En este artículo presentamos una posible vía para interpretar las nociones de posibilidad y necesidad desarrolladas en el seno de la lógica megárico-estoica como operadores modales aléticos. Se introducirá la semántica megárico-estoica como trasfondo metafísico de las definiciones de necesidad y posibilidad y se ofrecerán argumentos para abandonar las interpretaciones predominantes que incluyen variables temporales ad hoc. Tras proponer la lectura de las definiciones diodóricas desde una semántica modal relacional se señalará una serie de temas que merecen ser revisitados desde (...) esta óptica. (shrink)

(1) An introduction to the principles of conceptual modelling, combinatorial heuristics and epistemological history; (2) the examination of a number of perennial epistemological-methodological schemata: conceptual spaces and blending theory; ars inveniendi and ars demonstrandi; the two modes of analysis and synthesis and their relationship to ars inveniendi; taxonomies and typologies as two fundamental epistemic structures; extended cognition, cognitio symbolica and model-based reasoning; (3) Plato’s notions of conceptual spaces, conceptual blending and hypothetical-analogical models (paradeigmata); (4) Ramon Llull’s concept analysis and combinatoric (...) spaces; (5) Gottfried Leibniz’s development of compositional analysis and synthesis as a general modelling method and a paradigm for ars inveniendi; (6) Fritz Zwicky’s revival of the morphological method of analysis and construction, and its subsequent computerised applications. (shrink)

Modus ponens is the argument from premises of the form If A, then B and A to the conclusion B. Nearly all participants agree that the modus ponens conclusion logically follows when the argument appears in this Basic form. However, adding a further premise can lower participants’ rate of agreement—an effect called suppression. We propose a theory of suppression that draws on contemporary ideas about conditional sentences in linguistics and philosophy. Semantically, the theory assumes that people interpret an indicative conditional (...) as a context-sensitive strict conditional: true if and only if its consequent is true in each of a contextually determined set of situations in which its antecedent is true. Pragmatically, the theory claims that context changes in response to new assertions, including new conditional premises. Thus, the conclusion of a modus ponens argument may no longer be accepted in the changed context. Psychologically, the theory describes people as capable of reasoning about broad classes of possible situations, ordered by typicality, without having to reason about individual possible worlds. The theory accounts for the main suppression phenomena, and it generates some novel predictions that new experiments confirm. (shrink)

Arguably no concept is more fundamental to science than that of causality, for investigations into cases of existence, persistence, and change in the natural world are largely investigations into the causes of these phenomena. Yet the metaphysics and epistemology of causality remain unclear. For example, the ontological categories of the causal relata have been taken to be objects (Hume 1739), events (Davidson 1967), properties (Armstrong 1978), processes (Salmon 1984), variables (Hitchcock 1993), and facts (Mellor 1995). (For convenience, causes and effects (...) will usually be understood as events in what follows.) Complicating matters, causal relations may be singular (Socrates’s drinking hemlock caused Socrates’s death) or general (Drinking hemlock causes death); hence the relata might be tokens (e.g., instances of properties) or types (e.g., types of events) of the category in question. Other questions up for grabs are: Are singular causes metaphysically and/or epistemologically prior to general causes or vice versa (or neither)? What grounds the intuitive asymmetry of the causal relation? Are macro-causal relations reducible to micro-causal relations? And perhaps most importantly: Are causal facts (e.g., the holding of causal relations) reducible to non-causal facts (e.g., the holding of certain spatiotemporal relations)? (shrink)

Logic begins but does not end with the study of truth and falsity. Within truth there are the modes of truth, ways of being true: necessary truth and contingent truth. When a proposition is true, we may ask whether it could have been false. If so, then it is contingently true. If not, then it is necessarily true; it must be true; it could not have been false. Falsity has modes as well: a false proposition that could not have been (...) true is impossible or necessarily false; one that could have been true is merely contingently false. The proposition that some humans are over seven feet tall is contingently true; the proposition that all humans over seven feet tall are over six feet tall is necessarily true; the proposition that some humans are over seven feet tall and under six feet tall is impossible, and the proposition that some humans are over nine feet tall is contingently false. Of these four modes of truth, let us focus on necessity, plus a fifth: possibility. A proposition is possible if it is or could have been true; hence propositions that are either necessarily true, contingently true, or contingently false are possible. Notions that are similar to the modes of truth in being concerned with what might have been are called modal. Dispositions are modal notions, for example the disposition of fragility. Relatedly, there are counterfactual conditionals, for example “if this glass were dropped, it would break.” And the notion of supervenience is modal.1 But let us focus here on necessity and possibility. Modal words are notoriously ambiguous (or at least context-sensitive2). I may reply to an invitation to give a talk in England by saying “I can’t come; I have to give a talk in California the day before”. This use of “can’t” is perfectly appropriate. But it would be equally appropriate for me to say that I could cancel my talk in California (although that would be rude) and give the talk in England instead. What I cannot do is give both talks.. (shrink)

Like modal logic, temporal logic, and description logic, separation logic has become a popular class of logical formalisms in computer science, conceived as assertion languages for Hoare-style proof systems with the goal to perform automatic program analysis. In a broad sense, separation logic is often understood as a programming language, an assertion language and a family of rules involving Hoare triples. In this survey, we present similarities between separation logic as an assertion language and modal and temporal logics. Moreover, we (...) propose a selection of landmark results about decidability, complexity and expressive power. (shrink)

In 1934 Alonzo Church, Kurt Gödei, S. C. Kleene, and J. B. Rosser were all to be found in Princeton, New Jersey. In 1936 Church founded The Journal of Symbolic Logic. Shortly thereafter Alan Turing arrived for a two year visit. The United States had become a world center for cutting-edge research in mathematical logic. In this brief survey1 we shall examine some of the writings of American logicians during the 1920s, a period of important beginnings and remarkable insights as (...) well as of confused gropings.The publication of Whitehead and Russell's monumental Principia Mathematica [18] during the years 1910-1913 provided the basis for much of the research that was to follow. It also provided the basis for confusion that remained a factor during the period we are discussing. In 1908, Henri Poincaré, a famous skeptic where mathematical logic was concerned, wrote pointedly :It is difficult to admit that the word if acquires, when written ⊃, a virtue it did not possess when written if.Principia provided no very convincing answer to Poincaré. Indeed the fact that the authors of Principia saw fit to place their first two “primitive propositions”*1.1: Anything implied by a true proposition is true.*1.2: ⊢ p ⋁ p ⊃ punder one and the same heading suggest that they had thought of what they were doing as just such a translation as Poincare had derided. (shrink)

John Searle’s Chinese room argument is a celebrated thought experiment designed to refute the hypothesis, popular among artificial intelligence scientists and philosophers of mind, that “the appropriately programmed computer really is a mind”. Since its publication in 1980, the CRA has evoked an enormous amount of debate about its implications for machine intelligence, the functionalist philosophy of mind, theories of consciousness, etc. Although the general consensus among commentators is that the CRA is flawed, and not withstanding the popularity of the (...) systems reply in some quarters, there is remarkably little agreement on exactly how and why it is flawed. A newcomer to the controversy could be forgiven for thinking that the bewildering collection of diverse replies to Searle betrays a tendency to unprincipled, ad hoc argumentation and, thereby, a weakness in the opposition’s case. In this paper, treating the CRA as a prototypical example of a ‘destructive’ thought experiment, I attempt to set it in a logical framework, which allows us to systematise and classify the various objections. Since thought experiments are always posed in narrative form, formal logic by itself cannot fully capture the controversy. On the contrary, much also hinges on how one translates between the informal everyday language in which the CRA was initially framed and formal logic and, in particular, on the specific conception of possibility that one reads into the logical formalism. (shrink)

C. I. Lewis (I883-I964) was the first major figure in history and philosophy of logic—-a field that has come to be recognized as a separate specialty after years of work by Ivor Grattan-Guinness and others (Dawson 2003, 257).Lewis was among the earliest to accept the challenges offered by this field; he was the first who had the philosophical and mathematical talent, the philosophical, logical, and historical background, and the patience and dedication to objectivity needed to excel. He was blessed with (...) many fortunate circumstances, not least of which was entering the field when mathematical logic, after only six decades of toil, had just reaped one of its most important harvests with publication of the monumental Principia Mathematica. It was a time of joyful optimism which demanded an historical account and a sober philosophical critique. Lewis was one of the first to apply to mathematical logic the Aristotelian dictum that we do not understand a living institution until we see it growing from its birth. (shrink)

The logical properties of the 'if-then' connective of ordinary English differ markedly from the logical properties of the material conditional of classical, two-valued logic. This becomes apparent upon examination of arguments in conversational English which involve (noncounterfactual) usages of if-then'. A nonclassical system of propositional logic is presented, whose conditional connective has logical properties approximating those of 'if-then'. This proposed system reduces, in a sense, to the classical logic. Moreover, because it is equivalent to a certain nonstandard three-valued logic, its (...) decision procedure is almost as efficient as that of the classical logic. It therefore provides a rational and convenient system in which to formalize English arguments. (shrink)

Dugundji proved in 1940 that most parts of standard modal systems cannot be characterized by a single finite deterministic matrix. In the eighties, Ivlev proposed a semantics of four-valued non-deterministic matrices, in order to characterize a hierarchy of weak modal logics without the necessitation rule. In a previous paper, we extended some systems of Ivlev’s hierarchy, also proposing weaker six-valued systems in which the axiom was replaced by the deontic axiom. In this paper, we propose even weaker systems, by eliminating (...) both axioms, which are characterized by eight-valued non-deterministic matrices. In addition, we prove completeness for those new systems. It is natural to ask if a characterization by finite ordinary logical matrices would be possible for all those Ivlev-like systems. We will show that finite deterministic matrices do not characterize any of them. (shrink)

Recent experiments have shown that naive speakers find borderline contradictions involving vague predicates acceptable. In Cobreros et al. we proposed a pragmatic explanation of the acceptability of borderline contradictions, building on a three-valued semantics. In a reply, Alxatib et al. show, however, that the pragmatic account predicts the wrong interpretations for some examples involving disjunction, and propose as a remedy a semantic analysis instead, based on fuzzy logic. In this paper we provide an explicit global pragmatic interpretation rule, based on (...) a somewhat richer semantics, and show that with its help the problem can be overcome in pragmatics after all. Furthermore, we use this pragmatic interpretation rule to define a new consequence-relation and discuss some of its properties. (shrink)

In 1951 C.I.Lewis published a logic of general terms that he called the calculus of predicates. Although this system is of less significance than Lewis’s earlier work on proposition...

It is often thought that consciousness has a qualitative dimension that cannot be tracked by science. Recently, however, some philosophers have argued that this worry stems not from an elusive feature of the mind, but from the special nature of the concepts used to describe conscious states. Marc Champagne draws on the neglected branch of philosophy of signs or semiotics to develop a new take on this strategy. The term “semiotics” was introduced by John Locke in the modern period – (...) its etymology is ancient Greek, and its theoretical underpinnings are medieval. Charles Sanders Peirce made major advances in semiotics, so he can act as a pipeline for these forgotten ideas. Most philosophers know Peirce as the founder of American pragmatism, but few know that he also coined the term “qualia,” which is meant to capture the intrinsic feel of an experience. Since pragmatic verification and qualia are now seen as conflicting commitments, Champagne endeavors to understand how Peirce could (or thought he could) have it both ways. The key, he suggests, is to understand how humans can insert distinctions between features that are always bound. Recent attempts to take qualities seriously have resulted in versions of panpsychism, but Champagne outlines a more plausible way to achieve this. So, while semiotics has until now been the least known branch of philosophy ending in –ics, his book shows how a better understanding of that branch can move one of the liveliest debates in philosophy forward. (shrink)

Khemlani et al. (2018) mischaracterize logic in the course of seeking to show that mental model theory (MMT) can accommodate a form of inference (, let us label it) they find in a high percentage of their subjects. We reveal their mischaracterization and, in so doing, lay a landscape for future modeling by cognitive scientists who may wonder whether human reasoning is consistent with, or perhaps even capturable by, reasoning in a logic or family thereof. Along the way, we note (...) that the properties touted by Khemlani et al. as innovative aspects of MMT‐based modeling (e.g., nonmonotonicity) have for decades been, in logic, acknowledged and rigorously specified by families of (implemented) logics. Khemlani et al. (2018) further declare that is “invalid in any modal logic.” We demonstrate this to be false by our introduction (Appendix A) of a new propositional modal logic (within a family of such logics) in which is provably valid, and by the implementation of this logic. A second appendix, B, partially answers the two‐part question, “What is a formal logic, and what is it for one to capture empirical phenomena?”. (shrink)

In this essay we discuss Tarski's work on what he calledthe methodology of the deductive sciences, or more briefly, borrowing the terminology of Hilbert,metamathematics, The clearest statement of Tarski's views on this subject can be found in his textbookIntroduction to logic[41m].1Here he describes the tasks of metamathematics as “the detailed analysis and critical evaluation of the fundamental principles that are applied in the construction of logic and mathematics”. He goes on to describe what these fundamental principles are: All the expressions (...) of the discipline under consideration must be defined in terms of a small group of primitive expressions that seem immediately understandable. Furthermore, only those statements of the discipline are accepted as valid that can be deduced by precisely defined and universally accepted means from a small set of axioms whose validity seems evident. The method of constructing a discipline in strict accordance with these principles is known as thedeductive method, and the disciplines constructed in this manner are calleddeductive systems. Since contemporary mathematical logic is one of those disciplines that are subject to these principles, it itself is a deductive science. Tarski then goes on to say:“The view has become more and more common that the deductive method is the only essential feature by means of which the mathematical disciplines can be distinguished from all other sciences; not only is every mathematical discipline a deductive theory, but also, conversely, every deductive theory is a mathematical discipline”.This identification of mathematics with the deductive sciences is in our view one of the distinctive aspects of Tarski's work. Another characteristic feature is his broad view of what constitutes the domain of metamathematical investigations. A clue to this aspect of his work can also be found in Chapter 6 ofIntroduction to logic. After a discussion of the notions of completeness and consistency, he remarks that the investigations concerning these topics were among the most important factors contributing to a considerable extension of the domain of methodological studies, and caused even a fundamental change in the whole character of the methodology of deductive sciences. (shrink)

After the publication of Begriffsschrift, a conflict erupted between Frege and Schröder regarding their respective logical systems which emerged around the Leibnizian notions of lingua characterica and calculus ratiocinator. Both of them claimed their own logic to be a better realisation of Leibniz’s ideal language and considered the rival system a mere calculus ratiocinator. Inspired by this polemic, van Heijenoort (1967b) distinguished two conceptions of logic—logic as language and logic as calculus—and presented them as opposing views, but did not explain (...) Frege’s and Schröder’s conceptions of the fulfilment of Leibniz’s scientific ideal. -/- In this paper I explain the reasons for Frege’s and Schröder’s mutual accusations of having created a mere calculus ratiocinator. On the one hand, Schröder’s construction of the algebra of relatives fits with a project for the reduction of any mathematical concept to the notion of relative. From this stance I argue that he deemed the formal system of Begriffsschrift incapable of such a reduction. On the other hand, first I argue that Frege took Boolean logic to be an abstract logical theory inadequate for the rendering of specific content; then I claim that the language of Begriffsschrift did not constitute a complete lingua characterica by itself, more being seen by Frege as a tool that could be applied to scientific disciplines. Accordingly, I argue that Frege’s project of constructing a lingua characterica was not tied to his later logicist programme. (shrink)

It is well known that the formal system developed by Frege in Begriffsschrift is based upon the distinction between function and argument—as opposed to the traditional distinction between subject and predicate. Almost all of the modern commentaries on Frege's work suggest a semantic interpretation of this distinction, and identify it with the ontological structure of function and object, upon which Grundgesetze is based. Those commentaries agree that the system proposed by Frege in Begriffsschrift has some gaps, but it is taken (...) as an essentially correct formal system for second-order logic: the first one in the history of logic. However, there is strong textual evidence that such an interpretation should be rejected. This evidence shows that the nature of the distinction between function and argument is stated by Frege in a significantly different way: it applies only to expressions and not to entities. The formal system based on this distinction is tremendously flexible and is suitable for making explicit the logical structure of contents as well as of deductive chains. We put forward a new reconstruction of the function-argument scheme and the quantification theory in Begriffsschrift. After that, we discuss the usual semantic interpretation of Begriffsschrift and show its inconsistencies with a rigorous reading of the text. (shrink)

The aim of this paper is to clarify why propositional logic is Post complete and its weak completeness was almost unnoticed by Hilbert and Bernays, while first-order logic is Post incomplete and its weak completeness was seen as an open problem by Hilbert and Ackermman. Thus, I will compare propositional and first-order logic in the Prinzipien der Mathematik, Bernays’s second Habilitationsschrift and the Grundzüge der Theoretischen Logik. The so called “arithmetical interpretation”, the conjunctive and disjunctive normal forms and the soundness (...) of the propositional rules of inference deserve special emphasis. (shrink)

Jean van Heijenoort was best known for his editorial work in the history of mathematical logic. I survey his contributions to model-theoretic proof theory, and in particular to the falsifiability tree method. This work of van Heijenoort’s is not widely known, and much of it remains unpublished. A complete list of van Heijenoort’s unpublished writings on tableaux methods and related work in proof theory is appended.

The attempt to find usable diagrams fornterms of the sort devised by John Venn seems to have originated with Venn himself, who published diagrams for up to five classes (the fifth class, however, was shaped like a doughnut, and contained an area outside itself — like the hole in the doughnut). Venn then suggested that “if we wanted to use a diagram forsixterms (x, y, z, w, v, u) the best plan would probably be to taketwofive-term figures, one for theupart (...) and one for the non-upart of all the other combinations …” Such a method would, as Venn admits, be somewhat confusing to the eye, and thus fail to fulfil one of the major purposes of the diagram; nevertheless, it is a method which could be extended to covernclasses.Venn also suggests (but without illustration) another method for drawing diagrams fornterms: “… the rule of formation would be very simple. We should merely have to begin by drawing any closed figure, and then proceed to draw others subject to the one condition that each intersect once, and once only, all the existing subdivisions produced by those which had gone before.” Then, in a rather surprising footnote, he adds: “It will be found that when we adhere to continuous figures, instead of the discontinuous five-term figure given above, there is a tendency for the resultant outlines thus successively drawn to assume a comb-like shape after the first four or five. If we begin by circles or other rounded figures the teeth are curved, if by parallelograms then they are straight. (shrink)

The process of ‘logical differentiation’ was introduced by Peirce in 1870. Directly analogous to mathematical differentiation, it uses logical terms instead of mathematical variables. Here, this mysterious process receives new interpretations which serve to clarify Peirce’s use of logical terms. I introduce the logical terms, the operation of multiplication, the logical analogy to the binomial theorem, infinitesimal relatives, the concepts of numerical coefficients and the number associated with each term. I also analyse the algebraic development of ‘logical differentiation’ and consider (...) in depth one application of the process. (shrink)

For Ivor Grattan-Guinness on the occasion of his retirement. The work of Augustus De Morgan on symbolic logic in the mid-nineteenth century is familiar to historians of logic and mathematics alike. What is less well known is his work on probability and, more specifically, the use of probabilistic ideas and methods in his logic. The majority of De Morgan's work on probability was undertaken around 1837???1838, with his earliest publications on logic appearing from 1839, a period which culminated with the (...) publication of his Formal Logic in 1847. This article examines the overlap between his work on probability theory and logic during the earliest period of his interest in both. (shrink)

Die Gründe, die C. I. Lewis [5], [6] bewogen haben, neben der gewöhnlichen Implikation eine strikte Implikation einzuführen, sind bekannt. In der vorliegenden Arbeit wird aus ähnlichen Gründen eine strenge Implikation eingeführt, die jedoch einen engeren Begriff darstellt als die strikte Implikation. Mit einer Arbeit von Arnold Schmidt [7] hat meine nur geringe Berührungspunkte, da der Verfasser sich mit der strikten Implikation beschäftigt. Für diese wird ein relativ einfaches Axiomensystem angegeben und gezeigt, wie man durch geeignete Definitionen von Notwendigkeit und (...) Möglichkeit bei eventueller Hinzufügung von Zusatzaxiomen die bekannten Systeme des Modalitätenkalküls erhält. Den Hauptteil meiner Arbeit bildet die Einführung meines Implikationsbegriffs; im zweiten Teil der Arbeit komme ich dann auf dieser Basis auch zu einer Einführung der Modalitäten.Um die Art der Einführung des Kalküls zu motivieren, wollen wir kurz auf das eingehen, das uns mehr oder weniger deutlich vorschwebt, wenn wir von der strengen Implikation sprechen. Die strenge Implikation, die wir durchA → Bwiedergeben, soll ausdrücken, daß zwischenAundBein logischer Zusammenhang besteht, daß der Inhalt vonBein Teil des Inhaltes vonAist, oder wie wir es sonst ausdrücken wollen. Das hat mit der Richtigkeit oder Falschheit vonAundBnichts zu tun. So würde man die Allgemeingültigkeit einer FormelA→ (B → A) ablehnen, da sie den Schluß vonAaufB → Aeinschließt und da die Richtigkeit vonAnichts damit zu tun hat, ob zwischenBundAein logischer Zusammenhang besteht. (shrink)

Logical inferentialism maintains that the formal rules of inference fix the meanings of the logical terms. The categoricity problem points out to the fact that the standard formalizations of classical logic do not uniquely determine the intended meanings of its logical terms, i.e., these formalizations are not categorical. This means that there are different interpretations of the logical terms that are consistent with the relation of logical derivability in a logical calculus. In the case of the quantificational logic, the categoricity (...) problem is generated by the finite nature of the standard calculi and one direction in which it can be solved is to strengthen the deductive systems by adding infinitary rules (such as the ω-rule), i.e., to construct a full formalization. Another main direction is to provide a natural semantics for the standard rules of inference, i.e., a semantics for which these rules are categorical. My aim in this paper is to analyze some recent approaches for solving the categoricity problem and to argue that a logical inferentialist should accept the infinitary rules of inference for the first order quantifiers, since our use of the expressions “all” and “there is” leads us beyond the concrete and finite reasoning, and human beings do sometimes employ infinitary rules of inference in their reasoning. (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)

Robert Sinclair’s *Quine, Conceptual Pragmatism, and the Analytic-Synthetic Distinction* persuasively argues that Quine’s epistemology was deeply influenced by C. I. Lewis’s pragmatism. Sinclair’s account raises the question why Quine himself frequently downplayed Lewis’s influence. Looking back, Quine has always said that Rudolf Carnap was his “greatest teacher” and that his 1933 meeting with the German philosopher was his “first experience of sustained intellectual engagement with anyone of an older generation” (1970, 41; 1985, 97-8, my emphasis). Quine’s autobiographies contain only a (...) handful of biographical references to Lewis and he regularly soft-pedaled the latter’s influence in private correspondence. In this note, I discuss some archival evidence that helps us better understand Quine’s reluctance to acknowledge Lewis’s influence. I contextualize the relation between Lewis and Quine and argue that the latter viewed his teacher as a retrograde force in modern epistemology, impeding the more rigorous approach that Carnap had been developing in Europe. Next, I briefly discuss Lewis’s contribution to the development of scientific philosophy in the United States and argue that Quine underestimated his teacher’s role in this process. In doing so, I hope to show that Quine’s zealous commitment to Carnap’s approach negatively affected his assessment of Lewis’s influence, thereby supplementing Sinclair’s praiseworthy reconstruction with an explanation of why Quine himself underestimated Lewis’s role. (shrink)

This volume draws connections between Wittgenstein's philosophy and the work of Saul Kripke, especially his Naming and Necessity. Saul Kripke is regarded as one of the foremost representatives of contemporary analytic philosophy. His most important contributions include the strict distinction between metaphysical and epistemological questions, the introduction of the notions of contingent a priori truth and necessary a posteriori truth and original accounts of names, descriptions, identity, necessity and realism. The chapters in this book elucidate the relevant connections between Kripke's (...) work and Wittgenstein, specifically concerning the standard meter, contingent apriori and rule-following. The contributions shed light on how Kripke's philosophical outlook was influenced by Wittgenstein, and how mainstream analytic philosophy and Wittgensteinian philosophy can fruitfully engage with one another. Kripke and Wittgenstein will be of interest to philosophers working on Wittgenstein, Kripke, and the history of analytic philosophy. (shrink)

"This book is valuable as expounding in full a theory of meaning that has its roots in the work of Frege and has been of the widest influence.

This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they ?say? something), and they have subject matter (they are ?about? something). The asymmetry between establishing validity and establishing invalidity has long been noted: validity is established through an information-processing procedure exhibiting a step-by-step deduction of the conclusion from the premise-set. Invalidity is established by (...) exhibiting a countermodel satisfying the premises but not the conclusion. The process of establishing validity focuses on information content; the process of establishing invalidity focuses on subject matter. Corcoran's information-theoretic concept of logical consequence corresponds to the former. Tarski's model-theoretic concept of logical consequence formulated in his famous 1936 no-countermodels definition corresponds to the latter. Both are found to be indispensable for understanding the rationale of the deductive method and each complements the other. This study discusses the ontic question of the nature of logical consequence and the epistemic question of the human capabilities presupposed by practical applications of these two concepts as they make validity and invalidity accessible to human knowledge. (shrink)

a single life-span. Philosophers, then, do not see more or know more, and they do not see less or know less. They aim to see less detail and more of the abstract. Their details, if you like, are abstractions. Walking on God’s earth as a pedestrian, as a farmer working his fields or as a passer-by, one’s picture of one’s surroundings is every bit as intelligent as that of the pilot riding the sky. The views of the field are radically (...) different, however. One sees only a specific field and in all lively detail: the exact pattern of the land, or even the exact outline of a given leaf, grasshopper, grain of sand even. Acquaintance with minute detail is not without its price: details may stand in the way of conjuring the big picture. It may be difficult to compare whichever field one happens to be in with far off fields, with respect to their size or shape or any other quality. One may wish to inquire if far off fields were already planted, harvested, or even if they exist. A pedestrian may find it hard or even impossible to do so. The pedestrian view contains fine points that the pilot’s map never would, but it does not necessarily contain more information, for it lacks the general context. After all, there are only so many items that one can observe and account for at a single glance, a single map, a single book, a single life-span. (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)

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)

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.

In his Outlines of Pyrrhonism 2.110–113, Sextus Empiricus presents four different accounts of the conditional, presumably all from the Hellenistic period, in increasing logical strength. While the interpretation and provenance of the first three accounts is relatively secure, the fourth account has perplexed and frustrated interpreters for decades or longer. Most interpreters have ultimately taken a dismissive attitude towards the fourth account and discounted it as being of both little historical and logical interest. We argue that this attitude is unwarranted (...) and demonstrate that the conditional expressed in the fourth account can profitably be understood as a precursor of analytic entailment, familiar from containment logics such as those developed by Parry and Angell. Exploiting recent work by Fine, we present a formal truthmaker semantics for this conditional and show how it sheds light on a number of longstanding issues in the interpretation of this passage. (shrink)

The aim of this paper is to give a general solution to the paradoxes of the material conditional, including the paradoxes generated by embedded conditionals. The solution consists in a pragmatic reinterpretation of the formal languages of classical logic LK and relevant logic LR as presented in Paoli. In particular I argue that the material conditional in the classical logic LK captures the truth conditions of “if...then”, but ignores certain pragmatic enrichments that are associated to it, while relevant logic LR (...) can give a systematic diagnostic of the cases in which a conditional is pragmatically enriched and those cases in which it is not. This diagnostic shows the reason why the paradoxes seem unacceptable and why they are, nevertheless, truth-preserving. This reinterpretation will also cover the solution that Paoli gives to McGee’s paradoxes of the Modus Ponens. (shrink)

Quine has argued that modal logic began with the sin of confusing use and mention. Anderson and Belnap, on the other hand, have offered us a way out through a strategy of nominahzation. This paper reviews the history of Lewis's early work in modal logic, and then proves some results about the system in which "A is necessary" is intepreted as "A is a classical tautology.".

Discussions of the relation between pragmatism and logical empiricism tend to focus on the period when the logical empiricists found themselves in exile, mostly in the United States, and then attempt to gauge the actual extent of their convergence. My concern lies with the period before that and the question whether pragmatism had an earlier influence on the development of logical empiricism, especially on the thought of the former members of the “first” Vienna Circle. I argue for a substantially qualified (...) affirmative answer. (shrink)

The aim of the paper is to show that W. V. O. Quine's animadversions against modal logic did not get the same attention that is considered to be the case nowadays. The community of logicians focused solely on the technical aspects of C. I. Lewis’ systems and did not take Quine's arguments and remarks seriously—or at least seriously enough to respond. In order to assess Quine's place in the history, however, his relation to Carnap is considered since their notorious break (...) was about the status of extensionality and modal logic. Since much of the works about the history of analytic philosophy is centered on the relationship of Quine and Carnap, their break about modality deserves much more attention—it also sheds some light on why should anyone wonder about Quine's early arguments against modal logic. The paper ends with some further considerations regarding the early formation of modal logic and hitherto unconsidered problematic issues. (shrink)

The logical theory which is called thecalculus of (binary) relations, and which will constitute the subject of this paper, has had a strange and rather capricious line of historical development. Although some scattered remarks regarding the concept of relations are to be found already in the writings of medieval logicians, it is only within the last hundred years that this topic has become the subject of systematic investigation. The first beginnings of the contemporary theory of relations are to be found (...) in the writings of A. De Morgan, who carried out extensive investigations in this domain in the fifties of the Nineteenth Century. De Morgan clearly realized the inadequacy of traditional logic for the expression and justification, not merely of the more intricate arguments of mathematics and the sciences, but even of simple arguments occurring in every-day life; witness his famous aphorism, that all the logic of Aristotle does not permit us, from the fact that a horse is an animal, to conclude that the head of a horse is the head of an animal. In his effort to break the bonds of traditional logic and to expand the limits of logical inquiry, he directed his attention to the general concept of relations and fully recognized its significance. Nevertheless, De Morgan cannot be regarded as the creator of the modern theory of relations, since he did not possess an adequate apparatus for treating the subject in which he was interested, and was apparently unable to create such an apparatus. His investigations on relations show a lack of clarity and rigor which perhaps accounts for the neglect into which they fell in the following years. (shrink)

Philosophers like G.H. von Wright and D. Makinson have pointed to serious challenges regarding the foundations of deontic logic. In this paper, I suggest that to deal successfully with these challenges a reconsideration of the research program of the discipline is useful. Some problems that have troubled this particular field of logical study for decades may disappear or appear more tractable if we view them from the perspective of a language game introduced by D. Lewis involving three characters: the Master, (...) the Slave and the Kibitzer. The adoption of this perspective opens a natural approach to a new layout of the domain of deontic studies. I propose dividing deontic logic into six sub-areas which are distinguished by their focus on the different idioms typical of the individual players, by conceiving the language game as either being static or as dynamic and by the aims of the logical inquiry. What kind of insights the proposed perspective provides is illustrated by an analysis of the so-called Ross paradox—a problem that has troubled deontic logic since its origins and, though it was many times pronounced solved, still keeps coming back ‘alive and kicking’. (shrink)