This article deals with a set-theoretic interpretation of the Talmudic rules of General and Specific, known as Klal and Prat (KP), Prat and Klal (PK), Klal and Prat and Klal (KPK) and Prat and Klal and Prat (PKP).

In this paper we make some observations about Natural Deduction derivations [Prawitz, 1965, van Dalen, 1986, Bell and Machover, 1977]. We assume the reader is familiar with it and with proof-theory in general. Our development will be simple, even simple-minded, and concrete. However, it will also be evident that general ideas motivate our examples, and we think both our specific examples and the ideas behind them are interesting and may be useful to some readers. In a sentence, the bare technical (...) content of this paper is: Extending natural deduction with global well-formedness conditions can neatly and cheaply capture classical and intermediate logics. The interest here is in the ‘neatly’ and ‘cheaply’. By ‘neatly’ we mean ‘preserving proof-normalisation’,1 and ‘maintaining the subformula property’, and by ‘cheaply’ we mean ‘preserving the formal structure of deductions’ (so that a deduction in the original system is still, formally, a deduction in the extended system, and in particular it requires no extra effort to write just because it is in the extended system). To illustrate what we have in mind consider intuitionistic first-order logic (FOL) [van Dalen, 1986] as a paradigmatic example of a formal notion of deduction. A natural deduction derivation (or deduction) is an inductively defined tree structure where each node contains an instance of a formula. A deduction is valid when each successive node follows from its predecessors in accordance with some predetermined inference rules. A particular attraction of Natural Deduction is its clean and economical presentation. Here for example are deduction (fragments) proving A ∧ B from A and B, and ∀x. (P (x) ∧ Q(x)) from ∀x. P (x) and ∀x. Q(x): ∀x. P (x) (∀E). (shrink)

In this short paper I note that a key metatheorem does not hold for the bilateralist inferential framework: harmony does not entail consistency. I conclude that the requirement of harmony will not suffice for a bilateralist to maintain a proof theoretic account of classical logic. I conclude that a proof theoretic account of meaning based on the bilateralist framework has no natural way of distinguishing legitimate definitional inference rules from illegitimate ones (such as those for tonk). Finally, as an appendix (...) to the main argument, I propose an alternative non-bilateral formal solution to the problem of providing a proof-theoretic account of classical logic. (shrink)

In this paper I present a formalist philosophy mathematics and apply it directly to Arithmetic. I propose that formalists concentrate on presenting compositional truth theories for mathematical languages that ultimately depend on formal methods. I argue that this proposal occupies a lush middle ground between traditional formalism, fictionalism, logicism and realism.

This concise text treats logic as a tool, “generated so that half the work involved in thinking is done for you by somebody else (the rules and laws of the logic).” Gabbay explains in a clear and careful manner how formal features of, and formal relations between, ordinary declarative sentences are captured by the systems of propositional and predicate logic.

We build on an existing a term-sequent logic for the λ-calculus. We formulate a general sequent system that fully integrates αβη-reductions between untyped λ-terms into first order logic. We prove a cut-elimination result and then offer an application of cut-elimination by giving a notion of uniform proof for λ-terms. We suggest how this allows us to view the calculus of untyped αβ-reductions as a logic programming language (as well as a functional programming language, as it is traditionally seen).

Ways out: The proper relata of causal statements are large complexes of (macroscopic) conditions. For example: the spark, in the presence of oxygen together with flammable material and low humidity etc. The entailment is made virtue of the general macroscopic laws of flammable materials, humidity etc.

An intensional sentential context is one where such intersubstitutivity fails: It is informative that A It is known that A It is necessary that A The fact that A caused it to be that B Some authors characterise intensional contexts by the failure of the intersubstitution of co-referring terms. Such a characterisation is problematic.

A famous proposed solution to the one over many problem is found in Plato. For example, it appears in The Parmenedies and is introduce by Zeno arguing that . . . if being is many, it must be both like and unlike, and that this is impossible, for neither can the like be unlike, nor the unlike like-is that your position? and Socrates responds: do you not further think that there is an idea of likeness in itself, and another idea (...) of unlikeness, which is the opposite of likeness, and that in these two, you and I and all other things to which we apply the term many, participate-things which participate in likeness become in that degree and manner like; and so far as they participate in unlikeness become in that degree unlike, or both like and unlike in the degree in which they participate in both? And may not all things partake of both opposites, and be both like and unlike, by reason of this participation? Where is the wonder? and the discussion moves on to the interesting question that.. (shrink)

In this paper I present the strategy behind the proof-theoretic justification of logical inference. I then discuss how this strategy leads to the famous requirement that the inference rules for the logical constants should be in harmony. I argue that the proof-theoretic justification of the logical constants can be used to justify classical logic. To substantiate this I present a new normalisation theorem for first order classical logic involving Sheffer Stroke. The proof of this theorem can be modified to yield (...) a normalisation result for classical logic with conjunction, negation and the universal quantifier. (shrink)

This book is a specialized monograph on interpolation and definability, a notion central in pure logic and with significant meaning and applicability in all areas where logic is applied, especially computer science, artificial intelligence, logic programming, philosophy of science and natural language. Suitable for researchers and graduate students in mathematics, computer science and philosophy, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism, J. M. Dunn and G. Hardegree's Algebraic Methods (...) in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2, and David J. Pym and Eike Ritter's Reductive Logic and Proof Search: Proof theory, semantics and control. (shrink)

Using the framework of categorical logic, this paper analyzes and streamlines Gabbay's semantical proof of the Craig interpolation theorem for intuitionistic predicate logic. In the process, an apparently new and interesting fact about the relation of coherent and intuitionistic logic is found.

Agenda Relevance is the first volume in the authors' omnibus investigation of the logic of practical reasoning, under the collective title, A Practical Logic of Cognitive Systems. In this highly original approach, practical reasoning is identified as reasoning performed with comparatively few cognitive assets, including resources such as information, time and computational capacity. Unlike what is proposed in optimization models of human cognition, a practical reasoner lacks perfect information, boundless time and unconstrained access to computational complexity. The practical reasoner is (...) therefore obliged to be a cognitive economizer and to achieve his cognitive ends with considerable efficiency. Accordingly, the practical reasoner avails himself of various scarce-resource compensation strategies. He also possesses neurocognitive traits that abet him in his reasoning tasks. Prominent among these is the practical agent's striking (though not perfect) adeptness at evading irrelevant information and staying on task. On the approach taken here, irrelevancies are impediments to the attainment of cognitive ends. Thus, in its most basic sense, relevant information is cognitively helpful information. Information can then be said to be relevant for a practical reasoner to the extent that it advances or closes some cognitive agenda of his. The book explores this idea with a conceptual detail and nuance not seen the standard semantic, probabilistic and pragmatic approaches to relevance; but wherever possible, the authors seek to integrate alternative conceptions rather than reject them outright. A further attraction of the agenda-relevance approach is the extent to which its principal conceptual findings lend themselves to technically sophisticated re-expression in formal models that marshal the resources of time and action logics and label led deductive systems. Agenda Relevance is necessary reading for researchers in logic, belief dynamics, computer science, AI, psychology and neuroscience, linguistics, argumentation theory, and legal reasoning and forensic science, and will repay study by graduate students and senior undergraduates in these same fields. Key features: relevance action and agendas practical reasoning belief dynamics non-classical logics labelled deductive systems. (shrink)

Inductive Logic is number ten in the 11-volume Handbook of the History of Logic. While there are many examples were a science split from philosophy and became autonomous (such as physics with Newton and biology with Darwin), and while there are, perhaps, topics that are of exclusively philosophical interest, inductive logic — as this handbook attests — is a research field where philosophers and scientists fruitfully and constructively interact. This handbook covers the rich history of scientific turning points in Inductive (...) Logic, including probability theory and decision theory. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, cognitive psychology, and artificial intelligence, for whom the historical background of his or her work is a salient consideration. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:April 3, 2010 (11:17 am) C:\Users\Milt\Desktop\backup copy of Ken's G\WPData\TYPE2902\russell 29,2 050 red.wpd 1 Gabbay and Woods, eds., The Rise of Modern Logic from Leibniz to Frege, Vol. 3 of the Handbook of the History of Logic (Amsterdam, etc.: North-Holland, 2004). russell: the Journal of Bertrand Russell Studies n.s. 29 (winter 2009–10): 167–90 The Bertrand Russell Research Centre, McMaster U. issn 0036-01631; online 1913-8032 eviews BERTRAND RUSSELL IN (...) RECENT BOOKS ON LOGIC HISTORY Irving Anellis Peirce Edition / Indiana U.–Purdue U. Indianapolis, in 46202–5157, usa [email protected] Dov M. Gabbay and John Woods, eds. Logic from Russell to Church. Vol. 5 of The Handbook of the History of Logic. Amsterdam, etc.: North-Holland, 2009. Pp. xii + 1056. us$260; £157.99; ¤185.95. Leila Haaparanta, ed. The Development of Modern Logic. Oxford and New York: Oxford U. P., 2009. Pp. x + 904. us$150.00. The essay “Bertrand Russell’s Logic” by Andrew D. Irvine in Volume 5 of the Handbook of the History of Logic is one of the shortest in the volume (pp. 1–28). But something of Russell pervades many of the other essays as well, as we might expect given the centrality that Principia Mathematica has occupied in the historiography of logic through much of the twentieth century. This is especially evident in the lengthiest of the essays, Andrea Cantini’s “Paradoxes, Self-reference and Truth in the 20th Century” (pp. 875–1013), and more than tangentially in Dale Jacquette’s “Logic for Meinongian Object Theory Semantics” (pp. 29–76) and Michael Potter’s “The Logic of the Tractatusz” (pp. 255–304). In their preface, the editors of the Handbook explain (p. vii) the rationale for treating Frege and Russell in separate volumes.1 One reason is that it was a matter of simple conveniencez—zto keep the volume covering logic from Leibniz to Frege of manageable size. The other is historical: while admitting that Frege and Russell shared important goals and the philosophical position of logicism, it is noted that Frege’s most important work in logic and foundations was carried out prior to his learning of Russell’s paradox, whereas the entire corpus of Russell’s work, Principiaz included, was designed to a considerable extent in eTorts to deal with the paradox, as was much of the work in logic, set theory, and philosophy of logic that came as a consequence of the discovery of that paradox. Gabbay April 3, 2010 (11:17 am) C:\Users\Milt\Desktop\backup copy of Ken's G\WPData\TYPE2902\russell 29,2 050 red.wpd 168 Reviews and Woods also make explicit the philosophical diTerences between Frege and Russell regarding the epistemological status of axioms as one of the signiWcant diTerences between these two. More particularly, the editors assert (p. viii) that, in the very brief period between publication of the second volume of Frege’s Grundgesetze (1903) and Russell’s Principles of Mathematics (1903), there was a pronounced shift initiated in the goals of mathematical logic, away from naïve apriority of axiom systems, towards study of the limits and properties of formal systems.Thecontributors to these metalogical concerns, Herbrand, Hilbert, and Gödel, among others, are the subject of the bulk of the essays included in Volume 5 of the Handbook. The task that Irvine undertakes is initiated by the eTort to understand the unity of Russell’s technical body of work in mathematics, logic, metaphysics, and epistemology. Thus the discrepancy between the neo-Hegelian concept of relations, inherited from Bradley, with certain types of relationsz—zlogical, spatial, temporal, for examplez—zwith the mathematical understanding of relations, is given as the ground for Russell’s abandonment of idealism in favour of realism (pp. 4–5). For example, there is nothing inherent in two relata, say Abelard and Héloise, which contributes to the relation of love between the pair; rather, its root is found in the mental states of one or the other of the lovers, so that the relation itself is not an intrinsic property of one of the pair. Likewise, Russell reaches this position... (shrink)

The notion of negation is basic to any formal or informal logical system. When any such system is presented to us, it is presented either as a system without negation or as a system with some form of negation. In both cases, we are supposed to know intuitively whether there is no negation in the system or whether the form of negation presented in the system is indeed as claimed. To be more specific, suppose Robinson Crusoe writes a logical system (...) with Hilbert-type axioms and rules, which includes a unary connective ∗A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*A$$\end{document}. He puts the document in a bottle and lets it lose at sea. We find it and take a look. We ask: is the connective “∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}” a negation in the system? Yet the notion of what is negation in a formal system is not clear. When we see a unary connective ∗A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*A$$\end{document}, together with some other axioms for some additional connectives, how can we tell whether ∗A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*A$$\end{document} is indeed a form of negation of A? Are there some axioms which the connective “∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}” must satisfy in order to qualify ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document} as a negation? (shrink)

In this paper I outline the PAIR account of joint attention as a perceptual-practical, affectively charged intentional relation. I argue that to explain joint attention we need to leave the received understanding of propositions and propositional attitudes and the picture of content connected to it behind and embrace the notions of subject mode and position mode content. I also explore the relation between joint attention and communication.

Recent years have been blessed with an abundance of logical systems, arising from a multitude of applications. A logic can be characterised in many different ways. Traditionally, a logic is presented via the following three components: 1. an intuitive non-formal motivation, perhaps tie it in to some application area 2. a semantical interpretation 3. a proof theoretical formulation. There are several types of proof theoretical methodologies, Hilbert style, Gentzen style, goal directed style, labelled deductive system style, and so on. The (...) tableau methodology, invented in the 1950s by Beth and Hintikka and later per fected by Smullyan and Fitting, is today one of the most popular, since it appears to bring together the proof-theoretical and the semantical approaches to the pre of a logical system and is also very intuitive. In many universities it is sentation the style first taught to students. Recently interest in tableaux has become more widespread and a community crystallised around the subject. An annual tableaux conference is being held and proceedings are published. The present volume is a Handbook a/Tableaux pre senting to the community a wide coverage of tableaux systems for a variety of logics. It is written by active members of the community and brings the reader up to frontline research. It will be of interest to any formal logician from any area. (shrink)

Morals from Motives develops a virtue ethics inspired more by Hume and Hutcheson's moral sentimentalism than by recently-influential Aristotelianism. It argues that a reconfigured and expanded "morality of caring" can offer a general account of right and wrong action as well as social justice. Expanding the frontiers of ethics, it goes on to show how a motive-based "pure" virtue theory can also help us to understand the nature of human well-being and practical reason.

In the current paper, we re-examine how abstract argumentation can be formulated in terms of labellings, and how the resulting theory can be applied in the field of modal logic. In particular, we are able to express the (complete) extensions of an argumentation framework as models of a set of modal logic formulas that represents the argumentation framework. Using this approach, it becomes possible to define the grounded extension in terms of modal logic entailment.

The American Journal of Bioethics, Volume 12, Issue 7, Page 27-30, July 2012.

The Handbook of Formal Argumentation is a community effort aimed at providing a comprehensive and up-to-date view of the state of the art and current trends in the lively research field of formal argumentation. The first volume of the Handbook is organised into five parts, containing nineteen chapters in all, each written by leading experts in the field. The first part provides a general and historical perspective on the field. The second part gives a comprehensive coverage of the argumentation formalisms (...) available in the literature at various levels of abstraction. The third part is devoted to cover some of the many dialogical aspects of argumentation, while the fourth one deals with algorithmic, computational and implementation issues. Finally, the fifth part provides some deeper analyses on the previously introduced topics. The Handbook of Formal Argumentation is an open-ended initiative of which the present volume is the first outcome. Further volumes are planned to cover topics not included in the present one and the initiative is conceived to grow by the support and feeding it receives from the community members. (shrink)

This book fills a gap in studies of meaning by providing detailed case studies of attested corpus data on the meanings of words and phrases.

Theories of discourse bring to realism new ideas about how knowledge develops and how representations of reality are influenced. We gain an understanding of the conceptual aspect of social life and the processes by which meaning is produced. This collection reflects the growing interest realist critics have shown towards forms of discourse theory and deconstruction. The diverse range of contributions address such issues as the work of Derrida and deconstruction, discourse theory, Eurocentrism and poststructuralism. What unites all of the contributions (...) is a sense that it is essential to provide a realist alternative to the hitherto dominance of social constructionism, hermeneutics and postmodernism, over many of the issues discussed. By developing a realist perspective the different authors attempt to embed discourse within the structured nature of the reality of the world. Realism can situate language, discourse and ideology within context specific, or 'causally efficacious' circumstances. Realism can help to uncover issues of power, representation, and subjectivity and how discursive and other social practices produce real effects. This can help us understand the manner in which social structures are reproduced through various forms of ideology and discourse. And by knowing this, we can start to address questions concerning human emancipation and how the world is to be transformed. (shrink)

This paper studies general numerical networks with support and attack. Our starting point is argumentation networks with the Caminada labelling of three values 1=in, 0=out and ½=undecided. This is generalised to arbitrary values in [01], which enables us to compare with other numerical networks such as predator?prey ecological networks, flow networks, logical modal networks and more. This new point of view allows us to see the place of argumentation networks in the overall landscape of networks and import and export ideas (...) to and from argumentation networks. We make a special effort to make clear how general concepts in general networks relate to the special case of argumentation networks. We pay special attention to the handling of loops and to the special features of numerical support. We find surprising connections with the Dempster?Shafer rule and with the cross-ratio in projective geometry. This paper is an expansion of our 2005 paper and so we also consider higher level features such as numerical attacks on attacks, and propagation of numerical values.We conclude with a brief view of temporal numerical argumentation and with a detailed comparison with related papers published since 2005. (shrink)

The definitive work on the philosophical nature and impact of the theories of Charles Darwin, written by a well-known authority on the history and philosophy of Darwinism. Broadly explores the theories of Charles Darwin and Darwin studies Incorporates much information about modern Biology Offers a comprehensive discussion of Darwinism and Christianity – including Creationism – by one of the leading authorities in the field Written in clear, concise, user-friendly language supplemented with quality illustrations Examines the status of evolutionary theory as (...) a genuine theory and its implications for philosophy, epistemology and ethics Provides a strong understanding of the philosophical nature and impact of Darwin's thought Holds wide appeal for general audiences outside the world of academic philosophy Strongly supports Darwinism and fully explores modern naturalistic explanations of religion. (shrink)

Michael Quante focuses on what Hegel has to say about such central concepts as action, person and will, and then brings these views to bear on contemporary debates in analytic philosophy. This book enables professional analytic philosophers and their students to understand the significance of Hegel's philosophy to contemporary theory of action. As such, it will contribute to the ever-increasing erosion of the barrier between the continental and analytic approaches to philosophy.

We motivate and introduce a new method of abduction, Matrix Abduction, and apply it to modelling the use of non-deductive inferences in the Talmud such as Analogy and the rule of Argumentum A Fortiori. Given a matrix $${\mathbb {A}}$$ with entries in {0, 1}, we allow for one or more blank squares in the matrix, say a i,j =?. The method allows us to decide whether to declare a i,j = 0 or a i,j = 1 or a i,j =? (...) undecided. This algorithmic method is then applied to modelling several legal and practical reasoning situations including the Talmudic rule of Kal-Vachomer. We add an Appendix showing that this new rule of Matrix Abduction, arising from the Talmud, can also be applied to the analysis of paradoxes in voting and judgement aggregation. In fact we have here a general method for executing non-deductive inferences. (shrink)