20th Century Logic

We are much better equipped to let the facts reveal themselves to us instead of blinding ourselves to them or stubbornly trying to force them into preconceived molds. We no longer embarrass ourselves in front of our students, for example, by insisting that “Some Xs are Y” means the same as “Some X is Y”, and lamely adding “for purposes of logic” whenever there is pushback. Logic teaching in this century can exploit the new spirit of objectivity, humility, clarity, observationalism, (...) contextualism, and pluralism. Besides the new spirit there have been quiet developments in logic and its history and philosophy that could radically improve logic teaching. One rather conspicuous example is that the process of refining logical terminology has been productive. Future logic students will no longer be burdened by obscure terminology and they will be able to read, think, talk, and write about logic in a more careful and more rewarding manner. Closely related is increased use and study of variable-enhanced natural language as in “Every proposition x that implies some proposition y that is false also implies some proposition z that is true”. Another welcome development is the culmination of the slow demise of logicism. No longer is the teacher blocked from using examples from arithmetic and algebra fearing that the students had been indoctrinated into thinking that every mathematical truth was a tautology and that every mathematical falsehood was a contradiction. A fifth welcome development is the separation of laws of logic from so-called logical truths, i.e., tautologies. Now we can teach the logical independence of the laws of excluded middle and non-contradiction without fear that students had been indoctrinated into thinking that every logical law was a tautology and that every falsehood of logic was a contradiction. This separation permits the logic teacher to apply logic in the clarification of laws of logic. This lecture expands the above points, which apply equally well in first, second, and third courses, i.e. in “critical thinking”, “deductive logic”, and “symbolic logic”. (shrink)

Frege's definition of the real numbers, as envisaged in the second volume of Grundgesetze der Arithmetik, is fatally flawed by the inconsistency of Frege's ill-fated Basic Law V. We restate Frege's definition in a consistent logical framework and investigate whether it can provide a logical foundation of real analysis. Our conclusion will deem it doubtful that such a foundation along the lines of Frege's own indications is possible at all.

This paper is a short introduction to Carnap’s writings on semantics with an emphasis on the transition from the syntactic period to the semantic one. I claim that one of Carnap’s main aims was to investigate the possibility of the symmetry between the syntactic and the semantic methods of approaching philosophical problems, both in logic and in the philosophy of science. This ideal of methodological symmetry could be described as an attempt to obtain categorical logical systems, i.e., systems that allow (...) only the intended semantical interpretation. (shrink)

Tarski’s Convention T—presenting his notion of adequate definition of truth (sic)—contains two conditions: alpha and beta. Alpha requires that all instances of a certain T Schema be provable. Beta requires in effect the provability of ‘every truth is a sentence’. Beta formally recognizes the fact, repeatedly emphasized by Tarski, that sentences (devoid of free variable occurrences)—as opposed to pre-sentences (having free occurrences of variables)—exhaust the range of significance of is true. In Tarski’s preferred usage, it is part of the meaning (...) of true that attribution of being true to a given thing presupposes the thing is a sentence. Beta’s importance is further highlighted by the fact that alpha can be satisfied using the recursively definable concept of being satisfied by every infinite sequence, which Tarski explicitly rejects. Moreover, in Definition 23, the famous truth-definition, Tarski supplements “being satisfied by every infinite sequence” by adding the condition “being a sentence”. Even where truth is undefinable and treated by Tarski axiomatically, he adds as an explicit axiom a sentence to the effect that every truth is a sentence. Surprisingly, the sentence just before the presentation of Convention T seems to imply that alpha alone might be sufficient. Even more surprising is the sentence just after Convention T saying beta “is not essential”. Why include a condition if it is not essential? Tarski says nothing about this dissonance. Considering the broader context, the Polish original, the German translation from which the English was derived, and other sources, we attempt to determine what Tarski might have intended by the two troubling sentences which, as they stand, are contrary to the spirit, if not the letter, of several other passages in Tarski’s corpus. (shrink)

This paper raises some questions about the formalization of sentences containing ‘if’ or similar expressions. In particular, it focuses on three kinds of sentences that resemble conditionals in some respects but exhibit distinctive logical features that deserve separate consideration: whether-or-not sentences, biscuit conditionals, and concessive conditionals. As will be suggested, the examples discussed show in different ways that an adequate formalization of a sentence must take into account the content expressed by the sentence. This upshot is arguably what one should (...) expect on the view that logical form is determined by truth conditions. (shrink)

This chapter clarifies that it was the works Giuseppe Peano and his school that first led Russell to embrace symbolic logic as a tool for understanding the foundations of mathematics, not those of Frege, who undertook a similar project starting earlier on. It also discusses Russell’s reaction to Peano’s logic and its influence on his own. However, the chapter also seeks to clarify how and in what ways Frege was influential on Russell’s views regarding such topics as classes, functions, meaning (...) and denotation, etc., and summarizes the correspondence between Frege and Russell and the light it sheds on the philosophical logic of both. (shrink)

To believe that logic has no history might at first seem peculiar today. But since the early 20th century, this position has been repeatedly conflated with logical monism of Kantian provenance. This logical monism asserts that only one logic is authoritative, thereby rendering all other research in the field marginal and negating the possibility of acknowledging a history of logic. In this paper, I will show how this and many related issues have developed, and that they are founded on only (...) one prominent statement by Kant. I will argue, however, that this statement takes on a very different meaning in a broader context of the history and philosophy of science, and that Kant and his supporters never advocated the logical monism that they are still said to hold today. (shrink)

In the Tractatus, Wittgenstein conceives of tautology as 'saying nothing'. More precisely, he holds -- or so this essay contends -- that it says nothing in virtue of possessing a zero quantity of sense. Insofar as it is the limit of a series of propositions of diminishing quantity of sense, tautology resembles a degenerate conic section. But it also resembles the result of a summing together of equal and opposite linear vector quantities. Both of these models shape Wittgenstein's conception of (...) tautology in the Tractatus, though they are not fully reconciled. Many of the puzzling features of the Tractatus's view of logic arise from this failure to affect the needed reconciliation. These points are argued for by working through the development of Wittgenstein's views on these matters in the Wartime Notebooks and other pre-Tractatus writings. (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)

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)

Brouwer's philosophy of mathematics is usually regarded as an intra-subjective, even solipsistic approach, an approach that also underlies his mathematical intuitionism, as he strived to create a mathematics that develops out of something inner and a-linguistic. Thus, points of connection between Brouwer's mathematical views and his views about and the social world seem improbable and are rarely mentioned in the literature. The current paper aims to challenge and change that. The paper employs a socially oriented prism to examine Brouwer's views (...) on the construction, use, and practice of mathematics. It focuses on Brouwer's views on language, his social interactions, and the importance of group context as they appear in the significs dialogues. It does so by exploring the establishment and dissolution of the significs movement, focusing on Gerrit Mannoury's influence and relationship with Brouwer and analyzing several fragments from the significs dialogues while emphasizing the role Brouwer ascribed to groups in forming and sharing new ideas. The paper concludes by raising two questions that challenge common historical and philosophical readings of intuitionism. (shrink)

This chapter explores the metaphysical views about higher-order logic held by two individuals responsible for introducing it to philosophy: Gottlob Frege (1848–1925) and Bertrand Russell (1872–1970). Frege understood a function at first as the remainder of the content of a proposition when one component was taken out or seen as replaceable by others, and later as a mapping between objects. His logic employed second-order quantifiers ranging over such functions, and he saw a deep division in nature between objects and functions. (...) Russell understood propositional functions as what is obtained when constituents of propositions are replaced by variables, but eventually denied that they were entities in their own right. Both encountered contradictions when supposing there to exist as many objects as functions, and both adopted views about the meaningfulness of higher-order discourse that were difficult to state from within their own strictures. (shrink)

We give a new and elementary construction of primitive positive decomposition of higher arity relations into binary relations on finite domains. Such decompositions come up in applications to constraint satisfaction problems, clone theory and relational databases. The construction exploits functional completeness of 2-input functions in many-valued logic by interpreting relations as graphs of partially defined multivalued ‘functions’. The ‘functions’ are then composed from ordinary functions in the usual sense. The construction is computationally effective and relies on well-developed methods of functional (...) decomposition, but reduces relations only to ternary relations. An additional construction then decomposes ternary into binary relations, also effectively, by converting certain disjunctions into existential quantifications. The result gives a uniform proof of Peirce’s reduction thesis on finite domains, and shows that the graph of any Sheffer function composes all relations there. (shrink)

The historical consensus is that logical evidence is special. Whereas empirical evidence is used to support theories within both the natural and social sciences, logic answers solely to a priori evidence. Further, unlike other areas of research that rely upon a priori evidence, such as mathematics, logical evidence is basic. While we can assume the validity of certain inferences in order to establish truths within mathematics and test scientifi c theories, logicians cannot use results from mathematics or the empirical sciences (...) without seemingly begging the question. Appeals to rational intuition and analyticity in order to account for logical knowledge are symptomatic of these commitments to the apriority and basicness of logical evidence. This chapter argues that these historically prevalent accounts of logical evidence are mistaken, and that if we take logical practice seriously we fi nd that logical evidence is rather unexceptional, sharing many similarities to the types of evidence appealed to within other research areas. (shrink)

AbstractŁukasiewicz introduced a new methodological approach to the history of logic. It consists of the use of modern formal logic in the research of the history of logic. Although he was not the first to use formal logic in his historical research, Łukasiewicz was the first who used it consistently and formulated it as a requirement for a historian of logic. The aim of this paper is to present Łukasiewicz's contribution and the history of its formulation. In addition, the paper (...) will present its reception. There are two main themes in Łukasiewicz’s work on the history of logic. He discussed Stoic logic and Aristotle’s syllogistic. The second part of the paper focuses on current reception of Łukasiewicz’s contribution and on the question of whether there is a difference between knowledge about his contribution to Stoic logic and to Aristotle’s syllogistic, given that it has been hypothesised that Łukasiewicz’s contribution is currently more known among Aristotelian scholars than among those who focus on Stoic logic. Based on the papers that have appeared in the last five years at Web of Science and Scopus this hypothesis has not been confirmed. (shrink)

‘Feminist logic’ may sound like an impossible, incoherent, or irrelevant project, but it is none of these. We begin by delineating three categories into which projects in feminist logic might fall: philosophical logic, philosophy of logic, and pedagogy. We then defuse two distinct objections to the very idea of feminist logic: the irrelevance argument and the independence argument. Having done so, we turn to a particular kind of project in feminist philosophy of logic: Valerie Plumwood's feminist argument for a relevance (...) logic. Plumwood's work serves as our primary case study as we turn to the project of considering three different ways we might understand her argument and revisionist arguments like it: as a priori theorizing, as ameliorative conceptual engineering, or as instances of anti-exceptionalist approaches to logic. After arguing that the anti-exceptionalist approach seems to provide the most promising means of understanding the kind of project undertaken in a feminist challenge to classical logic, we briefly address the consequences that this might have for logic instruction. Here, we argue for the perhaps unexpected conclusion that feminist programs ought to offer more, not less, instruction in logic for those who take interest. (shrink)

In this short intellectual biography, we chronicle Saul Kripke’s involvement in the development of modal logic, focusing on the decade beginning in 1953 and ending in 1963, during which time he ranged in age from 12 to 23. We also describe the state of modal logic before Kripke, Kripke’s correspondence with other modal logicians, and Kripke’s early influential publications on the semantics of modal logic as well as several later and lesser known contributions.

This edited volume brings together papers by both eminent and rising scholars to celebrate Saul Kripke’s singular contributions to modal logic. Kripke’s work on modal logic helped usher in a new semantic epoch for the field and made facility with modal logic indispensable not only to technically oriented philosophers but to theoretical computer scientists and others as well. This volume features previously unpublished work of Kripke’s as well as a brief intellectual biography recounting the story of how Kripke became interested (...) in, and made his first contributions to, modal logic. However, the majority of the volume’s contributions are forward-looking, and produce new philosophical and technical insights by engaging with ideas tracing back to Kripke. (shrink)

This chapter responds to Saul Kripke’s critique of the idea of adopting an alternative logic. It defends an anti-exceptionalist view of logic, on which coming to accept a new logic is a special case of coming to accept a new scientific theory. The approach is illustrated in detail by debates on quantified modal logic. A distinction between folk logic and scientific logic is modelled on the distinction between folk physics and scientific physics. The importance of not confusing logic with metalogic (...) in applying this distinction is emphasized. Defeasible inferential dispositions are shown to play a major role in theory acceptance in logic and mathematics as well as in natural and social science. Like beliefs, such dispositions are malleable in response to evidence, though not simply at will. The chapter considers the Quinean objection that accepting an alternative logic involves changing the subject rather than denying the doctrine. The objection is shown to depend on neglect of the social dimension of meaning determination, akin to descriptivism about proper names and natural kind terms criticized by Kripke and Putnam. Normal standards of interpretation indicate that disputes between classical and non-classical logicians are genuine disagreements. (shrink)

Justification of theorems plays a vital role in any rational human activity. It is indispensable in science. The deductive method of justifying theorems is used in all sciences and it is the only method of justifying theorems in deductive disciplines. It is based on the notion of proof, thus it is a method of proving theorems. In the Warsaw School of Logic (WSL) – the famous branch of the Lvov-Warsaw School (LWS) – two types of the method: axiomatic deduction method (...) and natural deduction method were developed and practiced. In this paper, both of these methods are briefly discussed with an emphasis on their historical, groundbreaking significance for logic. The axiomatic method by means of rejection (proposed by Jan Łukasiewicz – a co-creator of the WSL), which is the method of the so-called rejection proof (rejection/refutation method) in logical systems and the proving method of generalized natural deduction, which is a hybrid deduction–refutation method of proving theorems, are also outlined in the paper. The author discusses their historical significance. This paper also contains a brief mention of the most significant results which the application of the discussed methods introduced into contemporary scientific research, not only logical one. (shrink)

The resemblance of the theory of formal consequence first offered by the fourteenth-century logician John Buridan to that later offered by Alfred Tarski has long been remarked upon. But it has not yet been subjected to sustained analysis. In this paper, I provide just such an analysis. I begin by reviewing today’s classical understanding of formal consequence, then highlighting its differences from Tarski’s 1936 account. Following this, I introduce Buridan’s account, detailing its philosophical underpinnings, then its content. This then allows (...) us to separate those aspects of Tarski’s account representing genuine historical advances, unavailable to Buridan, from others merely differing from—and occasionally explicitly rejected by—Buridan’s account. (shrink)

Reasoning over our knowledge bases and theories often requires non-deductive inferences, especially – but by no means only – when commonsense reasoning is the case, i.e. when practical agency is called for. This kind of reasoning can be adequately formalized via the notion of supraclassical consequence, a non-deductive consequence tightly associated with default and non-monotonic reasoning and featuring centrally in abductive, inductive, and probabilistic logical systems. In this paper, we analyze core concepts and problems of these systems in the light (...) of supraclassical consequence. (shrink)

The purpose of this paper is to analyse and compare two concepts which tend to be treated as synonymous, and to show the difference between them: these are critical thinking and logical culture. Firstly, we try to show that these cannot be considered identical or strictly equivalent: i.e. that the concept of logical culture includes more than just critical thinking skills. Secondly, we try to show that Christian philosophers, when arguing about philosophical matters and teaching philosophy to students, should not (...) focus only on critical thinking skills, but rather also consider logical culture. This, as we argue, may help to improve debate both within and outside of Christian philosophy. (shrink)

The paper is the Part IV of the large research, dedicated to both revision of the system of basic logical categories and generalization of modern predicate logic to functional logic. The topic of the paper is consideration of graphs of functions and relations as a derivative and definable category of ultra-Fregean logistics. There are two types of function specification: an operational specification, in which a function is first applied to arguments and then the value of the function is entered as (...) the result of such application, and a grammar specification, in which an object is first entered and then represented as a value of the function on the given arguments. Since function specifications are relations themselves (which was shown in the previous article of the series), this means that each function forms two relations: the obverse (for an operational specification) and the reverse one (for a grammatical specification). Hence, the definitions introduce two types of graphs: the obverse graph of a function, or a graph of the obverse relation formed by this function, is a set of sequences of such objects, the last of which is a value of this function on the sequence of other of these objects as arguments of the same function. Similarly, the reverse graph of a function, or the graph of the reverse relation formed by this function, is called a set of sequences of such objects, the first of which is the value of this function on the sequence of other of these objects as arguments of the same function. In the limiting case of 0-ary functions, both graphs of any such function coincide. Gottlob Frege discovered obverse graphs of functions and called them function value-ranges. He fundamentally emphasized the non-identity of functions and their graphs. Unfortunately, in set theory under the influence of Giuseppe Peano, an alternative approach was adopted, according to which functions and relations are identical to their graphs and, ultimately, to each other. We show that this approach leads to two unacceptable and, at the same time, mutually contradictory conclusions, and must therefore be rejected. (shrink)

In his paper on the incompleteness theorems, Gödel seemed to say that a direct way of constructing a formula that says of itself that it is unprovable might involve a faulty circularity. In this note, it is proved that ‘direct’ self-reference can actually be used to prove his result.

This paper collects and presents unpublished notes of Kurt Gödel concerning the field of many-valued logic. In order to get a picture as complete as possible, both formal and philosophical notes, transcribed from the Gabelsberger shorthand system, are included.

This paper presents the link between Arthur N. Prior and logicians that belonged to the Lvov-Warsaw School. Although certain members of the Lvov-Warsaw School influenced Prior’s views, the amount and the form of the impact are still under discussion. Prior also cooperated with some of them in the development of his systems of logic. This paper focuses on four main areas in which Prior admitted adopting ideas from the Lvov-Warsaw School: systems of propositional logic, the history of logic, modal and (...) temporal logic and ontology. Prior’s published works as well as his correspondence, which is stored in Prior’s Nachlass, are used to describe the influence. (shrink)

In his paper ‘The logic of temporal discourse’, Pavel Tichý pointed out that contemporary systems of logic were unable to sufficiently formalise tenses. He therefore suggested temporal specification in transparent intensional logic (TIL), a system of logic that he developed. Discussing contemporary systems of logic, Tichý also took into account the system of Arthur N. Prior, who developed the first systems of modern temporal logic, and his criticism was also addressed to Prior. Tichý only focused, however, on Prior’s early systems (...) of temporal logic. Patrick Blackburn recently raised the awareness that Prior also developed systems of hybrid logic in his latest periods. From the point of view of temporal specification, this system is particularly interesting as the system has greater expressive power than Prior’s early systems of temporal logic. It could also consequently deal with the problematic specifications of tenses that Pavel Tichý pointed out. It is not only the formal criterion that make Tichý and Prior’s approach suitable for comparison. Both logicians shared similar views on time and logic. All these convictions also influenced their systems of logic. The aim of this paper is to demonstrate that the temporal propositions that Tichý introduced as problematic could be formalised in Prior’s hybrid temporal logic. I will also compare formalisations in TIL and hybrid logic and Tichý and Prior’s views that influenced their systems of logic. (shrink)

This paper sketches the antilogism of Christine Ladd-Franklin and historical advancement about antilogism, mainly constructs an axiomatic system Atl based on first-order logic with equality and the wholly-exclusion and not-wholly-exclusion relations abstracted from the algebra of Ladd-Franklin, with soundness and completeness of Atl proved, providing a simple and convenient tool on syllogistic reasoning. Atl depicts the empty class and the whole class differently from normal set theories, e.g. ZFC, revealing another perspective on sets and set theories. Two series of Dotterer (...) and the theorem proved by Russinoff are re-characterized in antilogistic formulas corresponding to the square of opposition and twenty-four moods of syllogism. Then the restricted equivalence between two forms of antilogism and minimal inconsistency is built up, implying the internal connection between antilogisms and minimal inconsistency and leaving two related conjectures not proved. Besides, this paper provides a new explanation for the reason why contrariety, subcontrariety, and subalternation and the nine special moods require the non-empty assumption. (shrink)

In his Doppelvortrag, Edmund Husserl introduced two concepts of “definiteness” which have been interpreted as a vindication of his role in the history of completeness. Some commentators defended that the meaning of these notions should be understood as categoricity, while other scholars believed that it is closer to syntactic completeness. A detailed study of the early twentieth-century axiomatics and Husserl’s Doppelvortrag shows, however, that many concepts of completeness were conflated as equivalent. Although “absolute definiteness” was principally an attempt to characterize (...) non-extendible manifolds and axiom systems, an absolutely definite theory has a unique model and, thus, it is non-forkable and semantically complete. Non-forkability and decidability were formally delimited by Fraenkel and Carnap almost three decades later and, in fact, they mentioned Husserl as precursor of the latter. Therefore, this paper contributes to a reassessment of Husserl’s place in the history of logic. (shrink)

Cantor’s abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor’s proposal based upon the set theoretic framework of Bourbaki—called BK—which is a First-order set theory extended with Hilbert’s \-operator. Moreover, it is argued that the BK system and the \-operator provide a faithful reconstruction of Cantor’s insights on cardinal numbers. I will introduce first (...) the axiomatic setting of BK and the definition of cardinal numbers by means of the \-operator. Then, after presenting Cantor’s abstractionist theory, I will point out two assumptions concerning the definition of cardinal numbers that are deeply rooted in Cantor’s work. I will claim that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to those of Zermelo–von Neumann and Frege–Russell. On the basis of these similarities, I will make use of the BK framework in meeting Frege’s objections to Cantor’s proposal. A key ingredient in the defence of Cantorian abstraction will be played by the role of representative sets, which are arbitrarily denoted by the \-operator in the BK definition of cardinal numbers. (shrink)

A system of logic usually comprises a language for which a model-theory and a proof-theory are defined. The model-theory defines the semantic notion of model-theoretic logical consequence (⊨), while the proof-theory defines the proof- theoretic notion of logical consequence (or logical derivability, ⊢). If the system in question is sound and complete, then the two notions of logical consequence are extensionally equivalent. The concept of full formalization is a more restrictive one and requires in addition the preservation of the standard (...) meanings of the logical terms in all the admissible interpretations of the logical calculus, as it is proof-theoretically defined. Although classical first-order logic is sound and complete, its standard formalizations fall short to be full formalizations since they allow non-intended interpretations. This fact poses a challenge for the logical inferentialism program, whose main tenet is that the meanings of the logical terms are uniquely determined by the formal axioms or rules of inference that govern their use in a logical calculus, i.e., logical inferentialism requires a categorical calculus. This paper is the first part of a more elaborated study which will analyze the categoricity problem from its beginning until the most recent approaches. I will first start by describing the problem of a full formalization in the general framework in which Carnap (1934/1937, 1943) formulated it for classical logic. Then, in sections IV and V, I shall discuss the way in which the mathematicians B.A. Bernstein (1932) and E.V. Huntington (1933) have previously formulated and analyzed it in algebraic terms for propositional logic and, finally, I shall discuss some critical reactions Nagel (1943), Hempel (1943), Fitch (1944), and Church (1944) formulated to these approaches. (shrink)

The use of the symbol ∨for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead and (...) Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed. (shrink)

En The paradox of Addition and its dissolution (1969), Mario Bunge presenta algunos argumentos para mostrar que la Regla de Adición puede ocasionar paradojas o problemas semánticos. Posteriormente, Margáin (1972) y Robles (1976) mostraron que las afirmaciones de Bunge son insostenibles, al menos desde el punto de vista de la lógica clásica. Aunque estamos de acuerdo con las críticas de Margáin y Robles, no estamos de acuerdo en el diagnóstico del origen del problema y tampoco con la manera en la (...) que se proponen solucionarlo. En esta medida, en este texto mostraremos cómo la Regla de Adición sí trae problemas semánticos que pueden ser planteados desde diferentes perspectivas contemporáneas, como la extensionalidad del condicional, los principios proscriptivos y la validez conexiva. (shrink)

ABSTRACT This paper seeks to clarify Husserl’s critical remarks about Kant’s view of logic by comparing their respective views of logic. In his Formal and Transcendental Logic Husserl criticizes Kant for not asking transcendental questions about formal logic, but rather ascribing an ‘extraordinary apriority’ to it. He thinks the reason for Kant’s uncritical attitude to logic lies in Kant’s view of logic as directed toward the subjective, instead of being concerned with a ‘“world” of ideal Objects’. Whereas for Kant, general (...) logic is about laws of reasoning. Husserl thinks that formal logic should describe formal structures. Husserl claims that if Kant had had a more comprehensive concept of logic, he would have thought of raising critical questions about how logic is possible. This kind of criticism cannot itself use forms of judgments or syllogisms of logic, nor even the ‘inferential’ [schliessende] method more generally, but should be descriptive in nature. Husserl's transcendental phenomenology is the method for such criticism. The paper argues that this results in reflection, and possibly revision, of the logical principles with respect to the normative goals governing the investigation in question. (shrink)

This book is dedicated to V.A. Yankov’s seminal contributions to the theory of propositional logics. His papers, published in the 1960s, are highly cited even today. The Yankov characteristic formulas have become a very useful tool in propositional, modal and algebraic logic. The papers contributed to this book provide the new results on different generalizations and applications of characteristic formulas in propositional, modal and algebraic logics. In particular, an exposition of Yankov’s results and their applications in algebraic logic, the theory (...) of admissible rules and refutation systems is included in the book. In addition, the reader can find the studies on splitting and join-splitting in intermediate propositional logics that are based on Yankov-type formulas which are closely related to canonical formulas, and the study of properties of predicate extensions of non-classical propositional logics. The book also contains an exposition of Yankov’s revolutionary approach to constructive proof theory. The editors also include Yankov’s contributions to history and philosophy of mathematics and foundations of mathematics, as well as an examination of his original interpretation of history of Greek philosophy and mathematics. (shrink)

Susan Stebbing (1885–1943), the UK’s first female professor of philosophy, was a key figure in the development of analytic philosophy. Stebbing wrote the world’s first accessible book on the new polyadic logic and its philosophy. She made major contributions to the philosophy of science, metaphysics, philosophical logic, critical thinking, and applied philosophy. Nonetheless she has remained largely neglected by historians of analytic philosophy. This Element provides a thorough yet accessible overview of Stebbing’s positive, original contributions, including her solution to the (...) paradox of analysis, her account of the relation of sense-data to physical objects, and her anti-idealist interpretation of the new Einsteinian physics. Stebbing’s innovative work in these and other areas helped move analytic philosophy from its early phase to its middle period. (shrink)

Logicism is the view that mathematical truths are logical truths. But a logical truth is commonly thought to be one with a universally valid form. The form of “7 > 5” would appear to be the same as “4 > 6”. Yet one is a mathematical truth, and the other not a truth at all. To preserve logicism, we must maintain that the two either are different subforms of the same generic form, or that their forms are not at all (...) what they appear. The historical record shows that Russell pursued both these options, but that the struggle with the logical paradoxes pushed him away from the first kind of response and toward the second. An object cannot itself have a kind of inner logical complexity that makes a proposition have a different logical form merely in virtue of being about it, nor can their representatives in logical forms be single things different for different forms, at least not without postulating too many such objects and thereby creating Cantorian diagonal paradoxes. There are only apparent objects which are actually fragments of logical forms, different in different cases. (shrink)

We argue that traditional formulations of the reduction thesis that tie it to privileged relational operations do not suffice for Peirce’s justification of the categories and invite the charge of gerrymandering to make it come out as true. We then develop a more robust invariant formulation of the thesis, one that is immune to that charge, by explicating the use of triads in any relational operations. The explication also allows us to track how Thirdness enters the structure of higher order (...) relations and even to propose a numerical measure of it. Our analysis reveals new conceptual phenomena when negation or disjunction are used to compound relations. (shrink)

The Hilbert program was actually a specific approach for proving consistency, a kind of constructive model theory. Quantifiers were supposed to be replaced by ε-terms. εxA(x) was supposed to denote a witness to ∃xA(x), or something arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system S, each ε-term can be replaced by a numeral, making each line provable and true. This implies that S must not only be consistent, but also 1-consistent. Here we (...) show that if the result is supposed to be provable within S, a statement about all Pi-0-2 statements that subsumes itself within its own scope must be provable, yielding a contradiction. The result resembles Gödel's but arises naturally out of the Hilbert program itself. (shrink)

Cet ouvrage offre une introduction accessible à la théorie de la démonstration : il donne les détails des preuves et comporte de nombreux exemples et exercices pour faciliter la compréhension des lecteurs. Il est également conçu pour servir d’aide à la lecture des articles fondateurs de Gerhard Gentzen. L’ouvrage introduit également aux trois principaux formalismes en usage : l’approche axiomatique des preuves, la déduction naturelle et le calcul des séquents. Il donne une démonstration claire et détaillée des résultats fondamentaux du (...) domaine : traduction de l’arithmétique classique vers l’arithmétique intuitionniste, élimination des coupures, théorème de normalisation et conduit ensuite pas à pas le lecteur vers l’exposé de la célèbre preuve de cohérence de Gentzen pour l’arithmétique de Peano du premier ordre. Il comble ainsi une importante lacune éditoriale en présentant à la fois la théorie structurelle et la théorie ordinale de la démonstration. (shrink)

Since Kazimierz Twardowski introduced the notions of “symbolomania” and “pragmatophobia,” the relationship between logic and reality was the focus of the philosophers from the Lvov-Warsaw School — inter alia two prominent logicians of the group, Stanisław Leśniewski and Jan Łukasiewicz. Bolesław Sobociński has pointed out, however, that there was a contrast between their approach to logic and reality. Despite being members of the same philosophical group and even colleagues from the same department, their philosophical views on the position of logic (...) in reality differed considerably. Yet they both agreed that reality has a certain importance for logic and that logic could be valuable for reality. The aim of this paper is to introduce their divergent positions and describe in more detail how Leśniewski and Łukasiewicz understood the relationship between logic and the real world. (shrink)

In the nineteenth century, philosophy was at a crossroads. While the natural and technical sciences were developing in an unprecedented fashion, philosophy seemed to be stalled. Inspired by the progress of the natural sciences, many philosophers attempted to make such progress in philosophy and make philosophy a truly scientific discipline. This effort was also reflected in the philosophy of the Lvov-Warsaw school. While its founder, Kazimierz Twardowski, following his teacher Franz Brentano, promoted psychology as a method of scientific philosophy, one (...) of his first students, Jan Łukasiewicz, was convinced that mathematical logic was such a method. To use mathematical logic as a tool, Łukasiewicz had to, however, argue convincingly that logic is an independent science and hence is not a part of psychology, i.e., arguing for anti-psychologism in logic. He initially adopted the arguments provided by Husserl, then celebrated as a proponent of anti-psychologism, and Frege’s views. When Łukasiewicz developed, however, his systems of many-valued logic, he denied almost all the principles that characterise Husserl and Frege’s anti-psychologism, i.e., the objectivity of the laws of logic, the existence of apodictic propositions, and the distinction between a priori and empirical sciences. He was, however, a proponent of anti-psychologism up to the end of his life. The aim of my paper is to introduce Łukasiewicz’s unique concept of anti-psychologism that significantly affected the views of mathematical logic in the Lvov-Warsaw School, and the views of his colleagues which helped him develop the concept. (shrink)

At the beginning of modern logic, propositions were defined as unchangeable entities placed in a certain idealistic realm. These unchangeable propositions contain in themselves so-called indexical, i.e. the place, time and other circumstances of the utterance. This concept of the proposition, which is entitled eternalism, was and is still prevalent among analytic philosophers. Often even the term ‘proposition’ is identified with an idealistic entity located outside the real world. In my paper, I would like to focus on the concept of (...) propositions of two logicians who deviated from the standard understanding of propositions, Arthur N. Prior and Pavel Tichý. They were both proponents of temporalism, i.e. the view that propositions could change their truth-value over time. The paper will discuss the reasons why they were proponents of temporalism and compare their views. It claims that in Prior’s case, his metaphysical views were the main reasons he was a proponent of temporalism. In contrast, when Tichý presented his arguments for temporalism, he focused primarily on natural language. (shrink)

Charles Peirce incorporates modality into his Existential Graphs by introducing the broken cut for possible falsity. Although it can be adapted to various modern modal logics, Zeman demonstrates that making no other changes results in a version that he calls Gamma-MR, an implementation of Jan Łukasiewicz's four-valued Ł-modal system. It disallows the assertion of necessity, reflecting a denial of determinism, and has theorems involving possibility that seem counterintuitive at first glance. However, the latter is a misconception that arises from overlooking (...) the distinction between the intermediate truth values that are assigned to possibly true propositions as either X-contingent or Y-contingent. Any two propositions having the same ITV are possible together, while any two propositions having different ITVs, including those that are each other's negation, are possible individually yet not possible together. Porte shows that Ł-modal can be translated into classical logic by defining a constant for each ITV such that its implication of another proposition asserts the latter's possibility, while its conjunction with another proposition asserts the latter's necessity. These are expressed in the Alpha part of EG without broken cuts, simplifying derivations and shedding further light on Łukasiewicz's system, as long as graphs including either of the constants are properly interpreted. Ł-modal and Gamma-MR thus capture the two-sided nature of possibility as the limit between truth and falsity in Peirce's triadic conception. (shrink)