Propositional attitude sentences, such as (1) Pierre believes that snow is white, have proved to be formidably difficult to account for in a semantic theory. It is generally agreed that the that-clause ‘that snow is white’ purports to refer to the proposition that snow is white, but no agreement has been reached on what this proposition is. Sententialism is a semantic theory which tries to undermine the very enterprise of understanding what proposition is referred to in (1): according to sententialists, (...) in (1) reference is made to the sentence “Snow is white”. Sententialism is generally considered doomed. The two main reasons why are the famous translation argument, firstly suggested by Alonzo Church, and a problem raised by Stephen Schiffer. The purpose of this paper is to provide a unified solution to both criticisms. What I take to be the key ingredient sententialists may exploit is an observation that concerns the nature of languages and quotations: since quotation marks display the quoted material, if you are a speaker of the language the quoted material belongs to, you usually cannot but understand what is quoted. Moreover, I show that sententialists may appeal to that very observation also in order to answer another problem, pointed out by Kent Bach. I conclude that there are good reasons for resisting the temptation of introducing propositions in order to account for propositional attitude sentences. (shrink)

The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the ‘logic’ assumed rather than from Hume’s Principle. It is shown that Hume’s Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few rudimentary facts (...) of arithmetic are logically derivable from Hume’s Principle. And that hardly counts as a vindication of logicism. (shrink)

Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the idea arises of a dual (...) logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics. (shrink)

Following the development of the selectionist theory of the immune system, there was an attempt to characterize many biological mechanisms as being "selectionist" as juxtaposed to "instructionist." But this broad definition would group Darwinian evolution, the immune system, embryonic development, and Chomsky's language-acquisition mechanism as all being "selectionist." Yet Chomsky's mechanism (and embryonic development) are significantly different from the selectionist mechanisms of biological evolution or the immune system. Surprisingly, there is a very abstract way using two dual mathematical logics to (...) make the distinction between genuinely selectionist mechanisms and what are better called "generative" mechanisms. This note outlines that distinction. (shrink)

Contrary to common misconceptions, today's logic is not devoid of existential import: the universalized conditional ∀ x [S→ P] implies its corresponding existentialized conjunction ∃ x [S & P], not in all cases, but in some. We characterize the proexamples by proving the Existential-Import Equivalence: The antecedent S of the universalized conditional alone determines whether the universalized conditional has existential import, i.e. whether it implies its corresponding existentialized conjunction.A predicate is an open formula having only x free. An existential-import predicate (...) Q is one whose existentialization, ∃ x Q, is logically true; otherwise, Q is existential-import-free or simply import-free.How abundant or widespread is existential import? How abundant or widespread are existential-import predicates in themselves or in comparison to import-free predicates? We show that existential-import predicates are quite abundant, and no less so than import-free predicates. Existential.. (shrink)

In Writing the Book of the World, Ted Sider argues that David Lewis’s distinction between those predicates which are ‘perfectly natural’ and those which are not can be extended so that it applies to words of all semantic types. Just as there are perfectly natural predicates, there may be perfectly natural connectives, operators, singular terms and so on. According to Sider, one of our goals as metaphysicians should be to identify the perfectly natural words. Sider claims that there is a (...) perfectly natural first-order quantifier. I argue that this claim is not justified. Quine has shown that we can dispense with first-order quantifiers, by using a family of ‘predicate functors’ instead. I argue that we have no reason to think that it is the first-order quantifiers, rather than Quine’s predicate functors, which are perfectly natural. The discussion of quantification is used to provide some motivation for a general scepticism about Sider’s project. Shamik Dasgupta’s ‘generalism’ and Jason Turner’s critique of ‘ontological nihilism’ are also discussed. (shrink)

I examine the difficulties that several philosophers of language are liable to encounter in their attempts to provide an account of the connection between truth and assertion. I then attempt to provide an account of this connection. The analysis is concerned chiefly with difficulties which consist in elucidating the conceptual connection between truth and assertion in a way that respects certain linguistic intuitions while at the same time rendering the concept of truth amenable to a semantic interpretation. The proposed view (...) suggests one way in which we might go about meeting the theoretical demands implicit in addressing this concern, among others, demonstrating the extent to which a theory of truth should be regarded as belonging to the province of epistemology. Insofar as semantical considerations figure into such a theory, a more systematic investigation of the interface between epistemology and natural language semantics is recommended. The solution to many problems at this interface, I argue, lay in an analysis of judgment. (shrink)

Properties and facts play a central role within metaphysics, yet there is no widely accepted account of what constitutes a property or a fact. Traditional conceptions of these metaphysical notions raise serious philosophical puzzles, making the existence of each seem dubious. Drawing on the minimalist theory of truth, I argue in favor of a minimalist conception of properties and facts. A minimalist theory of properties and facts explains these matters in terms of the acceptance of trivial schemas. To make the (...) case that minimalism is a superior approach to properties and facts, I argue against the standard views in the philosophical literature. I argue that the minimalist approach to properties has advantages over realism, nominalism, and the trope theory. I argue that the minimalist approach to facts is superior to the standard treatment of facts on correspondence theories. Metaphysical minimalism, a minimalist metaphysics of properties and facts, is a distinct and superior alternative to the theories of properties and facts currently on offer. (shrink)

The triplet model treats a concept as complex structure that expresses three kinds of information. The first is about entities subsumed under a concept,their properties and relations. The second is about means and ways of representing the first information in intelligent systems. The third is about linkage between the first and second ones and methods of its constructing. The application of triplet models to generalization and development of concept models in philosophy, logic, cognitive psychology, cognitive science, linguistics, artificial intelligence has (...) been demonstrated. The componential structure of the triplet model and extended concept classification have been given. Theoretical considerations are combined with historical case-studies of concept history. For scientists and students with an interest in problems of knowledge structure and representation. Abstract and Extended Summary in English. (shrink)

The paper considers a commonly used axiomatization of the classical propositional logic and studies how different axiom schemata in this system contribute to proof complexity of the logic. The existence of a polynomial bound on proof complexity of every statement provable in this logic is a well-known open question.The axiomatization consists of three schemata. We show that any statement provable using unrestricted number of axioms from the first of the three schemata and polynomially-bounded in size set of axioms from the (...) other schemata, has a polynomially-bounded proof complexity. In addition, it is also established, that any statement, provable using unrestricted number of axioms from the remaining two schemata and polynomially-bounded in size set of axioms from the first scheme, also has a polynomially-bounded proof complexity. (shrink)

En su obra de 1884, Die Grundlagen der Arithmetik [Gl.], Frege establece el principio del contexto como uno de sus principios fundamentales y propone valerse de él para dar con el concepto adecuado de número. Así, ofrece una definición de tipo contextual de número cardinal, pero ante una objeción que le resulta insalvable se decide por una definición explicita. La cuestión acerca de cuál ha sido entonces el sentido de establecer con tanto énfasis el principio del contexto admite más de (...) una respuesta. Mi texto polemiza con la interpretación canónica que se ha dado al papel de este principio en Gl., acusándola de permanecer encandilada por lo que se lee de manera inmediata en desmedro de una evaluación más integral y crítica. En particular, defenderé que: i) a diferencia de lo afirmado en Dummett 1991, la definición contextual de número cardinal no es abandonada por Frege al dar su definición explícita de número, ii) el principio del contexto tiene en Gl. un papel erístico fundamental de eliminar posiciones rivales acerca de la naturaleza de los números, iii) dicha definición contextual de número tiene un papel heurístico en el contexto de descubrimiento, en el sentido de que ofrece una intuición clave que permite llegar a la formulación final, iv) a su vez, la definición contextual es una condición necesaria de la definición explícita en el contexto de justificación, siendo que esta última se ha construido a partir de la primera. Por todo ello, sostengo que la función del principio del contexto no es la que prioriza Dummett, i. e., la fallida función de resolver el problema ontológico de la naturaleza de los números, sino que es fundamentalmente metodológica. En efecto, es en su función erística y heurística en donde debe ubicarse su relevancia para la investigación llevada a cabo en Gl. In his masterpiece of 1884, Die Grundlagen der Arithmetik [Gl.] Frege presents the context principle as a fundamental one in his investigation à propos an accurate concept of number. Thus, resting on it he offers a contextual definition of cardinal number; but after analyzing some objections he realizes that this definition does not work and decides himself for an explicit one. The question about what is the purport of the context principle when something that appeared to be the consequence is put aside admits more than a single answer. In this paper I argue against the canonic interpretation for, even while keeping next to the Fregean text, it fails to offer an integral explanation of how the context principle works all along Gl. In particular, I will defend that: i) contrary with Dummett 1991, the contextual definition of cardinal number is not abandoned by Frege when giving his explicit definition of number, ii) the context principle has in Gl. a fundamental eristic role of crossing out rival positions about the nature of numbers, iii) such contextual definition of number has an heuristic role in the discovery context, in the sense it offers a key intuition towards the final formulation, iv) furthermore, the contextual definition is a necessary condition in the justification context for the latest has been built up through the first one. For all this, I maintain that the function of the context principle is not the one Dummett prioritizes, i. e., the failed function of solving the ontological problem of the nature of numbers; the function of the context principle is rather and fundamentally methodological. It is on its eristic and heuristic function that its relevance to the Gl. investigation must be looked for. (shrink)

In the first exposition of the doctrine of indeterminacy of translation, Quine asserted that the individuation and translation of truth-functional sentential connectives like 'and', 'or', 'not' are not indeterminate. He changed his mind later on, conjecturing that some sentential connectives might be interpreted in different non-equivalent ways. This issue has not been debated much by Quine, or in the subsequent literature, it is, as it were, an unsolved problem, not well understood. For the sake of the argument, I will adopt (...) Quine's background assumption that all the semantic features of a language can be reduced to the speakers' dispositions toward assent and dissent, as far as only the truth-conditional core of the meaning of sentences is concerned. I will put forward an argument to the effect that the speech dispositions of most, if not all, English (French, Italian, etc.) speakers constrain a unique translation of their connectives. This argument crucially relies on an empirical conjecture concerning the behaviour of these operators. (shrink)

This is a draft for a revised edition of Mark Sharlow's book "From Brain to Cosmos." It includes most of the material from the first edition, two shorter pieces pertaining to the book, and a detailed new introduction.

Propositionalism is the view that intentional attitudes, such as belief, are relations to propositions. Propositionalists argue that propositionalism follows from the intuitive validity of certain kinds of inferences involving attitude reports. Jubien (2001) argues powerfully against propositions and sketches some interesting positive proposals, based on Russell’s multiple relation theory of judgment, about how to accommodate “propositional phenomena” without appeal to propositions. This paper argues that none of Jubien’s proposals succeeds in accommodating an important range of propositional phenomena, such as the (...) aforementioned validity of attitude-report inferences. It then shows that the notion of a predication act-type, which remains importantly Russellian in spirit, is sufficient to explain the range of propositional phenomena in question, in particular the validity of attitude-report inferences. The paper concludes with a discussion of whether predication act-types are really just propositions by another name. (shrink)

To talk about simple concepts presupposes that the notion of concept has been aptly explicated. I argue that a most adequate explication should abandon the set-theoretical paradigm and use a procedural approach. Such a procedural approach is offered by Tichý´s Transparent Intensional Logic (TIL). Some main notions and principles of TIL are briefly presented, and as a result, concepts are explicated as a kind of abstract procedure. Then it can be shown that simplicity, as applied to concepts, is well definable (...) as a property relative to conceptual systems, each of which is determined by a finite set of simple (‘primitive’) concepts. Refinement as a method of replacing simple concepts by compound concepts is defined. (shrink)

Deliberate contextual vocabulary acquisition (CVA) is a reader’s ability to figure out a (not the) meaning for an unknown word from its “context”, without external sources of help such as dictionaries or people. The appropriate context for such CVA is the “belief-revised integration” of the reader’s prior knowledge with the reader’s “internalization” of the text. We discuss unwarranted assumptions behind some classic objections to CVA, and present and defend a computational theory of CVA that we have adapted to a new (...) classroom curriculum designed to help students use CVA to improve their reading comprehension. (shrink)

The metaphor of a branching tree of future possibilities has a number of important philosophical and logical uses. In this paper we trace this metaphor through some of its uses and argue that the metaphor works the same way in physics as in philosophy. We then give an overview of formal systems for branching possibilities, viz., branching time and (briefly) branching space-times. In a next step we describe a number of different notions of possibility, thereby sketching a landscape of possibilities. (...) In the final section of the paper we look at the place of branching-based possibilities in that larger landscape of possibilities. Our main message is that far from being an outlandish metaphysical extravagancy, branching-based possibilities are epistemically as well as metaphysically basic. (shrink)

This paper offers a critical assessment of the current state of the debate about the identity and individuality of material objects. Its main aim, in particular, is to show that, in a sense to be carefully specified, the opposition between the Leibnizian ‘reductionist’ tradition, based on discernibility, and the sort of ‘primitivism’ that denies that facts of identity and individuality must be analysable has become outdated. In particular, it is argued that—contrary to a widespread consensus—‘naturalised’ metaphysics supports both the acceptability (...) of non-qualitatively grounded (both ‘contextual’ and intrinsic) identity and a pluralistic approach to individuality and individuation. A case study is offered that focuses on non-relativistic quantum mechanics, in the context of which primitivism about identity and individuality, rather than being regarded as unscientific, is on the contrary suggested to be preferable to the complicated forms of reductionism that have recently been proposed. More generally, by assuming a plausible form of anti-reductionism about scientific theories and domains, it is claimed that science can be regarded as compatible with, or even as suggesting, the existence of a series of equally plausible grades of individuality. The kind of individuality that prevails in a certain context and at a given level can be ascertained only on the basis of the specific scientific theory at hand. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple (...) universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

This document consists primarily of an excerpt (chapter 5) from the author’s book From Brain to Cosmos. That excerpt presents an analysis of personal identity through time, using the concept of subjective fact that the author developed earlier in the book. (Readers unfamiliar with that concept are strongly advised to read chapters 2 and 3 of From Brain to Cosmos first. See the last page of this document for details on how to obtain those chapters.).

This document consists primarily of an excerpt (chapter 13) from the author’s book From Brain to Cosmos. In that excerpt, the author presents a study of the notion of truth using the concept of subjective fact developed earlier in the book. The author argues that mind-body materialism is compatible with certain forms of metaphysical idealism. The chapter closes with some remarks on relativism with regard to truth. (This document depends heavily upon the concept of subjective fact developed in From Brain (...) to Cosmos. Readers unfamiliar with that concept are strongly advised to read chapters 2 and 3 of From Brain to Cosmos first. See the last page of this document for details on how to obtain those chapters.). (shrink)

This document consists primarily of excerpts (chapters 5 and 10-12) from the author’s book From Brain to Cosmos. These excerpts address several traditional problems about the histories of conscious subjects, using the concept of subjective fact that the author developed earlier in the book. Topics include the persistence of conscious subjects through time, the unity or disunity of the self, and the possibility of splitting conscious subjects. (These excerpts depend heavily upon the author’s concept of subjective fact as developed in (...) From Brain to Cosmos. Readers unfamiliar with that concept are strongly advised to read chapters 2 and 3 of that book first. See the last page of this document for details on how to obtain those chapters.). (shrink)

This document consists primarily of an excerpt (chapter 4) from the author’s book From Brain to Cosmos. That excerpt presents a study of a specific problem about knowledge: the logical justification of one’s knowledge of the immediate past. (This document depends heavily upon the concept of subjective fact that the author developed in chapters 2 and 3 of From Brain to Cosmos. Readers unfamiliar with that concept are strongly advised to read those chapters first. See the last page of this (...) document for details on how to obtain those chapters.). (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)

Whether meaning is compositional has been a major issue in linguistics and formal philosophy of language for the last 2 decades. Semantic holism is widely and plausibly considered as an objection to the principle of semantic compositionality therein. It comes as a surprise that the holistic peculiarities of scientific language have been rarely addressed in formal accounts so far, given that semantic holism has its roots in the philosophy of science. For this reason, a model-theoretic approach to semantic holism in (...) the language of science is presented here. This approach preserves compositionality to a large extent. *Received September 2009; revised February 2010. †To contact the author, please write to: Seminar for Philosophy, Logic, and Theory of Science, Hauspostfach 49, Ludwig-Maximilians-Universität, Munich 80539, Germany; e-mail: [email protected]. (shrink)

Although the existence of correspondence between George Boole (1815?1864) and William Stanley Jevons (1835?1882) has been known for a long time and part was even published in 1913, it has never been fully noted; in particular, it is not in the recent edition of Jevons's letters and papers. The texts are transcribed here, with indication of their significance. Jevons proposed certain quite radical changes to Boole's system, which Boole did not accept; nevertheless, they were to become well established.

A first-order formulation of Leśniewski's ontology is formulated and shown to be interpretable within a free first-order logic of identity extended to include nominal quantification over proper and common-name concepts. The latter theory is then shown to be interpretable in monadic second-order predicate logic, which shows that the first-order part of Leśniewski's ontology is decidable.

This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)

The purpose of this article is to examine aspects of the development of the concept and theory of computability through the theory of recursive functions. Following a brief introduction, Section 2 is devoted to the presuppositions of computability. It focuses on certain concepts, beliefs and theorems necessary for a general property of computability to be formulated and developed into a mathematical theory. The following two sections concern situations in which the presuppositions were realized and the theory of computability was developed. (...) It is suggested in Section 3 that a central item was the problem of generalizing Gödel's incompleteness theorem. It is shown that this involved both the characterization of recursiveness and the attempt to clarify and formulate the notion of an effective process as it relates to the syntax of deductive systems. Section 4 concerns the decision problems which grew from the Hilbert program. Section 5 is devoted to the development of an informal' technique in the theory of computability often called ?argument by Church's thesis? (shrink)

This paper focuses on two notions of effectiveness which are not treated in detail elsewhere. Unlike the standard computability notion, which is a property of functions themselves, both notions of effectiveness are properties of interpreted linguistic presentations of functions. It is shown that effectiveness is epistemically at least as basic as computability in the sense that decisions about computability normally involve judgments concerning effectiveness. There are many occurrences of the present notions in the writings of logicians; moreover, consideration of these (...) notions can contribute to the clarification and, perhaps, solution of various philosophical problems, confusions and disputes. (shrink)

We provide for the first time an exact translation into English of the Polish version of Alfred Tarski's classic 1936 paper, whose title we translate as ?On the Concept of Following Logically?. We also provide in footnotes an exact translation of all respects in which the German version, used as the basis of the previously published and rather inexact English translation, differs from the Polish. Although the two versions are basically identical, to an extent that is even uncanny, we note (...) more than 400 differences. Several dozen of these are substantive differences due to revisions by Tarski to the Polish version which he did not incorporate in the German version. With respect to these revisions the Polish version should be regarded as more authoritative than the German. Hence scholars limited to an English translation should use ours. (shrink)

Everyday reasoning is replete with arguments which, though not logically valid, nonetheless harbor a measure of credibility in their own right. Here the claim that such arguments force us to acknowledge material validity, in addition to logical validity, is advanced, and criteria that attempt to unpack this concept are examined in detail. Of special concern is the effort to model these criteria on explications of logical validity that rely on notions of substitutivity and logical form. It is argued, however, that (...) such a parallel is not easily located and that it is uncertain that a construal of material validity can be fashioned after traditional accounts of logical validity. Attention is also given to the topics of enthymemes and to the proper domain of logic. (shrink)

An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match (...) with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question. (shrink)

Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, on the (...) epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (shrink)

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those (...) involving the distinction between characterizing a system and axiomatizing the truths of a system. (shrink)

Kant discovered a philosophical problem with mathematical proof. Despite being a priori , its methodology involves more than analytic truth. But what else is involved? This problem is widely taken to have been solved by Frege’s extension of logic beyond its restricted (and largely Aristotelian) form. Nevertheless, a successor problem remains: both traditional and contemporary (classical) mathematical proofs, although conforming to the norms of contemporary (classical) logic, never were, and still aren’t, executed by mathematicians in a way that transparently reveals (...) why these proofs—written in the vernacular to this very day—succeed in conforming to those norms. (shrink)

Frege's theory of indirect contexts and the shift of sense and reference in these contexts has puzzled many. What can the hierarchy of indirect senses, doubly indirect senses, and so on, be? Donald Davidson gave a well-known 'unlearnability' argument against Frege's theory. The present paper argues that the key to Frege's theory lies in the fact that whenever a reference is specified (even though many senses determine a single reference), it is specified in a particular way, so that giving a (...) reference implies giving a sense; and that one must be 'acquainted' with the sense. It is argued that an indirect sense must be 'immediately revelatory' of its reference. General principles for Frege's doctrine of sense and reference are sated, for both direct and indirect quotation, to be understood iteratively. I also discuss Frege's doctrine of tensed and first person statements in the light of my analysis. The views of various other authors are examined. The conclusion is to ascribe to Frege an implicit doctrine of acquaintance similar to that of Russell. (shrink)

Prior Analytics by the Greek philosopher Aristotle (384 – 322 BCE) and Laws of Thought by the English mathematician George Boole (1815 – 1864) are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle’s system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss (...) many other historically and philosophically important aspects of Boole’s book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole’s contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology. (shrink)

Gottlob Frege (1848-1925) was a German logician, mathematician and philosopher who played a crucial role in the emergence of modern logic and analytic philosophy. Frege's logical works were revolutionary, and are often taken to represent the fundamental break between contemporary approaches and the older, Aristotelian tradition. He invented modern quantificational logic, and created the first fully axiomatic system for logic, which was complete in its treatment of propositional and first-order logic, and also represented the first treatment of higher-order logic. In (...) the philosophy of mathematics, he was one of the most ardent proponents of logicism, the thesis that mathematical truths are logical truths, and presented influential criticisms of rival views such as psychologism and formalism. His theory of meaning, especially his distinction between the sense and reference of linguistic expressions, was groundbreaking in semantics and the philosophy of language. He had a profound and direct influence on such thinkers as Russell, Carnap and Wittgenstein. Frege is often called the founder of modern logic, and he is sometimes even heralded as the founder of analytic philosophy. (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)

We critically investigate and refine Dunn's relevant predication, his formalisation of the notion of a real property. We argue that Dunn's original dialectical moves presuppose some interpretation of relevant identity, though none is given. We then re-motivate the proposal in a broader context, considering the prospects for a classical formalisation of real properties, particularly of Geach's implicit distinction between real and ''Cambridge'' properties. After arguing against these prospects, we turn to relevance logic, re-motivating relevant predication with Geach's distinction in mind. (...) Finally we draw out some consequences of Dunn's proposal for the theory of identity in relevance logic. (shrink)

Predicates are term-to-sentence devices, and operators are sentence-to-sentence devices. What Kaplan and Montague's Paradox of the Knower demonstrates is that necessity and other modalities cannot be treated as predicates, consistent with arithmetic; they must be treated as operators instead. Such is the current wisdom.A number of previous pieces have challenged such a view by showing that a predicative treatment of modalities neednot raise the Paradox of the Knower. This paper attempts to challenge the current wisdom in another way as well: (...) to show that mere appeal to modal operators in the sense of sentence-to-sentence devices is insufficient toescape the Paradox of the Knower. A family of systems is outlined in which closed formulae can encode other formulae and in which the diagonal lemma and Paradox of the Knower are thereby demonstrable for operators in this sense. (shrink)

The literature on conditionals is rife with alternate formulations of the abstract semantics of conditional logic. Each formulation has its own advantages in terms of applications and generalizations; nevertheless, they are for the most part equivalent, in the sense that they underwrite the same range of logical systems. The purpose of the present note is to bring under this umbrella the partition semantics introduced by Brian Skyrms in (Skyrms, 1984).

This paper brings to light a new puzzle for Frege interpretation, and offers a solution to that puzzle. The puzzle concerns Frege’s judgement-stroke (‘|’), and consists in a tension between three of Frege’s claims. First, Frege vehemently maintains that psychological considerations should have no place in logic. Second, Frege regards the judgementstroke—and the associated dissociation of assertoric force from content, of the act of judgement from the subject matter about which judgement is made—as a crucial part of his logic. Third, (...) Frege holds that judging is an inner mental process, and that the distinction marked by the judgement-stroke, between entertaining a thought and judging that it is true, is a psychological distinction. I argue that what initially looks like confusion here on Frege’s part appears quite reasonable when we remind ourselves of the differences between Frege’s conception of logic and our own. (shrink)

The calculus of relations was created and developed in the second half of the nineteenth century by Augustus De Morgan, Charles Sanders Peirce, and Ernst Schröder. In 1940 Alfred Tarski proposed an axiomatization for a large part of the calculus of relations. In the next decade Tarski's axiomatization led to the creation of the theory of relation algebras, and was shown to be incomplete by Roger Lyndon's discovery of nonrepresentable relation algebras. This paper introduces the calculus of relations and the (...) theory of relation algebras through a review of these historical developments. (shrink)

The representation of quantification over relations in monadic third-order logic is discussed; it is shown to be possible in numerous special cases of foundational interest, but not in general unless something akin to the Axiom of Choice is assumed.