Books by Jacob Archambault
Consequences in medieval logic
Vivarium, 2018
A special issue of Vivarium on medieval contributions to the theory of consequence.
Proceedings of the Society of Medieval Logic and Metaphysics, 2018
Mereology is the metaphysical theory of parts and wholes, including their conditions of identity ... more Mereology is the metaphysical theory of parts and wholes, including their conditions of identity and persistence through change. Hylomorphism is the metaphysical doctrine according to which all natural substances, including living organisms, consist of matter and form as their essential parts, where the substantial form of living organisms is identified as their soul. The theories date to Plato and Aristotle and figure prominently in the history of philosophy up until the seventeenth century, where their influence wanes relative to a reductive materialism that culminates with deflationary accounts of objects and persons, where mere conglomerates constitute things and we are left to account for mental phenomena in terms of the powers of physical materials. In view of such difficulties, there is a renewed interest in hylomorphism, as its forms structure matter and can account for natural kinds, with their various capacities and powers. This volume presents medieval theories of hylomorphism and mereology, articulating the conceptual framework in which they developed and with an eye on their relevance today.
Dissertation by Jacob Archambault
The concept of formal consequence is at the heart of logic today, and by extension, plays an impo... more The concept of formal consequence is at the heart of logic today, and by extension, plays an important role in such diverse areas as mathematics, computing, philosophy, and linguistics.
In this dissertation, I trace the roots of this concept in medieval logic from Pseudo-Scotus and John Buridan back to the earliest treatises on consequences, and provide translations of the three earliest known treatises on consequences - two anonymous, a third by Walter Burley.
Chapter one introduces the reader to the dominant philosophical approaches to formal consequence from the turn of the twentieth century to today. After this, I introduce the account of formal consequence advanced by John Buridan, the medieval predecessor to the semantic account advocated by Tarski and his followers.
The second chapter provides a detailed contrast of Buridan's account of formal consequence with those of Tarski, on the one hand, and later classical logic, on the other.
Chapter three examines the account of formal consequence in Pseudo-Scotus. I show that Pseudo-Scotus' account is dependent on that of Buridan, and therefore most post-date it.
Chapter four examines the account of divided modal consequence in William of Ockham. I show that Ockham's divided modalities are not fully assimilable to narrow-scope propositions of classical modal logic; formalize Ockham's account in an extension of first-order modal logic with restricted quantification; and provide a complete account of relations between two-term divided modal propositions on Ockham's account.
Chapter five introduces Walter Burley's thinking about consequences, examining: Burley's division and enumeration of consequences; his distinction between principal and derivative rules licensing good consequences; the relation of the division of consequences into formal and material varieties to Burley's preferred division between natural and accidental consequences; the relation Burley's work bears to Buridan, to the Boethian reception of Aristotle's Topics, and to the earliest treatises on consequences.
The final chapter concludes: highlighting the characteristic marks of medieval and modern approaches to consequences relative to each other; summarizing the various developments that led to the adoption of the account of formal consequence epitomized in Buridan's work; and suggesting prospects for recovering the most promising aspects of the medieval treatments of the topic.
Journal articles by Jacob Archambault
Logica Universalis, 2019
A counterpossible conditional, or counterpossible for short, is a conditional proposition whose a... more A counterpossible conditional, or counterpossible for short, is a conditional proposition whose antecedent is impossible. The filioque doctrine is a dogma of western Christian Trinitarian theology according to which the Holy Spirit proceeds from the Father and the Son. The filioque doctrine was the principal theological reason for the Great Schism, the split between Eastern Orthodoxy and western Christianity, which continues today. In the paper, I review one of the earliest medieval defenses of the doctrine in Anselm of Canterbury, and I show that Anselm's treatment of counterpossible conditionals concerning the procession of the spirit from the son in Trinitarian theology represent an early foray into default logic. Thus, the mutual estrangement of eastern and western positions on the matter may not lie fundamentally in a change in dogma, but rather in a change in logic. Mathematics Subject Classification. Primary 01A35; Secondary 03B60, 03B80.
Vivarium, 2018
This paper summarizes medieval definitions and divisions of consequences and explains the import ... more This paper summarizes medieval definitions and divisions of consequences and explains the import of the medieval development of the theory of consequence for logic today. It then introduces the various contributions to this special issue of Vivarium on consequences in medieval logic.
Vivarium, 2018
With William of Ockham and John Buridan, Walter Burley is often listed as one of the most signifi... more With William of Ockham and John Buridan, Walter Burley is often listed as one of the most significant logicians of the medieval period. Nevertheless, Burley’s contributions to medieval logic have received notably less attention than those of either Ockham or Buridan. To help rectify this situation, the author here provides a comprehensive examination of Burley’s account of consequences, first recounting Burley’s enumeration, organization, and division of consequences, with particular attention to the shift from natural and accidental to formal and material consequence, and then locating Burley’s contribution to the theory of consequences in the context of fourteenth-century work on the subject, detailing its relation to the earliest treatises on consequences, then to Ockham and Buridan.
Logica Universalis, 2017
A consequence relation is monotonic iff for premise sets Γ, ∆ and conclusion ϕ, if Γ|- ϕ, Γ ⊆ ∆, ... more A consequence relation is monotonic iff for premise sets Γ, ∆ and conclusion ϕ, if Γ|- ϕ, Γ ⊆ ∆, then ∆ |- ϕ; and non-monotonic if this fails in some instance. More plainly, a consequence relation is monotonic when whatever is entailed by a premise set remains entailed by any of its supersets. From the High Middle Ages through the Early Modern period, consequence in theology is assumed to be monotonic. Concomitantly, to the degree the argument formulated by Anselm at Proslogion 2-4 is taken up by later commentators, it is accepted or rejected in accordance with a monotonic notion of consequence. Examining Anselm's use of parallelism in the Proslogion, I show Anselm embeds his famous argument in Proslogion 2-4 in a non-monotonic context. If this weren't so, Anselm would be contradicting himself when he adds God is greater than can be thought. The results found in this article challenge us to reconsider some deeply ingrained ideas governing the historiography of the long 12th century, particularly concerning how the theology of the later 11th through the 12th century relates to the scholasticism of the 13th.
Heythrop Journal, 2017
This paper approaches the question of Levinas' relation to philosophy by situating his understand... more This paper approaches the question of Levinas' relation to philosophy by situating his understanding of transcendence next to that of Leibniz. After offering some preliminary examples, I detail the structure of transcendence in the philosophies of Leibniz and Levinas, focusing on Leibniz's Principles of Nature and Grace and Levinas’ Essence and Disinterestedness. From here, I return to the question of whether Levinas’ thinking can be regarded as moving beyond philosophy as such. I conclude with some thoughts on what it would take for a thinking of transcendence to genuinely move past the maneuvers so characteristic of philosophical thinking.
Augustiniana, 2015
Abstract This paper compares the understanding of nature, will, and the Fall in Augustine and Max... more Abstract This paper compares the understanding of nature, will, and the Fall in Augustine and Maximus the Confessor, and finds their accounts to be identical on most points of substance, if not always in the terminology they use to express these points. On several points, they agree with each other against both Eastern and Western accounts as traditionally conceived. Given that these figures are often regarded as paradigmatic for Western and Eastern traditions of Christianity, respectively, this points to a need for a more nuanced account of the unity and divergences within and between Eastern and Western Christian traditions than that given to present.
Religious Studies, 2014
This article re-examines the applicability of Kant's dependency thesis to Aquinas’ cosmological p... more This article re-examines the applicability of Kant's dependency thesis to Aquinas’ cosmological proofs for the existence of God. The first part of the article provides a summary of Kant's dependency thesis, followed by a review of a defence of Aquinas by J. William Forgie. The second part of the article explains some of the logical apparatus upon which Aquinas’ argument hinges – specifically his understanding of the a priori/a posteriori distinction. I conclude by calling attention to certain distinct metaphysical assumptions within Kant's and Aquinas’ respective logical apparatus that would have to be addressed prior to the more specific question of whether Kant's critique of the cosmological argument is applicable to Aquinas.
Book chapters by Jacob Archambault
I. Logan, A. Forbes (eds.), Anselm of Canterbury: Nature, Order and the Divine, 2024
How should the phrase 'fides quaerens intellectum' be understood as a characterization of Anselm’... more How should the phrase 'fides quaerens intellectum' be understood as a characterization of Anselm’s Proslogion specifically and Anselm’s oeuvre more broadly? I argue that 'fides quaerens intellectum' should be understood not in accordance with any of the standard meanings we assign to the words ‘faith’ and ‘understanding’, but in accordance with their meaning in works used to teach grammar and dialectic at Bec in and around Anselm’s lifetime.
I begin by making the structures behind the usual understanding of ‘faith’ and ‘understanding’ explicit. From here, I detail the works used at Bec according to a 12th century library list, and provide justification for thinking these same works were used slightly earlier, in Anselm’s time. Next, I show how the terms 'fides' and 'intellectus' functioned in these works. Finally, I turn to the Proslogion and the exchange with Gaunilo to show how these considerations illuminate Anselm’s method in these works.
C. Normore, S. Schmid (eds.), Grounding in Medieval Philosophy (Springer), May 10, 2024
Developed out of earlier work on Aristotelian topics, syllogistic, and fallacies, by the early fo... more Developed out of earlier work on Aristotelian topics, syllogistic, and fallacies, by the early fourteenth century the medieval theory of consequence came to provide the first unified framework for the treatment of inference as such. With this development came the task of unifying the various justifications for inferences treated in earlier frameworks.
Prior to the appearance of theories of consequences, the task of providing a real foundation, or grounding, for good inferences is shared between theories of demonstration, such as those provided in commentaries on Aristotle’s Posterior Analytics, and theories of topical inference, passed on to the medievals via Boethius. But by the time of the earliest consequentiae, most consequences were grounded in the theory of supposition, which began its own development in the twelfth
century.
Secondary literature on supposition has generally held that in the most common form of supposition, personal supposition, a term is taken to stand for individuals falling under it. In this paper, I show that for the earliest consequentiae this is false: prior to William of Ockham’s work, personal supposition could also involve descent to concepts or types falling under a term, previously thought to be the exclusive provision of simple supposition. As such, a greater variety of ways of grounding consequence exists in the period than has hitherto been recognized.
J. Hochshild, T. Nevitt, A. Wood and G. Borbely, Metaphysics Through Semantics: The Philosophical Recovery of the Medieval Mind (Springer), 2023
The resemblance between Buridan's and Tarski's theories of formal consequence has long been remar... more The resemblance between Buridan's and Tarski's theories of formal consequence has long been remarked upon, so much so that it helped accelerate the rediscovery of Buridan as a major medieval figure in the latter half of the twentieth century. But while long noticed, it has not yet been subjected to sustained analysis.
In this article, I provide just such an analysis. I begin by reviewing today's understanding of classical consequence, then highlighting its differences from Tarski's 1936 account of formal consequence. Following this, I introduce Buridan's account of formal consequence, detailing its philosophical underpinnings, then its content. This then allows a clearer contrast between Buridan and Tarski's respective accounts of following formally.
Proceedings of the Society for Medieval Logic and Metaphysics, 2018
This paper contributes to an explanation of how a sea change in logic from the later 13th to the ... more This paper contributes to an explanation of how a sea change in logic from the later 13th to the early 14th century relates to certain metaphysical debates during the same period. The changes we refer to are:
i. The development of the concept of consequence, first as a replacement for the framework of Aristotle’s Topics, then as a broader category under which syllogistic, too, was subsumed.
ii. The division of consequences into natural and accidental, and the later replacement of this division by that into formal and material varieties.
iii. The accompanying growth of hylomorphism, and particularly mereological hylomorphism, in thinking about consequence.
Both the development of systematic treatises on consequence and the application of hylomorphic language to consequences first take place in the later middle ages, culminating in the work of the Parisian arts master John Buridan. Buridan is not the first Latin author to explicate a distinction between formal and material consequence; but it is Buridan’s version of the distinction that becomes widely adopted throughout continental Europe in the later middle ages, ultimately laying the groundwork for today’s model-theoretic accounts of formal consequence.
After setting forth the requisite definitions, I introduce Buridan’s distinction between formal and material consequence as presented in his Treatise on Consequences: first giving the generic structure of the distinction, then placing the distinction in its historical development. I then introduce Buridan’s metaphysical views: focusing on the question of whether several substantial forms may be present in the same subject; drawing from Buridan’s questions on Aristotle’s Physics and De Anima; and contrasting the views found there with the unicity theory of Aquinas. I then place Buridan’s logic specifically, and its development more broadly, against the backdrop of these developments in metaphysics over the same period. We find that Buridan’s logical hylomorphism reduplicates the peculiarities of his physical hylomorphism. In particular, both Buridan’s logical and physical hylomorphism are mereological in character, i.e. they both identify the form of a being (or consequence) with a proper part of the being itself - an identification explicitly rejected by Aristotle. We conclude that while Buridan’s mereological hylomorphism in physics did not necessitate his logical hylomorphism, it did help facilitate it.
M. Fichant, R. Pisano, A. R. E. Oliveira, and P. Bussotti (eds.), Leibniz and the Dialogue between Sciences, Philosophy and Engineering, 1646-2016: New Historical and Epistemological Insights, 2017
Abstract It is well-known that Leibniz rejected the accounts of gravitation offered by Newton and... more Abstract It is well-known that Leibniz rejected the accounts of gravitation offered by Newton and his followers. But literature on the topic up to present has been content either i) to explain this rejection as a consequence of Leibniz’s rejection of the intelligibility of action at a distance; or ii) to regard Leibniz’s rejection, given his acceptance of his own notion of force, as unjustified. This article provides a fuller account of why Leibniz regarded Newtonian gravitation as unintelligible.
The paper divides into two parts. The first part shows that neither of the above explanations suffices to explain Leibniz’s rejection of Newtonian gravitation. The second part explicates the notion of intelligibility standing behind Leibniz’s charge against the Newtonians. In this part, I begin by cataloguing a list of physical hypotheses that Leibniz takes to be unintelligible. I then move to the positive account of intelligibility Leibniz gives in his correspondence with Christian Wolff. Thereby, I show that the different Newtonian accounts of gravitation each undermine the Leibnizian theological tenet that this is the best of all possible worlds.
In preparation by Jacob Archambault
For S. L. Uckelman (ed.), The Cambridge Critical Guide to Anselm's De Grammatico and De Veritate
One of the more persistent interpretations of Leibniz's system of pre-established harmony is as a... more One of the more persistent interpretations of Leibniz's system of pre-established harmony is as a temporal dislocation of occasionalism: whatever God is always doing on the occasionalist account he need only have done once – at creation – on Leibniz's. In accordance with this interpretation, the difference between the systems of La Forge, Cordemoy, and Malebranche, on the one hand, and Leibniz's, on the other, is one of how involved God is in the world. Here, I show the difference between these systems has nothing to do with God's degree of involvement in the universe. Rather, the basic difference between the systems must be found in their differing accounts of the nature of substance.
This paper argues: first, that a presentist, powers based, diachronic account of modality can pro... more This paper argues: first, that a presentist, powers based, diachronic account of modality can provide a satisfactory account of our intuitions about modality as well as a compelling rebuttal to alternative accounts; second that taking this account seriously requires a more radically attenuated modal logic than even partisans of such an account realize. Attempting to navigate this problem, I provide a formal, non-normal semantics for a tensed modal logic, along with a philosophical interpretation of it.
Dispositionalism grounds modality in the powers of existing things. Eternalism assumes 'exists' r... more Dispositionalism grounds modality in the powers of existing things. Eternalism assumes 'exists' ranges equally over past, present, and future entities. In this article, I show that eternalism is incompatible with the most plausible understanding of what it is that dispositionalist modality explains: generation and corruption. As such, the dispositionalist is forced, against dispositionalism's traditional categorization as a form of actualism, to deny that everything exists.
Translations by Jacob Archambault
A short, early anonymous logical treatise on consequences, translated from the critical edition o... more A short, early anonymous logical treatise on consequences, translated from the critical edition of N. J. Green-Pedersen (Cahiers de l’Institut du Moyen-Âge Grec et Latin 35 (1980):, pp. 1-28). The ms. also contains Burley’s early treatise on consequences, and is dated by the scribe to 1302.
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Books by Jacob Archambault
Dissertation by Jacob Archambault
In this dissertation, I trace the roots of this concept in medieval logic from Pseudo-Scotus and John Buridan back to the earliest treatises on consequences, and provide translations of the three earliest known treatises on consequences - two anonymous, a third by Walter Burley.
Chapter one introduces the reader to the dominant philosophical approaches to formal consequence from the turn of the twentieth century to today. After this, I introduce the account of formal consequence advanced by John Buridan, the medieval predecessor to the semantic account advocated by Tarski and his followers.
The second chapter provides a detailed contrast of Buridan's account of formal consequence with those of Tarski, on the one hand, and later classical logic, on the other.
Chapter three examines the account of formal consequence in Pseudo-Scotus. I show that Pseudo-Scotus' account is dependent on that of Buridan, and therefore most post-date it.
Chapter four examines the account of divided modal consequence in William of Ockham. I show that Ockham's divided modalities are not fully assimilable to narrow-scope propositions of classical modal logic; formalize Ockham's account in an extension of first-order modal logic with restricted quantification; and provide a complete account of relations between two-term divided modal propositions on Ockham's account.
Chapter five introduces Walter Burley's thinking about consequences, examining: Burley's division and enumeration of consequences; his distinction between principal and derivative rules licensing good consequences; the relation of the division of consequences into formal and material varieties to Burley's preferred division between natural and accidental consequences; the relation Burley's work bears to Buridan, to the Boethian reception of Aristotle's Topics, and to the earliest treatises on consequences.
The final chapter concludes: highlighting the characteristic marks of medieval and modern approaches to consequences relative to each other; summarizing the various developments that led to the adoption of the account of formal consequence epitomized in Buridan's work; and suggesting prospects for recovering the most promising aspects of the medieval treatments of the topic.
Journal articles by Jacob Archambault
Book chapters by Jacob Archambault
I begin by making the structures behind the usual understanding of ‘faith’ and ‘understanding’ explicit. From here, I detail the works used at Bec according to a 12th century library list, and provide justification for thinking these same works were used slightly earlier, in Anselm’s time. Next, I show how the terms 'fides' and 'intellectus' functioned in these works. Finally, I turn to the Proslogion and the exchange with Gaunilo to show how these considerations illuminate Anselm’s method in these works.
Prior to the appearance of theories of consequences, the task of providing a real foundation, or grounding, for good inferences is shared between theories of demonstration, such as those provided in commentaries on Aristotle’s Posterior Analytics, and theories of topical inference, passed on to the medievals via Boethius. But by the time of the earliest consequentiae, most consequences were grounded in the theory of supposition, which began its own development in the twelfth
century.
Secondary literature on supposition has generally held that in the most common form of supposition, personal supposition, a term is taken to stand for individuals falling under it. In this paper, I show that for the earliest consequentiae this is false: prior to William of Ockham’s work, personal supposition could also involve descent to concepts or types falling under a term, previously thought to be the exclusive provision of simple supposition. As such, a greater variety of ways of grounding consequence exists in the period than has hitherto been recognized.
In this article, I provide just such an analysis. I begin by reviewing today's understanding of classical consequence, then highlighting its differences from Tarski's 1936 account of formal consequence. Following this, I introduce Buridan's account of formal consequence, detailing its philosophical underpinnings, then its content. This then allows a clearer contrast between Buridan and Tarski's respective accounts of following formally.
i. The development of the concept of consequence, first as a replacement for the framework of Aristotle’s Topics, then as a broader category under which syllogistic, too, was subsumed.
ii. The division of consequences into natural and accidental, and the later replacement of this division by that into formal and material varieties.
iii. The accompanying growth of hylomorphism, and particularly mereological hylomorphism, in thinking about consequence.
Both the development of systematic treatises on consequence and the application of hylomorphic language to consequences first take place in the later middle ages, culminating in the work of the Parisian arts master John Buridan. Buridan is not the first Latin author to explicate a distinction between formal and material consequence; but it is Buridan’s version of the distinction that becomes widely adopted throughout continental Europe in the later middle ages, ultimately laying the groundwork for today’s model-theoretic accounts of formal consequence.
After setting forth the requisite definitions, I introduce Buridan’s distinction between formal and material consequence as presented in his Treatise on Consequences: first giving the generic structure of the distinction, then placing the distinction in its historical development. I then introduce Buridan’s metaphysical views: focusing on the question of whether several substantial forms may be present in the same subject; drawing from Buridan’s questions on Aristotle’s Physics and De Anima; and contrasting the views found there with the unicity theory of Aquinas. I then place Buridan’s logic specifically, and its development more broadly, against the backdrop of these developments in metaphysics over the same period. We find that Buridan’s logical hylomorphism reduplicates the peculiarities of his physical hylomorphism. In particular, both Buridan’s logical and physical hylomorphism are mereological in character, i.e. they both identify the form of a being (or consequence) with a proper part of the being itself - an identification explicitly rejected by Aristotle. We conclude that while Buridan’s mereological hylomorphism in physics did not necessitate his logical hylomorphism, it did help facilitate it.
The paper divides into two parts. The first part shows that neither of the above explanations suffices to explain Leibniz’s rejection of Newtonian gravitation. The second part explicates the notion of intelligibility standing behind Leibniz’s charge against the Newtonians. In this part, I begin by cataloguing a list of physical hypotheses that Leibniz takes to be unintelligible. I then move to the positive account of intelligibility Leibniz gives in his correspondence with Christian Wolff. Thereby, I show that the different Newtonian accounts of gravitation each undermine the Leibnizian theological tenet that this is the best of all possible worlds.
In preparation by Jacob Archambault
Translations by Jacob Archambault