In this paper I argue that Hegel’s treatment of dialectical inferences, in particular of Plato’s dialectics in the Lectures on the History of Philosophy, belongs to the history of the logical rule that, from Gerolamo Cardano to Bertrand Russell, is known as consequentia mirabilis. In 1906 Russell formalises it as follows: and its correspondent positive form as My paper has two parts. First, I show that dialectical inferences, for Hegel, involve sentences of the form and. Hegel, following Plato, stresses (...) that these inferences are das Wunderbare, the marvellous element of dialectical reasoning. In this sense Hegel’s view belongs to the history of CM from a perspective that is both terminological and logical. Second, I stress the peculiarity of Hegel’s CM. In a classical setting, the conditional is admissible and from we can, by CM, derive, and from, by CM, infer. However, in Hegel’s analysis it is evident that we can neither infer from, nor from. Rather, in dialectical inferences what we can vali... (shrink)

The paper shows that, contrary to what has been held since the sixteenth-century mathematician Christoph Clavius, there is no application of consequentia mirabilis (CM) in Greek mathematical works. This is shown by means of a detailed discussion of the logical structure of the proofs where CM is allegedly employed. The point is further enlarged to a critical assessment of the unsound methodology applied by many interpreters in seeking for specific logical rules at work in ancient mathematical texts.

In Categories 12 (14b11–22), Aristotle famously claims that [1] true sentences and reality stand in a mutually implicative relationship, and that [2] reality causes the truth of sentences but not vice versa. In this paper, I first argue that Boethius’ reading of the above passage led medieval logicians to assess [1] and [2] within the framework of a theory of consequence. Then, I consider two important questions raised by Boethius and later logicians in relation to [1] and [2], and, namely, (...) whether the implication described in [1] really satisfies the requirement for consequence, and whether the truthmaking relation posited in [2] can be boiled down to the relation of consequence. Boethius himself, I argue, answered affirmatively to both questions and thus (much like some contemporary truthmaker theorists) committed himself to an account of truthmaking as entailment. I then consider criticisms to Boethius’ account found in the writings of William of Champeaux, two anonymous authors connected to the school of Jocelyn of Soissons, Peter Abelard, and the author of the Ars Meliduna. Here, I argue that the emergence of a concern for relevance drives the need for abandoning Boethius’ account as well as his conception of truthmaking as entailment. (shrink)

I examine the theory of consequentia of the medieval logician, John Buridan. Buridan advocates a strict commitment to what we now call proposition-tokens as the bearers of truth-value. The analysis of Buridan's theory shows that, within a token-based semantics, amendments to the usual notions of inference and consequence are made necessary, since pragmatic elements disrupt the semantic behaviour of propositions. In my reconstruction of Buridan's theory, I use some of the apparatus of modern two-dimensional semantics, such as two-dimensional matrices (...) and the distinction between the context of formation and the context of evaluation of utterances. (shrink)

The overarching aim of this excellent book is to demonstrate the common ground between medieval logic and logical theories of the twentieth century by analyzing some important medieval approaches to three important topics in medieval logic and then showing that in each case, once we determine what is really going on in the medieval theory, it can be formalized in such a way as to show how it resembles one or more developments in twentieth-century logical theory. Analysis in terms of (...) modern logical theory has a lot to offer the study of medieval logical theories, the author claims, and twentieth-century theory can learn some interesting lessons from examining its medieval counterparts. Much of the material in these specific discussions has been presented in earlier versions in the author's published articles. "The Philosophy of Formalization" , however, is new. English translations are used in the text, though the Latin is supplied in the footnotes. (shrink)

This book presents novel formalizations of three of the most important medieval logical theories: supposition, consequence and obligations. In an additional fourth part, an in-depth analysis of the concept of formalization is presented - a crucial concept in the current logical panorama, which as such receives surprisingly little attention.Although formalizations of medieval logical theories have been proposed earlier in the literature, the formalizations presented here are all based on innovative vantage points: supposition theories as algorithmic hermeneutics, theories of consequence analyzed (...) with tools borrowed from model-theory and two-dimensional semantics, and obligations as logical games. For this reason, this is perhaps the first time that these medieval logical theories are made fully accessible to the modern philosopher and logician who wishes to obtain a better grasp of them, but who has always been held back by the lack of appropriate ‘translations' into modern terms.Moreover, the book offers a reflection on the very nature of logic, a reflection that is prompted by the comparisons between medieval and modern logic, their similarities and dissimilarities. It is thus a contribution not only to the history of logic, but also to the philosophy of logic, the philosophy of language and semantics.The analysis of medieval logic is also relevant for the modern philosopher and logician in that, being the unifying methodology used across all disciplines at that time, logic really provided unity to science. It thus presents a unified model of scientific investigation, where logic plays the aggregating role. (shrink)

_ Source: _Volume 56, Issue 3-4, pp 272 - 291 This paper offers an analysis of Marsilius of Inghen’s definition of _consequentia_ and of his treatment of logical validity as presented in the first book of his treatise on _Consequentiae_. Comparing Marsilius of Inghen’s, John Buridan’s, and Albert of Saxony’s theories, the author argues that Marsilius’ account is based on a conception of consequence as a relation of entailment among propositions rather than as a type of conditional sentence and, thus, (...) moves the discussion away from the sentential level. Therefore, Marsilius’ theory represents an original and important contribution to fourteenth-century discussions on consequences. (shrink)

In 1898 C. S. Peirce declares that the medieval doctrine of consequences had been the starting point of his logical investigations in the 1860s. This paper shows that Peirce studied the scholastic theory of consequentiae as early as 1866–67, that he adopted the scholastics’ terminology, and that that theory constituted a source of logical doctrine that sustained Peirce for a lifetime of creative and original work.

In this paper, the author offers an introduction to Marsilius of Inghen’s treatment of expositiones of sentences de incipit and de desinit in his treatise on Consequentiae, with an analysis of the various modi exponendi presented by Marsilius and an edition of the text. The author argues that, in the split between physical and logical approaches to the issues arising in analyses of incipit and desinit, Marsilius’ theory presents some hybrid features, but tends towards the logical end of the spectrum.

In modern times the so?called consequentia mirabilis (if not-P, then P). then P) was first enthusiastically applied and commented upon by Cardano (1570) and Clavius (1574). Of later passages where it occurs Saccheri?s use (1697) has drawn a good deal of attention. It is less known that about the middle of the 17th century this remarkable mode of arguing became the subject of an interesting debate, in which the Belgian mathematician Andreas Tacquet and Christiaan Huygens were the main representatives (...) of opposite views concerning its probative force. In this article the several phases and moves of that debate are delineated. (shrink)

While knowledge of mere possibilities is difficult to understand, knowledge of possibilities that are actual seems unproblematic (as far as we know the actual world). The principle that what is actual is possible has been near-universally accepted. After summarizing some sporadic dissent, I present a proposal for how the validity of the principle might be restricted. While the principle certainly holds for sufficiently inclusive objective and epistemic possibilities, it may not hold when the accessibility of possibilities is contextually restricted.

Among the important conceptual innovations introduced in the second scholasticism era and motivated by theological debates following the Council of Trent were the theories of moral necessity and moral implication. As they were centred upon a view of moral necessity as a form of necessity weaker than physical necessity, and moral implication as weaker than physical implication, some interpretations of moral necessity encouraged the logic of statistical hypotheses and probability. Three branches of this debate are studied in this paper: the (...) explanation of moral necessity in terms of suppositio, the confrontation over the interpretation of moral necessity, and the theory of statistical quantification. (shrink)

In epoca medievale si è molto discusso su alcuni concetti chiave come prova o giustificazione. La teoria della prova contenuta nei trattati di logica medievale prende il nome tecnico di consequentia, che è un tipo di ragionamento fondato sul passaggio dalla concessione (o negazione) di uno o più enunciati denominati antecedenti alla concessione (o negazione) di uno o più enunciati denominati conseguenti. Questo tipo di teoria ha avuto un correlato a livello dei singoli termini che compongono l’enunciato all'interno della (...) teoria nota come suppositio terminorum. Prende, infatti, il nome di descensus, la regola che permette la verifica del passaggio da un termine maggiore (un termine comune) ad un termine minore (che denota individui). La stessa definizione di alcune ramificazioni della teoria della suppositio è col-legata alla possibilità di ammettere o meno il descensus cioè questa "discesa" da un termine comune ad una congiunzione o disgiunzione di termini singolari. Il presente lavoro intende descrivere la teoria del descensus come un tipo particolare di consequentia mostrandone le connessioni con le moderne tecniche logiche. I trattati che si affronteranno appartengono al XIV e al XV secolo, cioè ad un periodo caratterizzato da una sistemazione della materia che li rende confrontabili con un quadro teorico unitario. (shrink)

_ Source: _Volume 53, Issue 2-4, pp 353 - 371 While agreeing with Professor D’Ors’ thesis that the notion of logical consequence cannot be exhaustively characterized, I depart from Professor d’Ors’ conclusion that the very notion of good consequence is primitive and can only be identified with the set of acceptable rules of inference, and from his conviction that modal notions such as necessity and impossibility are equivocal and gain such clarity as they have by their interaction with rules of (...) inference. Inspired by this picture, Professor d’Ors undertook an examination of a number of medieval attempts to analyze the notion of consequence and tried to show how certain developments in the medieval history of logic made sense in the light of debate over such analyses. This paper examines a small fragment of Professor d’Ors programme and its relation to some aspects of Jean Buridan’s account of the consequence relation. (shrink)

In these pages, we expose the main traits of St. Albert the Great’s doctrine of providence and fate, considered by Palazzo the keystone of his philosophical system. To describe it we examine his systematic works, primarily his Summa of Theology. His discussion follows clearly the guidelines of the Summa of Alexander of Hales, in order to delve into the set of problems faced over the centuries by theological tradition. Albert also restates the reflections of different authors like Boethius or Saint (...) John of Damascus but, in his Summa he incorporates to his reflections also the noteworthy book of Nemesius of Emesa, De natura hominis, which includes some pages on providence. Albert gives his personal solution to the complex questions of providence, destiny and contingency of the world. His conception of providence is developed in the frame of the creative power of the almighty God. God’s knowledge is necessary and inerrant and his providential purposes are infallible, but that does not mean that every event is necessary. He does not communicate His own proprieties to the creatures. In order to understand this problem, Albert recalls the notion of hypothetical necessity coined by Boethius in an Aristotelian framework and the difference between 'necessitas consequentis' and 'necessitas consequentiae' proposed by Alexander of Hales. He also develops his account of providence, closely linked to the topic of fate. However, it would be exaggerated to deem his position deterministic. (shrink)

Leibniz' Metaphysik sieht sich mit folgender Aporie konfrontiert: (1) Notwendigerweise gilt: Gott verwirklicht die beste mögliche Wahl. (2) Notwendigerweise gilt: die mögliche Welt, die auf die Beschreibung der tatsächlichen Welt antwortet, ist die beste aller möglichen Welten. (3) Die Existenz der tatsächlichen Welt (als die von Gott erschaffene) ist nicht notwendig, sondern möglich. Diese drei Sätze - die alle von Leibniz anerkannt worden sind - sind gemeinsam allem Anschein nach inkonsistent. Um diese Inkonsistenz zu vermeiden, entwickelt Leibniz eine komplexe Strategie, (...) welche die beiden Unterscheidungen zwischen moralischer und metaphysischer Notwendigkeit einerseits und zwischen der Notwendigkeit der Folgerung (necessitas consequentiae) und der Notwendigkeit des Gefolgerten (necessitas consequentis) andererseits einbezieht. In dem er dieses herausarbeitet, entwickelt Leibniz eine komplexe Theorie, welche die Teilmöglichkeiten in den Zusammenhang eines unendlichen Analyseprozesses stellt. (shrink)

Construing miracles as “violations,” I argue that a law of nature must specify some kind of possibility. But we must have here a sense of possibility for which the ancient rule of logic---ab esse ad posse valet consequentia---does not hold. We already have one example associated with the concept of statute law, a law which specifies what is legally possible but which is not destroyed by a violation. If laws of nature are construed as specifying some analogous sense of (...) what is naturally possible, then they need not be invalidated by a (rare) violation, and Humean miracles remain a genuine possibility. (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)

Construing miracles as \textquotedblleft{}violations,\textquotedblright I argue that a law of nature must specify some kind of possibility. But we must have here a sense of possibility for which the ancient rule of logic---ab esse ad posse valet consequentia---does not hold. We already have one example associated with the concept of statute law, a law which specifies what is legally possible but which is not destroyed by a violation. If laws of nature are construed as specifying some analogous sense of (...) what is naturally possible, then they need not be invalidated by a (rare) violation, and Humean miracles remain a genuine possibility. (shrink)

This paper analyzes and evaluates Bolzano's remarks on the apagogic method of proof with reference to his juvenile booklet "Contributions to a better founded presentation of mathematics" of 1810 and to his ?Theory of science? (1837). I shall try to defend the following contentions: (1) Bolzanos vain attempt to transform all indirect proofs into direct proofs becomes comprehensible as soon as one recognizes the following facts: (1.1) his attitude towards indirect proofs with an affirmative conclusion differs from his stance to (...) indirect proofs with a negative conclusion; (1.2) by Bolzano's lights arguments via consequentia mirabilis only seem to be indirect. (2) Bolzano does not deny that indirect proofs can be perfect certifications (Gewissmachungen) of their conclusion; what he denies is rather that they can provide grounds for their conclusions. (2.1) They cannot do the latter, since they start from false premises and (2.2) since they make an unnecessary detour. (3) The far-reaching agreement between his early and late assessment of apagogical proofs (in the Beyträge of 1810 and the Wissenschaftslehre of 1837) is partly due to the fact that he develops his own position always against the background of Wolff's and Lambert's views. (shrink)

The paper focuses on the concepts of truth, truth-making and truth-preservation and their role in defining deductive validity as analysed by the late-medieval nominalist scholar Martin Le Maistre in his Tractatus consequentiarum. This treatise, examined from the point of view of fourteenth-century British and Parisian influences, can be characterised as a critical adoption of the previous logical tradition and as the analysis of validity in term of truth-preservation. Part of this analysis is a study of self-referential phenomena, in particular, of (...) self-referential inferences which are addressed in terms of a Bradwardinian implicit-meaning analysis of self-reference by Le Maistre. Also, his analysis of “consequentia formalis” summarises the fourteenth-century development of the discussion and compares alternative approaches towards formality. (shrink)

La teoria delle obbligazioni rappresenta un significativo esempio di come un insieme di tecniche logiche possa essere applicato, in maniera feconda, a contesti disputazionali in cui la dimensione dialogica e l’interazione tra soggetti coinvolti in un confronto dialettico hanno un ruolo di primo piano.L’attenzione che numerosi studiosi hanno manifestato nei confronti di questa parte della logica modernorum negli ultimi quarat’anni testimonia la profondità e la difficoltà dei problemi teorici che stanno alla base della teoria.Molte interpretazioni sono state avanzate circa le (...) motivazioni che spingono i logici medievali a scrivere trattati in cui il portato delle loro dottrine logiche – come, ad esempio, la teoria delle consequentiae - viene declinato ed elaboratamente applicato al gioco dialettico tra due disputanti. La gamma di tali letture comprende differenti ipotesi: la teoria delle obbligazioni potrebbe fornire un nucleo di regole per l’impostazione di un metodo assiomatico rudimentale così come rappresentare un tentativo di elaborare esercizi volti a valutare le capacità logiche degli studenti o, ancora, un gioco teorico, svincolato dall’effettiva applicazione a contesti concreti, la cui finalità sarebbe il mantenimento della coerenza logica.E’ stata sostenuta da più interpreti la funzione pedagogica svolta dalla teoria, al di là del fatto che si tratti o meno di un metodo per la conduzione effettiva di dispute reali. Un’altra linea interpretativa collega la teoria delle obbligazioni al contesto della letteratura dei sophismata e degli insolubilia, dal cui repertorio, in effetti, sono spesso tratti molti degli esempi che si incontrano nella lettura dei trattati de obligationibus. In questo senso, la teoria offrirebbe regole e schemi di risposta che un interlocutore deve utilizzare per risolvere o rispondere a enunciati problematici o paradossali. Non si può escludere, inoltre, che nella teoria si realizzi un intreccio delle differenti motivazioni sopra esposte: potrebbe trattarsi di uno strumento che viene sviluppato tanto per la risoluzione di sophismata e insolubilia quanto per indagare il rapporto tra linguaggio e metalinguaggio o altre questioni teoriche. Altri studiosi hanno infine sostenuto che lo scopo della teoria delle obbligazioni consisterebbe nell’elaborazione di un metodo adatto alla costruzione e alla discussione di esperimenti mentali e che questo costituirebbe una sorta di metametodologia della scienza medievale sulla linea del ragionamento secundum imaginationem. (shrink)

conditional with his discussions of the hypothetical proposition. Peirce spoke often of the consequentia de inesse ,1 the concept of which is intimately linked with the material, or "Philonian" conditional; indeed, we shall see him calling himself a Philonian. And it is not uncommon to hear Peirce—at least prior to the last decade of his life—declared a Philonian, whose fundamental analysis of the conditional was essentially the same as that of Philo (and of more modern types like Russell and (...) like Quine). (shrink)

In this paper I shall attempt to show that the difference which Boethius has in mind when distinguishing between conditional and simple necessity in the fifth book of Philosophiae Consolatio is not, as already Thomas Aquinas presumed, the difference between a necessitas consequentiae and the corresponding necessitas consequentis, but rather the difference between the necessity of a given state of affairs which is necessary in the sense that at the moment of its existence its present existence is unchangeable, and the (...) necessity of a given state of affairs which is necessary in the sense that even prior to the moment of its existence its future existence is unavoidable. (shrink)

In an earlier article (see J Gen Philos Sei (2010) 41: 341-355) I have compared Aristotle's syllogistic with Kant's theory of "pure ratiocination". "Ratiocinia pura" („reine Vernunftschlüsse") is Kant's designation for assertoric syllogisms Aristotle has called 'perfect'. In Kant's view they differ from non-pure ratiocinia precisely in that their validity rests only on the validity of the Dictum de omni et nullo (which, however, in Kant's view can be further reduced to more fundamental principles) whereas the validity of non-pure ratiocinia (...) additionally presupposes the validity of inferences which Kant calls consequentiae immediatae. I have argued that Kant's view is in some (not in all) essential features in accordance with Aristotle's view concerning perfect syllogisms and certainly leading to a tenable and interesting logical theory. As a result I have rejected not only the interpretation of Aristotle adopted by Theodor Ebert, but also the objections he has raised against Kant's logical theory. As far as Aristotle is concerned, Ebert has attempted to defend his position in the first part of his reply to my article published in J Gen Philos Sei (2009) 40: 357-365, and I have argued against this defence in issue 1 of the J Gen Philos Sei (2010) 41: 199-213 (cf. Ebert's answer in the same issue pp. 215-231). In the following discussion I deal with Eberts defence of his criticism of Kant published in the second part of his reply to my article (see J Gen Philos Sei (2009) 40: 365-372). I shall argue, that Kant's principle 'nota notae est nota rei ipsius' and his use of technical vocabulary stand up to the objections raised by Ebert. His attempts to prove that Kant's logical theory is defective are based on several misinterpretations. (shrink)

This paper examines Walter Chatton’s discussion of the problem of prophesied future contingents in his Lectura super Sententias. Faced with the challenge of reconciling the supposedly veridical character of divine prophecy with human freedom to do otherwise, Chatton casts the relation of prophecy to event in the form of a logical consequentia and formulates two rules which depend on the character of the antecedent in question. In the case of antecedents involving divine knowledge and related phenomena, the freedom of (...) the wayfarer to do otherwise is secured by the assumption that the consequence is necessary but the antecedent is semantically contingent. In the case of concrete utterances and phenomena, on the other hand, the wayfarer’s freedom is secured by the assumption that the consequence itself is contingent. Chatton’s treatment, while analytically subtle and rigorous, leaves a number of important questions unanswered, most notably that of the ontological openness or closure of the future. Nevertheless, it is interesting both in its own right and insofar as it provides a clarifying source for Chatton’s later discussion of the same topic in his Quodlibet. (shrink)

This paper analyzes and evaluates Bolzano's remarks on the apagogic method of proof with reference to his juvenile booklet ‘Contributions to a better founded presentation of mathematics’ of 1810 and to his ‘Theory of science’ (1837). I shall try to defend the following contentions: (1) Bolzanos’ vain attempt to transform all indirect proofs into direct proofs becomes comprehensible as soon as one recognizes the following facts: (1.1) his attitude towards indirect proofs with an affirmative conclusion differs from his stance to (...) indirect proofs with a negative conclusion; (1.2) by Bolzano's lights arguments via consequentia mirabilis only seem to be indirect. (2) Bolzano does not deny that indirect proofs can be perfect certifications (Gewissmachungen) of their conclusion; what he denies is rather that they can provide grounds for their conclusions. (2.1) They cannot do the latter, since they start from false premises and (2.2) since they make an unnecessary detour. Equation(3) The far-reaching agreement between his early and late assessment of apagogical proofs (in the Beyträge of 1810 and the Wissenschaftslehre of 1837) is partly due to the fact that he develops his own position always against the background of Wolff's and Lambert's views. (shrink)

This study based on manuscripts uncovers a network of Parisian Masters of Arts of the late fourteenth and early fifteenth centuries, especially in the period covered by the missing fifth Register of the English Nation of the Parisian Arts Faculty. Outstanding masters are Henry of Oy, Christian of Paris, John of Hokelem, James of Yvia, John of Paris and others. The life and work of John of Hokelem are critically studied here for the first time. The dominating figure is, however, (...) Marsilius of Inghen with his treatises on the Parva logicalia, especially the Suppositiones and the Consequentiae. Perhaps the list of works by Marsilius is not yet complete. Among the most important of the many topics is the semantics of terms like white, is and being. Among the wealth of interesting details are testimonies to Thomas of Cleve’s missing Logic, early occurrences of the names terministae vel... (shrink)

Aristotle's logical and metaphysical works contain elements of three distinct types of formal theory: an ontology, a theory of consequences, and a theory of reasoning. His formal ontology (unlike that of certain later thinkers) does not require all propositions of a given logical form to be true. His formal syllogistic (unlike medieval theories of consequences) was guided primarily by a conception of logic as a theory of reasoning; and his fragmentary theory of consequences exists merely as an adjunct to the (...) syllogistic. When theories of consequences took centre stage in the Middle Ages, the original motivation for the theory of the syllogism was forgotten. (shrink)