The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present (...) a Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \(\mathbb{DHMSH}\). It is deduced that the logic \(\mathcal{DHMSH}\) is algebraizable in the sense of Blok and Pigozzi, with the variety \(\mathbb{DHMSH}\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \(\mathcal{DHMSH}\) is dually isomorphic to the lattice of subvarieties of \(\mathbb{DHMSH}\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \(\mathcal{DHMSH}\) in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic \(\mathcal{DHMSH}\), corresponding to several important subvarieties of the variety \(\mathbb{DHMSH}\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras. (shrink)

A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to (...) a Łukasiewicz–Moisil Topos with an n-valued Łukasiewicz–Moisil Algebraic Logic subobject classifier description that represents non-random and non-linear network activities as well as their transformations in developmental processes and carcinogenesis. The unification of the theories of organismic sets, molecular sets and Robert Rosen’s (M,R)-systems is also considered here in terms of natural transformations of organismal structures which generate higher dimensional algebras based on consistent axioms, thus avoiding well known logical paradoxes occurring with sets. Quantum bionetworks, such as quantum neural nets and quantum genetic networks, are also discussed and their underlying, non-commutative quantum logics are considered in the context of an emerging Quantum Relational Biology. (shrink)

The Post, axled and Łukasiewicz–Moisil algebras are important lattices studied in algebraic logic. In this paper, we investigate a useful interpretation between these algebras and some rings. We give a term equivalence between Post algebras of order |$p$| and |$p$|-rings, |$p$| prime and lift this result to the axled Łukasiewicz–Moisil algebra |$L \cong B_s \times P$| and the ring |$\prod ^s F_2 \times \prod ^l F_p$|⁠, where |$B_s$| is a Boolean algebra of order |$2^s$|⁠, |$P$| a |$p$|-valued Post algebra (...) of order |$p^l$| and |$F_p$| is the prime field of order |$p$|⁠. (shrink)

A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Super-complex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item’s essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures that (...) emerge from the latter in living systems. Thus, several relational theories of living systems can be represented by natural transformations of organismic, relational structures. The ascent of man and other living organisms through adaptation, is viewed in novel categorical terms, such as variable biogroupoid representations of evolving species. Such precise but flexible evolutionary concepts will allow the further development of the unifying theme of local-to-global approaches to highly complex systems in order to represent novel patterns of relations that emerge in super- and ultra-complex systems in terms of compositions of local procedures. Solutions to such local-to-global problems in highly complex systems with ‘broken symmetry’ might be possible to be reached with the help of higher homotopy theorems in algebraic topology such as the generalized van Kampen theorems (HHvKT). Categories of many-valued, Łukasiewicz-Moisil (LM) logic algebras provide useful concepts for representing the intrinsic dynamic ‘asymmetry’ of genetic networks in organismic development and evolution, as well as to derive novel results for (non-commutative) Quantum Logics. Furthermore, as recently pointed out by Baianu and Poli (Theory and applications of ontology, vol 1. Springer, Berlin, in press), LM-logic algebras may also provide the appropriate framework for future developments of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. As shown in the preceding two papers in this issue, a paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics—which is presently unfolding in physics, mathematics, life and cognitive sciences—may be implemented through realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics. (shrink)

Involutive Stone algebras were introduced by R. Cignoli and M. Sagastume in connection to the theory of $n$-valued Łukasiewicz–Moisil algebras. In this work we focus on the logic that preserves degrees of truth associated to S-algebras named Six. This follows a very general pattern that can be considered for any class of truth structure endowed with an ordering relation, and which intends to exploit many-valuedness focusing on the notion of inference that results from preserving lower bounds of truth values, (...) and hence not only preserving the value $1$. Among other things, we prove that Six is a many-valued logic that can be determined by a finite number of matrices. Besides, we show that Six is a paraconsistent logic. Moreover, we prove that it is a genuine Logic of Formal Inconsistency with a consistency operator that can be defined in terms of the original set of connectives. Finally, we study the proof theory of Six providing a Gentzen calculus for it, which is sound and complete with respect to the logic. (shrink)

This fourth edition of one of the classic logic textbooks has been thoroughly revised by John Burgess. The aim is to increase the pedagogical value of the book for the core market of students of philosophy and for students of mathematics and computer science as well. This book has become a classic because of its accessibility to students without a mathematical background, and because it covers not simply the staple topics of an intermediate logic course such as Godel's (...) Incompleteness Theorems, but also a large number of optional topics from Turing's theory of computability to Ramsey's theorem. John Burgess has now enhanced the book by adding a selection of problems at the end of each chapter, and by reorganising and rewriting chapters to make them more independent of each other and thus to increase the range of options available to instructors as to what to cover and what to defer. (shrink)

It is fashionable now-a-days to regard Aristotle’s logic as being the skeleton in the closet of Aristotelian philosophy. As Miss Anscombe has acidly remarked, “Aristotle himself … misconceived the importance of the categorical syllogism, supposing that the theory of it gave him the key to the nature of scientific knowledge. He expresses this view in what I find his worst book: Book I of the Posterior Analytics.”.

Abstractabstract:In the following, I argue that Hegel took concepts—not propositions, judgments, or spatiotemporal objects—as the primary truth-bearer in his logic and attempt to offer a defensible interpretation of what it means for an individual concept (or "thought-determination") to be assessed as true or untrue. Along the way, I consider the shortcomings of several alternative interpretations of truth in Hegelian logic, paying particular attention to the now-common contention that a commitment to something like Frege's context principle prevents Hegel from (...) assessing concepts independently of the role that they play in judgments. (shrink)

This work treats the problem of axiomatizing the truth and falsity consequence relations, ⊨ t and ⊨ f, determined via truth and falsity orderings on the trilattice SIXTEEN 3 (Shramko and Wansing, 2005). The approach is based on a representation of SIXTEEN 3 as a twist-structure over the two-element Boolean algebra.

Moore's paradox pits our intuitions about semantic oddnessagainst the concept of truth-functional consistency. Most solutions tothe problem proceed by explaining away our intuitions. But``consistency'' is a theory-laden concept, having different contours indifferent semantic theories. Truth-functional consistency is appropriateonly if the semantic theory we are using identifies meaning withtruth-conditions. I argue that such a framework is not appropriate whenit comes to analzying epistemic modality. I show that a theory whichaccounts for a wide variety of semantic data about epistemic modals(Update Semantics) buys (...) us a solution to Moore's paradox as a corollary.It turns out that Moorean propositions, when looked at through the lenseof an appropriate semantic theory, are inconsistent after all. (shrink)

This article is concerned with some of the criteria which Hegel believes apply to a scientific treatment of logic. I briefly address criteria which I take Hegel to inherit from German rationalism before focusing on two fairly idiosyncratic criteria: the requirement that a science of logic exhibit a circular structure and that it begin with the concept of pure being. I offer an explanation of these criteria which understands them as motivated by anti-sceptical concerns, before arguing that Hegel’s (...) mature treatment of the latter criterion is problematically ambiguous. (shrink)

Query processing in ground definite deductive is known to correspond precisely to a linear programming problem. However, the “groundedness” requirement is a huge drawback to using linear programming techniques for logic program computations because the ground version of a logic program can be very large when compared to the original program. Furthermore, when we move from propositional logic programs to first-order logic programs, this effectively means that functions symbols may not occur in clauses. In this paper, (...) we develop a theory of “instantiate- by-need” that performs instantiations only when needed. We prove that this method is sound and complete when computing answer substitutions for non-ground logic programs including those containing function symbols. More importantly, when taken in conjunction with Colmerauer's result that unification can be viewed as linear programming, this means that resolution with unification can be completely replaced by linear programming as an operational paradigm. Additionally, our tree construction method is not rigidly tied to the linear programming paradigm-we will show that given any method M that can compute the set of atomic logical consequences of a propositional logic program, our method can use M to compute , the set of all atoms that are consequences of a first-order logic program. (shrink)

Purpose of this article is to study the phenomenon of desire in Ancient Chinese and ancient Indian society, to reveal a material basis for the appearance and formation of the specific ethical attitude towards desire in the philosophical reflection of ancient thinkers. To fulfil this purpose, we should study and analyse methodology of desire studies in philosophical and psychological literature, analyse the ethical attitude towards desire in religious and philosophical texts of Chinese and Indian thinkers, understand social and economic basis (...) of such an ethical attitude, that is to establish material basis of desire as the specific form of psychic activity and to recreate the logic of its development. Theoretical basis. Culturological and economic studies of orientalists and dialectical logic. Originality. The paper presents the analysis of the perspectives and drawbacks of different approaches in desire studies. Research of ethical attitude towards the desire in Ancient China and India allowed us to recreate the logic of social relations and the level of productive forces in that society. The ideological foundation of conservation and recreation of power and property relations was found in the religious doctrine of suppression of individual desires. The connection between desire formation and division of labour and its preservation in the religious, political and legal forms of social consciousness was shown. Methodological perspectives of dialectical logic were defined in the study of logical and historical coincidence in the further development of the phenomenon and the notion of desire. Conclusions. Analysis of modern philosophical and psychological literature has revealed methodological drawbacks in desire studies, as they are focused on studying desire’s notion, its representation or separate aspects of desire in individual psyche disregarding its inner logic and material basis of its becoming. Studies of the ethical attitude towards desire in Eastern culture have shown the consensus in desire suppression of individuals and similarities between Ancient Chinese and Indian cultures. There is shown significance and material basis of the transcendental law suppressing the desires even of the higher classes of the oriental society. The methodological approach stated in this article can be applied for further desire studies of Ancient Greece culture, Christian ethics and modern society. (shrink)

This paper studies the possible influences on Boole's logic of the writings related to the controversy over the quantification of the predicate between the philosopher William Hamilton and the mathematician Augustus De Morgan. As Boole himself testified in the introduction to his book The mathematical analysis of logic , this controversy was the external agent that stimulated him into writing up his earlier thoughts about a new conception of logic. But in addition to the external role that (...) was played by the controversy, it seems that other kinds of influences on Boole of Hamilton's and De Morgan's ideas about logic, which were made explicit in writings related to the controversy, can also be traced. (shrink)

My thesis is that you can lie with ‘ P therefore Q ’ without P or Q being lies. For you can lie by virtue of not believing that P supports Q. My thesis is reconciled with the principle that all lies are assertions through H. P. Grice’s account of conventional implicatures. These semantic cousins of conversational implicatures are secondary assertions that clarify the speaker’s attitude toward his primary assertions. The meaning of ‘therefore’ commits the speaker to an entailment thesis (...) even though the speaker does not enter that commitment into the text. Insincere conventional implicatures are akin to insincerely asserted footnotes. An absence of lies in the text is compatible with the presence of lies in the meta-text. (shrink)

This Handbook is a combination of authoritative exposition, comprehensive survey, and fundamental research exploring the underlying unifying themes in the various areas. The intended audience is graduate students and researchers in the areas of computing and logic, as well as other people interested in the subject. We assume as background some mathematical sophistication. Much of the material will also be of interest to logicians and mathematicians.

In this paper, I will show to what extent we can use our modern understanding of the Square of Opposition in order to make sense of Kant 's double standard solution to the cosmological antinomies. Notoriously, for Kant, both theses and antitheses of the mathematical antinomies are false, while both theses and antitheses of the dynamical antinomies are true. Kantian philosophers and interpreters have criticized Kant 's solution as artificial and prejudicial. In the paper, I do not dispute such claims, (...) but I show that our modern understanding of the Square of Opposition enables us to more naturally deliver the result Kant was aiming at. Accordingly, the paper does not pretend to be exegetically accurate. It is an attempt to revise the antinomies with the help of standard classical logic. And although such a revision entails some re-interpretation, in the end, it will actually help to unveil some of Kant 's thoughts. (shrink)

As its name implies, Vasubandhu’s _Vyākhyāyukti_ (VyY) explains the logic or methodology (_yukti_) of exegesis or sūtra interpretation (_vyākhyā_) and only survives in a Tibetan translation. In recent years, research on this treatise has been gradually accumulating. However, due to the difficulty of the Tibetan translation, some of the arguments therein have been misunderstood. In this article, after reviewing the general framework of Vasubandhu’s method of interpreting the sūtras, I will present a newly discovered parallel regarding his discussion of (...) the “purpose, _prayojana_” of the sūtras and reread it through a close philological examination of various sources in Sanskrit and Tibetan. Thus, this article will first elucidate the details of Vasubandhu’s explanation of the purpose of the Buddha’s preaching using synonyms and will clarify an aspect of his views on the Buddha’s word. In addition, concerning this “purpose,” I will elucidate the characteristics of the final chapter of this text, which provides stories about hearing the Buddha’s words with respect. By doing so, I would like to reveal the characteristics of the VyY as a manual for vivid preaching. (shrink)

I argue here that in the end Bourdieu's theory of practice fails to overcome the problem on which it expressly centers, namely, subject-object dualism. The failure is registered in his avowed materialism, which, though significantly "generalized," remains what it says: a materialism. In order to substantiate my criticism, I examine for their ontological presuppositions three areas of his theoretical framework pertaining to the questions of (1) human agency (as seen through the conceptual glass of the habitus), (2) otherness, and (3) (...) the gift. By scrutinizing Bourdieu's powerful and progressive social theory, with an eye to finding fault, I hope to show the need to take a certain theoretical action, one that is patently out of keeping with the usual self-presentation and self-understanding of social science. The action I have in mind is this: because the problem of subject-object dualism is in the first place a matter of ontology, in order successfully to address it there must take place a direct shift of ontological starting point, from the received starting point in Western thought to one that projects reality in terms of ambiguity that is basic. With this shift the dualism of subject and object dissolves by definition, leaving a social reality that, for reasons of its basic ambiguity, is best approached as a question of ethics before power. (shrink)

In this paper we show that, when analyzed with contemporary tools in logic—such as Dunn-style semantics, Reichenbach’s three-valued logic exhibits many interesting features, and even new responses to some of the old objections to it can be attempted. Also, we establish some connections between Reichenbach’s three-valued logic and some contra-classical logics.

The paper proposes a way to naturalise Charles S. Peirce’s conception of the scientific method, which he specified in terms of abduction, deduction and induction. The focus is on the central issue of the economy of research in abduction and self-correction by error reduction in induction. We show how Peirce’s logic of science receives support from modern breakthroughs in computational neuroscience, and more specifically from Karl Friston’s statements of active inference and the Free Energy Principle, namely the account of (...) how organisms’ capacity to decrease the discrepancy between the expected value and actual outcomes entails the minimisation of errors in their hypotheses about the world. A scientific account of organisms’ capacity to choose policies and form expectations is aligned with Peirce’s theories of abduction and induction, and especially with the economy of research. The upshot is the recovery of Peirce’s theory of the logic of science in the context of active inquiry. (shrink)

This chapter is about logic and the relationship between truth/thought and reality. The specific concern is the relation between ground and cause. On one view, the ground of an inference is identical with the respective cause. But there are cases where the ground is not some physical event, but rather something non-causal, as in judgements about similarity among objects in intrinsic respects. It is argued that cause is logically prior to ground. The world as a system is founded on (...) vast webs of causal relations, and these causal relations or larger causal structure have a temporal component; hence time figures in the structuring base of reality and system. A view about judgement and inference is then developed: inference is the scientist’s tool, whereas judgements are the material, just like the marble is the material from which the sculptor creates a statue. This leads to an objectivist theory of judgement. (shrink)

The main lines of interpretation offered to date of the Third Man Argument in Plato's Parmenides (132a1-b2) are considered and rejected. A new, set-theoretic, reconstruction of the argument is offered. It is concluded that the philosophical point of the argument is different from what it has been generally supposed to be: Plato is pointing out the logical shortcomings in his earlier formulated principle of One-Over-Many.

To oversimplify quite a bit, scholars’ presentation of Hegel's teleology constitutes a continuum according to how more-or-less secured the progress towards the goal is supposed to be, which tracks roughly the nature of the end and its necessity. In this article, rather than focus on the end and progress towards it, we will focus on the means and structure of teleological relationships on Hegel's account. This focus follows from an essential feature of Hegel's discussion of teleology in the Logic, (...) in which teleology is introduced to solve a problem in the individuation of entities. It will turn out that the fullest actualization of the end is in the durable means, which is also thereby individuated. And it will turn out that the paradigmatic historical means—the state—is tensed, as it were, between the end and its realization that makes it synchronically historical. This synchronic historicity is missing in the usual progressive and thus diachronic accounts of the teleological process of history. But first we step back even farther (at least historically). We begin by taking up the two most important philosophical accounts of teleology for Hegel, namely those of Aristotle and Kant. Then we go to Hegel's Logic for his reconstruction of teleological processes against the background of the explanatory need for individuation. We focus on four aspects of Hegel's account: that teleology is a structure of reciprocal interaction, that the purpose is an immanent governing principle, that change is the price of immanence, and that the durable means is the teleological object par excellence. Finally, we trace these features through Hegel's account of world history and conclude with some brief remarks on the historicity of the state described in the Philosophy of Right. (shrink)

This paper is part of a project to rescue fashion from the social sciences and restore it to philosophy. In Kawamura's Fashion-ology, power is understood solely as legal or institutional power. The work's strictly sociological approach means that, though the two are rightly distinguished, clothing continues to haunt the logic of fashion, and there is little reflection as to why the system of clothing and not some other commodity lends its name to cultural neomania in general. What is lacking (...) is an analysis of the specific nature of fashion's empowerment and disempowerment of subjectivity. For this we need to enlist Foucault's genealogy of power, which also has the advantage of explaining strategic resistance to institutional power. However, Foucault's account is itself unsatisfying, because it avoids offering a theory of power while nevertheless assuming one. Foucault's critique of power quietly assumes Nietzsche's metaphysic of power, the incessant, embodied movement of self-empowering and self-overpowering. The corporeal subject of this self-production is nothing but surface, one that extends to the surrounds of its world, and the ceaseless rhythm of its embodied self-presentation is fashion. Barthes insightfully observes that, were its rhythm to be disrupted by cultural developments, the intense jostling that resulted would be the beginning of a new history of fashion. Yet there is incoherence to this claim, for if the rhythm of fashion were to turn into the chaos of arrhythmia, then we might instead conclude that fashion was at an end. Barthes' uncertain prospect of a new beginning points to what I call, following Heidegger, the powerlessness of fashion. This is not intended as criticism, since this powerlessness is nothing negative but rather that which exceeds the limits of fashion while giving rise to it. But on questions concerning the historicality of its beginning and the finality of its self-understanding, the sociology of fashion can have nothing to say. (shrink)

We present a case study for the debate between the American and the Australian plans, analyzing a crucial aspect of negation: expressivity within a theory. We discuss the case of non-classical set theories, presenting three different negations and testing their expressivity within algebra-valued structures for ZF-like set theories. We end by proposing a minimal definitional account of negation, inspired by the algebraic framework discussed.

This paper is grounded on an exegetically creative reading of Kaplan's late approach to dthat-terms. I do have a few exegetic pretensions. In particular, I simply assume what I take to be the central tenets of Kaplan's theory of demonstratives between 1977 and 1989, and I develop them according to ideas suggested by certain passages in Afterthoughts. But my developments are also unashamedly creative. I recognize that I may overemphasize a few carefully chosen snippets and that, in doing so, I (...) may end up ignoring other important junctures. But I happily trade textual faithfulness for theoretical insight: if on the right track, my results hopefully cast a new light not only on Kaplan's take on ‘dthat’ but also on the fundamental ideas in his philosophical and semantic framework. (shrink)

Computability theory concerns information with a causal–typically algorithmic–structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals (...) which produced richness of local computability-theoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Post’s programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Post’s programme, and look at two interesting developments of Post’s approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computability-theoretic infrastructure. We will conclude by viewing Post’s programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. (shrink)

This section reproduces the part of Bibler’s book Thinking as Creative Work that discusses the subject as a “microsocium” combining the rational intellect, reason, intelligence, and intuitio...

Typical Gettieresque scenarios involve a subject, S, using a method, M, of believing something, p, where, normally, M is a reliable indicator of the truth of p, yet, in S’s circumstances, M is not reliable: M is deleteriously rigged. A different sort of scenario involves rigging that restores the reliability of a method M that is deleteriously rigged: M is restoratively rigged. Some theorists criticize the safe indication account of knowledge defended by Luper, Sosa, and Williamson on the grounds that (...) it treats such cases as knowledge. But other theorists also criticize the safe indication account because it treats the cases as examples of ignorance when they are really examples of knowledge. I answer both groups of critics by arguing that restorative rigging can enable us to know things, and restoratively rigged cases can meet the relevant conditions of the safe indication account. (shrink)