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)

In recent years, the e ffort to formalize erotetic inferences (i.e., inferences to and from questions) has become a central concern for those working in erotetic logic. However, few have sought to formulate a proof theory for these inferences. To fill this lacuna, we construct a calculus for (classes of) sequents that are sound and complete for two species of erotetic inferences studied by Inferential Erotetic Logic (IEL): erotetic evocation and regular erotetic implication. While an attempt has been made (...) to axiomatize the former in a sequent system, there is currently no proof theory for the latter. Moreover, the extant axiomatization of erotetic evocation fails to capture its defeasible character and provides no rules for introducing or eliminating question-forming operators. In contrast, our calculus encodes defeasibility conditions on sequents and provides rules governing the introduction and elimination of erotetic formulas. We demonstrate that an elimination theorem holds for a version of the cut rule that applies to both declarative and erotetic formulas and that the rules for the axiomatic account of question evocation in IEL are admissible in our system. (shrink)

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. (...) Next, the geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction. (shrink)

This paper aims to address the difficulties of high school students in bridging their computational understanding with their visualization skills in understanding the notion of the limits in their calculus class. This research used a pre-experimental one-group pretest-posttest design research on 62 grade 10 students enrolled in the Science, Technology, and Engineering Program (STEP) in one of the public high schools in Zamboanga del Sur, Philippines. A series of remedial sessions were given to help them understand the function values, (...) one-sided limits, limits of a function, and its discontinuities with the aid of the GeoGebra. Results revealed that there is a significant improvement in their mathematics performance before and after the remediation, t(61) = -19.99, p<.05. The real-time feedback provided by the drawing interface of GeoGebra has provided the students the insights into how objects behave in geometrical space, bridging their understanding in locating the values shown by the computational algebraic processes. Additionally, the significant difference in the achievement reveals a large effect size (Cohen’s =2.54) which could be accounted for by the remediation using interactive activities in the GeoGebra. (shrink)

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)

The basic idea is to put qualia into equations (broadly understood) to get what might as well be called qualations. Qualations arguably have different truth behaviors than the analogous equations. Thus ‘black’ has a different behavior than ‘ █ ’. This is a step in the direction of a ‘calculus of qualia’. It might help clarify some issues.

The idea of this paper is to put actual qualia into equations (broadly understood) to get what might be called qualations. Qualations arguably have different meanings and truth behaviors than the analogous equations. For example, the term ‘ black ’ arguably has a different meaning and behavior than the term ‘ █ ’. This is a step in the direction of a ‘calculus of qualia’ and of expanding science to include 1st-person phenomena.

This paper examines systematically which features of a life story (or history) make it good for the subject herself - not aesthetically or morally good, but prudentially good. The tentative narrative calculus presented claims that the prudential narrative value of an event is a function of the extent to which it contributes to her concurrent and non-concurrent goals, the value of those goals, and the degree to which success in reaching the goals is deserved in virtue of exercising agency. (...) The narrative value of a life is a simple sum of the values of individual events that comprise it. I claim that this view best explains and support common intuitions about the significance of the shape of a life. (shrink)

Building on the work of Peter Hinst and Geo Siegwart, we develop a pragmatised natural deduction calculus, i.e. a natural deduction calculus that incorporates illocutionary operators at the formal level, and prove its adequacy. In contrast to other linear calculi of natural deduction, derivations in this calculus are sequences of object-language sentences which do not require graphical or other means of commentary in order to keep track of assumptions or to indicate subproofs. (Translation of our German paper (...) "Ein Redehandlungskalkül. Ein pragmatisierter Kalkül des natürlichen Schließens nebst Metatheorie"; online available at http://philpapers.org/rec/CORERE.). (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)

In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic, an interesting variation upon the four-valued logic for first-degree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbert-style axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we will show that a (...) calculus for the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multiple-tree character—two proof trees may be needed to establish one proof. (shrink)

I am presenting a sequent calculus that extends a nonmonotonic consequence relation over an atomic language to a logically complex language. The system is in line with two guiding philosophical ideas: (i) logical inferentialism and (ii) logical expressivism. The extension defined by the sequent rules is conservative. The conditional tracks the consequence relation and negation tracks incoherence. Besides the ordinary propositional connectives, the sequent calculus introduces a new kind of modal operator that marks implications that hold monotonically. Transitivity (...) fails, but for good reasons. Intuitionism and classical logic can easily be recovered from the system. (shrink)

This essay provides a novel account of iterated epistemic states. The essay argues that states of epistemic determinacy might be secured by countenancing iterated epistemic states on the model of fixed points in the modal $\mu$-calculus. Despite the epistemic indeterminacy witnessed by the invalidation of modal axiom 4 in the sorites paradox -- i.e. the KK principle: $\square$$\phi$ $\rightarrow$ $\square$$\square$$\phi$ -- a hyperintensional epistemic $\mu$-automaton permits fixed points to entrain a principled means by which to iterate epistemic states and (...) account thereby for necessary conditions on self-knowledge. The hyperintensional epistemic $\mu$-calculus is applied to the iteration of the epistemic states of a single agent instead of the common knowledge of a group of agents, and is thus a novel contribution to the literature. (shrink)

The purpose of this paper is to outline an alternative approach to introductory logic courses. Traditional logic courses usually focus on the method of natural deduction or introduce predicate calculus as a system. These approaches complicate the process of learning different techniques for dealing with categorical and hypothetical syllogisms such as alternate notations or alternate forms of analyzing syllogisms. The author's approach takes up observations made by Dijkstrata and assimilates them into a reasoning process based on modified notations. The (...) author's model adopts a notation that addresses the essentials of a problem while remaining easily manipulated to serve other analytic frameworks. The author also discusses the pedagogical benefits of incorporating the model into introductory logic classes for topics ranging from syllogisms to predicate calculus. Since this method emphasizes the development of a clear and manipulable notation, students can worry less about issues of translation, can spend more energy solving problems in the terms in which they are expressed, and are better able to think in abstract terms. (shrink)

Hilbert's ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand (...) disjunctions of existential theorems obtained by this elimination procedure. (shrink)

It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church's “lambda calculus” for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903–1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the lambda calculus. Russell also anticipated Schönfinkel's combinatory logic approach of treating multiargument functions as functions having other functions (...) as value. Russell’s work in this regard seems to have been largely inspired by Frege’s theory of functions and “value-ranges”. This system was discarded by Russell due to his abandonment of propositional functions as genuine entities as part of a new tack for solving Russell’s paradox. In this article, I explore the genesis and demise of Russell’s early anticipation of the lambda calculus. (shrink)

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian (...) infinitesimal calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

It is fair to say that Georg Wilhelm Friedrich Hegel's philosophy of mathematics and his interpretation of the calculus in particular have not been popular topics of conversation since the early part of the twentieth century. Changes in mathematics in the late nineteenth century, the new set-theoretical approach to understanding its foundations, and the rise of a sympathetic philosophical logic have all conspired to give prior philosophies of mathematics (including Hegel's) the untimely appearance of naïveté. The common view was (...) expressed by Bertrand Russell: -/- The great [mathematicians] of the seventeenth and eighteenth centuries were so much impressed by the results of their new methods that they did not trouble to examine their foundations. Although their arguments were fallacious, a special Providence saw to it that their conclusions were more or less true. Hegel fastened upon the obscurities in the foundations of mathematics, turned them into dialectical contradictions, and resolved them by nonsensical syntheses. . . .The resulting puzzles [of mathematics] were all cleared up during the nineteenth century, not by heroic philosophical doctrines such as that of Kant or that of Hegel, but by patient attention to detail (1956, 368–69). (shrink)

A new proof style adequate for modal logics is defined from the polynomial ring calculus. The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra???Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S 5, and can be easily extended to other (...) modal logics. (shrink)

All first-order Gödel logics G_V with globalization operator based on truth value sets V C [0,1] where 0 and 1 lie in the perfect kernel of V are axiomatized by Ciabattoni’s hypersequent calculus HGIF.

Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this (...) problem by applying recent methods from sorted logics. This paper presents a tableau calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi. (shrink)

Drug regulation is fraught with inductive risk. Regulators must make a prediction about whether or not an experimental pharmaceutical will be effective and relatively safe when used by typical patients, and such predictions are based on a complex, indeterminate, and incomplete evidential basis. Such inductive risk has important practical consequences. If regulators reject an experimental drug when it in fact has a favourable benefit/harm profile, then a valuable intervention is denied to the public and a company’s material interests are needlessly (...) thwarted. Conversely, if regulators approve an experimental drug when it in fact has an unfavourable benefit/harm profile, then resources are wasted, people are needlessly harmed, and other potentially more effective treatments are underutilized. Given that such regulatory decisions have these practical consequences, non-epistemic values about the relative importance of these consequences impact the way such regulatory decisions are made (similar to the analysis of laboratory studies on the toxic effects of dioxins presented in Douglas [2000]). To balance the competing demands of the pertinent non-epistemic values, regulators must perform what I call an “inductive risk calculus.” At least in the American context this inductive risk calculus is not well-managed. (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 logical vocabulary for (...) expressing structural features of inference (§4). (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.

Application 1. The case against Materialism and Illusionism Application 2. Ineffability Application 3. Hard Problems Application 4. Knowledge Argument questions Application 5. Argument for A-theories of time Application 6. Possible qualia are necessary.

The deductive validity of arguments from analogy is formally demonstrable. After a brief survey of the historical development of doctrines relevant to this claim the present article analyzes the “analogy of proper proportionality”, which meets two requirements of valid deduction. First, the referents of analogues by proportionality must belong to a common genus. Here it must be cautioned, however, that the common genus does not constitute the basis of the deductive inference. Rather, it is a prerequisite for the second and (...) decisive requirement, that the different logical content to which an analogous middle term corresponds must exhibit the same proportional relation to this common genus. In Section II I translate a natural language argument with such an analogous middle term into the language of classical first-order predicate calculus, and show that its conclusion follows from its premises with the force of deductive necessity. The rule of inference justifying the analogical entailment of the conclusion from the premises functions much like the familiar modus ponens rule, and could be called “modus ponens analogice”. The validity of this rule ought to be judged on the same basis as that of the traditional modus ponens rule – immediate logical intuition. The present article endeavors to facilitate this logical intuition by making the logical structure of inference by analogy formally explicit. (shrink)

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The (...) previous attempts all required the brackets to take values in ℤ₂. But the usual QM brackets <ψ|ϕ> give the "overlap" between states ψ and ϕ, so for subsets S,T⊆U, the natural definition is <S|T>=|S∩T| (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell's Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over ℂ and QM/Sets over ℤ₂. (shrink)

Leibniz proposed the ‘Most Determined Path Principle’ in seventeenth century. According to it, ‘ease’ of travel is the end purpose of motion. Using this principle and his calculus method he demonstrated Snell’s Laws of reflection and refraction. This method shows that light follows extremal (local minimum or maximum) time path in going from one point to another, either directly along a straight line path or along a broken line path when it undergoes reflection or refraction at plane or spherical (...) (concave or convex) surfaces. The extremal time path avoided the criticism that Fermat’s least time path was subjected to, by Cartesians who cited examples of reflections at spherical surfaces where light took the path of longest time. Thereby it became the standard method of demonstration of Snell’s Laws. Ptolemy’s theorem is a fundamental theorem in geometry. A special case of it offers a method of finding the minimum sum of the two distances of a point from two given fixed points. We show in this paper that Leibniz’s calculus proof of Snell’s Laws violates Ptolemy’s theorem, whereby Leibniz’s proof becomes invalid. (shrink)

This article identifies and formalizes the logical features of analogous terms that justify their use in deduction. After a survey of doctrines in Aristotle, Aquinas, and Cajetan, the criteria of “analogy of proper proportionality” are symbolized in first-order predicate logic. A common genus justifies use of a common term, but does not provide the inferential link required for deduction. Rather, the respective differentiae foster this link through their identical proportion. A natural-language argument by analogy is formalized so as to exhibit (...) these criteria, thereby showing the validity of analogical deduction. (shrink)

The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.

Vector models of language are based on the contextual aspects of language, the distributions of words and how they co-occur in text. Truth conditional models focus on the logical aspects of language, compositional properties of words and how they compose to form sentences. In the truth conditional approach, the denotation of a sentence determines its truth conditions, which can be taken to be a truth value, a set of possible worlds, a context change potential, or similar. In the vector models, (...) the degree of co-occurrence of words in context determines how similar the meanings of words are. In this paper, we put these two models together and develop a vector semantics for language based on the simply typed lambda calculus models of natural language. We provide two types of vector semantics: a static one that uses techniques familiar from the truth conditional tradition and a dynamic one based on a form of dynamic interpretation inspired by Heim’s context change potentials. We show how the dynamic model can be applied to entailment between a corpus and a sentence and provide examples. (shrink)

We extend Barbanera and Berardi's symmetric lambda calculus [2] to second-order classical propositional logic and prove its strong normalization.

This study aimed to determine the relationship between academic performance and students’ perceptions in learning Calculus during distance learning modality. Learning Calculus is part of developing students' Mathematics skills and abilities towards enhancing STEM education in Senior High School. Thirty-five (35) STEM students were purposively sampled at a public school in Davao Region, Philippines. These students took up Pre-Calculus and Basic Calculus during pandemic. This study used a quantitative research design, particularly the descriptive-correctional method, to analyze (...) the collected data. Based on the result, the level of students’ perceptions was positive, while the level of academic performance was very satisfactory. Moreover, no significant correlation exists between students’ perceptions and academic performance in learning calculus on distance learning. This implies that students can perform well regardless of their perceptions in learning Calculus, in both Pre- Calculus and Basic Calculus. However, further assessment needs to be done in their Mathematics subjects at the Tertiary level since the modality used was not face-to-face. Appropriate action may be considered based on the assessment result, such as bridging courses or tutorial sessions. (shrink)

In this paper I will develop a lambda-term calculus, lambda-2Int, for a bi-intuitionistic logic and discuss its implications for the notions of sense and denotation of derivations in a bilateralist setting. Thus, I will use the Curry-Howard correspondence, which has been well-established between the simply typed lambda-calculus and natural deduction systems for intuitionistic logic, and apply it to a bilateralist proof system displaying two derivability relations, one for proving and one for refuting. The basis will be the natural (...) deduction system of Wansing's bi-intuitionistic logic 2Int, which I will turn into a term-annotated form. Therefore, we need a type theory that extends to a two-sorted typed lambda-calculus. I will present such a term-annotated proof system for 2Int and prove a Dualization Theorem relating proofs and refutations in this system. On the basis of these formal results I will argue that this gives us interesting insights into questions about sense and denotation as well as synonymy and identity of proofs from a bilateralist point of view. (shrink)

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or "toy" model of quantum mechanics over sets (QM/sets). There are two parts. The notion of an "event" is reinterpreted from being an epistemological state of indefiniteness to being an objective state of indefiniteness. And the mathematical framework of finite probability theory is recast as the quantum probability calculus for QM/sets. The point is (...) not to clarify finite probability theory but to elucidate quantum mechanics itself by seeing some of its quantum features in a classical setting. (shrink)

The aim of this paper is to explore the uses made of the calculus by Gilles Deleuze and G. W. F. Hegel. I show how both Deleuze and Hegel see the calculus as providing a way of thinking outside of finite representation. For Hegel, this involves attempting to show that the foundations of the calculus cannot be thought by the finite understanding, and necessitate a move to the standpoint of infinite reason. I analyse Hegel’s justification for this (...) introduction of dialectical reason by looking at his responses to Berkeley’s criticisms of the calculus. For Deleuze, instead, I show that the differential must be understood as escaping from both finite and infinite representation. By highlighting the sub-representational character of the differential in his system, I show how the differential is a key moment in Deleuze’s formulation of a transcendental empiricism. I conclude by dealing with some of the common misunderstandings that occur when Deleuze is read as endorsing a modern mathematical interpretation of the calculus. (shrink)

A simple interpretation of quantity calculus is given. Quantities are described as two-place functions from objects, states or processes (or some combination of them) into numbers that satisfy the mutual measurability property. Quantity calculus is based on a notational simplification of the concept of quantity. A key element of the simplification is that we consider units to be intentionally unspecified numbers that are measures of exactly specified objects, states or processes. This interpretation of quantity calculus combines all (...) the advantages of calculating with numerical values (since the values of quantities are numbers, we can do with them everything we do with numbers) and all the advantages of calculating with standardly conceived quantities (calculus is invariant to the choice of units and has built-in dimensional analysis). This also shows that the standard metaphysics and mathematics of quantities and their magnitudes is not needed for quantity calculus. At the end of the article, arguments are given that the concept of quantity as defined here is a pivotal concept in understanding the quantitative approach to nature. As an application of this interpretation of quantity calculus, an easy proof of dimensional homogeneity of physical laws is given. (shrink)

1. First section notes: subtraction -/- Evidently -/- (1) black – black = empty-set -/- because anything minus itself is the empty set, but -/- (2) █ – █ = █ + █ -/- because there are two instances of black on the left and two instances of black on the right...

The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses recently introduced infinite and infinitesimal numbers being in accordance with the principle ‘The part is less than the whole’ observed in the physical world around us. These numbers have a strong practical advantage with respect to traditional approaches: they are representable at a new (...) kind of a computer – the Infinity Computer – able to work numerically with all of them. An introduction to the theory of physical and mathematical continuity and differentiation (including subdifferentials) for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains is developed in the paper. This theory allows one to work with derivatives that can assume not only finite but infinite and infinitesimal values, as well. It is emphasized that the newly introduced notion of the physical continuity allows one to see the same mathematical object as a continuous or a discrete one, in dependence on the wish of the researcher, i.e., as it happens in the physical world where the same object can be viewed as a continuous or a discrete in dependence on the instrument of the observation used by the researcher. Connections between pure mathematical concepts and their computational realizations are continuously emphasized through the text. Numerous examples are given. (shrink)

It could be argued that -/- (1) red is not green -/- is contingently true, but -/- (2) █_1 is not █_2 -/- is necessarily true.

Resolving the Ross and Prior paradoxes proved to be a difficult task. Its first two stages, involving the identification of the true natures of the implication and truth values, are described in the articles "Solving the Paradox of Material Implication – 2024" and "Solving Jörgensen's Dilemma – 2024". This article describes the third stage, which involves the discovery of missing modal operators and the Classical Modal Calculus. Finally, procedures for solving both paradoxes are provided.

We give an argument for temporal A-theories based on the calculus of qualia.

We develop some of Williamson’s ideas regarding how propositional calculus aids in comprehending Aristotelian logic. Specifically, we enhance the utilisation of truth tables to examine the structure of opposition diagrams. Using ‘conditioned truth tables’, we establish logical dependency relationships between the truth values of different propositions. This approach proves effective in interpreting various texts of the Organon concerning the doctrine of opposition.

This book presents not only the mathematical concept of probability, but also its philosophical aspects, the relativity of probability and its applications and even the psychology of probability. All explanations are made in a comprehensible manner and are supported with suggestive examples from nature and daily life, and even with challenging math paradoxes.

In Hegel ou Spinoza,1 Pierre Macherey challenges the influence of Hegel’s reading of Spinoza by stressing the degree to which Spinoza eludes the grasp of the Hegelian dialectical progression of the history of philosophy. He argues that Hegel provides a defensive misreading of Spinoza, and that he had to “misread him” in order to maintain his subjective idealism. The suggestion being that Spinoza’s philosophy represents, not a moment that can simply be sublated and subsumed within the dialectical progression of the (...) history of philosophy, but rather an alternative point of view for the development of a philosophy that overcomes Hegelian idealism. Gilles Deleuze also considers Spinoza’s philosophy to resist the totalising effects of the dialectic. Indeed, Deleuze demonstrates, by means of Spinoza, that a more complex philosophy antedates Hegel’s, which cannot be supplanted by it. Spinoza therefore becomes a significant figure in Deleuze’s project of tracing an alternative lineage in the history of philosophy, which, by distancing itself from Hegelian idealism, culminates in the construction of a philosophy of difference. It is Spinoza’s role in this project that will be demonstrated in this paper by differentiating Deleuze’s interpretation of the geometrical example of Spinoza’s Letter XII (on the problem of the infinite) in Expressionism in Philosophy, Spinoza,2 from that which Hegel presents in the Science of Logic.3. (shrink)

El cálculo infinitesimal elaborado por Leibniz en la segunda mitad del siglo XVII tuvo, como era de esperarse, muchos adeptos pero también importantes críticos. Uno pensaría que cuatro siglos después de haber sido presentado éste, en las revistas, academias y sociedades de la época, habría ya poco qué decir sobre el mismo; sin embargo, cuando uno se acerca al cálculo de Leibniz –tal y como me sucedió hace tiempo– fácilmente puede percatarse de que el debate en torno al cálculo leibniziano (...) ha trascendido las fronteras temporales del siglo XVII y XVIII y aún sigue vigente, al menos en parte y sobre ciertos puntos específicos. Lo anterior resulta un tanto inquietante, entre otras cosas porque implica que hay que revisar el cálculo leibniziano para tratar de entender el motivo por el que se sigue debatiendo actualmente. El propósito de este artículo no es presentar las tesis principales del cálculo, tampoco hacer una defensa del mismo y mucho menos desarrollar una crítica de sus fundamentos matemáticos, epistémicos u ontológicos. Mi propósito es menos ambicioso, pretendo mostrar, en primer lugar, dos de las primeras objeciones que se formularon contra el cálculo infinitesimal y que fueron hechas por dos contemporáneos de Leibniz, a saber, Rolle y Nieuwentijit; por otro lado, y sirviéndome de lo anterior, quiero presentar dos comentaristas actuales que abordan el tema del cálculo. Lo anterior con el objetivo de ilustrar cuál es el estado de la cuestión, es decir, en qué momento está el debate sobre el cálculo infinitesimal y en qué se enfocan actualmente los comentaristas al hablar sobre él, pues al hacerlo –e incluso a veces sin pretenderlo– mantienen abierto el debate sobre los aciertos y desaciertos del método infinitesimal. (shrink)