The “official” history of connexive logic was written in 2012 by Storrs McCall who argued that connexive logic was founded by ancient logicians like Aristotle, Chrysippus, and Boethius; that it was further developed by medieval logicians like Abelard, Kilwardby, and Paul of Venice; and that it was rediscovered in the 19th and twentieth century by Lewis Carroll, Hugh MacColl, Frank P. Ramsey, and Everett J. Nelson. From 1960 onwards, connexive logic was finally transformed into non-classical calculi which partly concur with (...) systems of relevance logic and paraconsistent logic. In this paper it will be argued that McCall’s historical analysis is fundamentally mistaken since it doesn’t take into account two versions of connexivism. While “humble” connexivism maintains that connexive properties only apply to “normal” antecedents, “hardcore” connexivism insists that they also hold for “abnormal” propositions. It is shown that the overwhelming majority of the forerunners of connexive logic were only “humble” connexivists. Their ideas concerning connexive implication don’t give rise, however, to anything like a non-classical logic. (shrink)

Eine Einfiihrung in die epistemische Logik wurde schon vor unge fahr zwei Jahrzehnten durch Jaakko Hintikka geschrieben, und auf sein "Knowledge and Belief" nimmt das vorliegende Buch nicht nur durch den Titel "Glauben, Wissen und Wahrscheinlichkeit" Bezug. Es diirfte deshalb wohl angebracht sein, wenn der Autor eines Buches, das so deutlich die Nachfolgeschaft eines anderen beansprucht, sich dafiir rechtfertigt und erklart, worin sich sein Werk von dem des Vorgangers unterscheidet. Ein wichtiger Punkt, in dem die zweite iiber die erste Einleitung (...) in die epistemische Logik hinausgeht, wird schon durch die Titel erganzung "... und Wahrscheinlichkeit" signalisiert. Wahrend Hin tikkas erkenntnistheoretische Untersuchungen sich fast ausschlieB lich auf die Bestimmung einer Wissenslogik ("episteme") konzen trierten, werden hier unter Bezugnahme auf die Theorie subjektiver Wahrscheinlichkeit zusatzlich diverse Glaubenslogiken ("doxa") ent wickelt. Insbesondere wird ein Begriff "schwachen" Glaubens im Sinne des Fiir-wahrscheinlich-Haltens vom "starken" Glaubensbe griff im Sinne festester Oberzeugungen abgegrenzt, eine Unterschei dung, die im angelsachsischen Sprachraum bei der Diskussion der Logik von "belief" straflich vernachlassigt wurde. Die genaue Aus arbeitung dieses Unterschieds, die exakte semantische Charakteri sierung der entsprechenden doxastischen Modaloperatoren, und die Suche nach vollstandigen Axiomatisierungen korrespondierender Logiken fiihrte dabei in (mathematisches) Neuland: zur Theorie klassifikatorischer Wahrscheinlichkeit. Diese erste wesentliche Wei terentwicklung von "Knowledge and Belief" konnte man durch die Formel beschreiben: von rein epistemischer Logik hin zu epistemi scher und doxastischer Logik. (shrink)

It is often assumed that Aristotle, Boethius, Chrysippus, and other ancient logicians advocated a connexive conception of implication according to which no proposition entails, or is entailed by, its own negation. Thus Aristotle claimed that the proposition ‘if B is not great, B itself is great […] is impossible’. Similarly, Boethius maintained that two implications of the type ‘If p then r’ and ‘If p then not-r’ are incompatible. Furthermore, Chrysippus proclaimed a conditional to be ‘sound when the contradictory of (...) its consequent is incompatible with its antecedent’, a view which, in the opinion of S. McCall, entails the aforementioned theses of Aristotle and Boethius. Now a critical examination of the historical sources shows that the ancient logicians most likely meant their theses as applicable only to ‘normal’ conditionals with antecedents which are not self-contradictory. The corresponding restrictions of Aristotle’s and Boethius’ theses to such self-consistent antecedents, however, turn out to be theorems of ordinary modal logic and thus don’t give rise to any non-classical system of genuinely connexive logic,. (shrink)

One of the main controversies of the Logic Schools of the 12th century centered on the question: What follows from the impossible? In this paper arguments for two diametrically opposed positions are examined. The author of the ‘Avranches Text’ who probably belonged to the school of the Parvipontani defended the view that from an impossible proposition everything follows (‘Ex impossibili quodlibet’). In particular he developed a proof to show that by means of so-called ‘disjunctive syllogism’ any arbitrary proposition B can (...) be logically derived from a pair of contradictory propositions A and Not-A. The author of the Ars Meliduna instead argued that nothing follows from an impossible proposition (‘ex falso nihil sequitur’). This view is supported by various counterexamples which aimed to show that the admission of impossible premises would give rise to inconsistent conclusions. Upon closer analysis these inconsistencies do not, however, have the formal structure of a real contradiction like A and Not-A, but rather the structure of two rivalling conditionals like ‘If B then A’ and ‘If B then Not-A’. Hence these counterexamples rather have to be considered as refutations of the basic principles of ‘connexive logic’. (shrink)

The numerous modal systems between S4 and S5 are investigated from an epistemological point of view by interpreting necessity either as knowledge or as (strong) belief. It is shown that-granted some assumptions about epistemic logic for which the author has argued elsewhere-the system S4.4 may be interpreted as the logic of true belief, while S4.3.2 and S4.2 may be taken to represent epistemic logic systems for individuals who accept the scheme knowledge = true belief only for certain special instances. There (...) is strong evidence in favor of the assumption that S4.2 is the logic of knowledge. (shrink)

Buridan endorsed the principles that any impossible, and a fortiori any self-contradictory, proposition entails each proposition. These principles are usually referred to as ‘Ex impossibili quodlibet’ (EIQ) and ‘Ex contradictione quodlibet’ (ECQ). Buridan further considered the instance ECCQ according to which any proposition follows from the conjunction of two contradictory propositions. Buridan showed how ECCQ can be proven by means the usual laws of conjunction and disjunction. Furthermore, he discovered that EIQ can be derived from ECCQ by means of the (...) principle that if a conclusion q follows from two premises, one of which is necessary, then q follows from the other premise alone. Buridan thought that this principle would be provable; but his proof turned out to be circular since it presupposed EIQ. Nevertheless, EIQ is validated by the definition of a ‘bona consequentia’. Somewhat more exactly, according to Buridan one may distinguish between inferences which are formally valid, ‘materially’ valid, and valid ‘as of now’. It is shown that formally valid inferences satisfy ECQ while ‘materially’ valid inferences satisfy EIQ. Furthermore, inferences which are valid ‘as of now’ satisfy the principle ‘Ex falso quodlibet’ saying that any false proposition ‘as of now’ entails every proposition. (shrink)

Abelard: Logic This article describes and reconstructs Peter Abelard’s logic of the twelfth century. Much of what he regarded as logic is now classified as ontology or philosophical semantics. The article concentrates on his treatment of the relation of consequence. Abelard’s most important logical innovations consist of two points: The distinction between two kinds of … Continue reading Abelard: Logic →.

This paper scrutinizes Everett Nelson's conception of consistency by comparing it with the “standard” account of C. I. Lewis. This conflict surprisingly resembles a related controversy between the ancient logicians Chrysippus and Diodorus. Nelson's intuitions behind his peculiar conception of consistency are analysed and certain features of his logical system are critically examined. In particular, his objections against the law of the transitivity of implication and against the laws of conjunction and disjunction have to be discussed. Although Nelson's considerations contain (...) interesting ideas in favour of a relevantist conception of logic, they fail to convincingly support the hardcore connexive thesis that even a selfcontradictory proposition doesn't entail its own negation. (shrink)

The core idea of the ontological proof is to show that the concept of existence is somehow contained in the concept of God, and that therefore God’s existence can be logically derived—without any further assumptions about the external world—from the very idea, or definition, of God. Now, G.W. Leibniz has argued repeatedly that the traditional versions of the ontological proof are not fully conclusive, because they rest on the tacit assumption that the concept of God is possible, i.e. free from (...) contradiction. A complete proof will rather have to consist of two parts. First, a proof of premiseGod is possible. Second, a demonstration of the “remarkable proposition”If God is possible, then God exists. The present contribution investigates an interesting paper in which Leibniz tries to prove proposition. It will be argued that the underlying idea of God as a necessary being has to be interpreted with the help of a distinguished predicate letter ‘E’ as follows:\\). Proposition which Leibniz considered as “the best fruit of the entire logic” can then be formalized as follows:\) \rightarrow \, E)\). At first sight, Leibniz’s proof appears to be formally correct; but a closer examination reveals an ambiguity in his use of the modal notions. According to, the possibility of the necessary being has to be understood in the sense of something which possibly exists. However, in other places of his proof, Leibniz interprets the assumption that the necessary being is impossible in the diverging sense of something which involves a contradiction. Furthermore, Leibniz believes that an »impossible thing«, y, is such that contradictory propositions like \\) and \\) might both be true of y. It will be argued that the latter assumption is incompatible with Leibniz’s general views about logic and that the crucial proof is better reinterpreted as dealing with the necessity, possibility, and impossibility of concepts rather than of objects. In this case, the counterpart of turns out to be a theorem of Leibniz’s second order logic of concepts; but in order to obtain a full demonstration of the existence of God, the counterpart of, i.e. the self-consistency of the concept of a necessary being, remains to be proven. (shrink)

Buridan endorses the basic idea that q follows from p iff it is impossible that p is true but q is false. Since he also accepts the law that, if p is impossible, the conjunction (p ∧ q) must be impossible, he comes to regard the principle ‘Ex impossibili quodlibet’ (EIQ) as basically correct. However, his logic is based on a ‘nominalist’ view according to which propositions are tokens of spoken, written or thought language existing in space of time, and (...) the truth of such a proposition presupposes that it exists. This conception prompted Buridan to modify the definition of ‘consequence’, but none of his attempts was fully successful. Yet, a large part of his theory of consequences is based on the usual account of necessary truth-conservation. As regards principle EIQ, Buridan distinguishes two variants of impossibility. Proposition p is possible iff the state of affairs as described by p is possible; this doesn’t guarantee, however, that p is ‘possibly true’. E.g. ‘No proposition is negative’ is considered by Buridan as describing a possible state of affairs, but this proposition is not possibly true, because in order to be true, it would have to exist, and as soon as it comes into existence, it becomes false. Accordingly, EIQ requires that the antecedent p is really impossible and not only impossibly true. (shrink)

Against the prevailing opinion expressed, e.g., by L. Couturat it is argued that the so-called „intensional“ point of view which Leibniz mostly preferred to the nowadays usual extensional interpretation is neither „confuse et vague“ nor may it be made responsible for the alleged „échec final de son système“ . We present a precise definition of an „intensional“ semantics which reflects the Leibnizian ideas and which may be proven to be equivalent to standard extensional semantics.

It is well known that in his logical writings Leibniz typically disregarded the operation of disjunction, confining himself to the theory of conjunction ajid negation. Now, while this fact has been interpreted by Couturat and others as indicating a serious incompleteness of the Leibnizian calculus, it is shown in this paper that actually Leibniz's conjunction-negation logic, with 'est Ens', i. e. 'is possible' as an additional logical operator, is provably equivalent to Boolean algebra. Moreover, already in the Generales Inquisitiones of (...) 1686 Leibniz had established all basic principles that are necessary for a complete axiomatization of "Boolean" algebra. In this sense Leibniz should be acknowledged as the true inventor of the algebra of sets. (shrink)

Leibniz's development of a "calculus universalis" stands and falls with his theory of negation. During the entire period of the elaboration of the algebra of concepts, L1, Leibniz had to struggle hard to grasp the difference between propositional and conceptual negation. Within the framework of syllogistic, this difference seems to disappear because 'Omne A non B' may be taken to be equivalent to ‘Omne A est non-B’. Within the "universal calculus", however, the informal quantifier expression 'omne' is to be dropped. (...) Accordingly, ‘A non est B' expresses only the propositional negation of ‘A est B' and is hence logically weaker than ‘A est non-B'. Besides Leibniz's cardinal error of confusing propositional and conceptual negation the following issues are dealt with in this paper: -"Aristotelian" vs. "Scholastic" Syllogistic; – Metalinguistic theory of the truth-predicate; -Individual-concepts vs. concepts in general. (shrink)

After giving a short summary of the traditional theory of the syllogism, it is shown how the square of opposition reappears in the much more powerful concept logic of Leibniz. Within Leibniz’s algebra of concepts, the categorical forms are formalized straightforwardly by means of the relation of concept-containment plus the operator of concept-negation as ‘S contains P’ and ‘S contains Not-P’, ‘S doesn’t contain P’ and ‘S doesn’t contain Not-P’, respectively. Next we consider Leibniz’s version of the so-called Quantification of (...) the Predicate which consists in the introduction of four additional forms ‘Every S is every P’, ‘Some S is every P’, ‘Every S isn’t some P’, and ‘Some S isn’t some P’. Given the logical interpretation suggested by Leibniz, these unorthodox propositions also form a Square of Opposition which, when added to the traditional Square, yields a “Cube of Opposition”. Finally it is shown that besides the categorical forms, also the non-categorical forms can be formalized within an extension of Leibniz’s logic where “indefinite concepts” X, Y, Z\ function as quantifiers and where individual concepts are introduced as maximally consistent concepts. (shrink)

In “Lectio 55” of his Notule libri Priorum, Robert Kilwardby discussed various objections that had been raised against Aristotle’s Theses. The first thesis, AT1, says that no proposition q is implied both by a proposition p and by its negation, ∼p. AT2 says that no proposition p is implied by its own negation. In Prior Analytics, Aristotle had shown that AT2 entails AT1, and he argued that the assumption of a proposition p such that (∼p → p) would be “absurd”. (...) The unrestricted validity of AT1, AT2, however, is at odds with other principles which were widely accepted by medieval logicians, namely the law Ex Impossibili Quodlibet, EIQ, and the rules of disjunction introduction. Since, according to EIQ, the impossible proposition (p ∧ ∼p) implies every proposition, it also implies ∼(p∧∼p), in contradiction to AT2. Furthermore, by way of disjunction introduction, the proposition (p∨∼p) is implied both by p and by ∼p, in contradiction to AT1. Kilwardby tried to defend AT1, AT2 against these objections by claiming that EIQ holds only for accidental but not for natural implications. The second argument, however, cannot be refuted in this way because Kilwardby had to admit that every disjunction (p ∨ q) is naturally implied by its disjuncts. He therefore introduced the further requirement that, in order to constitute a genuine counterexample to AT1, (p → q) and (∼p → q) have to hold “by virtue of the same thing”. In a recently published paper, Spencer Johnston accepted this futile defence of AT1 and developed a formal semantics that would fit Kilwardby’s presumably connexive implication. This procedure, however, is misguided because the remaining considerations of Lesson 55  which were entirely ignored by Johnston  show that Kilwardby eventually recognized that AT2 is bound to fail. After all he concluded: “So it should be granted that from the impossible its opposite follows, and that the necessary follows from its opposite”. (shrink)

In many of his logical writings, G. W. Leibniz makes use of two kinds of symbols : while A, B, C, . . . stand for certain determinate or definite concepts, X, Y, Z, . . . are referred to as "indefinite concepts". We investigate the various rôles played by these variables and show i) that their most important function consists in serving as quantifiers ; ii) that Leibniz's elliptic representation of the quantifiers by means of two sorts of „indefinite (...) concepts” leads to certain difficulties; iii) that despite these problems Leibniz anticipated the most fundamental logical principles for the quantifiers and may thus be viewed as a forerunner of modern predicate logic. (shrink)

The early thirteenth century tract Ars Meliduna deals with the issue whether categorical propositions entail, or are entailed by, ‘continuing’ propositions, i.e. by implications. From the perspective of modern logic, with implication interpreted as a material, truth-functional connective, the first question has to be answered in the affirmative because, e.g. β entails (α ⊃ β). But conversely (α ⊃ β) ‘normally’ doesn’t entail the truth (or the falsity) of any of the components α, β; hence the second question should be (...) answered in the negative. These results cannot, however, be directly transferred to medieval logic because implication was there usually interpreted as a strict implication. Yet, within the framework of modal logic, one obtains parallel results. In particular, whenever β is necessary, then (α → β) must be true. Furthermore, if (α → β) is itself necessary, then it is entailed by α. Thus, in particular, ‘Socrates is a man’ entails the analytically true implication ‘If Socrates is a man, Socrates is an animal’, and the same premiss also entails the tautological consequent ‘If Socrates is white, Socrates is white’. (shrink)

In the first part of this paper a short review of the recently published 4th volume of Series 6 of the Akademie-Ausgabe of Leibniz’s Sämtliche Schriften und Briefe is given. This 3,000-page volume was edited by the Leibniz-Forschungsstelle in Münster, Germany. It contains unsurpassable, text-critical versions of more than 500 pieces which Leibniz composed between 1677 and 1690. One major topic dealt with in these essays is "Scientia Generalis, Characteristica, Calculus universalis". Here we find in particular various fragments of a (...) logical calculus that Leibniz developed around 1687. The main part of this paper presents a detailed reconstruction of this so-called "Plus-Minus-Calculus" which, by way of its somewhat unorthodox operators of "addition" and "subtraction", inclusion and identity, "communication", "commune" and "Nothing", provides an interesting alternative for the Boolean algebra of sets. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Possibility and Necessity in the Time of Peter Abelard by Irene BininiWolfgang LenzenIrene Binini. Possibility and Necessity in the Time of Peter Abelard. Investigating Medieval Philosophy Series. Leiden: Brill, 2021. Pp. xii + 326. Hardback, $166.00.This book is an impressive work written by a young Italian scholar who received her PhD only five years ago in Pisa. It is divided into three parts. Part 1 gives a survey (...) of "Early 12th-Century Debates on Modal Propositions"; part 2 is devoted to "Abelard's Modal Logic"; [End Page 327] part 3 deals with the views of "Abelard on the Metaphysics and Epistemology of Modalities." The author displays a profound knowledge of the relevant literature. The primary sources comprise twenty manuscripts preserved in various European libraries, plus forty-eight printed works (from Abelard to William of Sherwood). The list of secondary sources contains altogether 136 items from A (Astroh) to Z (Zagzebski), but, interestingly, it does not mention a monograph that, by its very title, claims to give an exposition of Abelard's logic, namely Maria Teresa Beonio-Brocchieri Fumagalli's La Logica di Abelardo (Firenze: La Nuova Italia, 1964).As far as I can judge (as a nonnative English-speaker), the linguistic quality of Binini's book is quite good. The explanations in the main text, including the translations from Latin sources, are easily comprehensible, and the many quotations (distributed over no less than 655 footnotes) are almost without any fault. Thus, the book rightly deserves inclusion in the prestigious series Investigating Medieval Philosophy. Yet, at least according to my taste, the book contains many redundancies. Again and again, the author first explains a certain issue at some length, then confirms the correctness of her interpretation by means of a longer (English) quotation, and finally reconfirms it by fully quoting the Latin text in the footnotes.As Binini explains in the book's concluding chapter, her aim is to present "an all-encompassing reflection on [Abelard's] works on modalities" and "a comparison between these works and the parallel reflections advanced by other masters and logicians in the early 12th century" (308). She undoubtedly achieves this goal. The comparison of Abelard's views on possibility and necessity with the views of other logicians is mainly carried out in chapters 1–3. Ancient and early medieval logicians such as, in particular, the Stoics and Boethius favored the basic idea that "something," p, is necessary if and only if p is always the case (or true). Accordingly, p is possible if and only if p is at least sometime the case. Abelard, however, was not willing to accept that each possibility has to become realized at some time. He instead offered a "nature-based characterization of possibility" that can "account for unrealized, past, counterfactual, and even unactualizable possibilities" (308). Abelard's idea is that something, x, is possibly a P if and only if the property of being P is compatible with the nature of x. Hence this view basically amounts to a de re conception of possibility that attributes modalities to things, while the temporal view attributes modal properties primarily to propositions (or states of affairs) and hence can be labeled de dicto (or de sensu).Let us now turn to Binini's twofold aim (as formulated on the back cover of her book) to present a "uniform and consistent" reassessment of Abelard's theory of modalities and of his modal logic. The former task is tackled in chapters 8–10. After summarizing, in chapter 8, Abelard's general views on "Nature and Modalities," chapter 9 is devoted to a detailed discussion of the "Many Senses of Possibility." In five consecutive sections, Binini investigates the conception of (1) possibility as "non-repugnancy with nature" (illustrated by the thesis that "any man can be a bishop" [235]); (2) unrealizable but "conceivable" possibilities (such as the existence of humans lacking the ability to laugh [240]); (3) determinate potencies (like that of an amputee who nevertheless "can walk" [248]); (4) past and counterfactual potencies (like that of a blind person who might be able to see); and (5) other puzzling de re possibilities (such as "irrational humans" [261]).The final... (shrink)

We first present an edition of the manuscript LH VII, B 2, 39 in which Leibniz develops a new formalism in order to give rigorous definitions of positive, of privative, and of primitive terms.This formalism involves a symbolic treatment of conceptual quantification which differs quite considerably from Leibniz’s “standard” theory of “indefinite concepts” as developed, e.g., in the “General Inquirles” In the subsequent commentary we give an interpretation and a critical evaluation of Leibniz’s symbolic apparatus. It turns out that the (...) definition of privative terms and primitive terms lead to certain inconsistencies which, however, can be avoided by slight modifications. (shrink)

In several works published between 1759 and 1782, Gottfried Ploucquet developed a logical system which deviates from traditional syllogistics in several respects. The most important features – which are critically examined in this paper – comprise.