An algebraic proof is presented for the finite strong standard completeness of the Involutive Uninorm Logic with Fixed Point ($${{\mathbf {IUL}}^{fp}}$$ IUL fp ). It may provide a first step towards settling the standard completeness problem for the Involutive Uninorm Logic ($${\mathbf {IUL}}$$ IUL, posed in G. Metcalfe, F. Montagna. (J Symb Log 72:834–864, 2007)) in an algebraic manner. The result is proved via an embedding theorem which is based on the structural description of (...) the class of odd involutive FL$$_e$$ e -chains which have finitely many positive idempotent elements. (shrink)

Two constructions for adding an involution operator to residuated ordered monoids are investigated. One preserves integrality and the mingle axiom x 2x but fails to preserve the contraction property xx 2. The other has the opposite preservation properties. Both constructions preserve commutativity as well as existent nonempty meets and joins and self-dual order properties. Used in conjunction with either construction, a result of R.T. Brady can be seen to show that the equational theory of commutative distributive residuated lattices (without (...) involution) is decidable, settling a question implicitly posed by P. Jipsen and C. Tsinakis. The corresponding logical result is the (theorem-) decidability of the negation-free axioms and rules of the logic RW, formulated with fusion and the Ackermann constant t. This completes a result of S. Giambrone whose proof relied on the absence of t. (shrink)

Axiomatizations are presented for fuzzy logics characterized by uninorms continuous on the half-open real unit interval [0,1), generalizing the continuous t-norm based approach of Hájek. Basic uninorm logic BUL is defined and completeness is established with respect to algebras with lattice reduct [0,1] whose monoid operations are uninorms continuous on [0,1). Several extensions of BUL are also introduced. In particular, Cross ratio logic CRL, is shown to be complete with respect to one special uninorm. A Gentzen-style (...) hypersequent calculus is provided for CRL and used to establish co-NP completeness results for these logics. (shrink)

In general, there is only one fuzzy logic in which the standard interpretation of the strong conjunction is a strict triangular norm, namely, the product logic. We study several equations which are satisfied by some strict t-norms and their dual t-conorms. Adding an involutive negation, these equations allow us to generate countably many logics based on strict t-norms which are different from the product logic.

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant $\overline{0}$ , namely $\neg \varphi$ is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that $\neg$ is an involutive (...) negation. However, for t-norms without non-trivial zero divisors, $\neg$ is Gödel negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation. (shrink)

In this paper we study intermediate logics between the logic G≤∼, the degree preserving companion of Gödel fuzzy logic with involution G∼ and classical propositional logic CPL, as well as the intermediate logics of their finite-valued counterparts G≤n∼. Although G≤∼ and G≤ are explosive w.r.t. Gödel negation ¬, they are paraconsistent w.r.t. the involutive negation ∼. We introduce the notion of saturated paraconsistency, a weaker notion than ideal paraconsistency, and we fully characterize the ideal and the (...) saturated paraconsistent logics between G≤n∼ and CPL. We also identify a large family of saturated paraconsistent logics in the family of intermediate logics for degree-preserving finite-valued Łukasiewicz logics. (shrink)

This text is designed to help students construct and evaluate arguments.

Abū al-Ḥajj¿j Yūsuf b. Muḥammad Ibn Ṭumlūs (Alhagiag Bin Thalmus, d. 620/1223) was a philosopher, physician and direct disciple of Ibn Rushd (Averroes, d. 595/1198), who lived and practiced rational sciences in Alzira and Marrakesh, a quarter of a century after the demise of his teacher. Ibn Ṭumlūs was not Ibn Rushd's only student who engaged in work on logic, but one of dozens of disciples, suggesting that the supposed simultaneous death of the latter's philosophy is "grossly exaggerated". As (...) a valuable window into the practice of logic in 13th century al-Andalus and the Maghreb, Ibn Ṭumlūs' Compendium on Logic (Al-Mukhtaṣar fī al-manṭiq) covers all the parts of "the expanded Organon", as it was known since al-F¿r¿bī (d. 339/951). The present volume offers a complete critical Arabic edition of this work, with an English and Arabic introduction, notes and indices. (shrink)

The goal of this paper is to introduce a new Gentzen formulation of the modal logic S5. The history of this problem goes back to the fifties where a counter-example to cut-elimination was given for an otherwise natural and straightforward formulation of S5. Since then, several cut-free Gentzen style formulations of S5 have been given. However, all these systems are technically involved, and furthermore, they differ considerably from Gentzen's original formulation of classical logic. In this paper we give (...) a new sequent system for S5 which is a straightforward and technically simple extension of Gentzen's original sequent system for classical logic. A characteristic feature is the notion of a connection in a proof. The new system satisfies cut-elimination as well as the subformula property. Cut-elimination is proved by giving an algorithm for eliminating cuts. The fact that we give an algorithm for eliminating cuts makes it clear that our system is susceptible to a formulae-as-types computational interpretation. (shrink)

Introductory-level undergraduate classes in Logic or Critical Thinking are a staple in the portfolio of many Philosophy programs. A standard approach to these classes is to include teaching and learning activities focused on formal deductive and inductive logic, sometimes accompanied by teaching and learning activities focused on informal fallacies or argument construction. In this article, I discuss a proposal to include an additional element within these classes—namely, teaching and learning activities focused on intellectual virtues. After clarifying the proposal, (...) I identify three reasons in favor of implementing it and I discuss how to implement it, focusing on questions about pedagogical strategies and pedagogical resources. (shrink)

Logical Forms explains both the detailed problems involved in finding logical forms and also the theoretical underpinnings of philosophical logic. In this revised edition, exercises are integrated throughout the book. The result is a genuinely interactive introduction which engages the reader in developing the argument. Each chapter concludes with updated notes to guide further reading.

Jaina interpretation of Tarkasaṅgraha of Annambhaṭṭa, work on Hindu logic ; includes Sanskrit commentary.

Compendium of Indian epistemology and logic.

A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal arithmetic, and the incompleteness theorems. Dr. Johnstone has (...) included numerous exercises designed to illustrate the key elements of the theory and to provide applications of basic logical concepts to other areas of mathematics. (shrink)

Experience, it is widely agreed, constrains our thinking and is also thoroughly theory-laden. But how can it constrain our thinking while depending on what it purports to constrain? To address this issue, I revisit and carefully analyze the account of observation provided by Norwood Russell Hanson, who introduced the term ‘theory-ladenness of observation’ in the first place. I show that Hanson’s account provides an original and coherent response to the initial question and argue that, if suitably developed, his account provides (...) a distinctive, powerful, and attractive alternative to relationalist and standard representationalist conceptions of experience. (shrink)

This robust, clear, and well-researched textbook for classes in logic introduces students to both formal logic and to the virtues of intellectual inquiry. Part 1 challenges students to develop the analytical skills of deductive and inductive reasoning, showing them how to identify and evaluate arguments. Part 2 helps students develop the intellectual virtues of the wise inquirer. The book includes helpful pedagogical features such as practice exercises and a concluding summary with definitions of key concepts for each chapter. (...) Resources for professors and students are available through Baker Academic's Textbook eSources. (shrink)

This careful and engaging introduction to Aristotle equips readers of ancient philosophy and classics with an intellectual map that will guide their further exploration within the terrains of Aristotelian philosophy and logic. The book does not seek to provide a verdict or to persuade the reader of the usefulness of Aristotle's ideas. Instead it offers a comprehensive introduction to key philosophical areas while situating the reader within the ongoing intellectual debates on Aristotle's significance and relevance. Crowley's book allows an (...) overview of Aristotle's contributions and includes discussions on the Organon, the epitome of his logical treatises, his works on biology, nature and causation, metaphysics, citizenship, ethics and eudaimonia. The book is a valuable tool for readers interested in becoming acquainted with the work of Aristotle but most importantly for those who are keen to understand his many interpretations. (shrink)

This book represents the first major attempt by any author to provide an integrated account of the evidence for bias in human reasoning across a wide range of disparate psychological literatures. The topics discussed involve both deductive and inductive reasoning as well as statistical judgement and inference. In addition, the author proposes a general theoretical approach to the explanations of bias and considers the practical implications for real world decision making. The theoretical stance of the book is based on a (...) distinction between preconscious heuristic processes which determine the mental representation of 'relevant' features of the problem content, and subsequent analytic reasoning processes which generate inferences and judgements. Phenomena discussed and interpreted within this framework include feature matching biases in propositional reasoning, confirmation bias, biasing and debiasing effects of knowledge on reasoning, and biases in statistical judgement normally attributed to 'availability' and 'representativeness' heuristics. In the final chapter, the practical consequences of bias for real life decision making are considered, together with various issues concerning the problem of 'debiasing'. The major approaches discussed are those involving education and training on the one hand, and the development of intelligent software and interactive decision aids on the other. (shrink)

We argue that abduction does not work in isolation from other inference mechanisms and illustrate this through an inference scheme designed to evaluate multiple hypotheses. We use game theory to relate the abductive system to actions that produce new information. To enable evaluation of the implications of this approach we have implemented the procedures used to calculate the impact of new information in a computer model. Experiments with this model display a number of features of collective belief-revision leading to consensus-formation, (...) such as the influence of bias and prejudice. The scheme of inferential calculations invokes a Peircian concept of ‘belief’ as the propensity to choose a particular course of action. (shrink)

Supercommentary on portion of Tattvacintāmaṇi of Gaṅgeśa, dealing with an argument liable to a valid objection (satpratipakṣa), one of the five forms of fallacious middle term hetvābhāsa.

In lively and readable prose, Arthur presents a new approach to the study of logic, one that seeks to integrate methods of argument analysis developed in modern “informal logic” with natural deduction techniques. The dry bones of logic are given flesh by unusual attention to the history of the subject, from Pythagoras, the Stoics, and Indian Buddhist logic, through Lewis Carroll, Venn, and Boole, to Russell, Frege, and Monty Python. A previous edition of this book appeared (...) under the title _Natural Deduction_. This new edition adds clarifications of the notions of explanation, validity and formal validity, a more detailed discussion of derivation strategies, and another rule of inference, Reiteration. (shrink)

A collection of interviews with influential theorists working in philosophy of logic. The questions: Why were you initially drawn to the philosophy of logic? What are your main contributions to the philosophy of logic? What is the proper role of philosophy of logic in relation to other disciplines, and to other branches of philosophy? What have been the most significant advances in the philosophy of logic? What are the most important open problems in philosophy of (...) logic, and what are the prospects for progress? Interviews with JC Beall, John Lane Bell, Johan van Benthem, Patricia A. Blanchette, Otávio Bueno, James Cargile, Mark Colyvan, Newton Carneiro Affonso da Costa, Pascal Engel, Susan Haack, Jaakko J. Hintikka, Dale Jacquette, Penelope Maddy, Lawrence S. Moss, Catarina Dutilh Novaes, Ahti-Veikko Pietarinen, Graham Priest, Stephen Read, Nicholas Rescher, Stewart Shapiro, Peter Simons, Timothy Williamson, Jan Woleński. (shrink)