This is the most comprehensive book ever published on philosophical methodology. A team of thirty-eight of the world's leading philosophers present original essays on various aspects of how philosophy should be and is done. The first part is devoted to broad traditions and approaches to philosophical methodology. The entries in the second part address topics in philosophical methodology, such as intuitions, conceptual analysis, and transcendental arguments. The third part of the book is devoted to essays about the interconnections between philosophy (...) and neighbouring fields, including those of mathematics, psychology, literature and film, and neuroscience. (shrink)

This paper explores the question of what makes diagrammatic representations effective for human logical reasoning, focusing on how Euler diagrams support syllogistic reasoning. It is widely held that diagrammatic representations aid intuitive understanding of logical reasoning. In the psychological literature, however, it is still controversial whether and how Euler diagrams can aid untrained people to successfully conduct logical reasoning such as set-theoretic and syllogistic reasoning. To challenge the negative view, we build on the findings of modern diagrammatic logic and introduce (...) an Euler-style diagrammatic representation system that is designed to avoid problems inherent to a traditional version of Euler diagrams. It is hypothesized that Euler diagrams are effective not only in interpreting sentential premises but also in reasoning about semantic structures implicit in given sentences. To test the hypothesis, we compared Euler diagrams with other types of diagrams having different syntactic or semantic properties. Experiment compared the difference in performance between syllogistic reasoning with Euler diagrams and Venn diagrams. Additional analysis examined the case of a linear variant of Euler diagrams, in which set-relationships are represented by one-dimensional lines. The experimental results provide evidence supporting our hypothesis. It is argued that the efficacy of diagrams in supporting syllogistic reasoning crucially depends on the way they represent the relational information contained in categorical sentences. (shrink)

Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...) foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummett’s program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+ε without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummett’s contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummett’s framework is improved by these results as they show that questions of realism and anti-realism are not an “all or nothing” matter, but that there are plausibly metaphysical stances between the poles of anti-realism and realism, because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as “is smart,” which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong logic. (shrink)

Commentators have not said much regarding Berkeley and Stoicism. Even when they do, they generally limit their remarks to Berkeley’s Siris (1744) where he invokes characteristically Stoic themes about the World Soul, “seminal reasons,” and the animating fire of the universe. The Stoic heritage of other Berkeleian doctrines (e.g., about mind or the semiotic character of nature) is seldom recognized, and when it is, little is made of it in explaining his other doctrines (e.g., immaterialism). None of this is surprising, (...) considering how Stoics are considered arch-materialists and determinists. My aim is to suggest that our understanding of Berkeley’s philosophy is improved significantly by acknowledging its underlying Stoic character. I argue that Berkeley proposes not only a semantic ontology based on assumptions of Stoic logic but also a doctrine in which perceptions or ideas are intelligible precisely because they are always embedded in the propositions of a discourse or language. (shrink)

This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. The paper also shows how the always-self-predicative universals of category theory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having (...) one theory (e.g., Frege's Paradise) where universals could be either self-predicative or non-self-predicative (instead of being always one or always the other). (shrink)

Explanations of former South African President Thabo Mbeki’s public and private views on the aetiology of the HIV/AIDS epidemic in the country remain partial at best without the recognition that the latter presuppose and imply a postmodernist/postcolonialist philosophy of science that erases the line separating the political from the scientific. Evidence from Mbeki’s public speeches, interviews, and private and anonymous writings suggests that it was postmodernist/postcolonialist theory that inspired him to doubt the “Western” scientific consensus on HIV/AIDS and to implement (...) a public health policy that dragged its feet on full roll-out of antiretroviral therapy, causing thousands of avoidable deaths. A weak reductio ad absurdum allows us to conclude from this premise that postmodernist/postcolonial critique of “Western” science ought to be shunned. A comparative argument from consequences further suggests that in a situation where a misguided health policy has lead to a humanitarian catastrophe, and where postmodernist/postcolonialist critique of science can and has been used to justify this policy, an alternative theory ought to be preferred on which such justification would not be possible. The paper closes with a call for a non-relativist alternative to postmodernist/postcolonialist philosophy of science, and evaluates the potential of recent developments in ‘Studies of Expertise” to yield such. (shrink)

Recent work on conditional reasoning argues that denying the antecedent [DA] and affirming the consequent [AC] are defeasible but cogent patterns of argument, either because they are effective, rational, albeit heuristic applications of Bayesian probability, or because they are licensed by the principle of total evidence. Against this, we show that on any prevailing interpretation of indicative conditionals the premises of DA and AC arguments do not license their conclusions without additional assumptions. The cogency of DA and AC inferences rather (...) depends on contingent factors extrinsic to, and independent of, what is asserted by DA and AC arguments. arguments do not license their conclusions without additional assumptions. The cogency of DA and AC inferences rather depends on contingent factors extrinsic to, and independent of, what is asserted by DA and AC arguments. (shrink)

In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...) in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917--18, and that Bernays's contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert's Program are the development of formal mathematical theories and proof-theoretical investigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann's 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to www . He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert's &egr;-substitution method. Detailed analyses of both proofs in Chapter 3 shed light on the development of finitism and proof theory in the 1920s as practiced in Hilbert's school. ;In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert's finitism. According to him, mathematical intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowledge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann's dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert's conception of finitism is concerned. Tait's conception, however, differs from Hilbert's in important respects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning. (shrink)

The dissertation is an investigation into the structure of Aristotle's modal propositions through careful attention to the text of the Prior Analytics. I take account not only of recent attempts to formalize Aristotle's modal syllogistic but also of the discussion that Aristotle himself provides about modal statements. I provide evidence that his modal propositions are to be construed in a de re manner and then go on to investigate the problems raised by a de re analysis, particularly those problems concerned (...) with Aristotle's modal conversion principles. A large part of my project is to show that these can be given a valid de re analysis that sits well with the results Aristotle sketches. Terms in modal syllogistic premises are shown to be restricted in ways that reflect Aristotle's early metaphysics and notions of predication. (shrink)

The studies that make up this book were written and brought together to honor the memory of Jan Pinborg. His unexpected death in 1982 at the age of forty-five shocked and saddened students of medieval philosophy everywhere and left them with a keen sense of disappoint ment. In his fifteen-year career Jan Pinborg had done so much for our field with his more than ninety books, editions, articles, and reviews and had done it all so well that we recognized him (...) as a leader and counted on many more years of his scholarship, his help, and his friendship. To be missed so sorely by his international colleagues in an academic field is a mark of Jan's achievement, but only of one aspect of it, for historians of philosophy are not the only scholars who have reacted in this way to Jan's death. In his decade and a half of intense productivity he also acquired the same sort of special status among historians of linguistics, whose volume of essays in his memory is being G. L. Bursill-Hall almost simultane published under the editorship of ously with this one. Sten Ebbesen, Jan's student, colleague, and successor as Director of the Institute of Medieval Greek and Latin Philology at the University of Copenhagen, has earned the gratitude of all of us by memorializing Jan 1 in various biographical sketches, one of which is accompanied by a 2 complete bibliography of his publications. (shrink)

Today it would be considered "bad Platonic metaphysics" to think that among all the concrete instances of a property there could be a universal instance so that all instances had the property by virtue of participating in that concrete universal. Yet there is a mathematical theory, category theory, dating from the mid-20th century that shows how to precisely model concrete universals within the "Platonic Heaven" of mathematics. This paper, written for the philosophical logician, develops this category-theoretic treatment of concrete universals (...) along with a new concept to abstractly model the functions of a brain. (shrink)

A dialectical contradiction can be appropriately described within the framework of classical formal logic. It is in harmony with the law of noncontradiction. According to our definition, two theories make up a dialectical contradiction if each of them is consistent and their union is inconsistent. It can happen that each of these two theories has an intended model. Plenty of examples are to be found in the history of science.

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

The essence of the argument of the beard (so-called by some logic textbooks) is the tactic used by a respondent to reply to a proponent, "The criterion you used to define a key term in your argument is vague, therefore your use of this term in your argument is illegitimate, and your argument is refuted." This familiar kind of argument tactic is similar to the much more famous heap (sorites) argument of Eubulides, closely associated with the slippery slope argument. This (...) article provides a system of classification for sorting out these three arguments, and related types of argument of interest in informal logic. (shrink)

Abstract. As a general theory of reasoning—and as a general theory of what holds true under every possible circumstance—logic is supposed to be ontologically neutral. It ought to have nothing to do with questions concerning what there is, or whether there is anything at all. It is for this reason that traditional Aristotelian logic, with its tacit existential presuppositions, was eventually deemed inadequate as a canon of pure logic. And it is for this reason that modern quantification theory, too, with (...) its residue of existentially loaded theorems and patterns of inference, has been claimed to suffer from a defect of logical purity. The law of non-contradiction rules out certain circumstances as impossible—circumstances in which a statement is both true and false, or perhaps circumstances where something both is and is not the case. Is this to be regarded as a further ontological bias? (shrink)

Inference versus consequence , an invited lecture at the LOGICA 1997 conference at Castle Liblice, was part of a series of articles for which I did research during a Stockholm sabbatical in the autumn of 1995. The article seems to have been fairly effective in getting its point across and addresses a topic highly germane to the Uppsala workshop. Owing to its appearance in the LOGICA Yearbook 1997 , Filosofia Publishers, Prague, 1998, it has been rather inaccessible. Accordingly it is (...) republished here with only bibliographical changes and an afterword. (shrink)

This paper examines the contemporary philosophical and cognitive relevance of Charles Peirce's diagrammatic logic of existential graphs (EGs), the ‘moving pictures of thought’. The first part brings to the fore some hitherto unknown details about the reception of EGs in the early 1900s that took place amidst the emergence of modern conceptions of symbolic logic. In the second part, philosophical aspects of EGs and their contributions to contemporary logical theory are pointed out, including the relationship between iconic logic and images, (...) the problem of the meaning of logical constants, the cognitive economy of iconic logic, the failure of the Frege–Russell thesis, and the failure of the Language of Thought hypothesis. (shrink)

A general survey of Frege's views on truth, the paper explores the problems in response to which Frege's distinctive view that sentences refer to truth-values develops. It also discusses his view that truth-values are objects and the so-called regress argument for the indefinability of truth. Finally, we consider, very briefly, the question whether Frege was a deflationist.

Toisaalta ennennäkemätön äärettömien joukko-opillisten menetelmien hyödyntäminen sekä toisaalta epäilyt niiden hyväksyttävyydestä ja halu oikeuttaa niiden käyttö ovat ratkaisevasti muovanneet vuosisatamme matematiikkaa ja logiikkaa. Tämän kehityksen vaikutus nykyajan filosofiaan on myös ollut valtaisa; merkittävää osaa siitä ei voi edes ymmärtää tuntematta sen yhteyttä tähän matematiikan ja logiikan vallankumoukseen. Lähestymistapoja, jotka tavalla tai toisella hyväksyvät äärettömän matematiikan ja perinteisten logiikan sääntöjen (erityisesti kolmannen poissuljetun lain) soveltamisen myös sen piirissä, on tullut tavaksi kutsua klassiseksi matematiikaksi ja logiikaksi erotuksena nämä hylkäävistä radikaaleista intuitionistisista ja (...) konstruktivistisista opeista. (shrink)

This book presents novel formalizations of three of the most important medieval logical theories: supposition, consequence and obligations. In an additional fourth part, an in-depth analysis of the concept of formalization is presented - a crucial concept in the current logical panorama, which as such receives surprisingly little attention.Although formalizations of medieval logical theories have been proposed earlier in the literature, the formalizations presented here are all based on innovative vantage points: supposition theories as algorithmic hermeneutics, theories of consequence analyzed (...) with tools borrowed from model-theory and two-dimensional semantics, and obligations as logical games. For this reason, this is perhaps the first time that these medieval logical theories are made fully accessible to the modern philosopher and logician who wishes to obtain a better grasp of them, but who has always been held back by the lack of appropriate ‘translations' into modern terms.Moreover, the book offers a reflection on the very nature of logic, a reflection that is prompted by the comparisons between medieval and modern logic, their similarities and dissimilarities. It is thus a contribution not only to the history of logic, but also to the philosophy of logic, the philosophy of language and semantics.The analysis of medieval logic is also relevant for the modern philosopher and logician in that, being the unifying methodology used across all disciplines at that time, logic really provided unity to science. It thus presents a unified model of scientific investigation, where logic plays the aggregating role. (shrink)

This paper examines how the work of Frege was known and received in Poland in the period 1910–1935 (with one exception concerning the later work of Suszko). The main thesis is that Frege's reception in Poland was perhaps faster and deeper than in other countries, except England, due to works of Russell and Jourdain. The works of Łukasiewicz, Leśniewski and Czeżowski are described.

Peter Geach describes the 'doctrine of distribution' as the view that a term is distributed if it refers to everything that it denotes, and undistributed if it refers to only some of the things that it denotes. He argues that the notion, so explained, is incoherent. He claims that the doctrine of distribution originates from a degenerate use of the notion of ?distributive supposition? in medieval supposition theory sometime in the 16th century. This paper proposes instead that the doctrine of (...) distribution occurs at least as early as the 12th century, and that it originates from a study of Aristotle's notion of a term's being ?taken universally?, and not from the much later theory of distributive supposition. A detailed version of the doctrine found in the Port Royal Logic is articulated, and compared with a slightly different modern version. Finally, Geach's arguments for the incoherence of the doctrine are discussed and rejected. (shrink)

This is a thorough treatment of first-order modal logic. The book covers such issues as quantification, equality (including a treatment of Frege's morning star/evening star puzzle), the notion of existence, non-rigid constants and function symbols, predicate abstraction, the distinction between nonexistence and nondesignation, and definite descriptions, borrowing from both Fregean and Russellian paradigms.

[full article, abstract in English; only abstract in Lithuanian] The paper deals with the problem of the “two Barbaras” in the Aristotelian modal syllogistic. The problem consists in Aristotle’s differing views on two at a first sight similar in nature syllogisms of mixed assertoric and necessary premises: Barbara LXL and Barbara XLL. The fact that Aristotle believed the first syllogism to be valid and the second one – not, has been received either 1) negatively, because both Barbaras have been held (...) to be invalid, or 2) negatively, because both Barbaras have been held to be valid, or 3) positively, by giving a reason why the two Barbaras differ. We commit ourselves to the position by proving that modal propositions for Aristotle have their modalities de dicto and that Aristotelian modal operators act according to their own separate rules where only the type of the modality of a major premise is relevant for the modal status of the conclusion. (shrink)

Conventional approaches to consciousness assume that our current science tells us within tolerable limits what physical nature is. Because nature so understood cannot explain consciousness as we seem to experience it ourselves, explaining consciousness becomes a problem. One solution is to rethink what consciousness is so that it becomes the sort of thing our current natural science could in principle explain. Whitehead takes the opposite approach, using the existence of consciousness as a clue to what nature must be if it (...) can generate something like consciousness. The justification for this approach can be found in Whitehead’s implicit indictment of descriptive phenomenology. According to Whitehead, the seemingly insoluble problem of explaining consciousness naturalistically is an artifact created by the assumption that consciousness faithfully samples the world, when in fact it obscures the very aspects of nature that are indispensable to understanding how anything, including consciousness itself, could emerge through a physical process. (shrink)

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

A shift from a metaphysical framework of substance to one of process enables an integrated account of the emergence of normative phenomena. I show how substance assumptions block genuine ontological emergence, especially the emergence of normativity, and how a process framework permits a thermodynamic-based account of normative emergence. The focus is on two foundational forms of normativity, that of normative function and of representation as emergent in a particular kind of function. This process model of representation, called interactivism, compels changes (...) in many related domains. The discussion ends with brief attention to three domains in which changes are induced by the representational model: perception, learning, and language. (shrink)

Recently it has become almost the received wisdom in certain quarters that Kripke models are appropriate only for something like metaphysical modalities, and not for logical modalities. Here the line of thought leading to Kripke models, and reasons why they are no less appropriate for logical than for other modalities, are explained. It is also indicated where the fallacy in the argument leading to the contrary conclusion lies. The lessons learned are then applied to the question of the status of (...) the formula. (shrink)

In 1880, when Oliver Wendell Holmes (later to be a Justice of the U.S. Supreme Court) criticized the logical theology of law articulated by Christopher Columbus Langdell (the first Dean of Harvard Law School), neither Holmes nor Langdell was aware of the revolution in logic that had begun, the year before, with Frege's Begriffsschrift. But there is an important element of truth in Holmes's insistence that a legal system cannot be adequately understood as a system of axioms and corollaries; and (...) this element of truth is not obviated by the more powerful logical techniques that are now available. (shrink)

Nykyaikaisen logiikan keskeisenä tutkimuskohteena ovat erilaiset formalisoidut teoriat. Erityisesti vuosisadan vaihteen aikoihin matematiikan perusteiden tutkimuksessa ilmaantuneiden hämmentävien paradoksien (Russell 1902, 1903) jälkeen (ks. kuitenkin jo Frege 1879, Dedekind 1888, Peano 1889; vrt. Wang 1957) keskeiset matemaattiset teoriat on pyritty tällaisten vaikeuksien välttämiseksi uudelleen muotoilemaan täsmällisesti keinotekoisessa symbolikielessä, jonka lauseenmuodostussäännöt on täsmällisesti ja yksikäsitteisesti määrätty. Edelleen teoriat on pyritty aksiomatisoimaan, ts. on pyritty antamaan joukko peruslauseita, joista kaikki muut - tai ainakin mahdollisimman monet - teorian todet lauseet voidaan loogisesti johtaa tarkoin (...) määrättyjen päättelysääntöjen mukaisesti (erityisesti Hilbert 1904, 1918, 1927; ks. myös Russell 1908, Zermelo 1908; vrt. Kleene 1952, §§ 14−15). (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

In the 'Critique of Pure Reason' Kant appears to characterize analytic judgments in four distinct ways: once in terms of “containment,” a second time in terms of “identity,” a third time in terms of the explicative–ampliative contrast, and a fourth time in terms of the notion of “cognizability in accordance with the principle of contradiction.” The paper asks: Which of these characterizations—or apparent characterizations—best captures Kant’s conception of analyticity in the first Critique? It suggests: “the second.” It argues, further, that (...) Kant’s distinction is intended to apply only to judgments of subject–predicate form, and that the fourth alleged characterization is not properly speaking a characterization at all. These theses are defended in the course of a more general investigation of the distinction’s meaning, its epistemology, and its tenability. (shrink)

The syntax of Frege's scientific language is commonly taken to be characterized by two oddities: the representation of the intended illocutionary role of sentences by a special sign, the judgement-stroke, and the treatment of sentences as a species of singular terms. In this paper, an alternative view is defended. The main theses are: the syntax of Frege's scientific language aims at an explication of the logical form of judgements; the judgement-stroke is, therefore, a truth-operator, not a pragmatic operator; in Frege's (...) first system, '⊦ Δ' expresses that the circumstance Δ is a fact, and in his second system that the truth-value - Δ is the True; in both systems, the judgement-stroke is construed as a sign sui generis, not as a genuine predicate; its counterpart in natural language is the syntactic "form of assertoric sentences", not the truth-predicate; neither in Frege's first nor in his second system sentences are treated as singular terms. (shrink)

This paper examines the question of the extensional correctness of Tarskian definitions of logical truth and logical consequence. I identify a few different informal properties which are necessary for a sentence to be an informal logical truth and look at whether they are necessary properties of Tarskian logical truths. I examine arguments by John Etchemendy and Vann McGee to the effect that some of those properties are not necessary properties of some Tarskian logical truths, and find them unconvincing. I stress (...) the point that since the hypothesis that Tarski's definitions are extensionally correct is deeply entrenched, the burden of proof is still on the shoulders of Tarski's critics, who have not lifted the burden. (shrink)

Some of the most important developments of symbolic logic took place in the 1920s. Foremost among them are the distinction between syntax and semantics and the formulation of questions of completeness and decidability of logical systems. David Hilbert and his students played a very important part in these developments. Their contributions can be traced to unpublished lecture notes and other manuscripts by Hilbert and Bernays dating to the period 1917-1923. The aim of this paper is to describe these results, focussing (...) primarily on propositional logic, and to put them in their historical context. It is argued that truth-value semantics, syntactic ("Post-") and semantic completeness, decidability, and other results were first obtained by Hilbert and Bernays in 1918, and that Bernays's role in their discovery and the subsequent development of mathematical logic is much greater than has so far been acknowledged. (shrink)

The paper studies the relation between systems of modal logic and systems of consequential implication, a non-material form of implication satisfying "Aristotle's Thesis" (p does not imply not p) and "Weak Boethius' Thesis" (if p implies q, then p does not imply not q). Definitions are given of consequential implication in terms of modal operators and of modal operators in terms of consequential implication. The modal equivalent of "Strong Boethius' Thesis" (that p implies q implies that p does not imply (...) not q) is identified. (shrink)

My first paper on the Is/Ought issue. The young Arthur Prior endorsed the Autonomy of Ethics, in the form of Hume’s No-Ought-From-Is (NOFI) but the later Prior developed a seemingly devastating counter-argument. I defend Prior's earlier logical thesis (albeit in a modified form) against his later self. However it is important to distinguish between three versions of the Autonomy of Ethics: Ontological, Semantic and Ontological. Ontological Autonomy is the thesis that moral judgments, to be true, must answer to a realm (...) of sui generis non-natural PROPERTIES. Semantic autonomy insists on a realm of sui generis non-natural PREDICATES which do not mean the same as any natural counterparts. Logical Autonomy maintains that moral conclusions cannot be derived from non-moral premises.-moral premises with the aid of logic alone. Logical Autonomy does not entail Semantic Autonomy and Semantic Autonomy does not entail Ontological Autonomy. But, given some plausible assumptions Ontological Autonomy entails Semantic Autonomy and given the conservativeness of logic – the idea that in a valid argument you don’t get out what you haven’t put in – Semantic Autonomy entails Logical Autonomy. So if Logical Autonomy is false – as Prior appears to prove – then Semantic and Ontological Autonomy would appear to be false too! I develop a version of Logical Autonomy (or NOFI) and vindicate it against Prior’s counterexamples, which are also counterexamples to the conservativeness of logic as traditionally conceived. The key concept here is an idea derived in part from Quine - that of INFERENCE-RELATIVE VACUITY. I prove that you cannot derive conclusions in which the moral terms appear non-vacuously from premises from which they are absent. But this is because you cannot derive conclusions in which ANY (non-logical) terms appear non-vacuously from premises from which they are absent Thus NOFI or Logical Autonomy comes out as an instance of the conservativeness of logic. This means that the reverse entailment that I have suggested turns out to be a mistake. The falsehood of Logical Autonomy would not entail either the falsehood Semantic Autonomy or the falsehood of Ontological Autonomy, since Semantic Autonomy only entails Logical Autonomy with the aid of the conservativeness of logic of which Logical Autonomy is simply an instance. Thus NOFI or Logical Autonomy is vindicated, but it turns out to be a less world-shattering thesis than some have supposed. It provides no support for either non-cognitivism or non-naturalism. (shrink)

In this paper, we investigate (1) what can be salvaged from the original project of "logicism" and (2) what is the best that can be done if we lower our sights a bit. Logicism is the view that "mathematics is reducible to logic alone", and there are a variety of reasons why it was a non-starter. We consider the various ways of weakening this claim so as to produce a "neologicism". Three ways are discussed: (1) expand the conception of logic (...) used in the reduction, (2) allow the addition of analytic-sounding principles to logic so that the reduction is not to "logic alone" but to logic and truths knowable a priori, and (3) revise the conception of "reducible". We show how the current versions of neologicism fit into this classification scheme, and then focus on a kind of neologicism which we take to have the most potential for achieving the epistemological goals of the original logicist project. We argue that that the "weaker" the form of neologicism, the more likely it is to be a new form of logicism, and show how our preferred system, though mathematically weak, is metaphysically and epistemogically strong, and can "reduce" arbitrary mathematical theories to logic and analytic truths, if given a legitimate new sense of "reduction". (shrink)

Dramatic changes or revolutions in a field of science are often made by outsiders or 'trespassers,' who are not limited by the established, 'expert' approaches. Each essay in this diverse collection shows the fruits of intellectual trespassing and poaching among fields such as economics, Kantian ethics, Platonic philosophy, category theory, double-entry accounting, arbitrage, algebraic logic, series-parallel duality, and financial arithmetic.

A comprehensive survey of Martin-Löf's constructive type theory, considerable parts of which have only been presented by Martin-Löf in lecture form or as part of conference talks. Sommaruga surveys the prehistory of type theory and its highly complex development through eight different stages from 1970 to 1995. He also provides a systematic presentation of the latest version of the theory, as offered by Martin-Löf at Leiden University in Fall 1993. This presentation gives a fuller and updated account of the system. (...) Earlier, brief presentations took no account of the issues related to the type-theoretical approach to logic and the foundations of mathematics, while here they are accorded an entire part of the book. Readership: Comprehensive accounts of the history and philosophy of constructive type theory and a considerable amount of related material. Readers need a solid background in standard logic and a first, basic acquaintance with type theory. (shrink)

In this paper we elaborate a conception of entailment based on what we call the Ackermann principle, which explicates valid entailment through a logical connection between sentences depending on their informational content. We reconstruct Dunn’s informational semantics for entailment on the basis of Restall’s approach, with assertion and denial as two independent speech acts, by introducing the notion of a ‘position description’. We show how the machinery of position descriptions can effectively be used to define the positive and the negative (...) information carried by sentences of a given language and to present a formalized version of the Ackermann principle as an inclusion relationship between the informational contents of the conclusions and the premises of a valid entailment. Being so interpreted, the entailment relation exhibits certain properties, including the property of transitivity. Whereas properties such as Anderson and Belnap’s variable sharing property or Parry’s proscriptive principle are normally presented as imposing a relevance requirement on valid entailment, the suggested formalization of the Ackermann principle supports all of Gentzen’s structural rules, including weakening, a rule that is normally given up in sequent-style proof systems for relevance logics. In this way we propose an Ackermann-inspired explication of the nature of entailment as a relation between the informational contents of sentences. (shrink)

This essay offers a conception of logic by which logic may be considered to be exceptional among the sciences on the backdrop of a naturalistic outlook. The conception of logic focused on emphasises the traditional role of logic as a methodology for the sciences, which distinguishes it from other sciences that are not methodological. On the proposed conception, the methodological aims of logic drive its definitions and principles, rather than the description of scientific phenomena. The notion of a methodological discipline (...) is explained as a relation between disciplines or practices. Logic serves as a methodological discipline with respect to any theoretical practice, and this generality, as well as logic’s reflexive nature, distinguish it from other methodological disciplines. Finally, the evolution of model theory is taken as a case study, with a focus on its methodological role. Following recent work by John Baldwin and Juliette Kennedy, we look at model theory from its inception in the mid-twentieth century as a foundational endeavour until developments at the end of the century, where the classification of theories has taken centre-stage. (shrink)

Based on historical comparisons among master-pupil chains and other aspects of social networks among philosophers, some prmciples are suggested regarding long-term intellectual change. The higher the eminence ofphilosophers, the more tightly they are connected to mtergenerational chains of other eminent philosophers, and to horizontal circles of the intellectual community. Intellectual creativity proceeds through the contemporaneous development of rival positions, dividing up the available attention space in the intellectual community. Strong thought-communities, those that have strong external support for their institutional base, (...) subdivide to maximize internal distinctiveness; weakly supported thought-communities disappear or amalgamate by syncretism. External conditions thus affect the content of ideas indirectly by affecting the space available in the mternal field of the intellectual community. The content of philosophies, the degree of abstraction and self-conscious reflection upon intellectual operations, depends on how many generations intellectual networks maintain continuity under conditions of creative rivalry. New positions are produced by competitive appropriation of prior ideas and by negation of preexisting positions along the lines of greatest organizational rivalry. (shrink)

This paper explores trivalent truth conditions for indicative conditionals, examining the “defective” truth table proposed by de Finetti and Reichenbach. On their approach, a conditional takes the value of its consequent whenever its antecedent is true, and the value Indeterminate otherwise. Here we deal with the problem of selecting an adequate notion of validity for this conditional. We show that all standard validity schemes based on de Finetti’s table come with some problems, and highlight two ways out of the predicament: (...) one pairs de Finetti’s conditional with validity as the preservation of non-false values, but at the expense of Modus Ponens; the other modifies de Finetti’s table to restore Modus Ponens. In Part I of this paper, we present both alternatives, with specific attention to a variant of de Finetti’s table proposed by Cooper and Cantwell. In Part II, we give an in-depth treatment of the proof theory of the resulting logics, DF/TT and CC/TT: both are connexive logics, but with significantly different algebraic properties. (shrink)

Controversy has surrounded the interpretation of the so-called 'Diodorean' and 'Chrysippean' conditionals of the Stoics. I critically evaluate and reject two interpretations of each of them: as expressing natural laws, and as strict conditionals. In doing so I engage with the work of authors such as Frede, Gould, Hurst, the Kneales, Mates, and Prior. I conclude by offering my own proposal for where these Stoic conditionals should be located on a 'ladder' of logical strength.

De acuerdo con los principales enfoques al respecto la lógica de los estoicos es principalmente un sistema deductivo, lo que, en términos actuales, ha sido visto como un sistema de lógica proposicional. La obra de Crísipo acerca de los cinco argumentos indemostrables constituye la principal base de dicho sistema. En este artículo se examina la naturaleza de dichos cinco indemostrables así como el llamando teorema de Antipatro y los esquemas básicos de inferencia, o zemas. Por otra parte y en particular, (...) sostengo que dichos argumentos indemostrables tienen otro importante rol para la filosofía y la historia de la lógica, cual es el de constituirse en una justificación de la deducción, un problema central en el presente. Se analiza, primero, la justificación por los indemostrables y por el principio de condicionalización y, segundo, la justificación cognitiva de la lógica implícita en un sistema estoico. (shrink)