The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...) development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

The aim of this paper is to clarify the truth-functional interpretation of the logical connective of the material implication. The importance of such clarification lies in the fact that it allows avoiding the supposed paradoxes introduced by C. I. Lewis (1918). I argue that an adequate understanding of the history and purposes of logic is enough to dissolve them away. The defense is based on an exposition of propositional compositionalism. To compare, I also present Stalnaker’s (1968) alternative that seeks to (...) explain the implicit concept of conditionality of conditionals assertions. In order to reinforce my aim I expound the twelve different meanings of ‘implication’ stated by Corcoran (1993) in order to hold that a proper understanding of these meanings allows us to avoid ambiguities and confusions. Finally, I expect to show that despite the possible alternatives, compositionalism is still the best explanation for the truth-functionality account of logical connectives. (shrink)

Neo-logicism is, not least in the light of Frege’s logicist programme, an important topic in the current philosophy of mathematics. In this essay, I critically discuss a number of issues that I consider to be relevant for both Frege’s logicism and neo-logicism. I begin with a brief introduction into Wright’s neo-Fregean project and mention the main objections that he faces. In Sect. 2, I discuss the Julius Caesar problem and its possible Fregean and neo-Fregean solution. In Sect. 3, I raise (...) what I take to be a central objection to the position of neo-logicism. In Sect. 4, I attempt to clarify how we should understand Frege’s stipulation that the two sides of an abstraction principle qua contextual definition of a term-forming operator shall be “gleichbedeutend”. In Sect. 5, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle: The number that belongs to the concept F is equal to the number that belongs to the concept G if and only if F and G are equinumerous. Section 6 is devoted to Frege’s two criteria of thought identity. In Sects. 7 and 8, I defend the position of the neo-logicist against an alleged “knock-down argument”. In Sect. 9, I comment on Frege’s description of abstraction in Grundlagen, §64 and the use of the terms “recarving” and “reconceptualization” in the relevant literature on Fregean abstraction and neo-logicism. I argue that Fregean abstraction has nothing to do with the recarving of a sentence content or its decomposition in different ways. I conclude with remarks on global logicism versus local logicisms. (shrink)

In this paper I describe the birth of Russell’s notion of a propositional function on 3 May 1902 and its immediate context and implications. In particular, I consider its significance in relation to the development of his views on analysis.

A theory of relations is presented that provides a detailed account of the logical structure of relational complexes. The theory draws a sharp distinction between relational complexes and relational states. A salient difference is that relational complexes belong to exactly one relation, whereas relational states may be shared by different relations. Relational complexes are conceived as structured perspectives on states ‘out there’ in reality. It is argued that only relational complexes have occurrences of objects, and that different complexes of the (...) same relation may correspond to the same state. (shrink)

This book is both an introduction to and a research work on Meinongianism. “Meinongianism” is taken here, in accordance with the common philosophical jargon, as a general label for a set of theories of existence – probably the most basic notion of ontology. As an introduction, the book provides the first comprehensive survey and guide to Meinongianism and non-standard theories of existence in all their main forms. As a research work, the book exposes and develops the most up-to-date Meinongian theory (...) (called modal Meinongianism), applies it to specific fields, and discusses its open problems. Part I of the book provides a historical introduction to, and critical discussion of, the dominant philosophical view of existence: the “Kantian-Fregean-Quinean” perspective. Part II is the full-fledged introduction to the Meinongian theories of existence as a real property of individuals: after starting with the so-called naïve Meinongian conception and its problems, it provides a self-contained presentation of the main neo-Meinongian proposals, and a detailed discussion of their strengths and weaknesses. Part III develops a specific neo-Meinongian theory of existence employing a model-theoretic semantic framework. It discusses its application to the ontology and semantics of fictional objects, and its open problems. The methodology of the book follows the most recent trends in analytic ontology. In particular, the meta-ontological point of view is largely privileged. (shrink)

This paper concerns the formal semantic analysis of imperative sentences. It is argued that such an analysis cannot be deferred to the semantics of propositions, under any of the three commonly adopted strategies: the performative analysis, the sentence radical approach to propositions, and the (nondeclarative) mood-as-operator approach. Whereas the first two are conceptually problematic, the third faces empirical problems: various complex imperatives should be analysed in terms of semantic operators over simple imperatives. One particularly striking case is the Dutch pluperfect (...) imperative. It is argued that this construction should be analysed as a genuine counterfactual imperative. On the constructive side, in the last part of the paper a formal semantic analysis of imperatives is presented, in the framework of Update Semantics. On this analysis, imperatives are sui generis semantic entities, on a par with propositions. The analysis also includes an account of the counterfactual imperatives. (shrink)

I present an argument for an interpretation of Kant's views on the nature of the ‘content [Inhalt]’ of ‘cognition [Erkenntnis]’. In contrast to one of the longest standing interpretations of Kant's views on cognitive content, which ascribes to Kant a straightforwardly psychologistic understanding of content, and in contrast as well to the more recently influential reading of Kant put forward by McDowell and others, according to which Kant embraces a version of Russellianism, I argue that Kant's views on this topic (...) are of a much more Fregean bent than has traditionally been admitted or appreciated. I conclude by providing a sketch of how a better grasp of Kant's views on cognitive content in general can help bring into sharper relief what is, and what is not, at stake in the recent debates over whether Kant accepts a particular kind of cognitive content—namely, non-conceptual content. (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 chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform (...) program. Finally, the fourth case study is paraconsistent logic, perhaps the most controversial of serious proposals. (shrink)

Variabilism is the view that proper names (like pronouns) are semantically represented as variables. Referential names, like referential pronouns, are assigned their referents by a contextual variable assignment (Kaplan 1989). The reference parameter (like the world of evaluation) may also be shifted by operators in the representation language. Indeed verbs that create hyperintensional contexts, like ‘think’, are treated as operators that simultaneously shift the world and assignment parameters. By contrast, metaphysical modal operators shift the world of assessment only. Names, being (...) variables, refer rigidly in the latter merely intensional contexts, but may vary their reference in hyperintensional contexts. This conforms to the intuition that the content of attitude ascriptions encapsulates referential uncertainty. Furthermore, names in hyperintensional contexts are ambiguous between de re* and de dicto* interpretations. This fact is used to account for asymmetric mistaken identity attributions (for example, Biron thinks Katherine is Rosaline, but he doesn’t think Rosaline is Katherine). -/- The variable theory compares favourably with its alternatives, including Millianism and descriptivism. Millians cannot account for the behaviour of names in hyperintensional contexts, while descriptivists cannot generate a necessary contrast between intensional and hyperintensional contexts. No other theory can capture the facts pertaining to the existentially bound use of names. (shrink)

I argued that Frege does not have a metatheory in the following sense: the justifications he offers for his basic laws and rules of inference neither employ nor require a truth-predicate or metalinguistic variables. In ‘Does Frege Use a Truth-predicate in his "Justification" of the Laws of Logic?’, Dirk Greimann disputes this. As Greimann interprets Frege, (i) Frege's remarks commit him to giving a metatheoretic justification of the basic laws and rules of his logic, and (ii) Frege actually gives such (...) a justification in the early sections of Grundgesetze—although the truth-predicate that Frege employs is a non-standard one: it is neither a predicate that holds of all and only true sentences nor a predicate that holds of all and only true thoughts. I argue that Greimann's interpretation is not, in the end, true to the text, and that his non-standard view of what is required of a Tarskian truth-predicate is ultimately not viable. (shrink)

One particular topic in the literature on Frege’s conception of sense relates to two apparently contradictory theses held by Frege: the isomorphism of thought and language on one hand and the expressibility of a thought by different sentences on the other. I will divide the paper into five sections. In (1) I introduce the problem of the tension in Frege’s thought. In (2) I discuss the main attempts to resolve the conflict between Frege’s two contradictory claims, showing what is wrong (...) with some of them. In (3), I analyze where, in Frege’s writings and discussions on sense identity, one can find grounds for two different conceptions of sense. In (4) I show how the two contradictory theses held by Frege are connected with different concerns, compelling Frege to a constant oscillation in terminology. In (5) I summarize two further reasons that prevented Frege from making the distinction between two conceptions of sense clear: (i) the antipsychologism problem and (ii) the overlap of traditions in German literature contemporary to Frege about the concept of value. I conclude with a hint for a reconstruction of the Fregean notion of ‘thought’ which resolves the contradiction between his two theses. (shrink)

The nature of changes in mathematics was discussed recently in Revolutions in Mathematics. The discussion was dominated by historical and sociological arguments. An obstacle to a philosophical analysis of this question lies in a discrepancy between our approach to formulas and to pictures. While formulas are understood as constituents of mathematical theories, pictures are viewed only as heuristic tools. Our idea is to consider the pictures contained in mathematical text, as expressions of a specific language. Thus we get formulas and (...) pictures into one linguistic framework, and so we are able to analyse their interplay in the course of history. (shrink)

Part 1 sets out the logical/semantical background to ‘On Denoting’, including an exposition of Russell's views in Principles of Mathematics, the role and justification of Frege's notorious Axiom V, and speculation about how the search for a solution to the Contradiction might have motivated a new treatment of denoting. Part 2 consists primarily of an extended analysis of Russell's views on knowledge by acquaintance and knowledge by description, in which I try to show that the discomfiture between Russell's semantical and (...) epistemological commitments begins as far back as 1903. I close with a non-Russellian critique of Russell's views on how we are able to make use of linguistic representations in thought and with the suggestion that a theory of comprehension is needed to supplement semantic theory. (shrink)

The basis of a rigorous formal axiomatization of quantum mechanics is constructed, built upon Dirac's bra-ket notation. The system is three-sorted, with separate variables for scalars, vectors and operators. First-order quantification over all three types of variable is permitted. Economy in the axioms is effected by, e.g., assigning a single logical function * to transform (i) a scalar into its complex conjugate, (ii) a ket vector into a bra and a bra into a ket, (iii) an operator into its adjoint. (...) The system is accompanied by a formal semantics. Further papers will deal with vector subspaces and projection operators, operators with continuous spectra, tensor products, observables, and quantum mechanical probabilities. (shrink)

Frege distinguished the thought qua logical content from the assertoric force attached to it when judged to be true. The gist of this distinction is captured by the so-called Frege-Geach point. Recently, several authors have drawn inspiration from Wittgenstein to reject this point and the distinction it is based on. This article proceeds from the observation that Wittgenstein himself did not reject the force-content distinction but urged us to reformulate it in a non-dualistic way. While drawing on Wittgensteinian lessons about (...) thought and its expression, the overall purpose of this paper is systematic, not exegetic: it seeks to contribute to the contemporary debate aboute force and content by arguing that this distinction should be redrawn in such a way as to exhibit force as internal to thought, namely, as that which provides for the unity of thought. To this end, it is investigated what it is for a thought to occur as a forceless part of a propositionally complex assertion (e. g. for p to occur as a part of the assertion that not p). (shrink)

A discussion of the connections and differences between completeness and representation theorems in logic, with examples drawn from classical and modal logic, the logic of friendliness, and nonmonotonic reasoning.

What is this discipline called history of philosophy? What standards are relevant to its assessment? There aren’t single, straightforward answers to these quest.

Logical guarantees of validity must be understood as subject to the proviso that no equivocation is committed. But we do not have a formal theory of equivocation. This paper attempts to formulate rules for responding to equivocal arguments in the context of dialogue. What occurs when one distinguishes meanings of an equivocal expression turns out to be rather different from definition.

Kripke’s Fregean quantification logic FQ fails to formalize the usual first-order logic with identity due to the interpretation of the conditional operator. Motivated by Kripke’s syntax and semantics, the three-valued Fregean quantification logic FQ3 is proposed. This three valued logic differs from Kleene and Łukasiewicz’s three-valued logics. The logic FQ3 is decidable. A sound and complete Hilbert-style axiomatic system for the logic FQ3 is presented.

Frege famously held that numbers play the role of objects in our language and thought, and that this role is on display when we use sentences like "The number of Jupiter's moons is four". I argue that this role is an example of a general pattern that also encompasses persons, times, locations, reasons, causes, and ways of appearing or acting. These things are 'objects' simply in the sense that they are answers to questions: they are the sort of thing we (...) search for and specify during investigation or inquiry. I support this epistemological conception of objects by studying specificational sentences, a class of sentences which includes Frege's original example. I defend an analysis of such sentences as question-answer pairs, and show how to formally represent this analysis using game-theoretical semantics. (shrink)

The goal of creating Artificial General Intelligence (AGI) – or in other words of creating Turing machines (modern computers) that can behave in a way that mimics human intelligence – has occupied AI researchers ever since the idea of AI was first proposed. One common theme in these discussions is the thesis that the ability of a machine to conduct convincing dialogues with human beings can serve as at least a sufficient criterion of AGI. We argue that this very ability (...) should be accepted also as a necessary condition of AGI, and we provide a description of the nature of human dialogue in particular and of human language in general against this background. We then argue that it is for mathematical reasons impossible to program a machine in such a way that it could master human dialogue behaviour in its full generality. This is (1) because there are no traditional explicitly designed mathematical models that could be used as a starting point for creating such programs; and (2) because even the sorts of automated models generated by using machine learning, which have been used successfully in areas such as machine translation, cannot be extended to cope with human dialogue. If this is so, then we can conclude that a Turing machine also cannot possess AGI, because it fails to fulfil a necessary condition thereof. At the same time, however, we acknowledge the potential of Turing machines to master dialogue behaviour in highly restricted contexts, where what is called “narrow” AI can still be of considerable utility. (shrink)

Though the divide between reason-based and causal-explanatory approaches in psychiatry and psychopathology is old and deeply rooted, current trends involving multi-factorial explanatory models and evidence-based approaches to interpersonal psychotherapy, show that it has already been implicitly bridged. These trends require a philosophical reconsideration of how reasons can be causes. This paper contributes to that trajectory by arguing that Donald Davidson’s classic paradigm of 1963 is still a valid option.

The paper attempts to analyze in some detail the main problems encountered in reasoning using diagrams, which may cause errors in reasoning, produce doubts concerning the reliability of diagrams, and impressions that diagrammatic reasoning lacks the rigour necessary for mathematical reasoning. The paper first argues that such impressions come from long neglect which led to a lack of well-developed, properly tested and reliable reasoning methods, as contrasted with the amount of work generations of mathematicians expended on refining the methods of (...) reasoning with formulae and predicate calculus. Next, two main groups of problems occurring in diagrammatic reasoning are introduced. The second group, called diagram imprecision, is then briefly summarized, its detailed analysis being postponed to another paper. The first group, called collectively the generalization problem, is analyzed in detail in the rest of the paper. The nature and causes of the problems from this group are explained, methods of detecting the potentially harmful occurrences of these problems are discussed, and remedies for possible errors they may cause are proposed. Some of the methods are adapted from similar methods used in reasoning with formulae, several other problems constitute new, specifically diagrammatic ways of reliable reasoning. (shrink)

This open access book is a superb collection of some fifteen chapters inspired by Schroeder-Heister's groundbreaking work, written by leading experts in the field, plus an extensive autobiography and comments on the various contributions by Schroeder-Heister himself. For several decades, Peter Schroeder-Heister has been a central figure in proof-theoretic semantics, a field of study situated at the interface of logic, theoretical computer science, natural-language semantics, and the philosophy of language. -/- The chapters of which this book is composed discuss the (...) subject from a rich variety of angles, including the history of logic, the proper interpretation of logical validity, natural deduction rules, the notions of harmony and of synonymy, the structure of proofs, the logical status of equality, intentional phenomena, and the proof theory of second-order arithmetic. All chapters relate directly to questions that have driven Schroeder-Heister's own research agenda and to which he has made seminal contributions. The extensive autobiographical chapter not only provides a fascinating overview of Schroeder-Heister's career and the evolution of his academic interests but also constitutes a contribution to the recent history of logic in its own right, painting an intriguing picture of the philosophical, logical, and mathematical institutional landscape in Germany and elsewhere since the early 1970s. The papers collected in this book are illuminatingly put into a unified perspective by Schroeder-Heister's comments at the end of the book. Both graduate students and established researchers in the field will find this book an excellent resource for future work in proof-theoretic semantics and related areas. (shrink)

In this work, it is shown that given a correct understanding of the nature of reference and of linguistic meaning generally, it is possible to produce non-revisionist analyses of the nature of -/- *Perceptual content, *Mental content generally, *Logical equivalence, *Logical dependence generally, *Counterfactual truth, *The causal efficacy of mental states, and *Our knowledge of ourselves and of the external world. -/- In addition, set-theoretic interpretations of several semantic concepts are put forth. These concepts include truth, falsehood, negation, and conjunction.

A constant of Frege’s writing is his rejection of indeterminate predicates as found in natural language. This paper follows Frege’s remarks on vagueness from the early "Begriffsschrift” to his mature works, drawing brief parallels with the main contemporary theories of vagueness. I critically examine Frege’s arguments for the inconsistency of natural language and argue that the inability to accommodate vagueness in his mature ontology is mainly due to heuristic rules of thumb which Frege took as essential, not to a deep (...) problem in his fundamental apparatus. (shrink)

In bilateral logic formulas are signed by + and –, indicating the speech acts assertion and denial. I argue that making an assumption is also speech act. Speech acts cannot be embedded within other speech acts. Hence we cannot make sense of the notion of making an assumption in bilateral logic. Attempts to solve this problem are considered and rejected.

The paper offers a historical survey of the emergence of logical formalization in twentieth-century analytically oriented philosophy of religion. This development is taken to have passed through three main ?stages?: a pioneering stage in the late nineteenth and early twentieth centuries (led by Frege and Russell), a stage of crisis in the 1920s and early 1930s (occasioned by Wittgenstein, logical positivists such as Carnap, and neo-Thomists such as Maritain), and a stage of rehabilitation in the 1930s, 1940s, and 1950s (led (...) by the Cracow Circle and Quine). (shrink)

Provides a comprehensive overview and introduction to the Routledge Handbook of Propositions.

In this paper I suggest an answer to the question of what Frege means when he says that his logical system, the Begrijfsschrift, is like the language Leibniz sketched, a lingua characteristica, and not merely a logical calculus. According to the nineteenth century studies, Leibniz's lingua characteristica was supposed to be a language with which the truths of science and the constitution of its concepts could be accurately expressed. I argue that this is exactly what the Begriffsschrift is: it is (...) a language, since, unlike calculi, its sentential expressions express truths, and it is a characteristic language, since the meaning of its complex expressions depend only on the meanings of their constituents and on the way they are put together. In fact it is in itself already a science composed in accordance with the Classical Model of Science. What makes the Begrijfsschrift so special is that Frege is able to accomplish these goals with using only grammatical or syncategorematic terms and so has a medium with which he can try to show analyticity of the theorems of arithmetic. (shrink)

Wilhelm von Humboldt (1767-1835) is rightly regarded as a thinker who extended the development of the so-called expressivist conception of language and world that Johann Georg Hamann (1730-1788) and especially Johann Gottfried Herder (1744-1803) initially articulated. Being immersed as Humboldt was in the intellectual climate of German Romanticism, he aimed not only to provide a systematic foundation for how he believed linguistic research as a science should be conducted, but also to attempt to rectify what he saw as the deficiencies (...) of Kant’s philosophical system. My aim in this paper is to show how an expressivist thinker like Humboldt has the conceptual resources from within his own framework and, perhaps surprisingly, with some help from the 20th century philosopher of language and mind, Donald Davidson, to reject a criticism commonly made against expressivist conceptions of language and world. This is the charge that this sort of expressivism threatens the objectivity of the world by emphasizing the role of language in the constitution and disclosure of the world. Cristina Lafont makes just this charge against Humboldt (and other philosophers in the German expressivist-hermeneutic tradition). Specifically, she argues that expressivist philosophers of language are all ultimately committed to some pernicious form of linguistic idealism and relativism. In this paper, I first present Humboldt’s reflections on language and give some textual evidence for why he is often read – mistakenly in my view – as a linguistic idealist and relativist. Second, I briefly sketch Lafont’s charge of linguistic idealism and relativism against Humboldt. Third and finally, I show how she misunderstands Humboldt’s expressivist conception of language and world by connecting my rebuttal to her criticism with Davidson’s argument that successful communication does not require the sharing of explicit rules or conventions that govern in advance the use and understanding of words. (shrink)

This is the first part of the essay devoted to the story of logicism, in particular to its Fregean version. Reviewing the classical period of Fregean studies, we first point out some critical moments of Frege‘s argumentation in the Grundla­gen, in order to be able later to differentiate between its salvageable and defec­tive features. We work on the presumption that there are no easy, catego­rical an­swers to questions like “Is logicism dead?“: Wittgenstein’s cri­tique of the foundational program as well as (...) the remarkable neo-Fregean dis­cov­eries of Boolos and Wright have to be confronted with the effects which the logi­cistic idea actually had on logico­matemati­cal practice. But that is another story, a sequel to this es­say, the purpose of which is systematic rather than criti­cal. (shrink)

Philosophy Compass, Volume 17, Issue 2, February 2022.

An act’s form of apprehension determines whether it is a perception, an imagination, or a signitive act. It must be distinguished from the act’s quality, which determines whether the act is, for instance, assertoric, merely entertaining, wishing, or doubting. The notion of form of apprehension is explained by recourse to the so-called content-apprehension model ; it is characteristic of the Logical Investigations that in it all objectifying acts are analyzed in terms of that model. The distinction between intuitive and signitive (...) acts is made, and the notion of saturation is described, by recourse to the notion of form of apprehension. (shrink)

The way Frege presented the Square of Opposition in a reduced form in 1879 and 1910 can be used to develop two distinct versions of the square: The traditional square that displays inferences and a “Table of Oppositions” displaying variations of negation. This Table of Oppositions can be further simplified and thus be made more symmetrical. A brief survey of versions of the square from Aristotle to the present shows how both aspects of the square have coexisted for a very (...) long time without ever being properly distinguished. (shrink)

The dilemma mentioned in the title have long been a puzzle to logic. To solve it, it was necessary to make some philosophical decisions. The article provides these decisions and discusses the way to solve the problem.

According to an influential view, when it comes to representing reality, some words are better suited for the job than others. This is elitism. There is reason to believe that the set of the best, or elite, words should not be redundant or arbitrary. However, we are often forced to choose between these two theoretical vices, especially in cases involving theories that seem to be mere notational variants. This is the riddle of redundancy: both redundancy and arbitrariness are vicious, but (...) there are cases in which one must be picked. Logical realists admit that there are some logical constants among the elite words. This leads to awkward questions, such as which among conjunction and disjunction is elite. In this paper, I show how the riddle of redundancy arises for logical realists, and offer a solution. This approach requires us to change how we represent negation. Instead of using some particular symbol, we represent negation by flipping formulae over the horizontal axis. (shrink)

Semantic analysis in early analytic philosophy belongs to a long tradition of adopting geometrical methodologies to the solution of philosophical problems. In particular, it adapts Descartes’ development of formalization as a mechanism of analytic representation, for its application in natural language semantics. This article aims to trace the mathematical roots of Frege, Russel and Carnap’s analytic method. Special attention is paid to the formal character of modern analysis introduced by Descartes. The goal is to identify the particular conception of “form” (...) developed by the analytic tradition, from Descartes to early analytic philosophy, and to determine its relation to similar notions, like ‘function’ and ‘syntax’. Finally, I focus on how Frege, Russell and Carnap’s methods of semantic analysis fit the general characterization of formal analysis previously developed. (shrink)