For those that may have missed this when it came out in late October: I recently guest-edited a volume of *Vivarium *on medieval theories of consequence. The volume covers major figures from Boethius to Marsilius of Inghen, and includes contributions on a wide variety of topics from Chris Martin, Joke Spruyt, Milo Crimi, Graziana Ciola, Bianca Bosman, and myself. The volume may be found here.

Here is an excerpt, from the introduction:

# 2 Medieval and modern definitions of consequence

In common English, ‘consequence’ usually refers to the result or outcome of an action, ‘inference’, to a subject’s act of asserting or coming to believe something on the basis of something else, and ‘implication’, to a suggestion communicated in a veiled manner through something else stated explicitly. In logicians’ English, these terms are naturally used interchangeably to refer to none of these things. In logic today, ‘consequence’ ‘inference’ and ‘implication’ refer to an ordered pair whose first element, called the *antecedent*, is usually a set (or multiset, or list)[1] of sentences, propositions, or even arguments,[2] and whose second element, called the *consequent*, is a single object of the same type. A consequent is said to *follow from* its antecedent, and an antecedent is said to *entail* its consequent.[3]

In medieval logic, ‘consequence’ (*consequentia*) usually refers to a relation between an antecedent and a consequent, variously described as a habit (*habitudo*), inference (*illatio*), or a following (*sequela*).[4] Some medieval logicians define a consequence according to its part of speech,[5] others in terms of its function,[6] and still others, seemingly throwing up their hands, regard it as a clustering of its various parts.[7] Others pass over the definition of consequence altogether and begin by listing good consequences or divisions of consequences.[8]

Modern logicians make a strong distinction between consequences and conditionals: a conditional is a connective appearing in formulae within a regimented language, called the *object language*, employed in a proof system. Consequence is a relation of following asserted to hold between [schemata for] formulae in the object language, and whose written expression does not appear in the object language but in a second, more expressive language called the *meta-language* (usually a natural language augmented with various mathematical symbols) which is used to evaluate the expressions of the object language.

Medieval logicians do not strongly distinguish consequences from conditionals, and certainly do not do so in the above way. Humanist scruples aside, medieval logicians worked within natural language. In accord with the range of source material from which it arises, the medieval concept of consequence comes to include conditional statements, categorical and hypothetical syllogisms, conversions, enthymemes, and other argument forms.[9]

None of this yet tells us what *counts* as a good consequence, either for medieval logicians or their modern counterparts. Today, the two most common ways of providing criteria for determining when a consequence exists are one, relying on the techniques of proof theory and called *proof-theoretic*, and the other relying on model theory, called *model-theoretic* or *semantic*.[10]

In the semantic approach to consequence, a consequence from a premise set to a conclusion – written – is valid if and only if every model of is, at the same time, a model of . In early model theory, e.g. that of Tarski, a model of a sentence [set of sentences ] in a recursively-defined language presupposes a division of the basic elements of into logical and non-logical kinds, and is a sequence of objects *satisfying* (roughly, making true) each *sentential function* obtained by uniformly replacing each non-logical element in the sentence [set of sentences ] with a variable – like variables replacing like constants, unlike replacing unlike. In classical model theory today, a model is a pair consisting of a (possibly infinite, possibly empty) set of objects , called the *domain*, and an interpretation that assigns non-logical constants in to elements in , and thereby provides the basis for recursively determining the truth value on of each sentence in . Modal, non-classical, and other model theoretic approaches to consequence generally arise by expanding the number or adjusting the interpretation of the logical constants of a language, and/or by modifying the notion of a model in interesting ways e.g. by the addition of Kripke frames in modal logic, or of further truth values in many-valued logics.

In modern proof-theoretic approaches to consequence, a consequence occurs from premises to conclusion in a proof system if and only if there exists a derivation of in from (open) assumptions .[11] Here, a proof system consists of a set of rules (and possibly, axioms) for obtaining certain formulae of a language from others,[12] and commonly consists of rules for introducing and eliminating logical connectives, along with structural rules governing matters like the introduction of assumptions and repetition of formulae. The definition of an open or closed assumption, and the corresponding notion of an open or closed argument, are given in a manner parallel to the treatment variables and formulae in syntax: just as a variable [formula of a language ] is said to be bound [closed] just if it occurs within the scope of a quantifier [all of its variables are bound], and free/unbound [open] otherwise, an assumption [argument] is said to be bound – that is, discharged – just if it occurs within the ‘scope’ of an inference [all of its assumptions are bound].[13] Consequently, the basic idea behind modern proof-theoretic approaches to consequence is that a consequence exists when, given a certain set of premises as inputs, there is a precise, rule-governed procedure for obtaining the consequent as output. The proof-theoretic approach to consequence traces its origins back to David Hilbert’s work on mathematical proof and Gerhard Gentzen’s on natural deduction, and has since been heavily influenced by the contributions of Dag Prawitz, Michael Dummett, and others.[14]

Medieval ways of explicating consequence have parallels to both of these approaches. Instead of models, some medieval accounts of consequence rely on a notion of *causes of truth*, which shares similarities with the modern notion of a truthmaker.[15] And instead of rule-governed proof systems, several medieval logicians appeal to inference-licensing rules called *maximal propositions* which arise out of discussions of topical argument.[16]

But what likely strikes the non-logician about these accounts is their abstraction from the actual content of an inference.[17] Since Tarski’s advances in model theory, for instance, classical models for standard formal representations of ‘Socrates runs’ have also served to model ‘Plato jumps’.[18] And since Hilbert’s advances in geometry, proof systems have been constructed with the intent of making the interpretation of the symbols occurring in them a matter of indifference. In accord with this characteristic, modern theories of consequence tend to focus almost entirely on formally valid inference.

Medieval approaches to consequence generally lack this feature. Rather, the earliest medieval criteria for consequence state that a consequence is good, true, valid, or holds when it is impossibile for the antecedent to be true and the consequent false, and later criteria for consequence generally consist of more sophisticated variations on this same theme – to avoid problems with self-falsifying propositions, for instance, several authors state a version on which a consequence is good when things cannot be *as the antecedent signifies* without being as the consequent signifies.[19] Another popular criterion, now called the ‘containment criterion’ and later used in characterizations of *formal* consequence, holds that a consequence is good when [the understanding of] the consequent is ‘contained’ in [the understanding of] the antecedent.[20] Some authors appeal to versions of both criteria, while others use one to the exclusion of the other. Both continue to be regarded as basic, intuitive criteria for consequence and to serve as the basis for its modern formalizations.[21]

# 4 Introduction to the articles

The articles collected in this issue survey a wide array of topics in the theory of consequence, beginning with the contributions of Boethius (which properly antedate the theory but are central to its later development) and following with discussions of subjects from the mid-twelfth through the later fourteenth century. Following the scholastic commitment to the question of the right order of reading,[1] the articles in this collection are ordered to facilitate reading from beginning to end, moving from material likely to be more basic for the understanding of other authors, more widely discussed in secondary literature, and more accessible from the standpoint of modern logic, to less familiar figures and broader thematic discussions.

Much of the basic material out of which the doctrine of consequence arose comes from Boethius. In ‘The Roots of the Notion of Containment in Theories of Consequence: Boethius on Topics, Containment, and Consequences’, Bianca Bosman addresses the question of whether and to what extent the containment criterion for consequence, common in both earlier discussions of natural consequence and later British discussions of formal consequence, is anticipated in the logical works of Boethius. Bosman argues that, while the containment criterion does draw on Boethian source texts, those sources are different from those standardly assumed. Bosman shows that the later criterion draws much from texts *not* devoted to conditionals, including Boethius’ discussions of *per se* predication in his treatment of the Porphyrian predicables, and of the *locus* *from a genus* in his commentary on Cicero’s *Topics*.

The best-known medieval accounts of consequences are those of William of Ockham and John Buridan. But the relation between these accounts remains obscure. In particular, Ockham classifies certain consequences as formal which Buridan admits only as material, and the exact reason for these differences has not been sufficiently explored. In ‘The Distinction between Formal and Material Consequences in Ockham and Buridan’, Milo Crimi provides a classification of consequences both figures treat as formal, those both treat as material, and those which Ockham calls formal and Buridan calls material. Crimi then shows that the taxonomical discrepancy between Ockham and Buridan’s accounts is not due to differences in their propositional hylomorphism, but to Ockham’s endorsement of relational characterizations of formal consequences.

One of the more outstanding continental authors writing on consequences after Buridan is Marsilius of Inghen, later founder and rector of the University of Heidelberg. Marsilius calls Buridan ‘my teacher’[2] and with Albert of Saxony Marsilius is traditionally regarded as a prominent member of a Buridanian school of logic. Though neither Marsilius nor Albert held such a relation to Buridan in any institutional sense,[3] their approach to consequences share some broad similarities when compared to that of later British writers, particularly in their use of a substitutional criterion for formal consequence. In ‘Marsilius of Inghen on the Definition of *consequentia*’, Graziana Ciola compares Marsilius’ account of consequences with those of Buridan and Albert, and finds that Marsilius diverges from Buridan and Albert in several important respects. Specifically, Buridan and Albert affirm, while Marsilius denies, that a consequence is a *propositio hypothetica*. Instead, Marsilius characterizes a consequence as an *oratio*, further distancing the theory of consequences from that of the conditional and more clearly establishing it as an entailment relation. In addition, Marsilius rejects, where Buridan and Albert accept, *ut nunc*, or as-of-now consequence. This rejection is found with some frequency among British and Italian logicians,[4] and the adoption of the position in Marsilius suggests the interaction between British and continental traditions may be more complex than currently recognized.[5]

With Ockham and Buridan, Walter Burley is often regarded as one of the most influential logicians of the later middle ages. One of the earliest extant *consequentiae* treatises, and the earliest with a known author, belongs to Burley. In addition, Burley is one of the few early authors to discuss both the natural/accidental division and formal/material division of consequences at some length. In ‘Consequence and Formality in the Logic of Walter Burley’, Jacob Archambault provides a comprehensive overview of Walter Burley’s account of consequences. After reviewing Burley’s division and enumeration of consequences, Archambault shows how Burley relates his own theory of natural and accidental consequence to the division into formal and material consequence found in Ockham. The article then compares Burley’s work to the earliest anonymous treatises on consequences and to Ockham and Buridan’s treatises on the subject. Archambault highlights Burley’s advances over the former treatises’ treatment of existential import in consequences, his disagreements with Ockham and Buridan on rules governing consequences, and his influence on the broader place of the study of consequences in logic.

Next, Joke Spruyt provides an overview of consequences in the thirteenth century. Successively examining thirteenth-century discussions of syllogisms, *syncategoremata*, and *sophismata*, Spruyt shows that across these genres, thirteenth-century work on consequences often assimilated the relation of a consequent to its antecedent(s) to that of an effect to its cause(s). Though thirteenth-century logicians typically regarded premises as causes not of being, but of following, the assimilation played an important role in thirteenth-century treatments of inferences from impossible antecedents or to necessary consequents. Many thirteenth-century logicians rejected the validity of these inferences, and those who admitted them to be valid in some respect did not regard them as unqualifiedly so.

This issue closes with Christopher Martin’s analysis of the development of the theory of natural consequence from Peter Abaelard to the turn of the fourteenth century. Martin argues that the early theory of natural consequence provides a medieval theory of relevant consequence, specifically one conforming to principles which today characterize connexive logic, and he shows the crucial role played by changes in the account of disjunction in the shift away from this relevantistic account of consequence. According to Martin, Abaelard distinguishes between extensionally-defined predicate disjunction for categorical propositions, on the one hand, and propositional disjunction, on the other, and employs an intensional account of propositional disjunction on which this kind of disjunction is equivalent to a conditional with the negation of the first disjunct as antecedent and the second disjunct as consequent. Like his account of the conditional, Abaelard’s account of propositional disjunction thus also conforms to connexive principles. As logic texts shifts from twelfth century manifestos for the doctrines of rival schools to more irenic thirteenth century textbooks, Abaelard’s distinction is lost, and largely replaced by an extensional account of propositional disjunction. But the need for a stronger form of consequence than that holding merely in virtue of a standard semantic requirement – namely, the impossibility of the antecedent holding with the consequent not holding – is found in authors through the thirteenth and into the fourteenth century, and was especially acute in a species of disputational exercises, or *obligation*, involving the positing of an impossible proposition, called *positio impossibilis*. It is in this disputational context, and specifically in the different treatments of impossible *positio* in Scotus and Ockham, that the seeds of Ockham’s alternative analysis of consequence, and the replacement of the earlier one, would be sown.

[1] A set is a grouping of elements without respect to their order or repetition. {A, B} and {B, A} and {A, A, B} all name the same set. A list respects both the order of elements and the number of times an element occurs. Hence, and are all different lists. Multisets are like sets but respect the number of times a given element occurs. Hence, though [A, B], and [B, A], and [A, A, B] all name the same set, only the first two refer to the same multiset. See David Ripley, ‘Comparing Substructural Theories of Truth’, *Ergo* 2 (2015), 300.

[2] James W. Garson, *What Logics Mean* (Cambridge, 2013).

[3] Both consequence and inference have been suggested as appropriate translations for the Latin *consequentia*. For discussion, see Peter King, ‘Consequence as Inference: Mediaeval Proof Theory 1300-1350’, in *Medieval Formal Logic: Obligations, Insolubles and Consequences*, ed. Mikko Yrjönsuuri (Dordrecht, 2001), 117–45; also Catarina Dutilh Novaes, ‘Buridan’s *Consequentia*: Consequence and Inference Within a Token-Based Semantics’, *History and Philosophy of Logic* 26 (2005), 277–97.

[4] Niels Jørgen Green-Pedersen, ‘Two Early Anonymous Tracts on Consequences’, *Cahiers de L’Institut Du Moyen-Âge Grec et Latin* 35 (1980), 1–28, 4: ‘Consequentia est habitudo inter antecedens et consequens. Antecedens est illud ad quod sequitur aliud. Consequens est illud quod sequitur ex alio’. W. K. Seaton, ‘An Edition and Translation of the Tractatus de Consequentiis of Ralph Strode’ (University of California at Berkeley, PhD Thesis, 1973), 1: ‘Consequentia dicitur illatio consequentis ex antecedente’. Rodolphus Anglicus: ‘Consequentia est quaedam habitudo vel sequela in qua consequens se habet ad antecedens’, in Niels Jørgen Green-Pedersen, ‘Early British Treatises on Consequences’, in *The Rise of British Logic: Acts of the Sixth European Symposium on Medieval Logic and Semantics, Balliol College, Oxford, 19-24 June 1983*, ed. P. Osmund Lewry (Toronto, 1985), 306.

[5] John Buridan, *Tractatus de Consequentiis*, ed. Hubert Hubien (Louvain, 1976) I, c. 3, 22.60-62: ‘Consequentia est propositio hypothetica ex antecedente et consequente composita, designans antecedens esse antecedens et consequens esse consequens’. Pseudo-Scotus, *Quaestiones Super Libros II Priorum Analyticorum*, Joannis Duns Scoti Doctoris Subtilis Ordinis Minorum Opera Omnia, vol. 2 (Paris, 1891) I. q. 10, 104-105: ‘Consequentia est propositio hypothetica, composita ex antecedente, et consequente, mediante conjunctione conditionali, vel rationali, quae denotat, quod impossibile est ipsis, scilicet antecedente, et consequente simul formatis, quod antecedens sit verum, et consequens falsum’.

[6] Niels Jørgen Green-Pedersen, ‘Bradwardine(?) On Ockham’s Doctrine of Consequences: An Edition’, *Cahiers de L’Institut Du Moyen-Âge Grec et Latin* (1982), 85–150, 92: ‘Circa definitionem nota quod consequentia est argumentatio composita ex antedente et consequente. ‘Argumentatio’ ponitur in definitione consequentiae, quia omnis consequentia sumitur ad aliquod argumentum producendum. ‘Composita’ dicitur, quia nullum incomplexum est consequentia. ‘Ex antecedente et consequente’ additur, quia in omni consequentia adminus requiruntur duae propositiones categoricae’.

[7] See no. 6 in Green-Pedersen, ‘Early British Treatises on Consequences’, 300: ‘Consequentia est quoddam aggregatum ex antecedente et consequente ad idem consequens cum nota consequentiae. Et sunt notae consequentiae ‘ergo’, ‘ideo’, ‘quia’, ‘igitur’, ‘idcirco’’. Also nos. 7, 9, and 15, ibid., 300–306.

[8] Green-Pedersen, ‘Two Early Anonymous Tracts on Consequences’ 11: ‘In omni consequentia bona quicquid sequitur ad consequens sequitur ad antecedens; ut sequitur ‘Socrates currit, ergo animal currit’ et sequitur ‘animal currit, ergo substantia currit’; ergo a primo ad ultimum sequitur ‘Socrates currit, ergo substantia currit’. William of Ockham, *Summa Logicae*, in *Opera Philosophica*, ed. Philotheus Boehner, Gedeon Gàl, and Stephen Brown, vol. 1 (St. Bonaventure, NY, 1974) III-3, c. 1, 587.4-9: ‘Habito de syllogismo in communi et de syllogismo demonstrativo, agendum est de argumentis et consequentiis quae non servant formam syllogisticam. Et primo ponam aliquas distinctiones quae sunt communes aliis consequentiis multis, quamvis non sint enthymemata, ex quibus omnibus faciliter patere poterit studioso quid de omnibus syllogismis non demonstrativis est tenendum’. Cf. Lorenzo Pozzi, *Le ’Consequentiae’ Nella Logica Medievale* (Padova, 1978), 262.

[9] Green-Pedersen, ‘Bradwardine(?) On Ockham’s Doctrine of Consequences’ 92: ‘Etiam ex ista definitione sequitur quod omnis argumentatio generaliter potest vocari consequentia, sive sit syllogistica sive inductiva sive exemplaris sive enthymematica’. William of Ockham, *Tractatus Minor Logicae*, in *Opera Philosophica: Opera Dubia et Spuria*, ed. Eligius M. Buytaert, Gedeon Gàl, and Joachim Giermek, vol. 7 (St. Bonaventure, NY, 1988), 1–57 V. c. 1, 31.4-5: ‘Sic syllogismus et inductio, conversio et multi alii modi considerandi sunt consequentiae formales’. Pseudo-Scotus, *Quaestiones Super Libros II Priorum Analyticorum* I. q. 20, 130: ‘Notandum est, quod quaedam est consequentia enthymematica, et quaedam syllogistica’.

[10] Cf. Alfred Tarski, ‘On the Concept of Following Logically’, trans. Magda Stroińska and David Hitchcock, *History and Philosophy of Logic* 23 (2002), 155–96; Dag Prawitz, ‘On the Idea of a General Proof Theory’, *Synthese* 27 (1974), 63–77.

[11] The definition is taken from Nissim Francez, ‘On Distinguishing Proof-Theoretic Consequence from Derivability’, *Logique et Analyse* 238 (2017), 152.

[12] Or, in the case of sequent calculi, arguments from arguments. The array of proof systems in logic today is vast, and the above description only captures a fraction of them.

[13] The definition given here is a simplification which leaves aside problems pertaining to the normal form of inferences. For fuller discussion, see Dag Prawitz, ‘Remarks on Some Approaches to the Concept of Logical Consequence’, *Synthese* 62 (1985), 153–71.

[14] Gerhard Gentzen, ‘Untersuchungen über Das Logische Schliessen’ *Mathematische Zeitschrift *39 (1935), 176-210; Prawitz, ‘On the Idea of a General Proof Theory’; Michael Dummett, *The Logical Basis of Metaphysics* (Cambridge, MA, 1991); Peter Schroeder-Heister, ‘Validity Concepts in Proof-Theoretic Semantics’, *Synthese* 148 (2006), 525–71; Curtis Franks, ‘Cut as Consequence’, *History and Philosophy of Logic* 31 (2010), 349–79.

[15] John Buridan, *Tractatus de Consequentiis* I. c. 2, 19-20. An early application of model-theoretic consequence has recently been traced to the 12th century Arabic logician Abū al-Barakāt. See Wilfrid Hodges, ‘Two Early Arabic Applications of Model-Theoretic Consequence’, *Logica Universalis* 12 (2018), 37–54.

[16] Walter Burleigh, *De Puritate Artis Logicae*, ed. Philotheus Boehner (St Bonaventure, NY, 1955), 76.5–7.

[17] Catarina Dutilh Novaes, ‘The Different Ways in Which Logic Is (Said to Be) Formal’, *History and Philosophy of Logic* 32 (2011), 303–32.

[18] Should this entail that ‘Plato jumps’ follows from ‘Socrates runs’? No: though every model of the former is a model of the latter, it is not so ‘at the same time’, and hence fails the semantic criterion for following.

[19] John Buridan, *Tractatus de Consequentiis* I, c. 3, 20-22, Pseudo-Scotus, *Quaestiones Super Libros II Priorum Analyticorum* I, q. 10, 103-105.

[20] Petrus Abaelardus, *Dialectica*, ed. Lambertus M. de Rijk (Assen, 1966), 283–84.

[21] José M. Sagüillo, ‘Logical Consequence Revisited’, *Bulletin of Symbolic Logic* 3 (1997), 216–41, 218-219; Kit Fine, ‘A Theory of Truthmaker Content I: Conjunction, Disjunction and Negation’, *Journal of Philosophical Logic* 46 (2017), 625–74; Volker Halbach, ‘The Substitutional Analysis of Logical Consequence’, *Noûs*, Forthcoming; Dag Prawitz, ‘The Fundamental Problem of General Proof Theory’, *Studia Logica*, forthcoming.

[1] Sten Ebbesen, ‘Ancient Scholastic Logic as the Source of Medieval Scholastic Logic’, in *The Cambridge History of Later Medieval Philosophy*, ed. Norman Kretzmann, Anthony Kenny, and Jan Pinborg (Cambridge, 1982), 104.

[2] Marsilius of Inghen, *Quaestiones Super Libros de Generatione et Corruptione* (Venice, 1501), fol. 106va.

[3] William J. Courtenay, ‘The University of Paris at the Time of Jean Buridan and Nicole Oresme’, *Vivarium* 42 (2004), 1–17; J. M. M. H. Thijssen, ‘The Buridan School Reassessed: John Buridan and Albert of Saxony’, *Vivarium* 42 (2004), 18–42.

[4] Green-Pedersen, ‘Bradwardine(?) On Ockham’s Doctrine of Consequences’ 92-93.

[5] Cf. Niels Jørgen Green-Pedersen, ‘Nicholas Drukken de Dacia’s Commentary on the Prior Analytics–with Special Regard to the Theory of Consequences’, *Cahiers de L’Institut Du Moyen-Âge Grec et Latin* 37 (1981), 46.