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)

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)

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)

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)

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)

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.

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)

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)

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)

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)

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)

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)

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)

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.

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)

This paper investigates structural properties of monotone function classes within the framework of three-valued logic (3VL), aiming to characterize dependencies and constraints that ensure structural finiteness and order-preserving properties. This research delves into characteristics of structurally finite classes of order-preserving 3VL map. Monotonicity plays a critical role in understanding functional behaviour, which is essential for structuring closed logical operations within $ P_{k} $. We define $ F $ as a closed class in $ P_{k} $, consisting only of mappings (...) that comply with specified operations and are strictly confined to $ P_{k} $. The notation $ F(n) $ represents the subset of mappings in $ F $ dependent on $ n $ variables, highlighting the limited scope of inputs and their respective outputs. Further analysis through $ CR(F) $ facilitates the identification of precursor subclasses within $ F $, which have not yet achieved closure under all necessary operations—a pivotal step in constructing intermediate mapping classes. Within the framework of 3VL, the study examines order-preserving maps $ f_{D}, f_{K}, f_{M^{(2)}}, f_{DM^{(2)}}, f_{KM^{(2)}} $, deliberately excluding the sets $ D, K, M^{(2)} $, where $ DM^{(2)} $ is defined as $ D \cap M^{(2)} $ and $ KM^{(2)} $ as $ K \cap M^{(2)} $. The lattice framework in $ P_{k}(1) $ for $ k = 3 $ systematically organizes these mapping classes based on their completeness properties, with particular emphasis on classes such as $ M $, which is characterized as a comprehensive set governed by the operations of maximum and minimum. Our theorems establish structural finiteness for classes of unary order-preserving maps, emphasizing their two-variable dependency in class construction. This study extends existing findings on single-variable monotonic functions within 3VL by integrating criteria of linear order. Key results include the classification of mapping classes $ F \subseteq M $ containing $ M^{(1)}_{3} $, categorized into distinct groups: $ M^{(1)}_{3}, KM^{(2)}, DM^{(2)}, M^{(2)}, K, D, M $. Inclusion relations are visualized diagrammatically, showing the hierarchical structure of these monotone classes, ultimately validating the structural finiteness of classes incorporating all unary order-preserving maps within the broader context of 3VL. (shrink)

In the fiqhī doctrine, the issue of nafy al-madārik does not find a place for itself with this very nomenclature, and in the classical approach, it is often referred to by different names, but we have found it appropriate to use this term in order to make the subject more understandable and to lead to a more practical use of the expression rather than a sentence. One of the main reasons why we find the subject worthy of research is that (...) there are almost no independent and original studies on it. Another reason was to determine the relationship between the two, since those who frequently and openly do not give it a chance examine it under the title of istishāb. In the process, our multidisciplinary reading tendency led us to realize how similar some of the terminological expressions that appear in modern logic are. In Islamic legal methodology, the method of “absence of evidence/nafy al-madārik”, which plays an important role in the knowledge of basic methods for experts who will deduce religious rulings from the relevant sources, is mostly discussed together with the evidence of istishāb and not analyzed in a separate context. In the absence of evidence that would constitute the ruling on any subject, jurists have mentioned this method, which states that there is no ruling because there is no evidence. The issue of the absence of evidence, which is usually analyzed by the Hanafīs under the heading of fasid evidence, was occasionally used by other jurists. If the absence of evidence does not result in the absence of the ruling, the scholars have resorted to the evidence of istishāb, which is the opinion that the ruling continues to exist. However, as we have mentioned, the mixed treatment of both methods has led to the problem of where they should be positioned in the hierarchical evidence platform and how they should be perceived. The fact that a sixth prayer was not commanded upon the necessity of the existence of the five daily prayers is one of the main examples of the method of nafy. Accordingly, it is possible to rule that the sixth prayer does not exist in the absence of any evidence showing that the legislator did not impose a new one on top of the five rak'ahs and vice versa. In terms of its scope, the issue of the absence of evidence is not only used in classical fiqh. In fact, the breadth of this subject should not be limited to Islamic sciences. In fact, the method of absence of evidence also guides the modern understanding of science. The method in question, which is mostly expressed as the absence of evidence, is rejected by a considerable number of popular acceptance. Science, so to speak, does not see the absence of evidence as a correct approach. A famous and striking example of the absence of evidence is as follows: “There is no evidence for the existence of aliens on Earth, therefore there are no aliens, or aliens do not exist, based on the lack of evidence for this”.In this study, first of all, the elements constituting the phrase “absence of evidence” are analyzed. The method of absence of evidence is discussed together with its debates and parties in terms of the meaning formed as a result of the examination. Its relationship with istishāb has been analyzed in terms of its relevance. In doing so, we have also included whether it is sufficient for the person who claims that the ruling does not exist due to the absence of the relevant evidence to prove his claim in a mere way. While trying to reveal all these, we have also tried to address how correct it is for the person who claims that there is no evidence for the relevant ruling in the method of absence of evidence to make a decision based solely on this. In addition, we have discussed the issue of absence of evidence together with some issues of various disciplines such as kalam and logic due to its relevance. As a result of the discussions, we tried to show that the method of nafy and istishab are not the same thing. Following these, some evaluations were made about the theory of nafys. Among these evaluations, there are some interpretations that are close to the Hanafīs' stance on this issue and some determinations about the fallacy of nafy in terms of the science of logic. As a result, an independent study on nafy al-madārīq has been presented to you, the reader, based on a logical approach to it. However, this attempt will not completely fill the gap in the literature on the subject. We hope that this study will serve as an opportunity for further research to fill the aforementioned gap, and that the subject will reach a wider audience. (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)

B a c k g r o u n d. The article examines the concept of human rights. This concept in modern philosophical discourse is both vague and imprecise. Although most authors recognize the lack of unanimity in the definition of the concept of human rights, few have tried to find out the reasons for this situation. The purpose of this study is to identify the sources of the problematic definition of the concept of human rights by means of a (...) logical analysis of the concept and to propose ways to overcome the identified difficulties in the interpretation of this concept. M e t h o d s. A philosophical hermeneutic approach combined with a comparative and terminological analysis of the concept of human rights was used in the study. R e s u l t s. Human rights discourse is fundamentally open: it is normatively undetermined, and anyone can join it. It is therefore inevitably polyphonic and multilingual. As a result, this discourse contains very different understandings of the most important thing: what human rights really are. Efforts to find a universal definition of the concept of human rights seem to be an attempt to unite under one roof things that are poorly or not at all combined: different ontological understandings, different scopes of the concept, and different logical characteristics. C o n c l u s i o n s. The difficulty of logical analysis of the concept of human rights, i.e. its assignment to a specific type, and its unambiguous interpretation and definition, is related to: 1) the lack of definition of the limits of scope of the concept of human rights in philosophical discourse and the insufficient definition of the limits of this concept as a legal notion; 2) the need to assess the legitimacy of the retrospective extrapolation of the term ‘human rights' to the entire body of works created before the adoption of the Universal Declaration of Human Rights, in which it was not used (in the Ukrainian context, this aspect is not obvious due to historical and linguistic peculiarities); 3) different ontological understanding of human rights by different actors in the discourse that makes it impossible to combine their positions under the same umbrella (e.g. according to one position, human rights are seen as anthropological attributes of a person; according to another, they are norms regulating social relations). The above testifies to the complexity of an unambiguous logical characteristic of the analyzed concept. It seems that thе situation does not allow for a single definition of the concept of "human rights". It is assumed that it is not a single concept, but a plurality of concepts. In this case, the position of the discourse participant in relation to a certain system of criteria and its justification may be a way to overcome the difficulties we have identified. (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)

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)

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)

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)

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)

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)

Self-reference and the Limits of Thought. This paper explores the connection between the natural language and a formal language from a particular point of view: self-referential constructions. Such constructions lead to some kind of limits of thought, either in the form of paradoxical constructions (Liar-type or Grelling-type), or in the form of the so called limitative theorems in mathematical logic (e.g. Gödel’s theorem). By deriving Gödel’s significant results from paradoxical constructions the limitative character of such self-referential (...) constructions is preserved, but they open the ways for a new representation of a great variety of arguments in the field of logic, mathematics and philosophy. (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)

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)

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)

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)

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)

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)

This book uses philosophical analysis to argue that there are tensions associated with using results of high stakes tests to predict students’ future potential. The implications of these issues for the interpretation of test scores in general are then elucidated before their connotations for academic selection are considered. After a brief overview of the history of academic selection in the United Kingdom, and a review of evidence pertaining to its consequences, it is argued that the practice of using the (...) results of contemporary high stakes tests to make important decisions about students incurs logical and moral problems that a conscientious educator cannot ignore. The gravity of the moral transgression depends on the purpose and significance of the test and, in the case of high stakes tests used for academic selection purposes, it is argued that, not only can the moral wrong be highly significant, but better solutions are within reach. (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.

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)

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 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)

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)

In Part I of this paper, we identified and compared various schemes for trivalent truth conditions for indicative conditionals, most notably the proposals by de Finetti and Reichenbach on the one hand, and by Cooper and Cantwell on the other. Here we provide the proof theory for the resulting logics DF/TT and CC/TT, using tableau calculi and sequent calculi, and proving soundness and completeness results. Then we turn to the algebraic semantics, where both logics have substantive limitations: DF/TT allows (...) for algebraic completeness, but not for the construction of a canonical model, while CC/TT fails the construction of a Lindenbaum-Tarski algebra. With these results in mind, we draw up the balance and sketch future research projects. (shrink)

Assuming the existence of a monster model, tameness, and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $\mu>\operatorname {LS}(\mathbf {K})$ be a regular stability cardinal and let $\chi $ be the local character of $\mu $ -nonsplitting. The following holds: 1.When $\mu $ -nonforking is restricted to $(\mu,\geq \chi )$ -limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension, and continuity. It also has local character $\chi $. This (...) generalizes Vasey’s result [37, Corollary 13.16] which assumed $\mu $ -superstability to obtain same properties but with local character $\aleph _0$.2.There is $\lambda \in [\mu,h(\mu ))$ such that if $\mathbf {K}$ is stable in every cardinal between $\mu $ and $\lambda $, then $\mathbf {K}$ has $\mu $ -symmetry while $\mu $ -nonforking in (1) has symmetry. In this case:(a) $\mathbf {K}$ has the uniqueness of $(\mu,\geq \chi )$ -limit models: if $M_1,M_2$ are both $(\mu,\geq \chi )$ -limit over some $M_0\in K_{\mu }$, then $M_1\cong _{M_0}M_2$ ;(b)any increasing chain of $\mu ^+$ -saturated models of length $\geq \chi $ has a $\mu ^+$ -saturated union. These generalize [31] and remove the symmetry assumption in [10, 38].Under $(<\mu )$ -tameness, the conclusions of (1), (2)(a)(b) are equivalent to $\mathbf {K}$ having the $\chi $ -local character of $\mu $ -nonsplitting.Grossberg and Vasey [18, 38] gave eventual superstability criteria for tame AECs with a monster model. We remove the high cardinal threshold and reduce the cardinal jump between equivalent superstability criteria. We also add two new superstability criteria to the list: a weaker version of solvability and the boundedness of the U-rank. (shrink)

Although the cognition is significant in strategic reasoning, its role has been weakly analyzed, because only the average intelligence is usually considered. For example, prisoner's dilemma in game theory, would have different outcomes for persons with different intelligence. I show how various levels of intelligence influence the quality of reasoning, decision, or the probability of psychosis. I explain my original methodology developed for my MA thesis in clinical psychology in 1998, and grant research in 1999, demonstrating the bias of the (...) classic IQ method, and how the intelligence limits thinking. Based on that I defined Personality Model, providing insight into understanding of psychosis (schizophrenia, bi-polar), which has not been explained yet by psychology or psychiatry. In addition, it enables to analyze and assess non-linear problems, utilizable in computer programming, visualization (animation) or other fields including Baduk game. I've already applied some principles in complex information system www.each.co.uk, and video-animations exhibited in London, Germany, Tokyo. I need to mention my experience in chess composition between 1994 and 2000, winning a few international prizes and inventing a special class of fairy rules redefining the mate. The chess composition principles or patterns show the way to organize logical series to higher advanced mechanisms (like calculus), applicable to other fields. One of such principles is a logical aesthetic innovation: new strategy, defined by Italian composers. Finally I show how the simple redefinition of the classic utility concept links economics and psychology to explain irrational / destructive behavior. All presented results (from the research) can be repeated. (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)

We give a new characterization of the hyperarithmetic sets: a set X of integers is recursive in e α if and only if there is a Turing machine which computes X and "halts" in less than or equal to the ordinal number ω α of steps. This result represents a generalization of the well-known "limit lemma" due to J. R. Shoenfield [Sho-1] and later independently by H. Putnam [Pu] and independently by E. M. Gold [Go]. As an application of this (...) result, we give a recursion theoretic analysis of clopen determinacy: there is a correlation given between the height α of a well-founded tree corresponding to a clopen game $A \subseteq \omega^\omega$ and the Turing degree of a winning strategy f for one of the players--roughly, f can be chosen to be recursive in 0 α and this is the best possible (see § 4 for precise results). (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)

We present a framework that provides a logic for science by generalizing the notion of logical (Tarskian) consequence. This framework will introduce hierarchies of logical consequences, the first level of each of which is identified with deduction. We argue for identification of the second level of the hierarchies with inductive inference. The notion of induction presented here has some resonance with Popper's notion of scientific discovery by refutation. Our framework rests on the assumption of a restricted class of structures (...) in contrast to the permissibility of classical first-order logic. We make a distinction between deductive and inductive inference via the notions of compactness and weak compactness. Connections with the arithmetical hierarchy and formal learning theory are explored. For the latter, we argue against the identification of inductive inference with the notion of learnable in the limit. Several results highlighting desirable properties of these hierarchies of generalized logical consequence are also presented. (shrink)

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)

In lieu of an abstract, here is a brief excerpt of the content:19. HUME AND BARKER ON THE LOGIC OF DESIGN I find myself in complete agreement with what I take to be the main thesis of Stephen Barker's paper. It is certainly a mistake to concentrate our attention on the negative critique which Hume directed at the modes of argument of his rationalist predecessors and contemporaries and directed even more at the mode of certain conviction with which they (...) presented conclusions arrived at by arguments that did not have a logically compelling force. Hume was very much concerned with the problem of just what kind of force an argument that was not logically compelling could have; and he was even more concerned (perhaps) with the problem of how the logical structure of non-compelling argument could best be exhibited so that the logic of the argument would not be confused with its force. He made significant contributions to both of these problems; and I think he knew what he was doing well enough not to deserve the charge of inconsistency which Barker seems inclined to bring against him. There are some questions which he did not address, for the very good reason that they could only arise, or at least they could only be formulated clearly, as a result of the clarification of our ideas and our situation that Hume himself achieved. His distinction between "relations of ideas" and "matters of fact" may be overly simplistic. But in all its simplicity this distinction is powerful enough to enable Hume to deal quite shortly with any claim that it is inconsistent to want to achieve this goal of knowing and saying what is true, just because we have shown that we have no logical guarantee that we shall achieve it. Christian orthodoxy has always maintained that we are all of us equally "sinners"; but no orthodox Christian moralist has ever maintained the Stoic paradox that all sins are equal. It is precisely because we know that we cannot achieve Stoic wisdom that we cannot possibly be tempted to 20. accept that paradox. Hume's achievement was to show that (in spite of the serpent's false promise) our situation with respect to knowledge has always been just as parlous as our situation with respect to action. But even in these unhappy circumstances it is not inconsistent to strive for whatever measure of truth we can achieve; and to strive at the same time for absolute logical clarity about the limits of our insight and about the comparative merits of all proposed approaches to the goal. But I must not devote my time to an issue that Barker wisely asks us to keep an open mind about. I want to discuss the form and substance of the arguments with which Barker supports the thesis that I do agree with, rather than take up an aside of his that I happen not to agree with. So then, first, let us consider the fault that he finds in the "fifth way" of Thomas. He says that it "begs the question". It "obscures the nexus of controversy" because the first premiss of the syllogism makes a crucial assumption that needs to be proved. Now I deny that the "fifth way" obscures the nexus of controversy in that way. Thomas (and all of his competent hearers) knew perfectly well that his first premiss had not been universally accepted. He was stating the view of Aristotle about the generally purposive character of nature, and he knew that it had been doubted. But he did not think that a rational believer could hesitate about the choice between Aristotle's position and that of the atheist defenders of chance. Anyone who did hesitate was a fool, and arguments could not touch him. That some men were fools, and could "say in their heart, 'There is no God'", was part of the "problem of evil" which we must consign trustfully to the inscrutable wisdom of God (save in so far as Scripture reveals to us the origin of and reason for our fallen condition). In Hume's world, Aristotle's conception of the orderliness of Nature had been supplanted by Newton's. Newton accepted the atomist view that "natural bodies... (shrink)