The fundamental problem of Christology is the apparent contradiction of Christ as recorded at Chalcedon. Christ is human and Christ is divine. Being divine entails being immutable. Being human entails being mutable. Were Christ two different persons there’d be no apparent contradiction. But Chalcedon rules as much out. Were Christ only partly human or only partly divine there’d be no apparent contradiction. But Chalcedon rules as much out. Were the very meaning of ‘mutable’ and/or ‘immutable’ other than what they are, (...) there’d be no apparent contradiction. But the meaning is what it is, and changing the meaning of our terms to avoid the apparent contradiction of Christ is an apparent flight from reality. What, in the end, is the explanation of the apparent contradiction of Christ? Theologians and philosophers have long advanced many consistency-seeking answers, all of which increase the metaphysical or semantical complexity of the otherwise strikingly simple but radical core of Christianity’s GodMan. In this paper, I put the simplest explanation on the theological table: namely, Christ appears to be contradictory because Christ is contradictory. This explanation may sound complicated to the many who are steeped in the mainstream account of logic according to which logic precludes the possibility of true contradictions. But the mainstream account of logic can and should be rejected. Ridding theology of the dogma of mainstream logic illuminates the simple though striking explanation of the apparent contradiction of Christ — namely, that Christ is a contradictory being. Just as the simplest explanation to the apparent roundness of the earth has earned due acceptance, so too should the simplest explanation of the apparent contradiction of Christ. (shrink)

Christine Ladd-Franklin is often hailed as a guiding star in the history of women in logic—not only did she study under C. S. Peirce and was one of the first women to receive a PhD from Johns Hopkins, she also, according to many modern commentators, solved a logical problem which had plagued the field of syllogisms since Aristotle. In this paper, we revisit this claim, posing and answering two distinct questions: Which logical problem did Ladd-Franklin solve in her thesis, and (...) which problem did she think she solved? We show that in neither case is the answer “a long-standing problem due to Aristotle.” Instead, what Ladd-Franklin solved was a problem due to Jevons that was first articulated in the nineteenth century. (shrink)

In 1672 John Eliot, English Puritan educator and missionary to New England, published _The Logick Primer: Some Logical Notions to initiate the INDIANS in the knowledge of the Rule of Reason; and to know how to make use thereof_. This roughly 80 page pamphlet introduces syllogistic vocabulary and reasoning so that syllogisms can be created from Biblical texts. The use of logic for proselytizing purposes is not distinctive: What is distinctive about Eliot's book is that it is bilingual, written in (...) both English and Massachusett (Wôpanâak), an Algonquian language spoken in eastern coastal and southeastern Massachusetts. It is one of the earliest bilingual logic textbooks and it is the first, and perhaps only, textbook in an indigenous American language. -/- In this paper, we (1) introduce John Eliot and the linguistic context he was working in; (2) introduce the contents of the _Logick Primer_---vocabulary, inference patterns, and applications; (3) discuss notions of 'Puritan' logic that inform this primer; and (4) address the importance of his work in documenting and expanding the Massachusett language and the problems that accompany his colonial approach to this work. (shrink)

Dialogue semantics for logic are two-player logic games between a Proponent who puts forward a logical formula φ as valid or true and an Opponent who disputes this. An advantage of the dialogical approach is that it is a uniform framework from which different logics can be obtained through only small variations of the basic rules. We introduce the composition problem for dialogue games as the problem of resolving, for a set S of rules for dialogue games, whether the set (...) of S-dialogically valid formulas is closed under modus ponens. Solving the composition problem is fundamental for the dialogical approach to logic; despite its simplicity, it often requires an indirect solution with the help of significant logical machinery such as cut-elimination. Direct solutions to the composition problem can, however, sometimes be had. As an example, we give a set N of dialogue rules which is well-justified from the dialogical point of view, but whose set N of dialogically valid formulas is both non-trivial and non-standard. We prove that the composition problem for N can be solved directly, and introduce a tableaux system for N. (shrink)

Though Arthur Prior is now best known for his founding of modern temporal logic and hybrid logic, much of his early philosophical career was devoted to history of logic and historical logic. This interest laid the foundations for both of his ground-breaking innovations in the 1950s and 1960s. Because of the important rôle played by Prior's research in ancient and medieval logic in his development of temporal and hybrid logic, any student of Prior, temporal logic, or hybrid logic should be (...) familiar with the medieval logicians and their work. In this article we give an overview of Prior's work in ancient and medieval logic. (shrink)

In, Kosterec attempts to provide ``model-theoretic proofs'' of certain theses involving the normal modal operators $\Diamond$ and $\square$ and the truth-in-fiction operator $F$ which he then goes on to show have counterexamples in Kripke models. He concludes from this that the embedding of modal logic under the truth-in-fiction operator is unsound. We show instead that it is the ``model-theoretic proofs'' that are themselves unsound, involving illicit substitution, a subtle error that nevertheless allows us to draw an important conclusion about intensional (...) contexts and semantic equivalences. (shrink)

This review article offers a discussion of some aspects of the historical and conceptual context when the term “ontology” (Lat. ontologia) was first introduced in the scholarly circles of the early 17th century. In particular, Barry Smith's (2022) analysis of the birth of ontology provides a springboard for some further remarks on the author of the work with the first known occurrence of the word “ontologia”, Jacob Lorhard, including an analysis of his relationship with earlier philosophers Petrus Ramus and Clemens (...) Timpler. (shrink)

Formal dialogue systems model rule-based interaction between agents and as such have multiple applications in multi-agent systems and AI more generally. Their conceptual roots are in formal theories of natural argumentation, of which Hamblin’s formal systems of argumentation in Hamblin (Fallacies. Methuen, London, 1970, Theoria 37:130–135, 1971) are some of the earliest examples. Hamblin cites the medieval theory of obligationes as inspiration for his development of formal argumentation. In an obligatio, two agents, the Opponent and the Respondent, engage in an (...) alternating-move dialogue, where the Respondent’s actions are governed by certain rules, and the goal of the dialogue is establishing the consistency of a proposition. We implement obligationes in the formal dialogue system framework of Prakken (Knowl Eng Rev 21(2):163–188, 2006) using Dynamic Epistemic Logic (van Ditmarsch et al. in Dynamic epistemic logic, Synthese Library Series. Springer, Berlin, 2007). The result is a new type of inter-agent dialogue, for consistency-checking, and analyzing obligationes in this way also sheds light on interpretational and historical questions concerning their use and purpose in medieval academia. (shrink)

Temporal logic as a modern discipline is separate from classical logic; it is seen as an addition or expansion of the more basic propositional and predicate logics. This approach is in contrast with logic in the Middle Ages, which was primarily intended as a tool for the analysis of natural language. Because all natural language sentences have tensed verbs, medieval logic is inherently a temporal logic. This fact is most clearly exemplified in medieval theories of supposition. As a case study, (...) we look at the supposition theory of Lambert of Lagny, extracting from it a temporal logic and providing a formalization of that logic. (shrink)

The struggle to delineate the relationship between theology and logic flourished in the thirteenth century and culminated in two condemnations in early 1277, one in Paris and the other in Oxford. To see how much and what kind of effect ecclesiastical actions such as condemnations and prohibitions to teach had on the development of logic in the Middle Ages, we investigate the events leading up to the 1277 actions, the condemned propositions, and the parts of these condemnations connected to modal (...) and temporal logic specifically. We show that because of the specific motivations late thirteenth-century and fourteenth-century logicians had when working in modal and temporal logic, the effect of the 1277 condemnations on the development of those branches was much smaller than might have been supposed. (shrink)

The medieval distinction between categorematic and syncategorematic words is usually given as the distinction between words which have signification or meaning in isolation from other words and those which have signification only when combined with other words . Some words, however, are classified as both categorematic and syncategorematic. One such word is Latin infinita ‘infinite’. Because infinita can be either categorematic or syncategorematic, it is possible to form sophisms using infinita whose solutions turn on the distinction between categorematic and syncategorematic (...) uses of infinita. As a result, medieval logicians were interested in identifying correct logical rules governing the categorematic and syncategorematic uses of the term. In this paper, we look at 13th–15th-century logical discussions of infinita used syncategorematically and categorematically. We also relate the distinction to other medieval distinctions with which it has often been conflated in modern times, and show how and where these conflations go wrong. (shrink)

In the 4th century BC, the Greek philosopher Diodoros Chronos gave a temporal definition of necessity. Because it connects modality and temporality, this definition is of interest to philosophers working within branching time or branching space-time models. This definition of necessity can be formalized and treated within a logical framework. We give a survey of the several known modal and temporal logics of abstract space-time structures based on the real numbers and the integers, considering three different accessibility relations between spatio-temporal (...) points. (shrink)

Recently logic has shifted emphasis from static systems developed for purely theoretical reasons to dynamic systems designed for application to real world situations. The emphasis on the applied aspects of logic and reasoning means that logic has become a pragmatic tool, to be judged against the backdrop of a particular application. This shift in emphasis is, however, not new. A similar shift towards “interactive logic” occurred in the high Middle Ages. We provide a number of different examples of “interactive logic” (...) in the Middle Ages, all species of the disputation game obligatio. These games display a recognition of the importance of interaction in logical contexts and the way that interactive logic differs from single-agent inference. (shrink)

We present Prior's discussion of a puzzle about valditity found in the writings of the fourteenth-century French logician Jean Buridan and show how Prior's study of this puzzle may have provided the conceptual inspiration for his development of hybrid logic.

The current trend in knowledge revision in the Dynamic Epistemic Logic tradition focuses on the addition of new knowledge, rather than the possibility of losing knowledge. Yet there are natural situations, such as an agent who does not want another agent to know that she knows a certain piece of information, where there is a need to be able to model the retraction of a proposition from a knowledge base. One situation where this is systematically required is the variant of (...) the medieval theory of obligationes known as dubitatio, where one of the agents in the dialogue is obliged to hold the initial proposition, the dubitatum, as doubtful, regardless of what other propositions she may concede or deny during the disputation. In this paper, we use dubitatio as a motivation for studying deceitful agents, and we discuss various ways that an agent can move from a model where f is known to one where f is not known. (shrink)

We give a simple definition of validity for syllogisms involving necessary and assertoric premises which validates all and only the Aristotelian apodeictic syllogisms.

In the early 1980s, Paul V. Spade advanced the thesis that obligational reasoning was counterfactual reasoning, based upon his interpretation of the obligationes of Walter Burley, Richard Kilvington, and Roger Swyneshed. Eleonore Stump in a series of contemporary papers argued against Spade’s thesis with respect to Burley and Swyneshed, provisionally admitting it for Kilvington with the caveat that Kilvington’s theory is by no means clear or non-idiosyncratic. In this paper, we revisit the connection between counterfactual reasoning and obligationes, focusing on (...) one particular treatise, the anonymous early twelfth-century Obligationes Parisienses edited by L.M. de Rijk in the late 70s. We show that while positio in this treatise does not involve counterfactual reasoning, the species sit verum or rei veritas apparently does, and it is precisely this which distinguishes the two species in this treatise. (shrink)

We argue that Kraay's "theistic multiverse" response to the objections to theism [Kraay 2011] is unsuccessful as it simply shifts the problems leveled against theism from the level of possible worlds to the level of possible universes. Furthermore, when we restate the objections at the level of possible universes, we can show how Kraay's conclusion about the uniqueness of the theistic multiverse is undermined.

Fictional discourse and fictional languages provide useful test cases for theories of meaning. In this paper, we argue against truth-conditional accounts of meaning on the basis of problems posed by language(s) of fiction. It is well-known how fictional discourse -- discourse about non-existent objects -- poses a problem for truth-conditional theories of meaning. Less well-considered, however, are the problems posed by fictional languages, which can be created to either be meaningful or not to be meaningful; both of these ultimately also (...) provide problems for a truth-conditional account of meaning, because it cannot account for the ways in which we use and evaluate such fictional languages. (shrink)

While for a long time the study of medieval logic focused on editorial projects and reconstructions of central medieval doctrines such as the theories of signification, supposition, consequences, and obligations, nowadays the spectrum of analysis has broadened and is increasingly informed by modern logical research, whose perspective is then applied to medieval logic. Promoting this tendency, logicians and researchers concerned with semantics in the Gesellschaft für Philosophie des Mittelalters und der Renaissance (GPMR) founded a working group bringing together medieval logic (...) and modern applied logic. The present volume is a seminal document of these interests and activities. It analyzes theories in medieval logic which are useful for solving questions of recent logic and explains crucial parts of medieval logic, philosophy, and theology by applying techniques of present-day logic. (shrink)

Review of Adamson Peter, Medieval Philosophy, History of Philosophy Without Any Gaps, volume 4, xxii+637pp. Reviewed by Sara L. Uckelman, Department of Philosophy, Durham University.

In this paper we look at the suitability of modern interval-based temporal logic for modeling John Buridan’s treatment of tensed sentences in his Sophismata. Building on the paper [Øhrstrøm 1984], we develop Buridan’s analysis of temporal logic, paying particular attention to his notions of negation and the absolute/relative nature of the future and the past. We introduce a number of standard modern propositional interval temporal logics to illustrate where Buridan’s interval-based temporal analysis differs from the standard modern approaches. We give (...) formal proofs of some claims in [Øhrstrøm 1984], and sketch how the standard modern systems could be defined in terms of Buridan’s proposals, showing that his logic can be taken as more basic. (shrink)

In this short note we respond to the claim made by Christopher Viger in [4] that Anselm’s so-called ontological argument falls prey to Russell’s paradox. We show that Viger’s argument is based on a flawed premise and hence does not in fact demonstrate what he claims it demonstrates.

In this paper we look at the suitability of modern interval-based temporal logic for modeling John Buridan’s treatment of tensed sentences in his Sophismata. Building on the paper, we develop Buridan’s analysis of temporal logic, paying particular attention to his notions of negation and the absolute/relative nature of the future and the past.We introduce a number of standard modern propositional interval temporal logics to illustrate where Buridan’s interval-based temporal analysis differs from the standard modern approaches. We give formal proofs of (...) some claims in, and sketch how the standard modern systems could be defined in terms of Buridan’s proposals, showing that his logic can be taken as more basic. In diesem Aufsatz betrachten wir das Geeignetsein einer modernen intervallbasierten temporalen Logik für die Modellierung von John Buridans Behandlung zeitabhängiger Sätze in seinen Sophismata. Aufbauend auf den Aufsatz stellen wir Buridans Analyse der temporalen Logik dar, mit besonderer Berücksichtigung seiner Begriffe der Negation und der absoluten/relativen Natur der Zukunft und der Vergangenheit. Wir führen einige standardmäßige moderne intervallbasierte temporale Aussagenlogiken ein, um zu illustrieren, wo Buridans intervallbasierte temporale Analyse von den standardmäßigen modernen Zugangsweisen abweicht. Wir geben formale Beweise einiger Behauptungen in an und skizzieren, wie die standardmäßigen modernen Systeme im Sinne der Vorschläge Buridans definiert werden könnten; wir zeigen damit, dass seine Logik als die grundlegendere aufgefasst werden kann. (shrink)