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)

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)

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)

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)

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)

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)

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)

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)

Leibniz: Logic The revolutionary ideas of Gottfried Wilhelm Leibniz (1646-1716) on logic were developed by him between 1670 and 1690. The ideas can be divided into four areas: the Syllogism, the Universal Calculus, Propositional Logic, and Modal Logic. These revolutionary ideas remained hidden in the Archive of the Royal Library in Hanover until 1903 when the French mathematician […].

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.

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)

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)

In the essay “Principia Calculi rationalis” Leibniz attempts to prove the theory of the syllogism within his own logic of concepts. This task would be quite easy if one made unrestricted use of the fundamental laws discovered by Leibniz, e.g., in the “General Inquiries” of 1686. In the essays of August 1690, Leibniz had developed some similar proofs which, however, he considered as unsatisfactory because they presupposed the unproven law of contraposition: “If concept A contains concept B, then conversely Non-B (...) contains Non-A”. The proof in “Principia Calculi rationalis” appears to reach its goal without resorting to this law. However, it contains a subtle flaw which results from failing to postulate that the ingredient concepts have to be “possible”, i.e. self-consistent. Once this flaw is corrected, it turns out that the proof – though formally valid – would not have been approved by Leibniz because, again, it rests on an unproven principle even stronger than the law of contraposition. (shrink)

Since the total utility of an action A, U, is taken as the decisive criterion for the moral quality of A in the sense that A1 is morally better than A2 iff U > U, utilitarian ethics usually requires that we should always chose that action A* by which the total utility is maximized. This requirement, however, is fundamentally mistaken since it entails that any morally good action A1 is forbidden as long as there exists another alternative A2 which is (...) better than A1. In other words: utilitarianism leaves no room for supererogation since it is always our duty to do the best! In the present paper the historical roots of this utilitarian misconception of moral permissibility are investigated. (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 his main work Summa Logicae written around 1323, William of Ockham developed a system of propositional modal logic which contains almost all theorems of a modern calculus of strict implication. This calculus is formally reconstructed here with the help of modern symbols for the operators of conjunction, disjunction, implication, negation, possibility, and necessity.

Singer's ,Practical Ethics, is based on a form of utilitarianism which takes into account the interests of a living being if and only if it displays a minimum of rationality and consciousness. Accordingly aborting a human fetus in an early stage of development is held to be morally acceptable, whereas killing chicken, pigs, and cattle for mere culinary pleasure is not. Singer's view on abortion are refuted because they only consider the actual properties of the fetus but ignore the quality, (...) of its future life. In general the ,principle of replaceability, must be rejected. And although making animals suffer certainly is immoral, mere killing does not necessarily do so great a harm to them that we have to become vegetarians. (shrink)

Aus epistemisch-logischer Sicht reduziert sich das Fundamentalismusproblem auf die Fragen, ob die Bedingung "a's Glaube, daß p, ist fundiert" notwendig bzw. — im Verband mit "p ist wahr" und "a ist davon überzeugt, daß p " — auch hinreichend dafür ist, daß a weiß, daß p. Drei Explikationsversuche der Fundiertheitsbedingung werden untersucht: während die ersten beiden, mit den üblichen epistemischen Termen 'wissen' und 'überzeugt sein' formulierten Varianten scheitern, erweist sich die dritte Definition mittels des Terms 'evident sein' als erfolgversprechend, sofern (...) "p ist evident für a" seinerseits durch die Bedingungen "a ist davon überzeugt, daß p" und "a kann sich bezüglich p nicht irren" definiert wird. Die genauere Bestimmung des 'kann' als (alethischen) Modaloperator ist ein offenes Problem. (shrink)

. In the 18th century, Gottfried Ploucquet developed a new syllogistic logic where the categorical forms are interpreted as set-theoretical identities, or diversities, between the full extension, or a non-empty part of the extension, of the subject and the predicate. With the help of two operators ‘O’ (for “Omne”) and ‘Q’ (for “Quoddam”), the UA and PA are represented as ‘O(S) – Q(P)’ and ‘Q(S) – Q(P)’, respectively, while UN and PN take the form ‘O(S) > O(P)’ and ‘Q(S) > (...) O(P)’, where ‘>’ denotes set-theoretical disjointness. The use of the symmetric operators ‘–’ and ‘>’ gave rise to a new conception of conversion which in turn lead Ploucquet to consider also the unorthodox propositions O(S) – O(P), Q(S) – O(P), O(S) > Q(P), and Q(S) > Q(P). Although Ploucquet’s critique of the traditional theory of opposition turns out to be mistaken, his theory of the “Quantification of the Predicate” is basically sound and involves an interesting “Double Square of Opposition”. (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.

Ausgehend von Descartes' Meditationen werden die Standard-Argumente für den — auf den Bereich empirischer Propositionen beschränkten — Skeptizismus untersucht. Der Versuch einer empirischen Begründung nach dem Schema "Wir haben uns alle schon einmal geirrt, also sollten wir es für möglich halten, daß wir uns immer irren" erweist sich dabei als zirkulär; der alternative, apriorische Begründungsversuch der Art "Es ist stets (logisch) möglich, daß wir uns irren, also wissen wir nichts mit absoluter Sicherheit" beruht dagegen auf einer Verwechslung der Begriffe logischer (...) Möglichkeit und doxastischer Möglichkeit und führt zu einer unhaltbaren Konzeption "notwendigen Wissens". (shrink)