy some general conditions of language adequacy. The adequacy ensures their unambiguous syntactic and semantic senses and mutual, syntactic, and semantic compatibility, correspondence guaranteed by the acceptance of a postulate of categorial compatibility syntactic and semantic categories of expressions of L. From this postulate, three principles of compositionality follow: one syntactic and two semantic already known to Frege. They are treated as conditions of homomorphism partial algebra of L into algebraic models of L: syntactic, intensional, and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, of course, an idealization, but only expressions with high degrees of precision of their senses, after due justification, may become theorems of science. (shrink)

One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative (...) results hold for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C1, each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics. (shrink)

We present a framework for epistemic logic, modeling the logical aspects of System 1 and System 2 cognitive processes, as per dual process theories of reasoning. The framework combines non-normal worlds semantics with the techniques of Dynamic Epistemic Logic. It models non-logically-omniscient, but moderately rational agents: their System 1 makes fast sense of incoming information by integrating it on the basis of their background knowledge and beliefs. Their System 2 allows them to slowly, step-wise unpack some of (...) the logical consequences of such knowledge and beliefs, by paying a cognitive cost. The framework is applied to three instances of limited rationality, widely discussed in cognitive psychology: Stereotypical Thinking, the Framing Effect, and the Anchoring Effect. (shrink)

In this article, I outline a logic of design of a system as a specific kind of conceptual logic of the design of the model of a system, that is, the blueprint that provides information about the system to be created. In section two, I introduce the method of levels of abstraction as a modelling tool borrowed from computer science. In section three, I use this method to clarify two main conceptual logics of information inherited from modernity: Kant’s (...) transcendental logic of conditions of possibility of a system, and Hegel’s dialectical logic of conditions of in/stability of a system. Both conceptual logics of information analyse structural properties of given systems. Strictly speaking, neither is a conceptual logic of information about the conditions of feasibility of a system, that is, neither is a logic of information as a logic of design. So, in section four, I outline this third conceptual logic of information and then interpret the conceptual logic of design as a logic of requirements, by introducing the relation of “sufficientisation”. In the conclusion, I argue that the logic of requirements is exactly what we need in order to make sense of, and buttress, a constructionist approach to knowledge. (shrink)

The logic of indicative conditionals remains the topic of deep and intractable philosophical disagreement. I show that two influential epistemic norms—the Lockean theory of belief and the Ramsey test for conditional belief—are jointly sufficient to ground a powerful new argument for a particular conception of the logic of indicative conditionals. Specifically, the argument demonstrates, contrary to the received historical narrative, that there is a real sense in which Stalnaker’s semantics for the indicative did succeed in capturing the (...) logic of the Ramseyan indicative conditional. (shrink)

Bilattices, introduced by Ginsberg as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the (...) perspective of abstract algebraic logic . We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú . We also prove some purely algebraic results concerning bilattices, for instance that the variety of distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices. (shrink)

This paper starts by indicating the analysis of Hempel's conditions of adequacy for any relation of confirmation (Hempel, 1945) as presented in Huber (submitted). There I argue contra Carnap (1962, Section 87) that Hempel felt the need for two concepts of confirmation: one aiming at plausible theories and another aiming at informative theories. However, he also realized that these two concepts are conflicting, and he gave up the concept of confirmation aiming at informative theories. The main part of the paper (...) consists in working out the claim that one can have Hempel's cake and eat it too - in the sense that there is a logic of theory assessment that takes into account both of the two conflicting aspects of plausibility and informativeness. According to the semantics of this logic, a is an acceptable theory for evidence β if and only if a is both sufficiently plausible given β and sufficiently informative about β. This is spelt out in terms of ranking functions (Spohn, 1988) and shown to represent the syntactically specified notion of an assessment relation. The paper then compares these acceptability relations to explanatory and confirmatory consequence relations (Flach, 2000) as well as to nonmonotonic consequence relations (Kraus et al., 1990). It concludes by relating the plausibility-informativeness approach to Carnap's positive relevance account, thereby shedding new light on Carnap's analysis as well as solving another problem of confirmation theory. (shrink)

I want to model a finite, fallible cognitive agent who imagines that p in the sense of mentally representing a scenario—a configuration of objects and properties—correctly described by p. I propose to capture imagination, so understood, via variably strict world quantifiers, in a modal framework including both possible and so-called impossible worlds. The latter secure lack of classical logical closure for the relevant mental states, while the variability of strictness captures how the agent imports information from actuality in the (...) imagined non-actual scenarios. Imagination turns out to be highly hyperintensional, but not logically anarchic. Section 1 sets the stage and impossible worlds are quickly introduced in Sect. 2. Section 3 proposes to model imagination via variably strict world quantifiers. Section 4 introduces the formal semantics. Section 5 argues that imagination has a minimal mereological structure validating some logical inferences. Section 6 deals with how imagination under-determines the represented contents. Section 7 proposes additional constraints on the semantics, validating further inferences. Section 8 describes some welcome invalidities. Section 9 examines the effects of importing false beliefs into the imagined scenarios. Finally, Sect. 10 hints at possible developments of the theory in the direction of two-dimensional semantics. (shrink)

We propose a nonmonotonic Description Logic of typicality able to account for the phenomenon of combining prototypical concepts, an open problem in the fields of AI and cognitive modelling. Our logic extends the logic of typicality ALC + TR, based on the notion of rational closure, by inclusions p :: T(C) v D (“we have probability p that typical Cs are Ds”), coming from the distributed semantics of probabilistic Description Logics. Additionally, it embeds a set of cognitive (...) heuristics for concept combination. We show that the complexity of reasoning in our logic is EXPTIME-complete as in ALC. (shrink)

In this paper, I attempt to clarify the heart of Dewey’s philosophy: his method (denotative method (DM) / pattern of inquiry (PI)). Despite the traditional understanding of Dewey as anti-foundationalist, I want to show that Dewey did have metaphysical foundations for his method: the principle of continuity or theory of emergentism. I also argue that Dewey’s metaphysical position is better named as ‘cultural emergentism’, rather than his own term ‘cultural naturalism’. What Dewey called ‘common sense’ in his Logic, (...) Husserl termed as the ‘life-world’ in his Crisis. I compare two perspectives of dealing with the phenomenon and conclude that for Dewey, the difference between natural sciences and the common sense inquiry is that of subject-matter but not of method. Thus, the goal is to find the unified method to be applied in both domains. Whereas Husserl was more pessimistic: for him, the difference was not only in subject-matter, but in the very methods. Following that discussion, I also attempt to reformulate the hard problem of consciousness in Deweyan terms. In the end, I compare Dewey’s DM / PI with Popper’s understandings of scientific method and conclude that there is no significant difference between the two and that Dewey’s method could also be looked at as hypothetic-deductive method, with the only difference in emphases. (shrink)

We aim to show that Kant’s theory of time is consistent by providing axioms whose models validate all synthetic a priori principles for time proposed in the Critique of Pure Reason. In this paper we focus on the distinction between time as form of intuition and time as formal intuition, for which Kant’s own explanations are all too brief. We provide axioms that allow us to construct ‘time as formal intuition’ as a pair of continua, corresponding to time as ‘inner (...) sense’ and the external representation of time as a line Both continua are replete with infinitesimals, which we use to elucidate an enigmatic discussion of ‘rest’ in the Metaphysical foundations of natural science. Our main formal tools are Alexandroff topologies, inverse systems and the ring of dual numbers. (shrink)

The paper surveys the currently available axiomatizations of common belief (CB) and common knowledge (CK) by means of modal propositional logics. (Throughout, knowledge- whether individual or common- is defined as true belief.) Section 1 introduces the formal method of axiomatization followed by epistemic logicians, especially the syntax-semantics distinction, and the notion of a soundness and completeness theorem. Section 2 explains the syntactical concepts, while briefly discussing their motivations. Two standard semantic constructions, Kripke structures and neighbourhood structures, are introduced in Sections (...) 3 and 4, respectively. It is recalled that Aumann's partitional model of CK is a particular case of a definition in terms of Kripke structures. The paper also restates the well-known fact that Kripke structures can be regarded as particular cases of neighbourhood structures. Section 3 reviews the soundness and completeness theorems proved w.r.t. the former structures by Fagin, Halpern, Moses and Vardi, as well as related results by Lismont. Section 4 reviews the corresponding theorems derived w.r.t. the latter structures by Lismont and Mongin. A general conclusion of the paper is that the axiomatization of CB does not require as strong systems of individual belief as was originally thought- only monotonicity has thusfar proved indispensable. Section 5 explains another consequence of general relevance: despite the "infinitary" nature of CB, the axiom systems of this paper admit of effective decision procedures, i.e., they are decidable in the logician's sense. (shrink)

This paper is concerned with a propositional modal logic with operators for necessity, actuality and apriority. The logic is characterized by a class of relational structures defined according to ideas of epistemic two-dimensional semantics, and can therefore be seen as formalizing the relations between necessity, actuality and apriority according to epistemic two-dimensional semantics. We can ask whether this logic is correct, in the sense that its theorems are all and only the informally valid formulas. This paper (...) gives outlines of two arguments that jointly show that this is the case. The first is intended to show that the logic is informally sound, in the sense that all of its theorems are informally valid. The second is intended to show that it is informally complete, in the sense that all informal validities are among its theorems. In order to give these arguments, a number of independently interesting results concerning the logic are proven. In particular, the soundness and completeness of two proof systems with respect to the semantics is proven (Theorems 2.11 and 2.15), as well as a normal form theorem (Theorem 3.2), an elimination theorem for the actuality operator (Corollary 3.6), and the decidability of the logic (Corollary 3.7). It turns out that the logic invalidates a plausible principle concerning the interaction of apriority and necessity; consequently, a variant semantics is briefly explored on which this principle is valid. The paper concludes by assessing the implications of these results for epistemic two-dimensional semantics. (shrink)

This essay was written for a special issue of Philosophical Topics on the links between Kant and analytic philosophy. It explores these links through consideration of: Wittgenstein’s Tractatus; the logical positivism endorsed by Ayer; and the (very different) variation on that theme endorsed by Quine. It is argued that in all three cases we see analytic philosophers trying to attain and express a general philosophical understanding of why the bounds of sense should be drawn where they should—but thereby confronting (...) the aporia, which Kant too confronted, that it is impossible to do this without transgressing those very bounds. It is suggested in conclusion that the problem is in trying to attain and express such an understanding; in other words, that such an understanding is attainable, but not expressible. (shrink)

The theory of granular partitions is designed to capture in a formal framework important aspects of the selective character of common-sense views of reality. It comprehends not merely the ways in which we can view reality by conceiving its objects as gathered together not merely into sets, but also into wholes of various kinds, partitioned into parts at various levels of granularity. We here represent granular partitions as triples consisting of a rooted tree structure as first component, a domain (...) satisfying the axioms of Extensional Mereology as second component, and a mapping (called ’projection’) of the first into the second as a third component. We define ordering relations among granular partitions the resulting structures are called partition frames. We then introduce an axiomatic theory which sentences are interpreted in partition frames. (shrink)

Through the reconstruction of Leibniz's theory of the degrees of knowledge, this e-book investigates and explores the intrinsic relationship of imagination with space and time. The inquiry into this relationship defines the logic of imagination that characterizes both human and non-human animals, albeit differently, making them two different species of imaginative animals. -/- Lucia Oliveri explains how the emergence of language in human animals goes hand in hand with the emergence of thought and a different form of rationality constituted (...) by logical inferences based on identity and contradiction, principles that are out of reach of the imagination. The e-book concludes that the presence of innate principles in human animals transforms the way in which they sense-perceive the world, thereby constantly increasing the distinction between human and non-human animals. -/- Keywords: human and non-human animals, Leibniz and Locke on ideas, Leibniz on bodies, Leibniz on conceivability, Leibniz on degrees of knowledge, Leibniz on degrees of perception, Leibniz on innate ideas, Leibniz on modality, Leibniz on similarity and congruence, Leibniz on space and time, Leibniz’s philosophy of language, theory of types. (shrink)

The article is centered on the question of how best to understand the logical pluralism/logical monism debate. A number of suggestions are brought up and rejected on the ground that they re...

I argue that mystical experience essentially involves two aspects: an element of direct encounter with God, and an element of union with God. The framework I use to make sense of is taken largely from William Alston’s magisterial book Perceiving God. While I believe Alston’s view is correct in many essentials, the main problem with the account is that it divorces the idea of encountering or perceiving God from the idea of being united with God. What I argue, on (...) the contrary, is that because our experience of God is an experience of a relationship-seeking, personal being, it brings with it an important element of union that Alston overlooks. (shrink)

In the paper, various notions of the logical semiotic sense of linguistic expressions – namely, syntactic and semantic, intensional and extensional – are considered and formalised on the basis of a formal-logical conception of any language L characterised categorially in the spirit of certain Husserl's ideas of pure grammar, Leśniewski-Ajdukiewicz's theory of syntactic/semantic categories and, in accordance with Frege's ontological canons, Bocheński's and some of Suszko's ideas of language adequacy of expressions of L. The adequacy ensures their unambiguous syntactic (...) and semantic senses and mutual, syntactic and semantic correspondence guaranteed by the acceptance of a postulate of categorial compatibility of syntactic and semantic (extensional and intensional) categories of expressions of L. This postulate defines the unification of these three logical senses. There are three principles of compositionality which follow from this postulate: one syntactic and two semantic ones already known to Frege. They are treated as conditions of homomorphism of partial algebra of L into algebraic models of L: syntactic, intensional and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, obviously, an idealisation. The syntactic and semantic unambiguity of its expressions is not, of course, a feature of natural languages, but every syntactically and semantically ambiguous expression of such languages may be treated as a schema representing all of its interpretations that are unambiguous expressions. (shrink)

Max Cresswell and Hilary Putnam seem to hold the view, often shared by classical logicians, that paraconsistent logic has not been made sense of, despite its well-developed mathematics. In this paper, I examine the nature of logic in order to understand what it means to make sense of logic. I then show that, just as one can make sense of non-normal modal logics (as Cresswell demonstrates), we can make `sense' of paraconsistent logic. (...) Finally, I turn the tables on classical logicians and ask what sense can be made of explosive reasoning. While I acknowledge a bias on this issue, it is not clear that even classical logicians can answer this question. (shrink)

There are at least two politically salient senses of “representation”—acting-for-others and portraying-something-as-something. The difference is not just semantic but also logical: relations of representative agency are dyadic (x represents y), while portrayals are triadic (x represents y as z). I exploit this insight to disambiguate constructivism and to improve our theoretical vocabulary for analyzing political representation. I amend Saward’s claims-based approach on three points, introducing the “characterization” to correctly identify the elements of representational claims; explaining the “referent” in pragmatic, not (...) metaphysical terms; and differentiating multiple forms of representational activity. This enables me to clarify how the represented can be both prior to representation and constituted by it, and to recover Pitkin’s idea that representatives ought to be “responsive” to the represented. These points are pertinent to debates about the role of representatives, the nature of representative democracy, and the dynamics of revolutionary movements. (shrink)

Bryant’s paper, "A Logic of Multiplicities: Deleuze, Immanence, and Onticology," is useful for showing how the historical legacy of hierarchy in its many philosophical forms is still present, important, and, in fact, required even by those such as Bryant who would seek to deconstruct or ignore it. The following response will discuss Bryant’s presentation of his alternative position and throughout point out: a) the straw-man versions of hierarchy that Bryant employs; b) why what Bryant claims to be inherent negatively (...) in hierarchy is not the case; c) how Bryant’s position actually relies upon hierarchy for its own explication, and finally; d) the various principles of hierarchical metaphysics that are required in order to make sense of experience and reality. These latter include the notions that i) there are many kinds of hierarchy; ii) Being and Unity are prior to Multiplicity; iii) relation without substance is incoherent; iv) hierarchy is not inherently tyrannical; v) the distinction between essentially and accidentally ordered causal series is a real and useful one; and vi) the existence of hierarchy is more self evident than is flat ontology. (shrink)

This paper aims at clarifying the role which the concept of 'divine sovereignty ' plays in the discussions which are taking place among Islamic thinkers (and others) concerning the possibility of democracy in an Islamic context. It argues that 'sovereignty ' has at least two meanings, one 'f'actual', the other 'normative'. The paper also argues that the second sense of 'sovereignty ' allows us to construe ta!k o{ 'divine sovereignty' as an attempt by Islamic thinkers to go beyond the (...) merely procedural views of democracy which focus on the familiar procedures o( power contestation, election and accountability. (shrink)

Whether mathematical truths are syntactical (as Rudolf Carnap claimed) or empirical (as Mill actually never claimed, though Carnap claimed that he did) might seem merely an academic topic. However, it becomes a practical concern as soon as we consider the role of questions. For if we inquire as to the truth of a mathematical statement, this question must be (in a certain respect) meaningless for Carnap, as its truth or falsity is certain in advance due to its purely syntactical (or (...) formal-semantical) nature. In contrast, for Mill such a question is as valid as any other. These differing views have their consequences for contemporary erotetic logic. (shrink)

Wittgenstein’s Tractatus construes the nature of reasoning in a manner which sharply conflicts with the conventional wisdom that logic is normative, not descriptive of thought. For although we sometimes seem to reason incorrectly, Wittgenstein denies that we can make logical mistakes (5.473). My aim in this paper is to show that the Tractatus provides us with good reasons to rethink some of the central assumptions that are standardly made in thinking about the relation between logic and thought. In (...) particular, the rejection of logical mistakes is to be understood in connection with Wittgenstein’s non-psychological approach to the thinking subject (5.641). On Wittgenstein's view, inference, understanding, and meaning are holistically related; cases of defective reasoning are to be explained in terms of a defective grasp of meaning which manifests in an indeterminate use of signs. Invalid reasoning therefore does not count for Wittgenstein as a species of reasoning, but rather as the mere illusion of reasoning. The rejection of logical mistakes thus gives voice to a radical disjunctivist approach. (shrink)

The notion of logical construction was used by Bertrand Russell in the early 20th century, which originally comes from A. N. Whitehead. Russell said that matter as a mind-independent thing can only be known by description. He also argued that matter is a logical construction of sense-data. However, this leads to an incoherent view of the direct or indirect connection between a mind and the external world. The problem examining is whether a collapsing house is a logical construction of (...) the sense-data of rumbling sounds and collapsing shapes. Using Russell's writings between 1911 and 1918, I will analyze how Russell characterized logical constructions. Finally, I will show Russell’s view about the relation of logical constructions to matter and sense-data. A careful interpretation of Russell's thoughts shows that the contents of the statements of the physical world are not constructions being equivalent to the contents of the sense-datum statements. (shrink)

For Kant, ‘reflection’ is a technical term with a range of senses. I focus here on the senses of reflection that come to light in Kant's account of logic, and then bring the results to bear on the distinction between ‘logical’ and ‘transcendental’ reflection that surfaces in the Amphiboly chapter of the Critique of Pure Reason. Although recent commentary has followed similar cues, I suggest that it labours under a blind spot, as it neglects Kant's distinction between ‘pure’ and (...) ‘applied’ general logic. The foundational text of existing interpretations is a passage in Logik Jäsche that appears to attribute to Kant the view that reflection is a mental operation involved in the generation of concepts from non-conceptual materials. I argue against the received view by attending to Kant's division between ‘pure’ and ‘applied’ general logic, identifying senses of reflection proper to each, and showing that none accords well with the received view. Finally, to take account of Kant's notio.. (shrink)

The paper shows how ideas that explain the sense of an expression as a method or algorithm for finding its reference, preshadowed in Frege’s dictum that sense is the way in which a referent is given, can be formalized on the basis of the ideas in Thomason (1980). To this end, the function that sends propositions to truth values or sets of possible worlds in Thomason (1980) must be replaced by a relation and the meaning postulates governing the (...) behaviour of this relation must be given in the form of a logic program. The resulting system does not only throw light on the properties of sense and their relation to computation, but also shows circular behaviour if some ingredients of the Liar Paradox are added. The connection is natural, as algorithms can be inherently circular and the Liar is explained as expressing one of those. Many ideas in the present paper are closely related to those in Moschovakis (1994), but receive a considerably lighter formalization. (shrink)

Jung's interpretative "matrix" seems to offer us the possibility to frame the social phenomenology concerning the loss of sense, with the consequent load of experience of widespread awkwardness, in a context of epoch-making, progressive, "one-dimensional" reduction of the symbolic. -/- This seems to us the fundamental matrix of the disastrous, schizoid conflict of the present day society: on one side a literalism in keeping with the logics of power and control, disheartening any possibility of individual and collective development and (...) wellbeing; on the other side the absolute impossibility to keep together the fragments of this vision of the world, anachronistic as it appears by now, in an epoch-making realizing of what could be defined as the violence of the "monotheism" of the reason. -/- The pervasive social influence we are subjected to and we suffer even in the ephemeral shelter of our privacy speaks the language of literalism, rejects the oxygenating receptiveness to the metaphor, prefers the poor inclusive significance of the meaning to the deeper (and potentially enlivening source of psychic welfare) but embarrassing liberty of the sense. (shrink)

Philosophers such as Nietzsche, Heidegger, Derrida, Deleuze and Gendlin pronounce that difference must be understood as ontologically prior to identity. They teach that identity is a surface effect of difference, that to understand the basis of logico-mathematical idealities we must uncover their genesis in the fecundity of differentiation. In this paper, I contrast Heidegger’s analyses of the present to hand logico-mathematical object, which he discuses over the course of his career in terms of the ‘as’ structure, temporalization and enframing , (...) with the approaches of Gendlin and Deleuze, supplementing this discussion with Husserl’s investigations of mathematical idealities. Deleuze and Gendlin distinguish between the representational power a logical pattern has in itself, apart from its virtual generative source, to exactly repeat itself, and the way this self-same pattern is generated and changed by the larger situational texture within which it is embedded. In so doing, they misconstrue the empty, meaningless temporalization of logical calculation as the explicitly preserving carrying-through of already instituted implicit sense. For Heidegger, by contrast, logical inference is less a supplement to or development of implicit experience than a narrowing of its scope , a deficient mode of handiness. Experiencing something as present to hand extension modifies the relevant usefulness of ‘as’ structured comportment by stripping away what is meaningful in our relation with beings, and in the process stripping away its intelligibility. Thus, contrary to the assertions of Deleuze and Gendlin, extensive repetition does not carry through intelligible, relevant meaning, it dissolves understanding into the nihilism of empty calculation. (shrink)

Logicism is, roughly speaking, the doctrine that mathematics is fancy logic. So getting clear about the nature of logic is a necessary step in an assessment of logicism. Logic is the study of logical concepts, how they are expressed in languages, their semantic values, and the relationships between these things and the rest of our concepts, linguistic expressions, and their semantic values. A logical concept is what can be expressed by a logical constant in a language. So (...) the question “What is logic?” drives us to the question “What is a logical constant?” Though what follows contains some argument, limitations of space constrain me in large part to express my Credo on this topic with the broad brush of bold assertion and some promissory gestures. (shrink)

This book was completed by the early 1960s and published in 1984 but it has not lost its topicality, for it contains an important re-assessment of the relations of two main streams of contemporary philosophy - the Analytical and the Dialectic. Adherents and critics of these traditions tend to assurnethat they are diametrically opposed, that their roots, concerns and approaches contradict each other, and that no reconciliation is possible. In contradistinction Russell derives both traditions from the common root of the (...) dissatisfaction with the arguments against speculative philosophy. These according to the author leave a lacuna - certain elementsof our Weltanschaaung have been removed, but they cannot be removed without replacement lest we have an incomplete world view, so incomplete in fact that it cannot be viable. According to Russell part of this vacuum is taken up by the analytical tradition but this tradition is not capable of taking up the remainder of it. That portion of the vacant space is however taken up by the dialectical tradi tion, which in turn cannot itself handle the whole of the problem. Thus the two reactions to the demise of speculative philosophy appear to be complementary in at least this sense. But the author goes further, for according to hirn the analytical arguments themselves clearly point to the emergence of dialectical problems, and the dialectical problems themselves need some such background to arise. (shrink)

Understanding Hegel's account of particularity has proven to be anything but straightforward. Two main accounts of particularity have been advanced: the particular as an example or instance and the particular as a subjective perspective on a universal concept. The problem with these accounts is that they reduce particularity either to singularity or to universality. As Derrida's analyses make apparent, the ‘structure of exemplarity’ in Hegel is quite intricate. Hegel uses ‘example’ in three senses: it means (1) ‘instance’, ‘illustration’, or (2) (...) ‘model’, ‘exemplary individual’, ‘paradigm’, or (3) a by-play (a meaning derived from Hegel's neologismbeiherspielen, in whichBeispielis understood quasi-etymologically as a ‘by-play’ of accidental moments). ABeispielin the first sense can be replaced by another instance in a free play (by-play). This play of accidental moments, however, is not entirely free; it generates a series (of replacements) that ultimately leads to an example in the second sense, to an exemplary individual. I argue that particularity can be taken as exemplarity of this kind, oscillating between a singular example and a universal paradigm. Within this by-play, the universal concept, its law, is supposed to be mediated and determined. However, out of the differences between the examples the by-play induces another law, the law of non-mediation, which may, in Derrida's view, actually negate the dialectical movement towards universality. I argue, utilizing Malabou's concept of plasticity, that this disruption may be recovered. This implies that each individual example within a series is a particular determination of the universal. Hence, we can take literally Hegel's claim that the movement of the concept is play. (shrink)

What justifies one interlocutor to challenge the conversational expectations of the other? Paul Grice approaches conversation as one instance of joint action that, like all such action, is governed by the Cooperative Principle. He thinks the expectations of the interlocutors must align, although he acknowledges that expectations can and do shift in the course of a conversation through a process he finds strange. Martin Heidegger analyzes discourse as governed by the normativity of care for self and for another. It is (...) the structure of care that warrants disrupting the presumed cooperative horizon of a conversation in order to occasion some new insight. The chapter expands Heidegger’s ontological conception of care to make sense of the exigencies of conversation. Conversation requires taking cognizance of (1) the human good, (2) the specifics of the conversational context, and (3) one’s responsibilities for the other. This threefold understanding can provide directives for subverting the interlocutor’s expectation for the purposes of a given conversation. (shrink)

Infectious logics are systems that have a truth-value that is assigned to a compound formula whenever it is assigned to one of its components. This paper studies four-valued infectious logics as the basis of transparent theories of truth. This take is motivated as a way to treat different pathological sentences differently, namely, by allowing some of them to be truth-value gluts and some others to be truth-value gaps and as a way to treat the semantic pathology suffered by at least (...) some of these sentences as infectious. This leads us to consider four distinct four-valued logics: one where truth-value gaps are infectious, but gluts are not; one where truth-value gluts are infectious, but gaps are not; and two logics where both gluts and gaps are infectious, in some sense. Additionally, we focus on the proof theory of these systems, by offering a discussion of two related topics. On the one hand, we prove some limitations regarding the possibility of providing standard Gentzen sequent calculi for these systems, by dualizing and extending some recent results for infectious logics. On the other hand, we provide sound and complete four-sided sequent calculi, arguing that the most important technical and philosophical features taken into account to usually prefer standard calculi are, indeed, enjoyed by the four-sided systems. (shrink)

A wide family of many-valued logics—for instance, those based on the weak Kleene algebra—includes a non-classical truth-value that is ‘contaminating’ in the sense that whenever the value is assigned to a formula φ⁠, any complex formula in which φ appears is assigned that value as well. In such systems, the contaminating value enjoys a wide range of interpretations, suggesting scenarios in which more than one of these interpretations are called for. This calls for an evaluation of systems with multiple (...) contaminating values. In this paper, we consider the countably infinite family of multiple-conclusion consequence relations in which classical logic is enriched with one or more contaminating values whose behavior is determined by a linear ordering between them. We consider some motivations and applications for such systems and provide general characterizations for all consequence relations in this family. Finally, we provide sequent calculi for a pair of four-valued logics including two linearly ordered contaminating values before defining two-sided sequent calculi corresponding to each of the infinite family of many-valued logics studied in this paper. (shrink)

This paper develops a criterion for sameness of Fregean senses. I consider three criteria: logical equivalence, intensional isomorphism, and epistemic equipollence. I reject the first two and argue for a version of the third.

The concept of the subject, of what Hegel calls absolute negativity, already appears early in the logic of being.1 Absolute negativity, negation of the negation, occurs throughout the logic as identity in difference understood as self-identification under different descriptions. First, the subject refers to itself merely under an incomplete description. Secondly, it refers to something other than itself under a second description which is logically required by the first. (For example, the description of being in general requires some (...) determinate description of being in particular). But this second description is dialectically excluded by the assumption that the first description is complete. Thirdly, the subject negates its negation of the other. It discovers itself in the other, under the other description, and thus comes to refer to itself less incompletely. This is Hegel in the analytic mode. (shrink)

The reasoning process of analogy is characterized by a strict interdependence between a process of abstraction of a common feature and the transfer of an attribute of the Analogue to the Primary Subject. The first reasoning step is regarded as an abstraction of a generic characteristic that is relevant for the attribution of the predicate. The abstracted feature can be considered from a logic-semantic perspective as a functional genus, in the sense that it is contextually essential for the (...) attribution of the predicate, i.e. that is pragmatically fundamental (i.e. relevant) for the predica-tion, or rather the achievement of the communicative intention. While the transfer of the predicate from the Analogue to the analogical genus and from the genus to the Primary Subject is guaranteed by the maxims (or rules of inference) governing the genus-species relation, the connection between the genus and the predicate can be complex, characterized by various types of reasoning patterns. The relevance relation can hide implicit arguments, such as an implicit argument from classification , an evaluation based on values, consequences or rules, a causal relation, or an argument from practical reasoning. (shrink)

Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence (...) the idea arises of a dual logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics. (shrink)

In previous articles, it has been shown that the deductive system developed by Aristotle in his "second logic" is a natural deduction system and not an axiomatic system as previously had been thought. It was also stated that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument expressible in the language of (...) the system is deducible by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotelian system. (shrink)