Modern Formal Logic Research Articles

What is the Principle of (Non-)Contradiction, Precisely? The Struggle at the Dawn of Formal Logic

The principle of contradiction, or non-contradiction, is traditionally included as one of the three fundamental principles of logic, together with the principle of identity and the principle of excluded middle. There is a consensus now regarding the shape of the principle of contradiction in modern formal logic. However, a deeper look at the history of its formulation reveals a much more complicated picture. We trace some of such developments from the beginning of the twentieth century when all sorts of formalisms were proposed, and even the name itself was up for debate. Our focal point is the proposals made by Christine Ladd-Franklin, which we describe against the background of other attempts at the time.

Mathematical Logic in the History of Logic: Łukasiewicz’s Contribution and Its Reception

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

Особенности толкования термина «следование» у И. Канта

Modern formal logic, which is based on Kant’s logical project, interprets logical consequence as formal, which leads to substantive paradoxes that combine any thoughts at all and so to the loss of consequence as such. Beginning with A. Tarski, modern history of logic brings the problem of logical consequence into the realmof search for the relation of consequence, or grounding. In his doctoral dissertation on the nature of logical formality J. MacFarlane claims that the paradoxes of formal theories of logical consequence stem from the loss of grounding by the transcendental system of logic in the post­Kantian logical tradition. Arguably, analysis of logical terminology of consequence in Kant’s seminal works — Critique of Pure Reason, Critique of the Power of Judgment — in comparison with the terminology of earlier works, Prolegomena and lectures on logic attributed to him will clarify the question of the relation oflogical consequence in the formal and non­formal sense. The key concept of consequence in Kant’s terminology is Folgerung, which denotes ‘following’ in logical and non­logical contexts. I have also analysed related concepts: Folge, Abfolge, folglich etc., established dif- ferences between logical terms with similar meaning ‘inference’ (Schluss) and ‘conclusion’ (Konklusion). Finally, I make an attempt to formulate the problem of logical consequence in formal logic through the logical terms Schlussfolge, Folgerung and Konsequenz. On the strength of my analysis I propose to consider Kant’s consequence (Folgerung) to be a concept of transcendental logic that reflects the relation of consequence and grounds formal consequence.

Łukasiewicz, Bocheński, and Feys: Their Impact on the Early Prior

We investigate how Arthur Prior became a tense-modal logician, and which moderns influenced him in his early thinking about modality. His unpublished manuscript The Craft of Formal Logic, written in the period 1949–51, is in effect a record of his rather isolated apprenticeship as he trained himself in formal logic, during the two years before the commencement of his torrent of publications on modality. We analyse sections of this rich record of his logical development, especially those dealing with modal logic, and we extract a detailed account of the pattern of influences exhibited in The Craft. The Craft reveals that Prior’s first encounters with modern symbolic modal logic were the pioneering explorations by Bocheński, Feys, and Lewis. Von Wright was also an early influence. It was through Bocheński’s writings that Prior learned of Łukasiewicz’s approach to modality, and Łukasiewicz’s work quickly became a beacon for Prior. The roles of Lewis and von Wright appear to have been smaller than those of Łukasiewicz, Bocheński, and Feys—hence our focus on these three figures. As well as biographical material on these three outstanding logicians, we include numerous previously unpublished passages from The Craft in order to establish the nature and extent of their impact on Prior.

Dzienniki, tom I, lata 1944–1960 [Diary, vol. 1, 1944–1960

<i>Dzienniki, tom I, lata 1944–1960</i> [Diary, vol. 1, 1944–1960

LUKASIEWICZ’S APPROACH TO SYLLOGISTIC: A CRITICAL ANALYSIS STUDY

There is no theory which has received a big interest historically as Aristotelian syllogistic, despite the criticisms to which the theory was exposed by philosophers and logicians like Francis Bacon and Jean Stuart mill in their philosophical and logical works, they considered it as an epistemological obstacle to the development of scientific knowledge, and there is a need to get over it to new method and process, but It is still an interesting subject of study and updated by many logicians to nowadays. The most prominent attempts: the Intentional approach opposite to the comprehensive one, then the approach of the logician and philosopher Ian Łukasiewicz, who tried to read the syllogistic theory with the use of accurate and rigorous analysis tool in classical logic, which is the calculi of propositions where he considered the real form of Aristotelian syllogism moods is conditional (If ... so), i.e., as computable logical laws not inferential rules (If ... then). In order to reach this purpose Łukasiewicz present his hypotheses, some of them are verified for the others he have had to make several interpretations to make his theory consistent. These interpretations took him away from the spirit of the theory and from the essence of what did Aristotle. This made his approach the subject of numerous and harsh criticisms. This is what we will try to show it through a critical analysis to some hypotheses which he presented in his book “Aristotle’s Syllogistic from the standpoint of Modern Formal Logic”.

Учить логике будущих учителей математики (часть III)

Introduction. The article is the third part of the author’s article under the same title published in previous issues of this journal. It discusses the problem of forming the logical competencies of future mathematics teachers both at the undergraduate level and at the master’s level. Theoretical analysis. In this case, logic is considered in three aspects – classical Aristotelian logic, modern mathematical logic and its application to Aristotelian logic, non-classical logics. This part of the article proposes a number of measures related to the formation of the logical and didactic training content for future mathematics teachers at the undergraduate and graduate levels in order to make this training a system-forming factor in the entire system of their training. At the same time, training at the master’s level must meet the principle of continuity and become a natural continuation of training at the undergraduate level. Conclusions. Such an approach will significantly increase the level of the logical and general professional competence of mathematics teachers who graduate from pedagogical and classical universities. Such teachers will be able to apply logical knowledge and skills in their pedagogical activities to educate logically and mathematically competent students at different levels of general education capable of further self-development and creative activity.

HERACLITUS, PARMENIDES AND ZENO:

This article tells the story of the proof known as reductio ad absurdum, through the biographies of Heraclitus, Parmenides and his disciple Zeno. Emphasis is placed on Zeno’s paradoxes and some classic demonstrations of modern mathematics, in which proof by reductio ad absurdum is used widely. In other words, we intend to discuss the logical-mathematical history of reductio ad absurdum, from the birth of logic in Heraclitus, Parmenides and Zeno and ipso facto to establish a relationship between Heraclitus, Parmenides and Zeno and modern mathematical logic. The method used was the bibliographic, referring mainly to secondary sources due to the strictly theoretical nature of the subject.

Prior’s Turn to Medieval Logic

Abstract The peculiar aspect of medieval logic, that the truth-value of propositions changes with time, gradually disappeared as Europe exited the Renaissance. In modern logic, it was assumed by W.V.O. Quine that one cannot appreciate modern symbolic logic if one does not take it to be tenseless. A.N. Prior’s invention of tense-logic challenged Quine’s view and can be seen as a turn to medieval logic. However, Prior’s discussion of the philosophical problems related to quantified tense-logic led him to reject essential aspects of medieval logic. This invites an evaluation of Prior’s formalisation of tense-logic as, in part, an argument in favour of the medieval view of propositions. This article argues that Prior’s turn to medieval logic is hampered by his unwillingness to accept essential medieval assumptions regarding facts about objects that do not exist. Furthermore, it is argued that presentists should learn an important lesson from Prior’s struggle with accepting the implications of quantified tense-logic and reject theories that purport to be presentism as unorthodox if they also affirm Quine’s view on ontic commitment. In the widest sense: philosophers who, like Prior, turn to the medieval view of propositions must accept a worldview with facts about individuals that, in principle, do not supervene (present tense) on being, for they do not yet exist.

A Few Important Landmarks in the Chinese Debates on Dialectical and Formal Logic from the 1930s

With the rise of the discourse on dialectical materialism in the late 1920s, ideas related to the Marxist notion of dialectical logic started to circulate in the Chinese intellectual world. Not long after the first public discussions on dialectical materialism started to emerge in the early 1930s, the discussants on both sides started to address the question of the Marxist notion of logic and its relationship with Western formal logic. Consequently, over the 1930s, a series of separate public debates ensued, in which dialectical logic contended against the “conventional” forms of logic, such as traditional Aristotelian and modern formal logic. This paper outlines the major landmarks within the public as well as internal Marxist debates on logic in the 1930s. The discussion starts with a general overview of the intellectual background of the debates, and proceeds by analysing the principal developments in them, starting with Ye Qing’s and Zhang Dongsun’s polemic about “dynamic logic” from 1933, and concluding with the internal Marxist discussions on the sublation of formal logic in the last years of the decade.

Łukasiewicz, determinism, and the four-valued system of logic

Abstract 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.

ЛОГІКА І ФІЛОСОФІЯ: ГРАНІ ВЗАЄМОДІЇ

In this paper, we argue in favour of the applicability of logical tools for the analysis of various philosophical issues. Different systems of the modern non-classical (philosophical) logic are considered, and a promising approach of research represented by informal logic. We conclude with some remarks about the prospects for a fruitful interaction of modern formal and informal logic with philosophical knowledge.

Research trends from computer and electronic sciences

Electronics as a technical and scientific discipline was born in 1904, based on the principles defined by John Ambrose Fleming (inventor of the diode) and André Marie Ampére (founder of electromagnetic theory), its evolution towards current electronics, has largely been possible thanks to the invention of the triode (device capable of converting alternating current into direct current) by Lee De Forest. On the other hand, computer science as born as a scientific discipline in the early 1940s, based on mathematical logic and algorithmic theory, the principles that sustain it go back to the works Gottlob Frege (father of modern mathematical logic) and Andréi Kolmogórov (founder of the theory of algorithm complexity). The complementarity between these two relevant disciplines has made possible the development of a wide diversity of technologies applied to studies in the care of health (e.g. biomechanics, computerized axial tomography and telemedicine) and to space exploration (e.g. communication mobile and GPS, robotics and nanotechnology), among others.

Aristotle’s Syllogistic as a Deductive System

Aristotle’s syllogistic is the first ever deductive system. After centuries, Aristotle’s ideas are still interesting for logicians who develop Aristotle’s work and draw inspiration from his results and even more from his methods. In the paper we discuss the essential elements of the Aristotelian system of syllogistic and Łukasiewicz’s reconstruction of it based on the tools of modern formal logic. We pay special attention to the notion of completeness of a deductive system as discussed by both authors. We describe in detail how completeness can be defined and proved with the use of an axiomatic refutation system. Finally, we apply this methodology to different axiomatizations of syllogistic presented by Łukasiewicz, Lemmon and Shepherdson.

ON THE DIAGONAL LEMMA OF GÖDEL AND CARNAP

AbstractA cornerstone of modern mathematical logic is the diagonal lemma of Gödel and Carnap. It is used in, for example, the classical proofs of the theorems of Gödel, Rosser, and Tarski. From its first explication in 1934, just essentially one proof has appeared for the diagonal lemma in the literature; a proof that is so tricky and hard to relate that many authors have tried to avoid the lemma altogether. As a result, some so-called diagonal-free proofs have been given for the above-mentioned fundamental theorems of logic. In this paper, we provide new proofs for the semantic formulation of the diagonal lemma, and for a weak version of the syntactic formulation of it.

Remarks on the Principle of Permanence of Forms

We discuss the role of a heuristic principle known as the Principle of Permanence of Forms in the development of mathematics, especially in abstract algebra. We try to find some analogies in the development of modern formal logic. Finally, we add a few remarks on the use of the principle in question in mathematical education.

Logic as a Blended Course

I present Modern Symbolic Logic, an introductory philosophy course in first-order logic, as a blended course. A blended course integrates online video learning with in-class activities, out of class supports, and deliverables into a cohesive and mutually supporting package. Blended courses are an enhancement on hybrid courses, which focus on online video learning but not on the additional supports needed for an effective learning experience. This paper has two central aims. The first is to present a blended course in action in order to address a need in the literature for detailed reports of blended classes. The second is to advance an iterative approach to blended course design that significantly lowers the bar of entry for instructions hoping to create a blended course.

Logic of theological debate in the Book of Job: Criticism of Maimonides’ interpretation in the Jewish biblical exegesis in the 14th century

The article discusses the critical understanding of Maimonides’ interpretation of the Book of Job, representing a theological debate on Divine Providence. Maimonides ascribes certain assessments on this issue to Job and his friends, linking their opinions to some theological or philosophical schools of the 10th–12th centuries, developed in the context of Arabic culture. A number of Jewish exegetes, who were working in a diverse cultural context and were writing in Hebrew, such as Joseph Ibn Kaspi and Gersonides, considered this distribution of opinions an anachronism. Being Maimonidean followers, they nevertheless criticized the distribution of opinions between Job and his friends proposed in The Guide of the Perplexed. The main argument of this criticism was logic. We will discuss the origins of such an approach and analyze some nuances of logical solutions, suggested by Maimonidean followers, juxtaposing them with a range of problems touched upon in The Guide of the Perplexed in regards to the Book of Job. It will be stressed among others, that the interpretation of Gersonides, coming from his statement on the logical completeness of the solution of the question of Divine Providence in the Book of Job, can be justified by the means of modern formal logic.

Frege’de Yargıların Formel Mantıksal Analizi ve Düşünceler Üzerine

Frege made a derivative of the distinction that he made between the meanings of senses (Sinn) and references (Bedeutung) on judgments, and named the judgments which he treats as hollow functions; ‘Unsaturated’ (ungesättigt) phrases and the phrases which completed an argument as, ‘Saturated’ (gesättigt) phrases. Saturated phrases consist of first and second level concepts and are considered to be meaningful expressions with an accurate value. Frege became the founder of modern symbolic logic with the language of formal logic he developed in ‘Concept Writing’, and based all this formal analysis on the distinction of ‘function (Funktion) – argument (Argument)’ and ‘concept – coverage’. The philosopher, who makes sense of the judgments as their ‘thoughts’; determines the logical value accuracy of the reference on the basis of this distinction, has adopted a contradictory acceptance in terms of both considering the judgments as a condition of being able to receive a truth value and reducing them to the senses of the sentences. Then, if the thoughts were indeed the sense of the sentences, as Frege thought, then it would not be possible to speak of sentences that were meaningful but lacked truthfulness. That is why, to overcome this distress, we propose to change the definition of thought in Frege's theory and to consider it as function values, not the sense of sentences. Likewise for us, thoughts are not the senses of the sentences, nor are they references. Contrariwise, thoughts are the function value of the sentences. To put it in a different way, if a sentence lacks function value, it can be meaningful but not true or false, and if the sentence has a function value, it can be true or false. Thus, for us, thoughts are neither the expression of the sentence in the language nor the logical value of the sentence; thoughts are function values that can be true or false.

A Grammar of Praxis: an Exposé of "A New Logical Foundation for Psychology", a Few Additions, and Replies to Alaric Kohler and Alexander Poddiakov.

Taking departure in the author's recent book "A New Logical Foundation for Psychology" the paper proposes a solution to the long-standing so-called crisis in psychology. The causes to this crisis are above all found in a ruling reductionist and mathematically supported mechanistic understanding of nature with roots in European Renaissance, and following that, of man's embedment in nature. This leaves no place for non-mechanistic relations to unique and irreplaceable persons and objects defining the human psyche or soul, and fundamental phenomena as love and grief are consequently not understood. No humanistic superstructure of language or systems of signs and concepts can repair this loss of a vital dimension in basic human practical relations to the world. However, it is just in modern mathematics and mathematical logic, that the reductions of mechanicism are surmounted, at the same time leaving a place for mechanicism within a broader conceptual frame and defining a rich practical basis for understanding the role of language and human concepts. The wider perspectives comprise a new union of natural and human sciences. Finally the paper presents replies to two important commentaries to the author's abovementioned book.