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.

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 infinite 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 infinite 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 infinite rules of inference in their reasoning. (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)

A German translation with 2017 postscript of Floyd, Juliet. 2012. "Wittgenstein's Diagonal Argument: A Variation on Cantor and Turing." In Epistemology versus Ontology, Logic, Epistemology: Essays in Honor of Per Martin-Löf, edited by P. Dybjer, S. Lindström, E. Palmgren and G. Sundholm, 25-44. Dordrecht: Springer Science+Business Media. An analysis of philosophical aspects of Turing's diagonal argument in his (136) "On computable numbers, with an application to the Entscheidungsproblem" in relation to Wittgenstein's writings on Turing and Cantor.

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.

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)

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 a...

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition for (...) granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined. (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)

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)

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)

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)

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)

Arthur Norman Prior was born on 4 December 1914 in Masterton, New Zealand. He studied philosophy in the 1930s and was a significant, and often provocative, voice in theological debates until well into the 1950s. He became a lecturer in philosophy at Canterbury University College in Christchurch in 1946 succeeding Karl Popper. He became a full professor in 1952. He left New Zealand permanently for England in 1959, first taking a chair in philosophy at Manchester University, and then becoming a (...) fellow of Balliol College, Oxford, in 1966. Prior died on 6 October 1969 in Trondheim, Norway. After Prior’s death, many logicians and philosophers have analysed and discussed his approach to formal and philosophical logic. In particular, his contributions to modal logic, tense-logic and deontic logic have been studied. -/- In 1957, A.N. Prior proposed the three-valued modal logic Q as a ‘correct’ modal logic from his philosophical motivations, see Prior (1957). Prior developed Q in order to offer a logic for contingent beings, in which one could intelligibly and rationally state that some beings are contingent and some are necessary, see Akama & Nagata (2005). According to Akama & Nagata (2005), Q has a natural semantics. In other words, from the philosophical point of view, Q can be regarded as an ‘actualist’ modal logic. -/- This review article is a developed description of, and discussion on, ‘The System Q’ that is the fifth chapter of Prior (1957). In addition, in his logical analysis of ‘Time & Existence’ (that is the eights chapter of Prior (1967)), Prior has worked on system Q. Thus, Prior (1967) has also been very useful for this article. This article analyses the logical structure of system Q in order to provide a more understandable description as well as logical analysis for today’s logicians, philosophers, and information-computer scientists. In the paper, the Polish notations are translated into modern notations in order to be more comprehensible and to support the developed formal descriptions as well as semantic analysis. (shrink)

In his 1918 logical atomism lectures, Russell argued that there are no molecular facts. But he posed a problem for anyone wanting to avoid molecular facts: we need truth-makers for generalizations of molecular formulas, but such truth-makers seem to be both unavoidable and to have an abominably molecular character. Call this the problem of generalized molecular formulas. I clarify the problem here by distinguishing two kinds of generalized molecular formula: incompletely generalized molecular formulas and completely generalized molecular formulas. I next (...) argue that, if empty worlds are logically possible, then the model-theoretic and truth-functional considerations that are usually given address the problem posed by the first kind of formula, but not the problem posed by the second kind. I then show that Russell’s commitments in 1918 provide an answer to the problem of completely generalized molecular formulas: some truth-makers will be non-atomic facts that have no constituents. This shows that the neo-logical atomist goal of defending the principle of atomicity—the principle that only atomic facts are truth-makers—is not realizable. (shrink)

In 1954, A. N. Prior discovered a way to formalize tense-logic—as such, there is no doubt that he is the father of modern tense-logic. Despite this, he considered his early teacher in p...

In several publications, Juliet Floyd and Hilary Putnam have argued that the so-called ‘notorious paragraph’ of the Remarks on the Foundations of Mathematics contains a valuable philosophical insight about Gödel’s informal proof of the first incompleteness theorem – in a nutshell, the idea they attribute to Wittgenstein is that if the Gödel sentence of a system is refutable, then, because of the resulting ω-inconsistency of the system, we should give up the translation of Gödel’s sentence by the English sentence “I (...) am unprovable”.I will argue against Floyd and Putnam’s use of the idea, and I will indirectly question its attribution to Wittgenstein. First, I will point out that the idea is inefficient in the context of the first incompleteness theorem because there is an explicit assumption of soundness in Gödel’s informal discussion of that theorem. Secondly, I will argue that of he who makes the observation that Floyd and Putnam think Wittgenstein has made about the first theorem, one will expect to see an analogous observation about Gödel’s second incompleteness theorem – yet we see nothing to that effect in Wittgenstein’s remarks. Incidentally, that never-made remark on the import of the second theorem is of genuine logical significance.‏. (shrink)

In his Lecture on Ethics, Wittgenstein mentions the difficulty a recipient of his philosophy has in ‘seeing both the road he is led and the goal which it leads to’. Oskari Kuusela's b...

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.

In 1928 Ivan Orlov published a remarkable paper which contains the first formulation of a relevant logic. The paper remained largely unknown to English-speakers until this discovery of relevant log...

Jan Łukasiewicz is known primarily as the founder of the three-valued system of logic. It is also generally renowned that his reason for introducing many-valued systems of logic was an attempt to refute determinism. When he developed the three-valued and n-valued logic, he employed these systems in his arguments against determinism. On the contrary, Łukasiewicz preferred the four-valued system of logic that is not suitable for a refutation of determinism in his latest period. It seems, however, that determinism still interested (...) Łukasiewicz, as he discussed its denial in his last work, the second edition of the book Aristotle’s syllogistic from the standpoint of modern formal logic. The aim of my paper is to introduce the issue and describe the relation of these two topics primarily in the latest period of Łukasiewicz’s work. There are several possible solutions to the issue. However, Łukasiewicz never addressed the issue clearly in his published works. Therefore, the solutions might be supported or contradicted by his unpublished works in the future. (shrink)

Abstract The aim of this paper is to take a look at Péter's talk "Rekursivität und Konstruktivität" delivered at the Constructivity in Mathematics Colloquium in 1957, where she challenged Church's Thesis from a constructive point of view. The discussion of her argument and motivations is then connected to her earlier work on recursion theory as well as her later work on theoretical computer science.

Frege's conception of the logical categories has vexed commentators for decades. In this dissertation, I argue that it revolves around two forms of internality. The first is the internality of its use in the expression of judgment to the sign. A proper understanding of that internality reveals how Frege's philosophical logic cannot be fit into the framework given by the contemporary syntax/semantics distinction. The second is the internality that obtains between the way in which Begriffsschrift signs stratify into different categories, (...) on the one hand, and the way in which the realm of Bedeutungen stratifies into logical categories, on the other hand. A proper understanding of this internality reveals how Frege's conception of the relation between language and the world is neither a form of realism nor a form of linguistic idealism. To come to understand Frege's conception of the logical categories, is to come to understand both these forms of internality by coming to understand how they hang together. (shrink)

In the years following Bertrand Russell's visit in China, fragments from his work on mathematical logic and the foundations of mathematics started to enter the Chinese intellectual world. While up...

Mathematical logic was first introduced to China in early 1920s. Although, the process of introduction was facilitated by the lectures of Bertrand Russel at Peking University in 1921 and continued...

Leśniewski is known for his pedantry and idiosyncratic notation, which make it extremely difficult to read and follow. As reading comes before understanding, this paper therefore attempts only one...

Hermann Weyl was one of the greatest mathematicians of the 20th century, with contributions to many branches of mathematics and physics. In 1918 he wrote a famous book, “Das Kontinuum”, on the foundations of mathematics. In that book he described mathematical analysis as a ‘house built on sand’, and tried to ‘replace this shifting foundation with pillars of enduring strength’. In this paper we reexamine and explain the philosophical and mathematical ideas that underly Weyl’s system in “Das Kontinuum”, and show (...) that they are still useful and relevant. We propose a precise formalization of that system, which is the first to be completely faithful to what is written in the book. Finally, we suggest that a certain set-theoretical modern system reflects better Weyl’s ideas than previous attempts (most notably by Feferman) of achieving this goal. (shrink)

This paper defines provably non-trivial theories that characterize Frege’s notion of a set, taking into account that the notion is inconsistent. By choosing an adaptive underlying logic, consistent sets behave classically notwithstanding the presence of inconsistent sets. Some of the theories have a full-blown presumably consistent set theory T as a subtheory, provided T is indeed consistent. An unexpected feature is the presence of classical negation within the language.

According to the Bad Company objection, the fact that Frege’s infamous Basic Law V instantiates the general definitional pattern of higher-order abstraction principles is a good reason to doubt the soundness of this sort of definitions. In this paper I argue against this objection by showing that the definitional pattern of abstraction principles – as extrapolated from §64 of Frege’s Grundlagen– includes an additional requirement (which I call the specificity condition) that is not satisfied by the Basic Law V while (...) is satisfied by other higher-order abstractions such as Hume’s Principle. I also show that the failure of this additional requirement in the case of Basic Law V is engendered by an essential feature of Frege’s conception of logic and thus that Frege himself should not have regarded the Basic Law V as a definition by abstraction. (shrink)

The paper offers a survey of four key moments in which symbolisms for quantification were first introduced: §§11–2 of Frege’s Begriffsschrift ; Peirce’s ‘Algebra of Logic’ ; Peano’s ‘St...

Val Plumwood’s 1993 paper, “The politics of reason: towards a feminist logic” (hence- forth POR) attempted to set the stage for what she hoped would begin serious feminist exploration into formal logic – not merely its historical abuses, but, more importantly, its potential uses. This work offers us: (1) a case for there being feminist logic; and (2) a sketch of what it should resemble. The former goal of Plumwood’s paper encourages feminist theorists to reject anti-logic feminist views. The paper’s (...) latter aim is even more challenging. Plumwood’s critique of classical negation (and classical logic) as a logic of domination asks us to recognize that particular logical systems are weapons of oppression. Against anti-logic feminist theorists, Plumwood argues that there are other logics besides classical logic, such as relevant logics, which are suited for feminist theorizing. Some logics may oppress while others may liberate. We provide details about the sources and context for her rejection of classical logic and motivation for promoting relevant logics as feminist. (shrink)

Volume 41, Issue 2, May 2020, Page 199-202.

This paper addresses an issue with the sign ‘⊢’ in Frege’s mature version of Begriffsschrift, i.e., the version in ‘Function and Concept’ and Grundgesetze. The sign is a performative for asserting in that writing down ‘⊢p’ is equivalent to asserting that p. Frege further says that writing ‘ p’ is also equivalent to identifying the reference of ‘p’ with the truth-value True. It looks as if he holds that asserting that p consists in identifying the True with the reference of (...) ‘p’. Frege’s commitment to it, however, seems to encounter a number of tensions. This paper aims to show that these tensions can be avoided by endorsing a non-assertive conception of identification under which making an identification is not making an identity assertion. The suggested reading leads to an entirely di erent understanding of the compositionality of the sign ‘⊢’ as well as Frege’s conception of assertion and judgment. (shrink)

This paper concerns Bertrand Russell’s changing views on negative judgement. ‘Negative judgement’ is considered in the context of three theories of judgement that Russell put forth at different times: a dual relation theory ; a multiple relation theory ; a psychological theory of judgement. Four issues are singled out for a more detailed discussion: quality dualism versus quality monism, that is, the question whether judgement comes in two kinds, acceptance and rejection, or whether there is only one judgement-quality ; the (...) structure of negative judging; the problem of truth-making for negative facts; the different roles of ‘fact’ in Russell’s theories of truth. What emerges from the discussion is a rough chronology of Russell’s views on negative judgement during the period from 1903 to 1948. (shrink)

In 1935/1936 Kurt Gödel wrote three notebooks on the foundations of quantum mechanics, which have now been entirely transcribed for the first time. Whereas a lot of the material is rather technical in character, many of Gödel's remarks have a philosophical background and concentrate on Leibnizian monadology as well as on vitalism. Obviously influenced by the vitalistic writings of Hans Driesch and his ‘proofs’ for the existence of an entelechy in every living organism, Gödel briefly develops the idea of a (...) computing machine which closely resembles Turing's groundbreaking conception. After introducing the notebooks on quantum mechanics, this article describes Gödel's vitalistic Weltbild and the ideas leading to the development of his computing machine. It investigates a notion of lawlike sequence which closely resembles Turing's concept of a computable number and which Gödel himself calls ‘problematic’, and compares it to the opposed concept of randomness, drawing upon the notion of program-size complexity. Finally, Gödel's machine is implemented in a dialect of the Lisp programing language. (shrink)

Charlotte Angas Scott (1858–1932) was an internationally renowned geometer, the first British woman to earn a doctorate in mathematics, and the chair of the Bryn Mawr mathematics department for forty years. There she helped shape the burgeoning mathematics community in the United States. Scott often motivated her research as providing a “geometric treatment” of results that had previously been derived algebraically. The adjective “geometric” likely entailed many things for Scott, from her careful illustration of diagrams to her choice of references (...) and citations. This article will focus on Scott’s striking and consistent use of geometric to describe a reality of dynamic points, lines, planes, and spaces that could be manipulated analogously to physical objects. By providing geometric interpretations of algebraic derivations, Scott committed to an early-nineteenth-century aesthetic vision of a “whole” analytical geometry that she adapted to modern research areas. (shrink)

In recent times there has been a shift in the interpretation of Aristotle’s logic. Many researchers have pointed out that the concept of existential import appears in Aristotle’s logic and philosophy, and that Aristotle worked with the concept of empty terms although his concept differs from that which is used in modern logic. Additionally, his search for the “culprits” of old and incorrect interpretation has been tied to the development of modern interpretation. Apart from the traditional concept of the logical (...) square and syllogistics, where the assumption of non-emptiness of terms was widely used, J. L. Łukasiewicz is also labeled as an offender. This article tries to show that Łukasiewicz was aware that Aristotle was also dealing with empty terms. His restriction of syllogistics to only non-empty terms therefore differs significantly from the approach of traditional logicians. (shrink)

Peirce is best known as the founder of pragmatism, but his dissatisfaction with how others understood and appropriated it prompted him to rename his own doctrine “pragmaticism” and to compose several variants of his original maxim defining it, as well as numerous restatements and elaborations. This paper presents an extensive selection of such formulations, followed by analysis and commentary demonstrating that for Peirce the ultimate meaning of an intellectual concept is properly expressed as a conditional proposition about the deliberate, self-controlled (...) conduct of its interpreters, not the law-governed behavior of its object. (shrink)

The article argues for a contextualised reading of Wittgenstein’s Tractatus. It analyses in detail the role that use and application play in the text and how that supports a conception of transcendentality of logic that allows for contextualisation. The article identifies a tension in the text, between the requirement that sense be determinate and the contextual nature of application, and suggests that it is this tension that is a major driver of Wittgenstein’s later ideas.

This is a companion article to the translation of ‘Zasada sprzeczności a logika symboliczna’, the appendix on symbolic logic of Jan Łukasiewicz's 1910 book O zasadzie sprzeczności u Arytotelesa (On the Principle of Contradiction in Aristotle). While the appendix closely follows Couturat's 1905 book L'algebra de la logique (The Algebra of Logic), footnotes show that Łukasiewicz was aware of the work of Peirce, Huntington and Russell (before Principia Mathematica). This appendix was influential in the development of the Polish school of (...) logic, directly inspiring Stanisław Leśniewski and Leon Chwistek and more widely by serving as a text of the new symbolic logic. This appendix was an important source of the dominant algebraic logic in Poland, but also indicates that Łukasiewicz appreciated Russell's axiomatic approach to logic. (shrink)

Is a mathematical theorem proved because provable, or provable because proved? If Brouwer’s intuitionism is accepted, we’re committed, it seems, to the latter, which is highly problematic. Or so I will argue. This and other consequences of Brouwer’s attempt to found mathematics on the intuition of a move of time have heretofore been insufficiently appreciated. Whereas the mathematical anomalies of intuitionism have received enormous attention, too little time, I’ll try to show, has been devoted to some of the temporal anomalies (...) that Brouwer has invited us to introduce into mathematics. (shrink)