History of Logic

While there are good reasons to think that Hegel would not engage with modern scepticism in the Science of Logic, this article argues that he nevertheless does so in a way that informs the text's conception of logic as the latter pertains to metaphysics. Hegel engages with modern scepticism's general concerns that philosophy should begin without unexamined presuppositions and should come to attain not only knowledge of truth, but corresponding second-order knowledge: knowledge of knowing truth. These concerns inform two needs (...) that Hegel formulates for first philosophy, which logic—by unifying with metaphysics, which is traditionally synonymous with first philosophy—is to satisfy. However, logic, for the Logic, is unified with metaphysics as a science of absolute knowing, the form of thinking involved in traditional metaphysics. As such, logic, for the Logic, is neither anti-metaphysical nor reducible to metaphysics, but is rather a science of metaphysical thinking, which, for Hegel, includes metaphysics. The article emphasizes how Hegel's construal of logic as a science of absolute knowing avoids running into the ‘swimming problem’ that Hegel raises against, broadly, epistemological forms of first philosophy. (shrink)

Aristotle's Organon in Old and New Logic 1800–1950 explores the reception and interpretation of Aristotle's logic over the last two centuries. The volume covers seminal works during this period by logicians, historians of logic, and historians of philosophy, including John Lloyd Akrill, Francesco Barone, Günther Patzig, Enrico Berti, and Mario Mignucci. Contributors consider the reception of the Organon in old logic and chart the appearance of formal approaches to logic beginning with Boole. This in-depth study of Aristotelianism also covers logic (...) in Kant and Hegel, alongside the problems and projects of interpreting Aristotle in the new logic after Boole and Frege. The background of modern debates concerning induction and abduction provides further insight into Aristotelian logic during the period. By filling gaps in our understanding of Aristotelian logic, this book provides a fundamental missing link in 21st century studies of the history of Aristotelianism. It brings together scholars of both ancient and modern logic to understand the interpretation of ancient logic before and after the development of the modern, algebraic approach to logic. (shrink)

We present a study of unpublished fragments of Jan F. Drewnowski’s manuscript from the years 1922–1928, which contains his own axiomatics for mereology. The sources are transcribed and two versions of mereology are reconstructed from them. The first one is given by Drewnowski. The second comes from Leśniewski and was known to Drewnowski from Leśniewski’s lectures. Drewnowski’s version is expressed in the language of ontology enriched with the primitive concept of a (proper) part, and its key axiom expresses the so-called (...) weak super-supplementation principle, which was named by Drewnowski “the postulate of the existence of subtractions”. Leśniewski’s axiomatics with the primitive concept of an ingrediens contains the axiom expressing the strong super-supplementation principle. In both systems the collective class of objects from the range of a given non-empty concept is defined as the upper bound of that range. From a historical point of view it is interesting to notice that the presented version of Leśniewski’s axiomatics has not been published yet. The same applies to Drewnowski’s approach. We reconstruct the proof of the equivalence of these two systems. Finally, we discuss questions stemming from their equivalence in frame of elementary mereology formulated in a modern way. (shrink)

The notion of the extension of a concept has been used in logic for a long time. It is usually considered to be closely connected to the intuitive notion of a set and thus seems as though it should be embedded into set theory. However, there are significant differences between this “logical” concept of set and the notion of set (class) as defined via standard axiomatic systems of set theory; it may, therefore, be quite misleading to consider the two concepts (...) as being continuous with each other. When we look at the writings of Gottlob Frege and consider the development of his attitude to extensions, we can see what the differences consist in and which of the two notions is more apt to be used in foundations of logic. Frege himself eventually rejected sets entirely. (shrink)

The literary culture of late antiquity established a list of questions to be answered before studying an author or a text. Among other types of introductory sets, we find the six didascalica used by Boethius in his commentaries on Aristotle’s Organon. Twelfth-century commentaries inherited these requirenda, although each master felt free to modify and rearrange traditional elements. Within the logical commentaries, the Abelardian commentaries Logica ingredientibus and Logica nostrorum petitioni sociorum show some peculiarities, such as the modus tractandi; this feature (...) is interestingly similar to the Notae Dunelmenses. With regard to Alberic and his school, we have, on the one hand, the information preserved in the H17 commentary and, on the other, the commentaries on the Categories of the Mont Sainte-Geneviève school. The authors of these commentaries were not mere executors of the master’s will, but their texts show an internal coherence in the presentation of the elements causa/utilitas/finis; moreover, they inherited Abelard’s modus tractandi. The most widespread element of all twelfth-century logical commentaries, however, is the entry materia; its origin is not rooted in Boethius’s commentaries, but it is typical of the literary tradition of the accessus. (shrink)

‘Byzantine logic diagrams’ have been used since at least late antiquity, and became popular in Europe in the 16th century. However, since the criticism of W. Hamilton and J. Venn in the 19th century, Byzantine diagrams have been largely dismissed as obsolete. This paper challenges this prevailing view. Initially, we provide a comprehensive overview of the current state of research pertaining to these diagrams, illustrating their applicability in analyzing assertoric syllogisms. Subsequently, we propose that the expressive capacity of these diagrams (...) extends far beyond their traditional use. In particular, we demonstrate that the expressivity of these diagrams is not limited to checking the validity of syllogisms. Rather, we show that there are various extensions of Byzantine diagrams, including opposition relations, modal logic, particular instances, n-term diagrams and logical connectives. These extensions are compositional and can therefore be combined to create very complex Byzantine diagrams that go far beyond what has been commonly believed since Hamilton and Venn. (shrink)

This paper discusses the relation between the natural deduction rules of deduction in sequent format and the provability valuation starting from Garson’s Local Expression Theorem, which is meant to establish that the natural deduction rules of inference enforce exactly the classical meanings of the propositional connectives if these rules are taken to be locally valid, i.e. if they are taken to preserve sequent satisfaction. I argue that the natural deduction rules for disjunction are in no better position than the axiomatic (...) calculi in uniquely determining the intended meaning of disjunction when the local models are used, if a satisfied sequent embeds, as a logical inferentialist should require, a formal derivability relation. This happens because, when governed by these rules and without additional semantic assumptions, the disjunction sign still expresses a non-extensional connective, i.e. a connective that, properly understood, has no unique logical characteristic. However, this is not a dead end for the logical inferentialists since both a multiple conclusions formalization of the disjunction operator and a bilateralist one do succeed in restoring the standard meaning of disjunction. (shrink)

Heart disease remains one of the leading causes of mortality worldwide. Early prediction and diagnosis are critical in preventing severe outcomes and improving the quality of life for patients. This project focuses on developing a robust heart disease prediction system using machine learning techniques. By analyzing a comprehensive dataset consisting of various patient attributes such as age, sex, blood pressure, cholesterol levels, and other medical parameters, the system aims to predict the likelihood of a patient having heart disease. The project (...) employs various machine learning algorithms such as Logistic Regression, Decision Trees, Support Vector Machines (SVM), and Random Forests to classify the data and provide an accurate prediction. The system's performance is evaluated using metrics like accuracy, precision, recall, and F1-score, ensuring that it can offer reliable results in real-world applications. Furthermore, feature selection techniques are applied to identify the most significant factors contributing to heart disease, thus improving the model's interpretability. The proposed solution is intended to aid healthcare professionals by providing early alerts and recommendations, ultimately facilitating timely interventions. (shrink)

The project titled "Efficient Machine Learning Algorithm for Future Gold Price Prediction" focuses on the development of a machine learning model that can accurately predict future gold prices using historical data and various economic indicators. Gold has long been regarded as a safe-haven asset, and its price is influenced by multiple factors, including global economic conditions, inflation rates, interest rates, and geopolitical events. This research aims to design and implement a robust machine learning model that can analyze complex patterns and (...) trends in gold prices to generate reliable predictions. The project employs a range of machine learning techniques such as Linear Regression, Random Forest, and Support Vector Machines (SVM) to build the prediction model. It utilizes key features like historical price data, currency exchange rates, commodity prices, and stock market trends as input variables. The performance of the model is evaluated using standard metrics such as accuracy, precision, recall, and RMSE (Root Mean Squared Error). The final model is optimized for accuracy and efficiency, making it a useful tool for investors and financial analysts who seek to forecast gold prices and make informed investment decisions. (shrink)

I argue that discussions of Frege’s conception of elucidation have suffered from a conflation of two distinct issues: elucidation of primitive scientific terms, and elucidation of the logical categories. The former seeks to bring us to grasp the Bedeutung of terms that stand at the beginning of the chain of definitions of a scientific system. The latter cannot be understood on the model of securing agreement in Bedeutung at all. I show how existing discussions of Fregean elucidation insufficiently take this (...) difference into account. I adumbrate what I take to be a more accurate understanding of Fregean elucidation of the logical categories, starting from the observation that Frege, when engaged in such elucidation, consistently reverts to talking about signs. Frege, I argue, takes signs to possess logical features, and it is these logical features which his elucidations are meant to help us to grasp. For Frege, the nature of the logical categories lies in the signs. In this way, I argue, Frege’s approach to elucidating the logical categories is incompatible with a realist conception according to which there is a logical order of reality that is prior to the logical order of language. (shrink)

This manuscript presents an English translation of the work titled “Begründung Einer Strengen Implikation” by the German logician and mathematician Wilhelm Ackermann (1896-1962), first published in June 1956.

Johannes Rudbeckius (1581–1646) var en av de främsta kyrkoledarna i stormaktstidens Sverige. Under studietiden i Wittenberg influerades han av ”luthersk skolastik”, enligt vilken förnuft och uppenbarelse är förenliga som kunskapskällor och logik är ett instrument för teologi. -/- I boken visar Tero Tulenheimo hur Rudbeckius lade vikt vid logik som stöddisciplin för teologi. -/- Inledningsvis behandlas frågan om omfattningen av Rudbeckius teologiska produktion. Här argumenterar Tulenheimo emot Bengt Hägglunds tes, enligt vilken Rudbeckius skulle ha författat en viktig opublicerad handskrift om (...) dogmatik. -/- Med hänvisning till Rudbeckius publicerade dissertationer lyfter Tulenheimo fram Rudbeckius reflektioner över språkanvändning, diskussion om förnuftsargument för uppenbarelsesanningar samt begreppsanalytiska frågeställningar. (shrink)

The present paper investigates whether strictly classical inferences contribute to the formalization of (genuine) paradoxes within natural deduction. Tennant's criterion for paradoxicality relies on the generation of an infinite reduction sequence, which distinguishes genuine paradoxes from mere inconsistencies. His methodological conjecture posits that genuine paradoxes are never strictly classical and can be derived without classical inferences such as the Law of Excluded Middle, Dilemma, Classical Reductio, and Double Negation Elimination. -/- It appears that there were two reasons for Tennant's proposal (...) of the methodological conjecture. The one is that strictly classical inferences hinder the generation of an infinite reduction sequence and the other is that strictly classical inferences have no role in the formalization of genuine paradoxes. -/- This paper raises questions about these two reasons. Focusing on the liar paradox, it will be argued that strictly classical inferences do not interfere with the generation of an infinite reduction sequence and that the liar sentence may implicitly entail strictly classical inferences. Should this analysis hold, it would call into question not only Tennant’s motivation for advancing the methodological conjecture, but also challenge his contention that genuine paradoxes—exemplified by the liar paradox—are never constructed with reliance on strictly classical inferences. (shrink)

Arthur Schopenhauer was one of the first logicians of the 19th century to develop a visual representation of the sorites with which it is possible to depict multiple terms in one diagram. With the help of an example from Seneca, he explains how the sorites diagram works and provides two reading interpretations for the diagram. Since Schopenhauer’s sorites diagram in particular or sorites diagrams in general have never been dealt with in research before, this paper will introduce and analyse Schopenhauer’s (...) sorites diagram and its different reading possibilities. Thereby, it will not only be shown how the sorites diagram can help to dissolve the sorites back into individual syllogisms, but also other topics such as the reading directions, the extension and intension of the sorites diagram will be discussed in more detail. It turns out that Schopenhauer’s sorites was just the foundation for further works in his philosophical system. (shrink)

I analyze Peirce’s reply in the 1903 Lowell Lectures to the “defendant argument” and show how his response provides a key to interpreting his later philosophy of logic and his views on the normative role of deductive logic in inquiry. I argue that in Peirce’s discussion of self-control in reasoning and evaluation of reasoning, we find an underappreciated position on logical revision and how to understand rational choice between deductive theories. To defend this point, I reconstruct Peirce’s reply by providing (...) an interpretation of his comments on higher-order criticism and the soundness of our reasoning, arguing that the resulting position’s ability to separate questions involving the role of logic in inquiry from questions better suited to the mathematical study of a logical theory provides a fruitful pragmatist reframing of problems in the epistemology of logic. (shrink)

Neil Tennant was the first to propose a proof-theoretic criterion for paradoxicality, a framework in which a paradox, formalized through natural deduction, is derived from an unacceptable conclusion that employs a certain form of id est inferences and generates an infinite reduction sequence. Tennant hypothesized that any derivation in natural deduction that formalizes a genuine paradox would meet this criterion, and he argued that while the liar paradox is genuine, Russell's paradox is not. -/- The present paper delves into Tennant's (...) conjecture for genuine paradoxes and suggests that to validate the conjecture, one of two issues must be addressed. The first issue is the need for a philosophical consensus on the identification of a genuine paradox in an informal sense. The second issue is the requirement for a uniform approach to formalize paradoxes in natural deduction. If either of these issues is addressed, the conjecture could be validated, or at the very least, it could hold philosophical importance in delineating the proof-theoretic features of paradoxicality. (shrink)

In this paper, I reflect on the idea, hinted at by Kant in a footnote to §16 of the B- Deduction that is not often discussed (KrV B 134n.), that transcendental logic is the ground of logic as a whole. This has important repercussions for the way we should see the role of transcendental logic with respect to the question of truth as well as the nature and scope of transcendental logic in relation to cognition, and in relation to general (...) or formal logic as such. To illustrate one of the ways in which transcendental logic is fundamental to our way of thinking, I address an issue that is brought up by Kant’’s counterfactual claim at B 132 that if a representation is not accompanied by an ‘‘I’’ thought, it is ‘‘either impossible’’ or at least ‘‘nothing for me’’, suggesting to some recent commentators that by the former Kant means the impossibility of thinking contradictory thoughts. Unlike these commentators, I do not think Kant is saying here that we cannot think contradictory thoughts. To believe he is betrays a misunderstanding of the metaphysical nature of transcendental logic, or so I shall argue. It is because transcen- dental logic is a metaphysical logic that Kant can claim that transcendental logic grounds even the whole of logic, including the possibility of thinking contradictory thoughts. (shrink)

Copied in 80 manuscripts and printed in 25 editions Logica Parva was the most widely read logic book in 15th century Italy. By transmitting Oxford logic to Italy it influenced the course of science, philosophy and theology in the Renaissance.

"From the 1930 Reiser edition (emended second impression) of the Ars logica, itself comprising the first two parts of the five part Cursus philosophicus of 1631-1635, by the same author.".

Gödel’s ontological proof is by now well known based on the 1970 version, written in Gödel’s own hand, and Scott’s version of the proof. In this article new manuscript sources found in Gödel’s Nachlass are presented. Three versions of Gödel’s ontological proof have been transcribed, and completed from context as true to Gödel’s notes as possible. The discussion in this article is based on these new sources and reveals Gödel’s early intentions of a liberal comprehension principle for the higher order (...) modal logic, an explicit use of second-order Barcan schemas, as well as seemingly defining a rigidity condition for the system. None of these aspects occurs explicitly in the later 1970 version, and therefore they have long been in focus of the debate on Gödel’s ontological proof. (shrink)

This paper argues for the significance of Kaplan's logic LD in two ways: first, by looking at how logic got along before we had LD, and second, by using it to bring out the similarity between David Hume's thesis that one cannot deduce claims about the future on the basis of premises only about the past, and the so‐called "essentiality" of the indexical.

Aristotle counts as the founder of formal logic. The logic he develops dominated until Frege and others introduced a new logic. This new logic is taken to be more powerful and better capable of capturing inference patterns. The new logic differs from Aristotelian logic in significant respects. It has been argued by Fred Sommers and Hanoch Ben-Yami that the new logic is not well equipped as a logic of natural language, and that a logic closer to Aristotle's is better suited (...) for this task. Each of them developed their own formalism - Sommers in form of term logic, Ben-Yami in form of his Quantified Argument Calculus (QUARC). I discuss Aristotle's logic - a term logic - and attempt a comparison between Aristotelian logic and (i) the new logic, (ii) Sommers' term logic, and (iii) Ben-Yami's QUARC. I consider differences between the systems, and show how they are related to and diverge from the new logic. (shrink)

The systematic account of deductive reasoning and the development of a formal logic to reveal the principles of such reasoning began with Aristotle's syllogistic. It was a term logic, a logic that dominated the field until the rise of modern predicate logic at the end of the Nineteenth century. That system quickly supplanted the old logic of terms. However, in the middle of the Twentieth century Fred Sommers took up the challenge to build a revised and strengthened term logic, one (...) fit to challenge the hegemony of predicate logic. The aim was to devise a formal logic that could better serve as the logic of natural language. -/- In recent years a new group of logicians have taken this version of term logic into many new directions. This book presents here for the first time some of the best examples of their work with new essays by philosophers, logicians, mathematicians, computational theorists, and historians of logic. The topics addressed include relative terms, logical copulation, Aristotelian diagrams, modality, truth, semantics, epistemic term logic, non-classical quantifiers, and more. (shrink)

In this paper, I outline a pragmatist epistemology of logic inspired by later work of Charles S. Peirce that shares many features with an anti-exceptionalism about logic but, I argue, can better respond to a key problem that plagues the anti-exceptionalist. I first lay out what I take to be the tenets of anti-exceptionalism, discussing some difficulties in formulating the position that make it difficult to definitively label the position discussed here. I then analyze a key problem for the anti-exceptionalist, (...) the background logic problem, and suggest that when framed as a problem for certain types of inquiry, it shares a presupposition about the priority of logic with other problems in the epistemology of logic. Turning to a discussion of Peirce from the 1903 Lowell Lectures, I then outline a pragmatism about logic that rejects the priority claims underlying the problems for inquiry in logic and, I argue, can resolve the background logic problem. (shrink)

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.

In this paper I provide an analysis of an anonymous text which appeared at Sondershausen and Mühlhausen in 1687: Initiatio philosophi sive Dubitatio Cartesiana, ad indubiam philosophiam viam monstrans, iuxta mentem Renati des Cartes, Nobilis Galli, utraque methodo explicata, titled after Johannes Clauberg’s homonymous 1655 treatise. It consisted of (1) an abridgement of his Paraphrasis in Renati Des Cartes Meditationes (1658), and (2) a demonstration more geometrico of the necessity of methodical doubt as the beginning of philosophy, partially based on (...) Clauberg’s own 1655 Initiatio. I place the publication of the 1687 text in the context of the attacks on Cartesianism taking place in German areas especially after 1677 and the appearance of Spinoza’s texts, focusing on the ways in which Clauberg’s philosophy was adapted to the rebuking of such attacks and to the re-establishment of a Cartesian presence in Germany. (shrink)

Justification of theorems plays a vital role in any rational human activity. It is indispensable in science. The deductive method of justifying theorems is used in all sciences and it is the only method of justifying theorems in deductive disciplines. It is based on the notion of proof, thus it is a method of proving theorems. In the Warsaw School of Logic (WSL) – the famous branch of the Lvov-Warsaw School (LWS) – two types of the method: axiomatic deduction method (...) and natural deduction method were developed and practiced. In this paper, both of these methods are briefly discussed with an emphasis on their historical, groundbreaking significance for logic. The axiomatic method by means of rejection (proposed by Jan Łukasiewicz – a co-creator of the WSL), which is the method of the so-called rejection proof (rejection/refutation method) in logical systems and the proving method of generalized natural deduction, which is a hybrid deduction–refutation method of proving theorems, are also outlined in the paper. The author discusses their historical significance. This paper also contains a brief mention of the most significant results which the application of the discussed methods introduced into contemporary scientific research, not only logical one. (shrink)

In this paper we present an overview, with historical and critical remarks, of two articles by S. Jaśkowski ([20, 21] 1948 and [22, 23] 1949), which contain the oldest known formulation of a paraconsistent logic. Jaśkowski has built the logic – he termed discussive (D2) – by defining two new connectives and by introducing a modal translation map from D2 systems into Lewis’ modal logic S5. Discussive systems, for their formal details and their original philosophical justification, have attracted discrete attention (...) among experts. Indeed, in what follows, after having introduced Jaśkowski methodology of building D2 and his main philosophical motivations for providing such a system, we will explore some of the main contributions to the development of D2. (shrink)

Bolzano reduced inferential validity of the inference (from premise judgements to conclusion judgment) to the holding of logical consequence between the propositions (in themselves) that serve as contents of the respective judgements. This explicit reduction of inferential validity among judgements to logical consequence among propositions (or, alternatively, to logical truth of certain implicational propositions) has been largely taken over by current logical theory, say, by Wittgenstein’s Tractatus, by Hilbert and Ackermann, by Quine, and by Tarski also. Frege, though, stands out (...) among those who did not adopt such an account. Under the Bolzano reduction also some blind inferences, with no epistemic support, are deemed valid, which is unacceptable. The Completeness Theorem reduces inferential validity to the logical truth of a certain well-formed formula, whence the unacceptable blindness phenomena are retained. Accordingly, the interest of the Completeness Theorem for logic, construed as the theory of inference, is nugatory. (shrink)