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)

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.

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)

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 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.

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)

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).

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.

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)

This commentary on Philip Kitcher’s book What’s the Use of Philosophy? addresses two questions. First, must philosophers be methodologically self-conscious to do good work? Second, is there value in the questions pursued in the traditional areas of analytic philosophy?

Wedin addresses the debate over whether nonsubstantial individuals, that inhere in a subject but are not said of a subject, i.e. accidents, such as the pallor of Socrates, are nonrecurring particulars or a kind of determinate universal. Wedin examines the secondary literature on this topic and divides it into two schools of thought, determined by the contributions of J.L. Ackrill and G.E.L. Owen. According to Ackrill, individuals in non‐substance categories are particular to the substance they are in; Owen critiques Ackrill's (...) view, and proposes that these items can recur in more than one subject and hence are a sort of universal. Wedin finds Owen's thesis unsatisfactory, even after supplementing it with an improved version due to Michael Frede; instead, Wedin argues for a revised version of Ackrill's interpretation of nonsubstantial individuals as nonrecurrent particulars. According to Wedin, Aristotle is committed to individuals only—e.g. to Socrates and to the particular bit of pallor in him: this conclusion has an important bearing on the ontological status of individuals and on the primacy of substance to nonsubstantial items. (shrink)

Wedin considers the problem of the compatibility of the Categories account of primary substance with the theory of substantial form of the Metaphysics. Wedin collects from the secondary literature the most important arguments for incompatibilism, and offers some proposals for restoring their harmony. While admitting the evident differences in the way Aristotle treats the question of substance in each treatise, Wedin is keen to argue that these differences are not sufficient to conclude that the treatises are incompatible. Wedin singles out (...) for particular attention, and criticism, Michael Frede and Gunther Patzig's account of Metaphysics Zeta, according to which the primary substance of the Metaphysics and the basic subject of predications is the form. Wedin considers two ‘philosophical’ arguments and one ‘Aristotelian’ argument for the subjecthood of forms, and argues that the textual evidence does not support the claim that form is the subject of accidents. (shrink)

Michael J. Zimmerman explores whether and how our ignorance about ourselves and our circumstances affects what our moral obligations and moral rights are. He rejects objective and subjective views of the nature of moral obligation, and presents a new case for a 'prospective' view.

The aging of attentional and emotional functions has been extensively studied but relatively independently. Therefore, the relationships between aging and the interactions of attentional and emotional processes remain elusive. This study aimed to determine how age affected the interactions between attentional and emotional processes during adulthood. One-hundred forty adults aged 18–79 performed the emotional variant of the Attention Network Test, which probed alerting, orienting, and executive control in the presence and absence of threatening faces. During this task, contexts with varying (...) levels of task preparatory processes were created to modulate the effect of threatening faces on attention, and functional near-infrared spectroscopy (fNIRS) was used to examine the neural underpinnings of the behavioural effects. The behavioural results showed that aging was associated with a significant decline in alerting efficiency, and there was a statistical trend for age-related deficits in executive control. Despite these age differences, age did not significantly moderate the interactions among attentional networks or between attention and emotion. Additionally, the fNIRS results showed that decreased frontal cortex functioning might underlie the age-related decline in executive control. Therefore, while aging has varying effects on different attentional networks, the interactions of attentional and emotional processes remain relatively unaffected by age. (shrink)

Wedin considers the relation between the ontological commitment in the Categories and the semantical theory of underlying ontological configurations for standard categorical statements. According to Wedin, Aristotle's fourfold division of beings, which divides things according to whether they are, or are not, said of, and/or present in a subject, is a meta‐ontology that is concerned with beings per se, i.e. the fundamental things that are. Wedin explains that the primacy of c‐substance involves an asymmetry in the relation between c‐substance and (...) everything else, or between the substantial and the nonsubstantial. This is because there is an ontological dependence of non‐substances, and also of secondary substances, on substances: everything is either said of, or in, primary substance. The argument of this chapter, then, is that the world of the Categories is a world of individuals: all ontological relations needed for semantics can be accommodated by appeal to individuals only. (shrink)

Wedin argues that Aristotle makes form the substance of c‐substances because it is the essence of the c‐substance. Much of this chapter consists of a careful examination of a passage in Metaphysics Zeta 4, which Wedin calls the ‘New Primacy Passage’, that is crucial to Wedin's overall thesis, because here Aristotle appeals to a notion of definitional primacy, as opposed to the ontological primacy of the Categories. Z.4 focuses on this claim that form must be essence: Wedin argues that essence (...) is a certain kind of form, a form of a genus ; this does not mean ‘species’, as some commentators have thought. This is because the species is a combination of form and matter, whereas the genous eidos, Wedin argues, is the formal part only of the definition that applies to the thing. On Wedin's account, Aristotle introduces the notion of explanatory primacy of form in Z.4. (shrink)

Wedin offers an interpretation of Metaphysics Zeta 13, a very important and difficult chapter, where Aristotle apparently denies that substance is a universal, having, on most accounts, already claimed that form is substance, and that form is a universal. This interpretation of the argument of Z.13, Wedin argues, threatens the possibility of attaining a definition of substance, and places in doubt what has gone before in the treatise. According to Wedin, what Aristotle is concerned with in Z.13 is not the (...) denial that substance is a universal but rather with the denial that anything predicated universally of a c‐substance is the substance‐of the c‐substance, which Wedin calls ‘Weak proscription’. In Wedin's account of Z.13, then, Aristotle is continuing to pursue issues first raised in Z.10–11 that have to do with the relations between form and definition, and form and the universal. Wedin also argues that the dilemma at the conclusion of Z.13, regarding the definability of substance, is resolved in Z.16, with the notion of ‘Dual Complexity’, according to which the parts of a form exist potentially in the form, just as the parts of the c‐substance exist potentially in the c‐substance. (shrink)

Chapters 10 and 11 are critical to the argument of Metaphysics Zeta: these chapters are concerned with the purification of form. Z.10 introduces the apparatus of part and whole and consists of an argument to the end that form and its parts have priority over the other internal structural components of c‐substances, i.e. matter and the compound of form and matter; while in Z.11 Aristotle argues that form and its parts cannot involve any admixture of matter. Wedin argues that the (...) causal role of form presupposes its purity: if form is to explain why a certain portion of matter constitutes a c‐substance of a given kind, then matter—whether this latter is understood as functional, or remnant matter—cannot already be contained in the form. This is the ‘purity thesis’, as Wedin calls it: matter cannot be included in the definitions of natural organisms, Wedin points out, because that would make the explanandum a part of the principle that explains it. Wedin also argues that form must be sufficiently complex to provide definitions of the natures of particular compounds, which are complex, even though form itself is a kind of universal. (shrink)

In the Metaphysics, Aristotle often says that ‘form is substance’: in this chapter, Wedin argues that ‘substance’ in this context means the ‘substance‐of’ c‐substances. Wedin begins by examining Aristotle's use, and retention, of the framework of the Categories in Metaphysics Zeta, before turning to discuss Z.3, which is crucial to understanding the relation between the Categories and Metaphysics theories of substance, because it is usually thought that here Aristotle departs from the substance of the Categories. Wedin denies that Z.3 involves (...) a rejection of the Categories subjecthood criterion for primary substance: Aristotle rejects only the claim that one thing is the substance of another simply if it is the primary underlying subject of the other. Wedin argues that in Z.3 Aristotle introduces the idea of the ‘substance‐of’ a thing, and identifies as candidates for the substance of c‐substances the internal structural components form, matter, and the compound of form and matter. (shrink)

Wedin discusses Aristotle's claims in Metaphysics Zeta 6 that the essence of a thing is to be sought among its per se attributes, and that each thing that is primary and spoken of per se, e.g. primary substance, is the same as its essence. Wedin argues that the Zeta 6 Thesis, i.e. that the essence of a thing is the thing's immediate essence, is a crucial requirement of the explanatory role of essence as the substance of c‐substances. According to Wedin, (...) Aristotle introduces this requirement in order to establish that a thing's essence cannot depend on another essence. Form must be the essence; if form cannot explain the nature of the thing of which it is the form, there would be the threat of a regress of explanatory forms. At the end of the chapter, Wedin considers the problem that the identity of a thing and its essence is a requirement that can be met by non‐substance items, but he denies that this leads to a dilution of the conception of substance. (shrink)

Aristotelian virtue ethics does not treat motives or even character as the grounding basis for the rest of ethics, but the present agent‐based approach does. However, there are objections to agent‐basing that need to be considered. Having answered those objections, the chapter discusses three major forms of agent‐based virtue ethics: a somewhat less than plausible ”morality as inner strength” ; ”morality as universal benevolence” ; and ”morality as caring”. Any agent‐based morality does well to treat overall motivation, rather than occasional (...) single motives, as the fundamental basis for ethical characterizations. (shrink)

A commonsense virtue ethics can contribute to moral education by pointing out the importance of self‐regarding virtues. That importance is often ignored or neglected in school ”values” curricula.

Even though commonsense virtue ethics is less unified than utilitarianism, various forms of pluralism are inherent in utilitarianism.

Kantian and commonsense moral thinking are incoherent because self‐other asymmetry does not cogently combine with the belief that we owe more to people the closer they are to us in familial or personal terms. The latter is commonsensically explained by the claim that it is natural or inevitable that we should care about those closer to us more than about those less close to us, but this seemingly plausible assumption tends to undercut the justification that is typically and intuitively offered (...) for self‐other asymmetry, namely, that we naturally care more about ourselves than about others, so that there is no need to assume any duty to avoid harming oneself. A commonsense ethics of virtue avoids this and other incoherences or tensions in Kantian and commonsense moral thinking. (shrink)

Commonsense views about practical rationality are self‐other asymmetric in a way diametrically opposed to the asymmetry involved in commonsense or Kantian morality. What is likely to harm others does not count as irrational in the same fundamental way that what is likely to harm oneself does. Commonsense or Kantian morality is agent sacrificingly asymmetrical, whereas commonsense rationality is agent favouringly asymmetrical. This means that these two parts of ordinary thinking tug in opposite directions, but a virtue‐ethical approach that focuses exclusively (...) on commonsense thinking about what is admirable or counts as a virtue lacks this sort of tension, and such greater unity counts in its favor. (shrink)

Is an agent‐based sentimentalist virtue ethics of caring or benevolence sufficiently action‐guiding, given the focus on the inner life rather than external factors? The answer is that such forms of ethics are not meant to be practical in this sense, because a focus on what is right or obligatory takes the agent away from a praiseworthy focus on the good of other individuals. The ideal agent is deeply connected with and directly concerned about the welfare of others, and such a (...) person does not conscientiously worry about whether he or she is acting rightly or virtuously. This conclusion allows us to understand moral growth and moral conflict – though in a way different from the way in which the Kantian tradition, with its emphasis on conscientiousness, does. (shrink)