Leibnizian-Newtonian calculus was a theory that dealt with geometrical objects; the figure continued to play one of the fundamental roles it had played in Greek geometry: it susbstituted a part of reasoning. During the eighteenth century a process of de-geometrization of calculus took place, which consisted in the rejection of the use of diagrams and in considering calculus as an 'intellectual' system where deduction was merely linguistic and mediated. This was achieved by interpreting variables as universal quantities (...) and introducing the notion of function, which replaced the study of curves. However, the emancipation of calculus from its basis in geometry was not comprehensive. In fact, the geometrical properties of curves were attributed de facto to functions and thus eighteenth-century calculus continued implicitly to use principles borrowed from geometry. There was therefore no transition to a purely syntactical theory based on axiomatically introduced terms, a shift which only took place subsequently in modern times. (shrink)

As with mathematics, logic is easier to do if its symbols and their rules are better. In a graphic way, the logic symbols introduced in thís paper show their truth-table values, their composite truth-functions, and how to say them as either ?or? or ?if ? then? propositions. Simple rules make the converse, add or remove negations, and resolve propositions.

We investigate incomplete symbols, i.e. definite descriptions with scope-operators. Russell famously introduced definite descriptions by contextual definitions; in this article definite descriptions are introduced by rules in a specific calculus that is very well suited for proof-theoretic investigations. That is to say, the phrase ‘incomplete symbols’ is formally interpreted as to the existence of an elimination procedure. The last section offers semantical tools for interpreting the phrase ‘no meaning in isolation’ in a formal way.

The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal (...) class='Hi'>calculus. It was also central for the rigorization movement in mathematics in the late nineteenth century, as well as for the mathematics of modern physics in the 20th century.However, the nature of symbolic mathematics has been concealed and confused due to the strong influence of the heritage from the Euclidean and Aristotelian traditions. This essay sheds some light on what has been concealed by approaching some of the crucial issues from a historical perspective. Furthermore, I argue that the conception of modern mathematics as symbolic mathematics was essential to Wittgenstein’s approach to the foundations and nature of mathematics. This connection between Wittgenstein’s thought and symbolic mathematics provides the resources for countering the still prevalent view that he defended an uttrely idiosyncratic conception, disconnected from the progress of serious science. Instead, his project can be seen as clarifying ideas that have been crucial to the development of mathematics since early modernity. (shrink)

Among the more important changes in this revised edition: the incompleteness of the first set of natural deduction rules is proved; many proofs are shortened and simplified, especially in the development of the first-order functional calculus; there is a more lucid exposition of the quantification rules; more exercises are provided, with answers given for a number of them. The changes are all improvements, but none of them are of a sufficiently radical nature to be likely to alter anyone's original (...) opinion of Copi's book—A. E. J. (shrink)

In this research I will focus on a basis for a formal model based on an alternative kind of logic invented by Hans Götzsche: Occurrence Logic (Occ Log), which is not based on truth values and truth functionality. Also, I have taken into account tense logic developed and elaborated by A. N. Prior. In this article I will provide a conceptual and logical foundation for formal Occurrence Logic based on symbolic logic and will illustrate the most important relations between (...) symbolic logic and Occurrence Logic. There will be a special need to focus on arguments based on symbolic occurrence logic. The interrelationships and dependencies between symbolic logic and Occ Log could be conducive to an approach that can express and analyse ‘the occurrences of the world’s descriptions based on various logical principles at different moments’. Symbolic Occ Log checks the occurrence conditioning and the priority of occurrences by employing occurrence values. I shall conclude that the desired approach could gradually support me in providing a truth-functional independent first-order predicate calculus that could work on semantic analysis of different propositions within world descriptions. (shrink)

The canonical history of mathematics suggests that the late 19th-century “arithmetization” of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving a (...) theorem, which reveals a reliance on dynamic conceptual resources. The second is a cognitive-historical case study of an incident in 19th-century mathematics that suggests a functional role for such dynamism in the reasoning of the renowned mathematician Augustin Cauchy. Taken together, these two studies indicate that essential concepts in calculus that have been defined entirely in abstract, static terms are nevertheless conceptualized dynamically, in both contemporary and historical practice. (shrink)

ABSTRACT In a recent essay, Sören Stenlund tries to align Wittgenstein’s approach to the foundations and nature of mathematics with the tradition of symbolic mathematics. The characterization of symbolic mathematics made by Stenlund, according to which mathematics is logically separated from its external applications, brings it closer to the formalist position. This raises naturally the question whether Wittgenstein holds a formalist position in philosophy of mathematics. The aim of this paper is to give a negative answer to this (...) question, defending the view that Wittgenstein always thought that there is no logical separation between mathematics and its applications. I will focus on Wittgenstein’s remarks about arithmetic during his middle period, because it is in this period that a formalist reading of his writings is most tempting. I will show how his idea of autonomy of arithmetic is not to be compared with the formalist idea of autonomy, according to which a calculus is “cut off” from its applications. The autonomy of arithmetic, according to Wittgenstein, guarantees its own applicability, thus providing its own raison d’être. RESUMO Em um recente artigo, Sören Stenlund procura alinhar a abordagem de Wittgenstein em relação aos fundamentos e à natureza da matemática com a tradição da matemática simbólica. A caracterização da matemática simbólica feita por Stenlund, de acordo com a qual a matemática é logicamente separada de suas aplicações externas, a aproxima da posição formalista. Isto naturalmente levanta a questão de se Wittgenstein defende uma posição formalista em filosofia da matemática. O objetivo deste artigo é dar uma resposta negativa a esta questão, ao defender que Wittgenstein sempre pensou não haver separação lógica entre a matemática e suas aplicações. Atenção especial será dada às observações de Wittgenstein sobre a aritmética pertencentes ao seu período intermediário, pois é neste período que uma leitura formalista de seus escritos é mais sedutora. Mostrarei como sua ideia de autonomia da aritmética não deve ser comparada à ideia formalista de autonomia, segundo a qual um cálculo é “cortado” de suas aplicações. A autonomia da aritmética, para Wittgenstein, garante ela própria sua aplicabilidade, provendo, assim, sua própria raison d’être. (shrink)

One might intuitively expect that logic would lend itself to programmed teaching. This text shows that it does. The authors have provided a carefully worked out program for the propositional calculus. Considerable emphasis is placed on the intuitive plausibility of moves. The student is first introduced to Principia Mathematica notation, then shown the advantages of Polish, which is used throughout the remainder of the text. The program includes techniques for discovering proofs and a thorough discussion of validity.—R. J. W.

We present a sequent calculus of a paraconsistent logic QMPT0, which has the paraconsistent-type excluded middle law (PEML) as an initial sequent. Our system shows that the presence of PEML is essentially important for QMPT0. It also has special rules when the set of constant symbols is finite. We also discuss the cut-elimination property of our system.

Cognitive technology is an increasingly important form of technology that can deal with meaning by either replicating or simulating human cognition. Cognitive technology can make use of information technology, but it strives to go beyond mere information processing by recognizing, changing, and creating meaning. This presents us with a two-sided challenge: On the one hand, cognitive technology is challenged to ‘understand’ meaning in ordinary language. And on the other, it challenges us to rethink fundamental questions of human cognition and sense-making. (...) Both challenges demand a better understanding of the difference between the technical transformation of symbols and the understanding of meaning in the ordinary sense. After explaining the topic in relation to both the insights and the limitations of the reflections by Turing, Searle, and Heidegger, this paper primarily builds on Wittgenstein’s contributions to a better understanding of the difference between two conceptions of meaning and their implications for technical replication and simulation. The paper shows that Wittgenstein developed his early calculus account of meaning into that of language games and that language games not only come in many different varieties, but are also much more flexible than calculi. Of particular interest will be the difference between rigid and creative rule-following. Creative rule-following involves an intricate interplay of very different bodily, mental, and cultural constituents, so that its simulation is not merely a technical problem but also requires clarification of a number of profound philosophical questions. It will become clear that the challenge of cognitive technology shows up at unexpected places and that is much bigger than usually assumed. (shrink)

A sequent calculus is given in which the management of weakening and contraction is organized as in natural deduction. The latter has no explicit weakening or contraction, but vacuous and multiple discharges in rules that discharge assumptions. A comparison to natural deduction is given through translation of derivations between the two systems. It is proved that if a cut formula is never principal in a derivation leading to the right premiss of cut, it is a subformula of the conclusion. (...) Therefore it is sufficient to eliminate those cuts that correspond to detour and permutation conversions in natural deduction. (shrink)

Gottfried Leibniz embarked on a research program to prove all the Aristotelic categorical syllogisms by diagrammatic and algebraic methods. He succeeded in proving them by means of Euler diagrams, but didn’t produce a manuscript with their algebraic proofs. We demonstrate how key excerpts scattered across various Leibniz’s drafts on logic contained sufficient ingredients to prove them by an algebraic method –which we call the Leibniz-Cayley (LC) system– without having to make use of the more expressive and complex machinery of first-order (...) quantificational logic. In addition, we prove the classic categorical syllogisms again by a relational method –which we call the McColl-Ladd (ML) system– employing categorical relations studied by Hugh McColl and Christine Ladd. Finally, we show the connection of ML and LC with Boolean algebra, proving that ML is a consequence of LC, and that LC is a consequence of the Boolean lattice axioms, thus establishing Leibniz’s historical priority over George Boole in characterizing and applying (a sufficient fragment of) Boolean algebra to effectively tackle categorical syllogistic. (shrink)

A new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀ x∀ yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.

The λμ-calculus is an extension of the λ-calculus that has been introduced by M Parigot to give an algorithmic content to classical proofs. We show that Bohm's theorem fails in this calculus.

"We detail Abramsky's 'proofs-as-processes" paradigm for interpreting classical linear logic (CCL) [11] into a 'synchronous' version of the [pi]-calculus recently proposed by Milner [24]. The translation is given at the abstract level of proof structures. We give a detailed treatment of information flow in proof-nets and show how to mirror various evaluation strategies for proof normalization. We also give Soundness and Completeness results for the process-calculus translations of various fragments of CLL. The paper also gives a self-contained introduction (...) to some of the deeper proof-theory of CLL, and its process interpretation.". (shrink)

A revised edition of the author's 1948 introductory text, the present version differs only in minor points from the original work. Truth functions are discussed in some detail, and a propositional calculus derived from the five axioms of Principia Mathematica is presented. Quantification, however, is less extensively treated. The "traditional Aristotelian" interpretation of categorical propositions is contrasted with the non-existential interpretation of modern logic; a theory of the syllogism is offered; and various methods for evaluating the validity of syllogisms (...) are given. The last section treats of E. V. Huntington's formulation of the- class calculus. The work is designedly an introductory text and omits discussion of the theory of types, any metatheoretic considerations of formal systems, and any consideration of the relation of logic to mathematics. There are exercises at the end of each chapter.--J. D. (shrink)

In his main work Summa Logicae written around 1323, William of Ockham developed a system of propositional modal logic which contains almost all theorems of a modern calculus of strict implication. This calculus is formally reconstructed here with the help of modern symbols for the operators of conjunction, disjunction, implication, negation, possibility, and necessity.

After the publication of Begriffsschrift, a conflict erupted between Frege and Schröder regarding their respective logical systems which emerged around the Leibnizian notions of lingua characterica and calculus ratiocinator. Both of them claimed their own logic to be a better realisation of Leibniz’s ideal language and considered the rival system a mere calculus ratiocinator. Inspired by this polemic, van Heijenoort (1967b) distinguished two conceptions of logic—logic as language and logic as calculus—and presented them as opposing views, but (...) did not explain Frege’s and Schröder’s conceptions of the fulfilment of Leibniz’s scientific ideal. -/- In this paper I explain the reasons for Frege’s and Schröder’s mutual accusations of having created a mere calculus ratiocinator. On the one hand, Schröder’s construction of the algebra of relatives fits with a project for the reduction of any mathematical concept to the notion of relative. From this stance I argue that he deemed the formal system of Begriffsschrift incapable of such a reduction. On the other hand, first I argue that Frege took Boolean logic to be an abstract logical theory inadequate for the rendering of specific content; then I claim that the language of Begriffsschrift did not constitute a complete lingua characterica by itself, more being seen by Frege as a tool that could be applied to scientific disciplines. Accordingly, I argue that Frege’s project of constructing a lingua characterica was not tied to his later logicist programme. (shrink)

R. G. Collingwood saw one of the main tasks of philosophers and of historians of human thought in uncovering what he called the ultimate presuppositions of different thinkers, of different philosophical movements and of entire eras of intellectual history. He also noted that such ultimate presuppositions usually remain tacit at first, and are discovered only by subsequent reflection. Collingwood would have been delighted by the contrast that constitutes the overall theme of the essays collected in this volume. Not only has (...) this dichotomy ofviews been one ofthe mostcrucial watersheds in the entire twentieth-century philosophical thought. Not only has it remained largely implicit in the writings of the philosophers for whom it mattered most. It is a truly Collingwoodian presupposition also in that it is not apremise assumed by different thinkers in their argumentation. It is the presupposition of a question, an assumption to the effect that a certain general question can be raised and answered. Its role is not belied by the fact that several philosophers who answered it one way or the other seem to be largely unaware that the other answer also makes sense - if it does. This Collingwoodian question can be formulated in a first rough approximation by asking whether language - our actual working language, Tarski's "colloquiallanguage" - is universal in the sense of being inescapable. This formulation needs all sorts of explanations, however. (shrink)

LK is much more difficult than NK, and to make matters worse, Gentzen's intention is still unclear when it comes to that system (LK). -/- After long surveys conducted from Vol.1 to Vol.4, all of which shares the common title What is LK?, this fifth volume finally handles the solved problems in the realm of propositional logic in LK.

Most accounts, including leading textbooks, credit Arthur Norman Prior with the invention of temporal (tense logic). However, (i) Jerzy Łoś delivered his version of temporal logic in 1947, several years before Prior; (ii) Henrk Hiż’s review of Łoś’s system in Journal of Symbolic Logic was published as early as 1951; (iii) there is evidence to the effect that, when constructing his tense calculi, Prior was aware of Łoś’s system. Therefore, although Prior is certainly a key figure in the history (...) tense logic, as well as modal logic in general, it should be accepted both in the literature that temporal logic was invented by Jerzy Łoś. (shrink)

The Quantified argument calculus (Quarc) has received a lot of attention recently as an interesting system of quantified logic which eschews the use of variables and unrestricted quantification, but nonetheless achieves results similar to the Predicate calculus (PC) by employing quantifiers applied directly to predicates instead. Despite this noted similarity, the issue of the relationship between Quarc and PC has so far not been definitively resolved. We address this question in the present paper, and then expand upon that (...) result. Utilizing recent developments in structural proof theory, we develop a G3-style sequent calculus for Quarc and briefly demonstrate its structural properties. We put these properties to use immediately to construct direct proofs of the meta-theoretical properties of the system. We then incorporate an abstract (and, as we shall see, logical) predicate into the system in a way that preserves all the structural properties. This allows us to identify a system of Quarc which is deductively equivalent to PC, and also yields a constructive method of demonstrating the Craig interpolation theorem (which speaks in favor of the aforementioned predicate being logical). We further generalize this extension to develop a bivalent system of Quarc with defining clauses that still maintains all the desirable properties of a good proof system. (shrink)

I develop a formal logic in which quantified arguments occur in argument positions of predicates. This logic also incorporates negative predication, anaphora and converse relation terms, namely, additional syntactic features of natural language. In these and additional respects, it represents the logic of natural language more adequately than does any version of Frege’s Predicate Calculus. I first introduce the system’s main ideas and familiarize it by means of translations of natural language sentences. I then develop a formal system built (...) on these principles, the Quantified Argument Calculus or Quarc. I provide a truth-value assignment semantics and a proof system for the Quarc. I next demonstrate the system’s power by a variety of proofs; I prove its soundness; and I comment on its completeness. I then extend the system to modal logic, again providing a proof system and a truth-value assignment semantics. I proceed to show how the Quarc versions of the Barcan formulas, of their converses and of necessary existence come out straightforwardly invalid, which I argue is an advantage of the modal Quarc over modal Predicate Logic as a system intended to capture the logic of natural language. (shrink)