Because of the “all-or-none” character of nervous activity, neural events and the relations among them can be treated by means of propositional logic. It is found that the behavior of every net can be described in these terms, with the addition of more complicated logical means for nets containing circles; and that for any logical expression satisfying certain conditions, one can find a net behaving in the fashion it describes. It is shown that many particular choices among possible (...) neurophysiological assumptions are equivalent, in the sense that for every net behaving under one assumption, there exists another net which behaves under the other and gives the same results, although perhaps not in the same time. Various applications of the calculus are discussed. (shrink)

John Venn has the “uneasy suspicion” that the stagnation in mathematical logic between J. H. Lambert and George Boole was due to Kant’s “disastrous effect on logical method,” namely the “strictest preservation [of logic] from mathematical encroachment.” Kant’s actual position is more nuanced, however. In this chapter, I tease out the nuances by examining his use of Leonhard Euler’s circles and comparing it with Euler’s own use. I do so in light of the developments in logical calculus (...) from G. W. Leibniz to Lambert and Gottfried Ploucquet. While Kant is evidently open to using mathematical tools in logic, his main concern is to clarify what mathematical tools can be used to achieve. For without such clarification, all efforts at introducing mathematical tools into logic would be blind if not complete waste of time. In the end, Kant would stress, the means provided by formal logic at best help us to express and order what we already know in some sense. No matter how much mathematical notations may enhance the precision of this function of formal logic, it does not change the fact that no truths can, strictly speaking, be revealed or established by means of those notations. (shrink)

A three-valued propositional logic is presented, within which the three values are read as ?true?, ?false? and ?nonsense?. A three-valued extended functional calculus, unrestricted by the theory of types, is then developed. Within the latter system, Bochvar analyzes the Russell paradox and the Grelling-Weyl paradox, formally demonstrating the meaninglessness of both.

Many years after the publication of “A Logical Calculus of the Ideas Immanent in Nervous Activity,” Warren McCulloch gave Walter Pitts credit for contributing his knowledge of modular mathematics to their joint project. In 1941 I presented my notions on the flow of information through ranks of neurons to Rashevsky’s seminar in the Committee on Mathematical Biology of the University of Chicago and met Walter Pitts, who then was about seventeen years old. He was working on a mathematical (...) theory of learning and I was much impressed. He was interested in problems of circularity, how to handle regenerative nervous activity in closed loops....For two years Walter and I worked on these problems whose solution depended upon modular mathematics of which I knew nothing, but Walter did.. In this paper, we will fill in some of the details regarding Pitts’s interest in problems of circularity, regenerative activity in closed loops of neurons, and modular mathematics, and the way in which they relate to “A Logical Calculus.”. (shrink)

By way of a close reading of Boole and Frege’s solutions to the same logical problem, we highlight an underappreciated aspect of Boole’s work—and of its difference with Frege’s better-known approach—which we believe sheds light on the concepts of ‘calculus’ and ‘mechanization’ and on their history. Boole has a clear notion of a logical problem; for him, the whole point of a logical calculus is to enable systematic and goal-directed solution methods for such problems. Frege’s (...) Begriffsschrift, on the other hand, is a visual tool to scrutinize concepts and inferences, and is a calculus only in the thin sense that every possible transition between sentences is fully and unambiguously specified in advance. While Frege’s outlook has dominated much of philosophical thinking about logical symbolism, we believe there is value—particularly in light of recent interest in the role of notations in mathematics and logic—in reviving Boole’s idea of an intrinsic link between, as he put it, a ‘calculus’ and a ‘directive method’ to solve problems. (shrink)

[This résumé was published in English in Matematicheskii Sbornik along with the article.]The present paper contains an investigation of a three-valued logical calculus (the system) previously described by the author [Recueil Mathématique 4 (46), 2 (1938)].

Despite its significance in neuroscience and computation, McCulloch and Pitts's celebrated 1943 paper has received little historical and philosophical attention. In 1943 there already existed a lively community of biophysicists doing mathematical work on neural networks. What was novel in McCulloch and Pitts's paper was their use of logic and computation to understand neural, and thus mental, activity. McCulloch and Pitts's contributions included (i) a formalism whose refinement and generalization led to the notion of finite automata (an important formalism in (...) computability theory), (ii) a technique that inspired the notion of logic design (a fundamental part of modern computer design), (iii) the first use of computation to address the mind–body problem, and (iv) the first modern computational theory of mind and brain. (shrink)

We provide a logical matrix semantics and a Gentzen-style sequent calculus for the first-degree entailments valid in W. T. Parry’s logic of Analytic Implication. We achieve the former by introducing a logical matrix closely related to that inducing paracomplete weak Kleene logic, and the latter by presenting a calculus where the initial sequents and the left and right rules for negation are subject to linguistic constraints.

Dieser Aufsatz zeigt die Möglichkeiten auf, aus den von Leibniz im April 1679 geschriebenen logischen Abhandlungen ein numerisches Modell für die durch Widerspruchsfreiheit bestimmten Modalbegriffe zu lesen. Dazu wird als erstes die Definition der Modalisatoren durch die Widerspruchsfreiheit betrachtet und die Probleme‚ die das Leibniz'sche ‘continere’ oder ‘implicare contradictionem’ auslöst. Danach werden die in den Schriften von 1679 entwickelten numerischen Modelle untersucht‚ besonders dasjenige‚ das sich auf den Mechanismus des charakteristischen Zahlenpaares stützt. Dieses Modell wird für den Aufbau formaler Definitionen (...) von Möglichkeit‚ Unmöglichkeit‚ Kompatibilität und Inkompatibilität benutzt. Schließlich werden kurz einige spezifische Probleme angeführt‚ die die vorgeschlagene Interpretation offenläßt: die Darstellung der einfachen Begriffe innerhalb des numerischen Kalküls und die Beziehung zwischen den Wortpaaren Kompatibilität-Inkompatibilität und Kompossibilität-Inkompossibilität. Was diese letzteren betrifft‚ wird die Bedeutung betont‚ die eine diachronische Betrachtung der Leibnizschen Lehre für ein besseres Verständnis der theoretischen Zusammenhänge hat‚ in denen diese Bezeichnungen auftreten. (shrink)

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The purpose of this paper is to introduce a bi-intuitionistic sequent calculus and to give proofs of admissibility for its structural rules. The calculus I will present, called SC2Int, is a sequent calculus for the bi-intuitionistic logic 2Int, which Wansing presents in [2016a]. There he also gives a natural deduction system for this logic, N2Int, to which SC2Int is equivalent in terms of what is derivable. What is important is that these calculi represent a kind of bilateralist (...) reasoning, since they do not only internalize processes of verifcation or provability but also the dual processes in terms of falsifcation or what is called dual provability. In [Wansing, 2017] a normal form theorem for N2Int is stated, here, I want to prove a cut-elimination theorem for SC2Int, i.e., if successful, this would extend the results existing so far. (shrink)

Expressivism in logic is the view that logical vocabulary plays a primarily expressive role: that is, that logical vocabulary makes perspicuous in the object language structural features of inference and incompatibility (Brandom, 1994, 2008). I present a precise, technical criterion of expressivity for a logic (§2). I next present a logic that meets that criterion (§3). I further explore some interesting features of that logic: first, a representation theorem for capturing other logics (§3.1), and next some novel (...) class='Hi'>logical vocabulary for expressing structural features of inference (§4). (shrink)

In this paper we introduce a Gentzen calculus for (a functionally complete variant of) Belnap's logic in which establishing the provability of a sequent in general requires \emph{two} proof trees, one establishing that whenever all premises are true some conclusion is true and one that guarantees the falsity of at least one premise if all conclusions are false. The calculus can also be put to use in proving that one statement \emph{necessarily approximates} another, where necessary approximation is a (...) natural dual of entailment. The calculus, and its tableau variant, not only capture the classical connectives, but also the `information' connectives of four-valued Belnap logics. This answers a question by Avron. (shrink)