In this paper, I would like to point out some problems of the presently reigning functional concept of the logical form of sentences, which presents itself as the final answer to the question of true logical form of sentences and, with this, as the basic scheme of logic. I believe that the present conception of the logical form of sentences is a historical result, which in many ways surpasses all former concepts of logical form in the history of (...) class='Hi'>logic, but which seems not the final concept of the logical form. It contains some immanent limitations which are, in my opinion, linked mainly to the ‘functional’ concept of elementary sentences, which is the foundation of all other logical structures of sentences. (shrink)

In this book, Livingston develops the political implications of formal results obtained over the course of the twentieth century in set theory, metalogic, and computational theory. He argues that the results achieved by thinkers such as Cantor, Russell, Godel, Turing, and Cohen, even when they suggest inherent paradoxes and limitations to the structuring capacities of language or symbolic thought, have far-reaching implications for understanding the nature of political communities and their development and transformation. Alain Badiou's analysis of logical-mathematical (...) structures forms the backbone of his comprehensive and provocative theory of ontology, politics, and the possibilities of radical change. Through interpretive readings of Badiou's work as well as the texts of Giorgio Agamben, Jacques Lacan, Jacques Derrida, Gilles Deleuze, and Ludwig Wittgenstein, Livingston develops a formally based taxonomy of critical positions on the nature and structure of political communities. These readings, along with readings of Parmenides and Plato, show how the formal results can transfigure two interrelated and ancient problems of the One and the Many: the problem of the relationship of a Form or Idea to the many of its participants, and the problem of the relationship of a social whole to its many constituents. (shrink)

Boolos's proof of incompleteness is extended straightforwardly to yield simple “diagonalization-free” proofs of some classical limitative theorems of logic.

Tagmemic theory as a semiotic theory can be used to analyze multiple systems of logic and to assess their strengths and weaknesses. This analysis constitutes an application of semiotics and also a contribution to understanding of the nature of logic within the context of human meaning. Each system of logic is best adapted to represent one portion of human rationality. Acknowledging this correlation between systems and their targets helps explain the usefulness of more than one system. Among (...) these systems, the two-valued system of classical logic takes its place. All the systems of logic can be incorporated into a complex mathematical model that has a place for each system and that represents a larger whole in human reasoning. The model can represent why tight formal systems of logic can be applied in some contexts with great success, but in other contexts are not directly applicable. The result suggests that human reasoning is innately richer than any one formal system of logic. (shrink)

Carnap’s Logical Syntax of Language was one of the first philosophical applications of the results in logical metatheory that appeared in the early 1930s. In using these results, Carnap claimed that he stood in general agreement with Wittgenstein’s Tractatus, but had overcome the limits on the scope of logic that Wittgenstein believed he had found. I argue that Carnap had in fact presupposed a conception of linguistic meaning fundamentally at odds with that presented in the Tractatus, and (...) that he had no non-questioning-begging way of defending this presupposition. As a result Carnap had not so much extended the conception of language and logic present in the Tractatus as he had fundamentally reoriented it, and this reorientation continued with Carnap’s later move to semantics. (shrink)

Contemporary accounts of logic and language cannot give proper treatments of plural constructions of natural languages. They assume that plural constructions are redundant devices used to abbreviate singular constructions. This paper and its sequel, "The logic and meaning of plurals, II", aim to develop an account of logic and language that acknowledges limitations of singular constructions and recognizes plural constructions as their peers. To do so, the papers present natural accounts of the logic and meaning of (...) plural constructions that result from the view that plural constructions are, by and large, devices for talking about many things (as such). The account of logic presented in the papers surpasses contemporary Fregean accounts in its scope. This extension of the scope of logic results from extending the range of languages that logic can directly relate to. Underlying the view of language that makes room for this is a perspective on reality that locates in the world what plural constructions can relate to. The papers suggest that reflections on plural constructions point to a broader framework for understanding logic, language, and reality that can replace the contemporary Fregean framework as this has replaced its Aristotelian ancestor. (shrink)

Logical reconstruction is a fundamental philosophical method for achieving clarity concerning the prerequisites, presuppositions and the logical structure of natural language arguments. The scope and limits of this method have become visible not least through its intense application to Anselm of Canterbury’s notorious proofs for the existence of God. This volume collects, on the one hand, reconstructions of Anselmian arguments that take account of the problems of reconstruction and, on the other hand, theoretical reflections on reconstruction with a view to (...) Anselm. We hope that this will allow the reader to critically assess the merits of the theoretical accounts in the light of the reconstructions and the merits of the reconstructions in the light of the theoretical accounts. In general, by considering the example of an outstanding thinker in one or the other way, this volume aims at advancing the methodology and the practice of logical reconstruction. The following remarks explain and motivate this undertaking: What purposes does logical reconstruction serve? What purposes does the reflection on logical reconstruction serve? In which ways can the practice and theory of reconstruction profit particularly from studying Anselm’s reasoning? How can the resulting problems be addressed? – Finally, a short overview over the contributions to this volume is offered. (shrink)

What is the relation of logic to thinking? My dissertation offers a new argument for the claim that logic is constitutive of thinking in the following sense: representational activity counts as thinking only if it manifests sensitivity to logical rules. In short, thinking has to be minimally logical. An account of thinking has to allow for our freedom to question or revise our commitments – even seemingly obvious conceptual connections – without loss of understanding. This freedom, I argue, (...) requires that thinkers have general abilities to respond to support and tension among their thoughts. And these abilities are constituted by following logical rules. So thinkers have to follow logical rules. But there isn’t just one correct logic for thinking. I show that my view is consistent with logical pluralism: there are a range of correct logics, any one of which a thinker might follow. A logic for thinking does, however, have to contain certain minimal principles: Modus Ponens and Non-Contradiction, and perhaps others. We follow logical rules by exercising logical capacities, which display a distinctive first-person/third-person asymmetry: a subject can find the instances of a rule compelling without seeing them as instances of a rule. As a result, there are two limits on illogical thinking. First, thinkers have to tend to find instances of logical rules compelling. Second, thinkers can’t think in obviously illogical ways. So thinking has to be logical – but not perfectly so. When we try to think, but fail, we produce nonsense. But our failures to think are often subjectively indistinguishable from thinking. To explain how this occurs, I offer an account of nonsense. To be under the illusion that some nonsense makes sense is to enter a pretence that the nonsense is meaningful. Our use of nonsense within the pretence relies on the role of logical form in understanding. Finally, while the normativity of logic doesn’t fall directly out of logical constitutivism, it’s possible to build an attractive account of logical normativity which has logical constitutivism as an integral part. I argue that thinking is necessary for human flourishing, and that this is the source of logical normativity. (shrink)

Quantum theory is our deepest theory of the nature of matter. It is a theory that, notoriously, produces results which challenge the laws of classical logic and suggests that the physical world is illogical. This book gives a critical review of work on the foundations of quantum mechanics at a level accessible to non-experts. Assuming his readers have some background in mathematics and physics, Peter Gibbins focuses on the questions of whether the results of quantum theory require (...) us to abandon classical logic and whether quantum logic can resolve the paradoxes produced by quantum mechanics. He argues that quantum logic does not dispose of the problems faced by classical logic, that no reasonable interpretation of quantum mechanics in terms of 'hidden variables' can be found, and that after all these years quantum mechanics remains a mystery to us. Particles and Paradoxes provides a much-needed and valuable introduction to the philosophy of quantum mechanics and, at the same time, an example of just what it is to do the philosophy of physics. (shrink)

It is a celebrated result of McKinsey and Tarski [28] thatS4is the logic of the closure algebraΧ+over any dense-in-itself separable metrizable space. In particular,S4is the logic of the closure algebra over the realsR, the rationalsQ, or the Cantor spaceC. By [5], each logic aboveS4that has the finite model property is the logic of a subalgebra ofQ+, as well as the logic of a subalgebra ofC+. This is no longer true forR, and the main result of (...) [5] states that each connected logic aboveS4with the finite model property is the logic of a subalgebra of the closure algebraR+.In this paper we extend these results to all logics aboveS4. Namely, for a normal modal logicL, we prove that the following conditions are equivalent: (i)Lis aboveS4, (ii)Lis the logic of a subalgebra ofQ+, (iii)Lis the logic of a subalgebra ofC+. We introduce the concept of a well-connected logic aboveS4and prove that the following conditions are equivalent: (i)Lis a well-connected logic, (ii)Lis the logic of a subalgebra of the closure algebra$\xi _2^ + $over the infinite binary tree, (iii)Lis the logic of a subalgebra of the closure algebra${\bf{L}}_2^ + $over the infinite binary tree with limits equipped with the Scott topology. Finally, we prove that a logicLaboveS4is connected iffLis the logic of a subalgebra ofR+, and transfer our results to the setting of intermediate logics.Proving these general completeness results requires new tools. We introduce the countable general frame property (CGFP) and prove that each normal modal logic has the CGFP. We introduce general topological semantics forS4, which generalizes topological semantics the same way general frame semantics generalizes Kripke semantics. We prove that the categories of descriptive frames forS4and descriptive spaces are isomorphic. It follows that every logic aboveS4is complete with respect to the corresponding class of descriptive spaces. We provide several ways of realizing the infinite binary tree with limits, and prove that when equipped with the Scott topology, it is an interior image of bothCandR. Finally, we introduce gluing of general spaces and prove that the space obtained by appropriate gluing involving certain quotients ofL2is an interior image ofR. (shrink)

In this sequel to "The logic and meaning of plurals. Part I", I continue to present an account of logic and language that acknowledges limitations of singular constructions of natural languages and recognizes plural constructions as their peers. To this end, I present a non-reductive account of plural constructions that results from the conception of plurals as devices for talking about the many. In this paper, I give an informal semantics of plurals, formulate a formal characterization of (...) truth for the regimented languages that results from augmenting elementary languages with refinements of basic plural constructions of natural languages, and account for the logic of plural constructions by characterizing the logic of those regimented languages. (shrink)

Dung’s (1995) argumentation framework takes as input two abstract entities: a set of arguments and a binary relation encoding attacks between these arguments. It returns acceptable sets of arguments, called extensions, w.r.t. a given semantics. While the abstract nature of this setting is seen as a great advantage, it induces a big gap with the application that it is used to. This raises some questions about the compatibility of the setting with a logical formalism (i.e., whether it is possible to (...) instantiate it properly from a logical knowledge base), and about the significance of the various semantics in the application context. In this paper we tackle the above questions. We first propose to fill in the previous gap by extending Dung’s (1995) framework. The idea is to consider all the ingredients involved in an argumentation process. We start with the notion of an abstract monotonic logic which consists of a language (defining the formulas) and a consequence operator. We show how to build, in a systematic way, arguments from a knowledge base formalised in such a logic. We then recall some basic postulates that any instantiation should satisfy. We study how to choose an attack relation so that the instantiation satisfies the postulates. We show that symmetric attack relations are generally not suitable. However, we identify at least one ‘appropriate’ attack relation. Next, we investigate under stable, semi-stable, preferred, grounded and ideal semantics the outputs of logic-based instantiations that satisfy the postulates. For each semantics, we delimit the number of extensions an argumentation system may have, characterise the extensions in terms of subsets of the knowledge base, and finally characterise the set of conclusions that are drawn from the knowledge base. The study reveals that stable, semi-stable and preferred semantics either lead to counter-intuitive results or provide no added value w.r.t. naive semantics. Besides, naive semantics either leads to arbitrary results or generalises the coherence-based approach initially developed by Rescher and Manor (1970). Ideal and grounded semantics either coincide and generalise the free consequence relation developed by Benferhat, Dubois, and Prade (1997), or return arbitrary results. Consequently, Dung’s (1995) framework seems problematic when applied over deductive logical formalisms. (shrink)

Generalizing on some arguments due to Arthur Prior and Dmitry Mirimanoff, we provide some further limitative results on what can be thought.

Is formal logic a failure? It may be, if we accept the context-independent limits imposed by Russell, Frege, and others. In response to difficulties arising from such limitations I present a Toulmin-esque social recontextualization of formal logic. The results of my project provide a positive view of formal logic as a success while simultaneously reaffirming the social and contextual concerns of argumentation theorists, critical thinking scholars, and rhetoricians.

In recent years the question of whether adding the limited principle of omniscience, LPO, to constructive Zermelo–Fraenkel set theory, CZF, increases its strength has arisen several times. As the addition of excluded middle for atomic formulae to CZF results in a rather strong theory, i.e. much stronger than classical Zermelo set theory, it is not obvious that its augmentation by LPO would be proof-theoretically benign. The purpose of this paper is to show that CZF+RDC+LPO has indeed the same strength (...) as CZF, where RDC stands for relativized dependent choice. In particular, these theories prove the same Π02 theorems of arithmetic. (shrink)

According to the received view formal semantics applies to natural language to some extent only. It is so because natural language is inherently indefinite, in particular, its expressions are ambiguous, vague and admits departures from syntactic rule. Moreover, intensional contexts occur in ordinary language—it results in limitations of the principle of compositionality. The ordinary conversation appeals to various principles, for instance, Grice’s maxims which exceed logical formalism. Thus, ordinary language cannot be fully formalized. On the other hand, if L (...) is a formal language, its metalanguage ML, must be partially informal—for instance, it contains, terms of ordinary mathematics, especially set theory. Even, if, for instance, due to the technique of aritmetization, ML can be represented in L, such a representation is only local. In fact, this view can be derived from some Tarski’s remarks on the role played by natural language. It is usually assumed that the universality of natural language, is the source of troubles associated with antinomies. It is so this circumstance requires a solution, for example by distinguishing levels of language. However, even if antinomies are excluded, what is informal is prior with respect to what is formal. It shows that formal semantics has limitations even with respect to formalized languages. (shrink)

This paper introduces a logic game which can be used to demonstrate the working of Boolean connectives. The simplicity of the system turns out to lead to some interesting meta-theoretical properties, which themselves carry a philosophical import. After introducing the system, we demonstrate an interesting feature of it—that it, while being an accurate model of propositional logic Booleans, does not contain any tautologies nor contradictions. This result allows us to make explicit a limitation of application of propositional (...) class='Hi'>logic to those sentences with relatively stable truth values. (shrink)

While stigmatisation is universal, stigma research in low- and middle-income countries (LMIC) is limited. LMIC stigma research predominantly concerns health-related stigma, primarily regarding HIV/AIDS or mental illness from an adult perspective. While there are commonalities in stigmatisation, there are also contextual differences. The aim of this study in DR Congo (DRC), as a formative part in the development of a common stigma reduction intervention, was to gain insight into the commonalities and differences of stigma drivers (triggers of stigmatisation), facilitators (factors (...) positively or negatively influencing stigmatisation), and manifestations (practices and experiences of stigmatisation) with regard to three populations: unmarried mothers, children formerly associated with armed forces and groups (CAAFAG), and an indigenous population. Group exercises, in which participants reacted to statements and substantiated their reactions, were held with the ‘general population’ (15 exercises, n = 70) and ‘populations experiencing stigma’ (10 exercises, n = 48). Data was transcribed and translated, and coded in Nvivo12. We conducted framework analysis. There were two drivers mentioned across the three populations: perceived danger was the most prominent driver, followed by perceived low value of the population experiencing stigma. There were five shared facilitators, with livelihood and personal benefit the most comparable across the populations. Connection to family or leaders received mixed reactions. If unmarried mothers and CAAFAG were perceived to have taken advice from the general population and changed their stereotyped behaviour this also featured as a facilitator. Stigma manifested itself for the three populations at family, community, leaders and services level, with participation restrictions, differential treatment, anticipated stigma and feelings of scapegoating. Stereotyping was common, with different stereotypes regarding the three populations. Although stigmatisation was persistent, positive interactions between the general population and populations experiencing stigma were shared as well. This study demonstrated utility of a health-related stigma and discrimination framework and a participatory exercise for understanding non-health related stigmatisation. Results are consistent with other studies regarding these populations in other contexts. This study identified commonalities between drivers, facilitators and manifestations—albeit with population-specific factors. Contextual information seems helpful in proposing strategy components for stigma reduction. (shrink)

Lorenzo Valla's rhetorical reform of logic resulted in important changes in sixteenth-century mathematical sciences, and not only in mathematical education and in the use of mathematics in other sciences, but also in mathematical theory itself. Logic came to be identified with dialectic, syllogisms with enthymemes and necessary truth with the limit case of probable truth. Two main ancient authorities mediated between logical and mathematical concerns: Cicero and Proclus. Cicero's 'common notions' were identified with Euclid's axioms, so that mathematics (...) could be viewed as core knowledge shared by all humankind. Proclus' interpretation of Euclid's axioms gave rise to the idea of a universal human natural light of reasoning and of a mathesis universalis as a basic mathematics common to both arithmetic and geometry and as an art of thinking interpretable as algebra. (shrink)

In recent years, there has been a growing interest in truthmaker semantics as a framework for understanding a range of phenomena in philosophy and linguistics. Despite this interest, there has been limited study of the various logics that arise from the semantics. This paper aims to address this gap by exploring numerous ‘truthmaker logics’ and proving their compactness and decidability. This is in continuation with the inquiry of Fine and Jago (2019), who proved compactness and decidability for a particular kind (...) of truthmaker logic. The key results going into this are (1) ‘standard translations’ into first-order logic; (2) a truthmaker analogue of the finite model property; and (3) a proof showing that truthmaker consequence on semilattices coincides with truthmaker consequence on complete lattices. Finally, the connection with modal logic is examined. Specifically, it is illustrated how endowing truthmaker semantics with classical negation results in modal information logics. (shrink)

For the given logical calculus we investigate the proportion of the number of true formulas of a certain length n to the number of all formulas of such length. We are especially interested in asymptotic behavior of this fraction when n tends to infinity. If the limit exists it is represented by a real number between 0 and 1 which we may call the density of truth for the investigated logic. In this paper we apply this approach to the (...) intuitionistic logic of one variable with implication and negation. The result is obtained by reducing the problem to the same one of Dummett's intermediate linear logic of one variable (see [2]). Actually, this paper shows the exact density of intuitionistic logic and demonstrates that it covers a substantial part (more than 93%) of classical prepositional calculus. Despite using strictly mathematical means to solve all discussed problems, this paper in fact, may have a philosophical impact on understanding how much the phenomenon of truth is sporadic or frequent in random mathematics sentences. (shrink)

In this thesis I investigate the notion of experience in German Idealist Philosophy. I focus on the exploration of an alternative to the transcendental model notion of experience through Schelling's insight into the notion of logogrif. The structural division of this project into two sections reflects the two theoretical standpoints of this project, namely the logic and the logogrif of experience. The first section - the logic of experience - explores the notion of experience provided in Kant's Critique (...) of Pure Reason, Critique of Judgement and Fichte's Science of Knowledge. I argue that Kant's fundamental question about the possibility of synthetic a priori judgements succeeds in thematising the aporia of cognitive experience but results in a subject-oriented, representational model which radically confines the notion of experience to the constitutive laws of the understanding or to the normative precepts of Reason. Experience is founded upon a sharp division between faith and knowledge, will and logic, desire and reflection, absolute and finitude. Fichte's endeavour to articulate a non-representational account of experience, does not succeed in extricating itself from the representational model, so long as experience is reduced to the ever-producing deeds of the self-positing ego. Despite the serious differences between Kant's and Fichte's notions of experience, both accounts, so long as they unfold from a transcendental standpoint, attempt to resolve experience into conceptual laws or determinations of the ego's absolute will. Experience is transformed into an object of the subject's cognitive or volitional faculties. The paradoxes of man's interaction with the world are intended to be accommodated either by the law-giving spontaneity of the understanding and the Architectonic of Pure Reason or by the overpowerful primordial act of the self-positing ego. This implies the conceptualisation of the self in terms of constant identity-through-time, or sheer self-determination. However, this conceptualisation remains at the normative or prescriptive level, which in turn is projected upon the world. The latter, though appears as the subject's property, essentially remains alien and opaque, confirming the radical limitations of the ego rather than its order-giving authority. Moreover, this notion of experience is ultimately founded upon a radical expulsion of the divine from the world, the de-spiritualisation of the sensual and the de-sensualisation of the spiritual, the sharp juxtaposition between absolute and finitude. This results in a self-defeating subjectivity, whose firm identity and rule-giving authority does not rescue it from its perennial unattainability to 'organise the conditioned' or 'conquer the unconditioned'. In Kant's and Fichte's thought, however, I detect elements that potentially transgress their transcendental account of experience. These are found in Kant's concept of spontaneity and free play between understanding and imagination, and Fichte's concept of productivity. I argue that these elements lose their potential dynamism, so long as they are absorbed by the transcendental demands for the solution of the aporias of logic. However, these elements point to the need for a radical re-conceptualisation of the notion of experience. This is provided by means of Schelling's logogriflic approach, which constitutes the theme of the second section. The second section - the logogrif of experience - attempts to articulate a different approach towards the notion of experience, through an exploration of Schelling's versatile and provocative thought. This section focuses on Schelling's original insight into the notion and act of logogrif, which opens the dialogue between logos and mythos, cosmic becoming and human soul, cosmic imagination and human reflection, faith and knowledge. This section attempts to illuminate Schelling's fascinating philosophical investigations and discoveries that have been rather overlooked, possibly, due to Hegel's overwhelming critique. This section, after a brief critical examination of the Identity Philosophy, attempts to elucidate Schelling's notion of experience through his middle works, Of Human Freedom, Ages of the World, The Deities of Samothrace, which are treated as a self-developing trilogy. Schelling re-addresses the aporias of logic not as part of Reason's self-interrogation but as part of the cosmic paradoxes and living experiences. In this way, Schelling resets the scene of the debate on the conditions of possibility for cognitive experience by putting on the stage the enigmas of the cosmos and life rather than the Tribunal of Reason. Logic itself is conceived as a potency in the cosmic becoming, and consequently can no longer attempt to establish the transcendental conditions for the possibility of cognitive experience. Cosmic becoming, in which man is an active part, is conceived as the process of the movement, the interaction, the transformations and transmutations of multiple potencies. These, far beyond any mechanical conceptualisation, appear as self-moving and yet interdependent, unknown yet familiar, inscrutable and yet manifest powers, describing the mystery of life itself. The latter is depicted as an ever-recurrent act of longing for self-expression as active unity. Experience is conceived as the lived process of a network of living potencies, which may not only resist rational powers but may also puzzle and seize them. In this context, reflection acquires a plastic dimension, as opposed to its rigidity in the representational model of experience. Reflection depicts cosmic longing's self-formation, whose man is part. This self-bending formation partially illuminates the nature of longing, and from this standpoint is the logic of the longing. However, this formation is movable, transmutable and mostly ineffable, and from this standpoint is the logic of a riddle: a logogrif. Logogrif is the transitive term that attempts to describe the transition of experience from its enacted phase to its allusive conceptual utterance, and in this sense the term itself participates in both phases, as both form of thought and form of life. The logogrific approach to experience in turn transposes us as from the realm of pure concepts to the realm of the mystery of life, from pure thought to acts of longing, from the Architectonic of Pure Reason to Cosmic Theurgy. The latter term attempts to grasp the paradox and dynamism of cosmic and non-cosmic becoming by means of multiple, vanishing and ever-recurring, transmutable potencies, or in Schelling's terms 'the magic of insoluble life'. Schelling's logogrific account consists in a powerful voice for the re-enchantment of the world, the introduction into the notion of experience of the imminence of the divine. This is not suggested in terms of the adoption of old religious doctrines but by means of the discovery and re-discovery of the theurgy of life, through the intensification of our artistic mood, the creative expansion of our deeds. This notion of experience allows for the reconsideration of the notion of the self, in terms of a dynamic, conflictual process between conscious and unconscious powers and the critical revaluation of the accounts of subjectivity which reduce it to the sphere of self-consciousness. The thesis concludes with the need for an investigation into the relation between logos and mythos, which only tangentially has been introduced by the present project. In this context it will be possible to re-appraise the potential that the logogrific approach opens for an alternative to both logical and traditional mythological patterns of thinking. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Spectres of Pessimism: A Cultural Logic of the Worst by Mark SchmittJohn StoreyMark Schmitt. Spectres of Pessimism: A Cultural Logic of the Worst. Cham: Palgrave Macmillan, 2023. 147 pp., hardcover, $44.99. ISBN 9783031253508.[End Page 256]What I have called radical utopianism was an important concept for two of the founding figures of British cultural studies, E. P. Thompson and Raymond Williams.1 In 1976, in the revised edition (...) of William Morris: Romantic to Revolutionary, Thompson introduced into English Miguel Abensour's concept of the "education of desire."2 This has had a profound impact on what has become known as utopian studies but has had hardly any influence on cultural studies. Ruth Levitas together with Thompson (from whom she borrowed the concept) have probably done more than most to make Abensour's formulation popular in critical work on utopianism.3Abensour, however, was almost certainly drawing on part of a point made by William Morris in a letter written in 1884. Morris writes of the teaching of desire and political organization as the two main components necessary to bring about radical change: "The means whereby this is brought about is first, educating people into desiring it, next organising them into claiming it."4 But he also writes of another form of education, what he called "the education of discontent."5 In "Art Under Plutocracy," a lecture given in Oxford in 1883, he explained, "It is my business here to-night and everywhere to foster your discontent … [and] to help in educating that discontent into hope, that is into the demand for the new birth of society."6 He made a similar point in a lecture given in Manchester earlier in 1883:My business herein is to spread discontent. I do not think that this is an unimportant office; for, as discontent spreads, the yearning for bettering the state of things spreads with it, and the longing of many people, when it has grown deep and strong, melts away resistance to change in a sure, steady, unaccountable manner.7The role of the education of discontent is to undermine our sense of the inevitability of the here and now, to make change conceivable and to believe that it is possible, promoting what Raymond Williams calls an "active wanting"—"Possibility, seriously considered, is … not what with luck might happen. It is what we can believe in enough to want, and then, by active wanting, make possible."8 For Morris, the production of utopian desire, as a result of the interrogation of the here and now from the perspective of an alternative social space, would, according to Thompson, "help people to find out their wants, to encourage them to want more, to challenge them to want differently, and to envisage a society of the future in which people, freed at last of necessity, might choose between different wants."9 [End Page 257]Radical utopianism allows us to dare to dream that the way things are is not inevitable—that is, a world arranged in the interest of the powerful few. It broadens our vision of the world and opens up new ways of seeing. As the familiar is defamiliarized, what we have learned to think of as immutable suddenly looks changeable and the capitalist dogma There Is No Alternative can begin to give way to the enabling slogan Another World Is Possible. It allows us to anticipate the possibility of a world in which the words of Percy Bysshe Shelley, to slightly misquote them, have a revolutionary insistence: "We are many, they are few."Radical utopianism seeks to free imagination from the limits placed on it by a society run in the interests of the powerful few and to know that these limits are historical. While we seem able to see the past as historical and the future as history waiting to happen, the present seems outside history: in a word, "natural." Radical utopianism makes the present historical. Understood in this way, it is not an attempt to produce blueprints of the perfect society. Given the historical variability of human nature, it is impossible to predict what a perfect society might be like. Instead, radical utopianism is... (shrink)

We extend the results of Laver on using inverse limits to reflect large cardinals of the form, there exists an elementary embedding Lα → Lα. Using these inverse limit reflection embeddings directly and by broadening the collection of U-representable sets, we prove structural results of L under the assumption that there exists an elementary embedding j : L → L. As a consequence we show the impossibility of a generalized inverse limit X-reflection result for X ⊆ Vλ+1, thus (...) focusing the study of L generalizations on L. (shrink)

After the publication of Marshall’s theorem (2009), it has been widely accepted that the intrinsic/extrinsic distinction cannot be analyzed in broadly logical terms, but instead requires appealing to more robust metaphysical notions like grounding, naturalness or duplication. However, this is not so. Instead of showing the limitations of Marshall’s still impressive result, I will present here a broadly logical definition of the intrinsic/extrinsic distinction, and show that it is extensional adequate regardless of our preferred conception of property identity.

ABSTRACT George Gargov was an active pioneer in the ‘Sofia School’ of modal logicians. Starting in the 1970s, he and his colleagues expanded the scope of the subject by introducing new modal expressive power, of various innovative kinds. The aim of this paper is to show some general patterns behind such extensions, and review some very general results that we know by now, 20 years later. We concentrate on simulation invariance, decidability, and correspondence. What seems clear is that ‘modal (...) logic’ as a genre of logical systems has a much wider scope than originally conceived, and that we have not reached its limits yet. (shrink)

The first beginning logic text to employ the tree method--a complete formal system of first-order logic that is remarkably easy to understand and use--this text allows students to take control of the nuts and bolts of formal logic quickly, and to move on to more complex and abstract problems. The tree method is elaborated in manageable steps over five chapters, in each of which its adequacy is reviewed; soundness and completeness proofs are extended at each step, and (...) the decidability proof is extended at the step from truth functions to the logic of nonoverlapping quantifiers with a single variable, after which undecidability is demonstrated by example. The first three chapters are bilingual, with arguments presented twice, in logical notation and in English. The last three chapters consider the discoveries defining the scope and limits of formal methods that marked logic’s coming of age in the 20th century: Godel’s completeness and incompleteness theorems for first and second-order logic, and the Church-Turing theorem on the undecidability of first-order logic. This new edition provides additional problems, solutions to selected problems, and two new Supplements: Truth-Functional Equivalence reinstates material on that topic from the second edition that was omitted in the third, and Variant Methods, in which John Burgess provides a proof regarding the possibility of modifying the tree method so that it will always find a finite model when there is one, and another, which shows that a different modification—once contemplated by Jeffrey--can result in a dramatic speed--up of certain proofs. (shrink)

Gödel’s incompleteness applies to any system with recursively enumerable axioms and rules of inference. Chaitin’s approach to Gödel’s incompleteness relates the incompleteness to the amount of information contained in the axioms. Zurek’s quantum Darwinism attempts the physical description of the universe using information as one of its major components. The capacity of quantum Darwinism to describe quantum measurement in great detail without requiring ad-hoc non-unitary evolution makes it a good candidate for describing the transition from quantum to classical. A baby-universe (...) diffusion model of cosmic inflation is analyzed using quantum Darwinism. In this model cosmic inflation can be approximated as Brownian motion of a quantum field, and quantum Darwinism implies that molecular interaction during Brownian motion will make the quantum field decohere. The quantum Darwinism approach to decoherence in the baby-universe cosmic-inflation model yields the decoherence times of the baby-universes. The result is the equation relating the baby-universe’s decoherence time with the Hubble parameter, and that the decoherence time is considerably shorter than the cosmic inflation period. (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)

The human in continuous century envisage the skepticism. When the human envisage the deficiency of his knowledge, will be in trouble of skepticism, when the knowledge of human fundamentally is doubted, all internal or external impressions will be doubted, so the man envisage the unlimited skepticism. But is it possible and logical? The possibility of it is a psychological question too, but my effort is the epistemological surveying of it. We can survey this question in two ways. One way is (...) justifier reasons of skepticism. Are there sufficient for unlimited skepticism? Every deduction is based on preludes and if they’re uncertain the result will be uncertain too. There any deduction can’t justify unlimited skepticism. The other way is the surveying results of this skepticism. We have beliefs thatnever can doubt them like our existence. This inability express that the unlimited skepticism doesn’t agree the logic of our subject. Moreover the unlimited skepticism brings the mixed continuums. If we doubt anything, doubt this doubting too and this involving haven’t extremity, the demoniac hang that can’t be accepted by our. (shrink)

Semantic paradoxes pose a real threat to logics that attempt to be capable of expressing their own semantic concepts. Particularly, Curry paradoxes seem to show that many solutions must change our intuitive concepts of truth or validity or impose limits on certain inferences that are intuitively valid. In this way, the logic of a universal language would have serious problems. In this paper, we explore a different solution that tries to avoid both limitations as much as possible. Thus, we (...) argue that it is possible to capture the naive concepts of truth and validity without losing any of the valid inferences of classical logic. This approach is called the Buenos Aires plan. We present the logic of truth and validity, $$\mathsf {STTV}_{\omega }$$ based on the hierarchy of logics $$\textsf{ST}_{\omega }$$, whose validity predicate has the same semantic conditions as the material conditional. We argue that $$\mathsf {STTV}_{\omega }$$ is capable of blocking the problematic results while keeping the deductive power of classical logic as much as possible and offering an adequate semantic theory. On the other hand, one could object that it is not possible to reason with $$\mathsf {STTV}_{\omega }$$ because it is not closed under its logical principles. We respond to this objection and argue that the local characterization of validity shows how to make inferences using the logic $$\textsf{ST}_{\omega }$$. (shrink)

The discovery of the importance of mereology follows and does not precede the formalisation of the theory. In particular, it was only after the construction of an axiomatic theory of the part-whole relation by the Polish logician Stanisław Leśniewski that any attempt was made to reinterpret some periods in the history of philosophy in the light of the theory of parts and wholes. Secondly, the push for formalisation - and the individuation of mereology as a specific theoretical field - arise (...) from several particular theories, first from biology and then from psychology. Mereological considerations were explicitly developed within the school of Brentano, by Brentano himself- starting from the problem of the unity of consciousness - and by his students, particularly Stumpl, Twardowski and the Gestaltists. The most complete result of this tradition is undoubtedly the third of Husserl's Logical Investigations, called "the single most important contribution to realist (Aristotelian) ontology in the modern period". (shrink)

Process logic described by A. V. Smirnov is based on materials from the arab-muslim cul­ture as a whole. Process logic is contrasted with the substance logic which forms the foundation of European culture as a whole. It has been proven theoretically that any situation could be interpreted using any logic of sense. The article provides an empirical illustration of the use of the process logic in a consultation of a preschool educational in­stitution psychologist on problems (...) connected with a mental disorder of one of the pupils of the institution. On the basis of this single example it is demonstrated how process logic allows the shift of the focus of attention retaining the rational basis of the actions. Situa­tions relating to the health are difficult to describe using substantial logic as the course of the disease depends on the actions of the patient and on the actions of people in his/her immediate surroundings. This limits the usability of the law of identity which is funda­mental to substantial logic. The disease is “not that very same” disease – it is changing as the result of the actions. This creates a persistent uncertainty which is of remarkable im­portance in pediatric healthcare, where the actions of the parents are complicating the sit­uation much more significantly compared to the adult healthcare. In contrast to this, process logic allows us to observe the course of the ailment as such, as being a process, which eliminates the uncertainty. The process determines the actions regarding the ail­ment as well as their evaluation. If an action is changing the course of the ailment to the better we consider it successful. In the case described, the logic of understanding the situation used by the consultant has changed the client’s logic of understanding the situation. The person stopped seeing the problem as something rigid and insurmount­able. The change of the logic didn’t require any special training of the client. It follows that the shift of understanding from the substance logic to the process logic can help to solve practical pediatric problems. (shrink)

The aim of the article is to present the results of research conducted as part of the project Polonica Philosophica Orientalia: namely, to give an overview of the most important logical and metaphysical treatises written in Vilnius between the sixteenth and eighteenth centuries that are currently accessible in some of the Vilnius libraries. Although the research focused primarily on the Vilnius University Library and its resources, some interesting results were also obtained while researching the Wróblewski Library of the (...) Lithuanian Academy of Sciences. Most sources containing information about the state and the nature of metaphysical and logical investigations conducted in Vilnius during the period discussed are the manuscripts of lecture courses delivered for the students of Vilnius University and other schools. The analysis of those manuscripts has revealed a strong and long-lasting influence of scholastic Aristotelianism in both logic and metaphysics. Namely, up to the mid-eighteenth century, the main questions discussed in logic were Aristotelian syllogism, the square of opposition and the medieval topics of semantical antinomies, modalities and supposition theory, while investigations into metaphysical issues were limited to commentaries on Aristotle’s theoretical treatises without seriously questioning their main theses. In the second part of the eighteenth century, a moderate shift towards subjectivism and psychologism was observed in both of these disciplines. This was influenced by modern epistemologists. Logicians and metaphysicians of Vilnius University developed an interest in the topics of human understanding and cognition. However, this shift did not take a radical form – while the majority of eighteenth-century authors sought a combination of scholastic and modern ideas, a critical and even hostile stance towards modern philosophy and science is still noticeable in many of the analysed sources. An overview of the research supported by statistical data is followed by some preliminary remarks on how and where investigation of this topic might be continued in the future. (shrink)

How do people reason about their opponent in turn-taking games? Often, people do not make the decisions that game theory would prescribe. We present a logic that can play a key role in understanding how people make their decisions, by delineating all plausible reasoning strategies in a systematic manner. This in turn makes it possible to construct a corresponding set of computational models in a cognitive architecture. These models can be run and fitted to the participants’ data in terms (...) of decisions, response times, and answers to questions. We validate these claims on the basis of an earlier game-theoretic experiment about the turn-taking game “Marble Drop with Surprising Opponent”, in which the opponent often starts with a seemingly irrational move. We explore two ways of segregating the participants into reasonable “player types”. The first way is based on latent class analysis, which divides the players into three classes according to their first decisions in the game: Random players, Learners, and Expected players, who make decisions consistent with forward induction. The second way is based on participants’ answers to a question about their opponent, classified according to levels of theory of mind: zero-order, first-order and second-order. It turns out that increasing levels of decisions and theory of mind both correspond to increasing success as measured by monetary awards and increasing decision times. Next, we use the logical language to express different kinds of strategies that people apply when reasoning about their opponent and making decisions in turn-taking games, as well as the ‘reasoning types’ reflected in their behavior. Then, we translate the logical formulas into computational cognitive models in the PRIMs architecture. Finally, we run two of the resulting models, corresponding to the strategy of only being interested in one’s own payoff and to the myopic strategy, in which one can only look ahead to a limited number of nodes. It turns out that the participant data fit to the own-payoff strategy, not the myopic one. The article closes the circle from experiments via logic and cognitive modelling back to predictions about new experiments. (shrink)

This book is a thorough study of the question posed by Kant, For what can a human being rationally hope? It offers a detailed commentary on Kant's seminal work, Religion Within the Limits of Reason Alone, as well as an original development of the logic of three of Kant's basic ideas: ambivalence, ignorance, and hope. Sophisticated analytic techniques, including symbolic logic, are applied to this conceptual matrix. The result is a striking case for the transformation of world society (...) into a Kingdom of Ends of individuals and a peaceful League of Nations. (shrink)

The dissertation considers both metaphysical and logical issues related to the vagueness of natural language. The principle metaphysical claim is that the vagueness of language is, at least in some cases, a direct result of indeterminacy in the subject matter of the language, rather than any sort of flaw of the language. A limited defense of this claim is given, as well as criticism of alternative views. ;A number of logical issues are addressed. First, the relationship between the notion of (...) determinacy and the idea of an unsharp line is considered, and it is suggested that the relationship is not as simple as it may at first seem, and that the idea of the unsharp line may be irreducible. Next, it is urged that there is a methodological fork in the road for the systematic treatment of vague language, including formal semantics. On one path, we accept certain intuitively puzzling propositions, exemplified by "This is red or it is not the case that this is red, though it is indeterminate which." On the other path, we reject classical principles of reasoning in our own reasoning both in and about vague language. Some limited arguments are given for taking the former path, and a formal system relevant to this path is motivated and discussed. Finally, both the metaphysical and logical work of the prior parts of the dissertation are brought to bear on the subject of the indeterminacy of identity. (shrink)

The logical omniscience problem, whereby standard models of epistemic logic treat an agent as believing all consequences of its beliefs and knowing whatever follows from what else it knows, has received plenty of attention in the literature. But many attempted solutions focus on a fairly narrow specification of the problem: avoiding the closure of belief or knowledge, rather than showing how the proposed logic is of philosophical interest or of use in computer science or artificial intelligence. Sentential epistemic (...) logics, as opposed to traditional possible worlds approaches, do not suffer from the problems of logical omniscience but are often thought to lack interesting epistemic properties. In this paper, I focus on the case of rule-based agents, which play a key role in contemporary AI research but have been neglected in the logical literature. I develop a framework for modelling monotonic, nonmonotonic and introspective rule-based reasoners which have limited cognitive resources and prove that the resulting models have a number of interesting properties. An axiomatization of the resulting logic is given, together with completeness, decidability and complexity results. (shrink)

Current orthodoxy in modal logic and metaphysics has it that actuality is non-contingent in the following sense: for all p, if actually, p, then necessarily, actually, p. Call this thesis (Actuality) Necessitism and its negation (Actuality) Contingentism. Thus, according to Contingentism, there is at least one proposition p which is actually true but which could have been actually false. In another paper, one of us (Glazier 2023) has recently defended Contingentism. The present paper explores the logic of actuality (...) under Contingentism. After informally introducing Contingentism and outlining what we take to be the main motivations for accepting it, we consider how contingent actuality should be taken to be. We argue that the nature, or essence, of actuality imposes certain limits on the extent to which actuality could have been different, and these then inform our view of the logic of actuality. Turning to the formal study of the logic of contingent actuality, we specify a formal language and a possible world semantics for that language and distinguish a number of natural notions of validity and consequence to which it gives rise. We present axiomatizations of the different resulting logics and examine the variety of iterated modalities in our systems. We establish soundness and completeness results. (shrink)

This essay explores Hegel’s treatment of Carl Friedrich Gauss’s mathematical discoveries as examples of “Analytic Cognition.” Unfortunately, Hegel’s main point has been virtually lost due to an editorial blunder tracing back almost a century, an error that has been perpetuated in many subsequent editions and translations.The paper accordingly has three sections. In the first, I expose the mistake and trace its pervasive influence in multiple languages and editions of the Wissenschaft der Logik. In the second section, I undertake to explain (...) the mathematical significance of Gauss’s discoveries. In the third section, I take a look at the deeper implications of Hegel’s treatment of Gauss’s work as a window onto the nature and limitations of analytic cognition. In conclusion, I seek to explain how the linear method embodied in deductive reason leads by its own inner principle, according to Hegel, to its dialectical Aufhebung (sublation). The result is a kind of deliberately circular reasoning that he describes as “the Absolute Idea.”. (shrink)

For decades, much literature on causality has focused on causal processes and causal reasoning in the natural sciences. According to a relatively new trend however, such research on causality remains insufficient because of its refusal to accept a certain degree of pluralism within the concept, a pluralism that is evident in how we use ideas of cause and effect in everyday life. I will build on work in this latter trend, following philosophers like G. E. M. Anscombe and N. Cartwright. (...) I explore the limits of the concept of causality by determining the extent to which our ideas can remain consistent as we stretch this concept along two dimensions, one concerning the maximizing of the effect and the other the maximizing of explanatory depth. Dealing with the cause of the universe, such an investigation touches upon some issues in current empirical cosmology and revisits some classic arguments regarding philosophy of religion and the Platonic notion of participation. The results indicate that the use of current conceptual, logical and analytic tools can deliver new insights that are useful especially for those interested in how causality in the natural sciences links with causality in everyday life. (shrink)

The Logic of Causation: Definition, Induction and Deduction of Deterministic Causality is a treatise of formal logic and of aetiology. It is an original and wide-ranging investigation of the definition of causation (deterministic causality) in all its forms, and of the deduction and induction of such forms. The work was carried out in three phases over a dozen years (1998-2010), each phase introducing more sophisticated methods than the previous to solve outstanding problems. This study was intended as part (...) of a larger work on causal logic, which additionally treats volition and allied cause-effect relations (2004). The Logic of Causation deals with the main technicalities relating to reasoning about causation. Once all the deductive characteristics of causation in all its forms have been treated, and we have gained an understanding as to how it is induced, we are able to discuss more intelligently its epistemological and ontological status. In this context, past theories of causation are reviewed and evaluated (although some of the issues involved here can only be fully dealt with in a larger perspective, taking volition and other aspects of causality into consideration, as done in Volition and Allied Causal Concepts). Phase I: Macroanalysis. Starting with the paradigm of causation, its most obvious and strongest form, we can by abstraction of its defining components distinguish four genera of causation, or generic determinations, namely: complete, partial, necessary and contingent causation. When these genera and their negations are combined together in every which way, and tested for consistency, it is found that only four species of causation, or specific determinations, remain conceivable. The concept of causation thus gives rise to a number of positive and negative propositional forms, which can be studied in detail with relative ease because they are compounds of conjunctive and conditional propositions whose properties are already well known to logicians. The logical relations (oppositions) between the various determinations (and their negations) are investigated, as well as their respective implications (eductions). Thereafter, their interactions (in syllogistic reasoning) are treated in the most rigorous manner. The main question we try to answer here is: is (or when is) the cause of a cause of something itself a cause of that thing, and if so to what degree? The figures and moods of positive causative syllogism are listed exhaustively; and the resulting arguments validated or invalidated, as the case may be. In this context, a general and sure method of evaluation called ‘matricial analysis’ (macroanalysis) is introduced. Because this (initial) method is cumbersome, it is used as little as possible – the remaining cases being evaluated by means of reduction. Phase II: Microanalysis. Seeing various difficulties encountered in the first phase, and the fact that some issues were left unresolved in it, a more precise method is developed in the second phase, capable of systematically answering most outstanding questions. This improved matricial analysis (microanalysis) is based on tabular prediction of all logically conceivable combinations and permutations of conjunctions between two or more items and their negations (grand matrices). Each such possible combination is called a ‘modus’ and is assigned a permanent number within the framework concerned (for 2, 3, or more items). This allows us to identify each distinct (causative or other, positive or negative) propositional form with a number of alternative moduses. This technique greatly facilitates all work with causative and related forms, allowing us to systematically consider their eductions, oppositions, and syllogistic combinations. In fact, it constitutes a most radical approach not only to causative propositions and their derivatives, but perhaps more importantly to their constituent conditional propositions. Moreover, it is not limited to logical conditioning and causation, but is equally applicable to other modes of modality, including extensional, natural, temporal and spatial conditioning and causation. From the results obtained, we are able to settle with formal certainty most of the historically controversial issues relating to causation. Phase III: Software Assisted Analysis. The approach in the second phase was very ‘manual’ and time consuming; the third phase is intended to ‘mechanize’ much of the work involved by means of spreadsheets (to begin with). This increases reliability of calculations (though no errors were found, in fact) – but also allows for a wider scope. Indeed, we are now able to produce a larger, 4-item grand matrix, and on its basis find the moduses of causative and other forms needed to investigate 4-item syllogism. As well, now each modus can be interpreted with greater precision and causation can be more precisely defined and treated. In this latest phase, the research is brought to a successful finish! Its main ambition, to obtain a complete and reliable listing of all 3-item and 4-item causative syllogisms, being truly fulfilled. This was made technically feasible, in spite of limitations in computer software and hardware, by cutting up problems into smaller pieces. For every mood of the syllogism, it was thus possible to scan for conclusions ‘mechanically’ (using spreadsheets), testing all forms of causative and preventive conclusions. Until now, this job could only be done ‘manually’, and therefore not exhaustively and with certainty. It took over 72’000 pages of spreadsheets to generate the sought for conclusions. This is a historic breakthrough for causal logic and logic in general. Of course, not all conceivable issues are resolved. There is still some work that needs doing, notably with regard to 5-item causative syllogism. But what has been achieved solves the core problem. The method for the resolution of all outstanding issues has definitely now been found and proven. The only obstacle to solving most of them is the amount of labor needed to produce the remaining (less important) tables. As for 5-item syllogism, bigger computer resources are also needed. (shrink)

In the Preface of the Tractatus Wittgenstein presents his proposal of “drawing limits” separating sense from nonsense as a way to get rid of philosophical problems caused by “misunderstandings of the logic of our language.” Such limits, we will later discover, will be drawn by means of a method which allows one to determine whether a given projection of a strings of signs was made in accordance with the rules of logical syntax, or else violated them, thus generating metaphysical (...) propositions. Notwithstanding its centrality for the Tractatus, the idea of drawing such limits seems to be in tension with Wittgenstein’s actual procedure in most of the book, which from its very first numbered proposition introduces “metaphysical” theses again and again in order to achieve the results programmatically indicated in the Preface—hence the need for the self-undoing message of 6.54, urging the reader to recognize those propositions “as nonsensical, when he has used them—as steps—to climb up beyond them.” That tension creates some of the most challenging questions in the debate about how to read the Tractatus—questions such as: how are we supposed to use Wittgenstein’s propositions as “steps in a ladder”? What exactly does “throwing the ladder away” amount to? And what does it mean to “see the world aright” after “overcoming” those propositions? This paper attempts to answer those questions by means of a close reading of an exemplary set of propositions dealing with solipsism and the limits of language. Although limited in scope, the hope is that such reading might stand as a test case for parallel readings of other parts of the book. (shrink)

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it (...) to the study of deduction, without providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

There is a well-known gap between metamathematical theorems and their philosophical interpretations. Take Tarski’s Theorem. According to its prevalent interpretation, the collection of all arithmetical truths is not arithmetically definable. However, the underlying metamathematical theorem merely establishes the arithmetical undefinability of a set of specific Gödel codes of certain artefactual entities, such as infix strings, which are true in the standard model. That is, as opposed to its philosophical reading, the metamathematical theorem is formulated (and proved) relative to a specific (...) choice of the Gödel numbering and the notation system. A similar observation applies to Gödel’s and Church’s theorems, which are commonly taken to impose severe limitations on what can be proved and computed using the resources of certain formalisms. The philosophical force of these limitative results heavily relies on the belief that these theorems do not depend on contingencies regarding the underlying formalisation choices. The main aim of this paper is to provide metamathematical facts which support this belief. While employing a fixed notation system, I showed in previous work (_Review of Symbolic Logic_, 2021, 14(1):51–84) how to abstract away from the choice of the Gödel numbering. In the present paper, I extend this work by establishing versions of Tarski’s, Gödel’s and Church’s theorems which are invariant regarding _both_ the notation system and the numbering. This paper thus provides a further step towards absolute versions of metamathematical results which do not rely on contingent formalisation choices. (shrink)

We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II1factor as Fraïssé limits of suitable classes of structures. Moreover by means of Fraïssé theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.

Algorithmic chemistries are often based on a fixed formalism which limits the fragment of chemistry expressible in the domain of the models. This results in limited applicability of the models in contemporary mathematical chemistry and is due to the poor fit between the logic used for model construction and the system being modeled. In this paper, I propose a system-oriented methodology which selects a formalism through a mapping of chemical transformation rules to proof-theoretic structural rules. Using a formal (...) specification framework from the field of artificial chemistry, expressive adequacy is ensured by the choice of logic being based on the system properties to be modeled. To illustrate the methodology, a case study is provided that shows how the proposed approach selects linear logic for modeling resource sensitivity and the proof-theoretic interpretation facilitates translation to a programming language. Since the method results in a plurality of models, I conclude with a discussion on how the proposal contributes to multi-model paradigms in computational pharmacology. (shrink)

The notion of strength has featured prominently in recent debates about abductivism in the epistemology of logic. Following Williamson and Russell, we distinguish between logical and scientific strength and discuss the limits of the characterizations they employ. We then suggest understanding logical strength in terms of interpretability strength and scientific strength as a special case of logical strength. We present applications of the resulting notions to comparisons between logics in the traditional sense and mathematical theories.

The paper tackles two problems. The first one is to grasp the real meaning of Jerzy Kalinowski’s theory of normative sentences. His formal system K 1 is a simple logic formulated in a very limited language . While presenting it Kalinowski formulated a few interesting philosophical remarks on norms and actions. He did not, however, possess the tools to formalise them fully. We propose a formulation of Kalinowski’s ideas with the use of a set-theoretical frame similar to the one (...) presented by Krister Segerberg in his A Deontic Logic of Action. At the same time we enrich the language used by Kalinowski with more operators on actions and present an adequate axiomatisation of the resulting system. That allows us to disclose some unrevealed aspects of Kalinowski’s theory. The most important one is a relation between acts which we call moral indiscernibility. Our second problem is a proper understanding of moral indiscernibility. We show how a repertoire of agent’s actions, defined with the use of simple observable elements of actions, can be filtrated by the relation of moral indiscernibility. That allows us to understand the consequences of Kalinowski’s claim that not doing something good is always bad. (shrink)

One of the main criteria for an adequate semantic theory is that it solve the problem of substitution into intensional contexts, otherwise known as Frege's Puzzle. Given common-sense assumptions about how natural language functions, a contradiction arises in explaining attitude reports. For example, Lisa might believe that Twain is tall, but not believe that Clemens is tall. Lisa is perhaps unaware that the names "Twain" and "Clemens" corefer. But Twain's being tall is just Clemens' being tall, so one and the (...) same state of affairs is either believed or not. The tasks are to specify what Lisa believes and to accurately and consistently describe the situation given certain foundational semantic principles. ;I argue that of three main semantic frameworks: Davidsonian Semantics, Model-Theoretic Semantics and Propositional Semantics , PS has the best chance to provide a complete, consistent and natural solution. On my view, dynamic propositions are complex states of affairs where things stand in the propositional relation of having expressions of a certain type which pick them out as semantic values and where those expressions stand in appropriate syntactic relations. The propositional relation crucially involves restricted quantification over expression types which are logical features of the language. The sentences "Twain is tall" and "Clemens is tall" express different DP's because even though their constituents are identical, they stand in distinct propositional relations in virtue of the different syntactic expressions. This explains how Lisa could believe one and not the other. These fine-grained propositions successfully serve as both the truth-conditions of sentences and the objects of propositional attitudes. The solution leads to the surprising result that the question of whether singular terms are Fregean or directly referential is actually irrelevant to Frege's Puzzle. ;While DP's are naturalistic and theoretically well-motivated, they face apparent difficulty accounting for translation and being the objects of mental content. The logical form of DP's limits their possible interpretation and thus sheds light on the conceptual framework of its user. I argue that DP's are not beliefs, but cognitive models that serve as interpretations of beliefs in communication. (shrink)