There are four historical modal paradigms in ancient philosophy: thefrequency interpretation of modality, the model of possibility as apotency, the model of antecedent necessities and possibilities withrespect to a certain moment of time (diachronic modalities), and themodel of possibility as non-contradictoriness. None of these habits ofthought, which were well known in early medieval times through theworks of Boethius, was based on the idea of modality as involvingreference to simultaneous alternatives. This new paradigm wasintroduced into Western thought in early twelfth-century discussionsinfluenced by Augustine’s theological conception of God asacting by choice between alternative histories.

While the new idea of associating modal terms with simultaneousalternatives was used also in thirteenth-century theology, it was notoften discussed in philosophical contexts at that time. The increasingacceptance of Aristotle’s philosophy gave support to traditionalmodal paradigms, as is seen in Robert Kilwardby’s influentialcommentary on Aristotle’sPrior Analytics, in whichmodal syllogistic is treated as an essentialist theory of thestructures of being. There were analogous discussions of philosophicaland theological modalities in Arabic philosophy. Arabic modal theoriesinfluenced Latin discussions mainly through the translations ofAverroes’s works.

John Duns Scotus developed the model of modality as alternativenessinto a detailed theory. A logical possibility is something to which tobe is not repugnant, though it may not be compossible with otherpossibilities. Scotus’s modal semantics influenced earlyfourteenth-century philosophy and theology in many ways.Thirteenth-century essentialist assumptions were dropped from modalsyllogistic, the Aristotelian version of which was regarded as afragmentary theory without a sufficient explication of the variousfine structures of modal propositions. While the Aristotelianfrequency interpretation of modality was not found relevant in latemedieval logic, it continued to play a role in natural philosophy.

• 1. Aspects of Ancient Modal Paradigms
• 2. Early Medieval Developments
• 3. Modalities in Thirteenth-Century Logical Treatises
• 4. Fourteenth-Century Discussions
• Bibliography
  • Primary sources
  • Secondary sources
• Academic Tools
• Other Internet Resources
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1. Aspects of Ancient Modal Paradigms

In speaking about the general features of the universe, ancientphilosophers were inclined to think that all generic possibilitieswill be actualized, a habit of thinking called the principle ofplenitude by Arthur O. Lovejoy (1936). Correspondingly, it was naturalfor them to think that the invariant structures of reality arenecessary. This line of thought is found, e.g., in Plato’sdoctrine of the ideas which are exhaustively imitated in the world bythe Demiurge, in Aristotle’s theory of the priority of actualityover potentiality, in the Stoic doctrine of the rational world-orderand the eternal cosmic cycles, as well as in Plotinus’smetaphysics of emanation (Knuuttila 1993).

In these approaches to the constituents of the universe, modal notionscould be understood in accordance with the so-called‘statistical’ or ‘temporal frequency’ model ofmodality where the meaning of modal terms is spelled out extensionallyas follows: what is necessary is always actual, what is impossible isnever actual and what is possible is at least sometimes actual. Theterm ‘statistical interpretation of modality’ wasintroduced into the modern discussion by Oscar Becker (1952), and ithas been applied since in descriptions of certain ways of thinking inthe history of philosophy as well, particularly by Jaakko Hintikka(1973).

Even though Aristotle did not define modal terms with the help ofextensional notions, examples of this way of thinking can be found inhis discussion of eternal beings, the natures of things, the types ofevents, and generic statements about such things. (See, for example,Metaphysics. IX.10.) Modal terms refer to the one and onlyworld of ours and classify the types of things and events on the basisof their actuality. This paradigm suggests that actualization is thegeneral criterion of the genuineness of possibilities, but thedeterministic implications of this view compelled Aristotle to seekways of speaking about unrealized singular possibilities. DiodorusChronus (fl. 300 BCE) was a determinist who found no problem in thisway of thinking. Some commentators have argued that Aristotle’sviews showing similarities to the statistical model are based onspecial metaphysical and ontological doctrines and not on hisunderstanding of modal terms themselves. However, it is not clear thatAristotle had any such distinction in mind. (For differentinterpretations and evaluations of the role of this model inAristotle, see Hintikka 1973, Sorabji 1980, Seel 1982, Waterlow 1982a,van Rijen 1989, Gaskin 1995.) InPosterior Analytics I.6,Aristotle states that certain predicates may belong to their subjectsat all times without belonging to them necessarily. Some ancientcommentators took this to mean that Aristotle operated with adistinction between strong essentialper se necessities andweak accidental necessities in the sense of non-essential invariances,such as inseparable accidents (see also Porphyry,Isagoge3.5–6), and that this distinction played an important role inhis modal syllogistic. See the commentaries on thePriorAnalytics by Alexander of Aphrodisias (36.25–32;201.21–24) and Philoponus (43.8–18; 126.7–29);Flannery 1995, 62–65, 99–106. This was also the view ofAverroes and some Latin authors in the Middle Ages. (See below,section 3.)

Another Aristotelian modal paradigm was that of possibility as apotency. InMet. V.12 and IX.1 potency is said to be theprinciple of motion or change either as the activator or as thereceptor of a relevant influence. (For agent and patient inAristotle’s natural philosophy in general, see Waterlow 1982b.)The types of potency-based possibilities belonging to a species arerecognized as possibilities because of their actualization — nonatural potency type remains eternally frustrated. Aristotle says thatwhen the agent and the patient come together and there is nohindrance, the one must act and the other must be acted on(Met. IX.5).

InDe Caelo I.12 Aristotle supposes,perimpossibile, that a thing has contrary potencies, one of which isalways actualized. He argues that the alleged eternally unactualizedpotencies are not potencies at all because they cannot be assumed tobe realized at any time without contradiction. Aristotle applies herethe model of possibility as non-contradictoriness which is defined inPrior Analytics I.13 as follows: when a possibility isassumed to be realized, it results in nothing impossible. In speakingabout the assumed non-contradictory actualization of a possibility,Aristotle thinks that it is realized in real history. Aristotle usesinDe Caelo I.12 and in some other places (for example,Met IX.4,An. pr. I.15) areductio argumentwhich consists of assuming the denial of what is to be proved andshowing that this leads to a contradiction. The argument has createdmuch controversy about how possibilities are supposed to obtain. SeeJudson 1983; Rini 2011, 135–156; Rosen and Malink 2012; Smith2016.)

Aristotle refers to potencies in criticizing some of hiscontemporaries who claimed that only that which takes place ispossible (Met. IX.3). The model of possibility as potencyprima facie allowed him to speak about all kinds ofunrealized singular possibilities by referring to passive or activepotencies, but taken separately they represent partial possibilitieswhich do not guarantee that their actualization can take place. Moreis required for a real singular possibility, but when the furtherrequirements are added, such as a contact between the active andpassive factor and the absence of an external hindrance, the potencymodel suggests that the potency can really be actualized only when itis actualized (Met. IX.5,Phys. VIII.1). It ispossible that this led Aristotle to define motion(kinêsis) as the actuality of the potentiality (of theend)qua potentiality (Phys. III.1), but this didnot explain the possibility of beginning (Hintikka et al. 1977).

In discussing future contingent statements in In Chapter 9 ofDeinterpretatione, Aristotle says that what is, necessarily is whenit is, but he then qualifies this necessity of the present with theremark that it does not follow that what is actual is necessarywithout qualification. If he meant that the temporal necessity of apresent event does not imply that such an event necessarily takesplace in circumstances of that type, this is an unsatisfactory‘statistical’ attempt to avoid the problem thatchangeability as a criterion of contingency makes all temporallydefinite singular events necessary (Hintikka 1973). Anotherinterpretation is that Aristotle wanted to show that the necessity ofan event at a certain time does not imply that it would have beenantecedently necessary. Aristotle discusses such singular diachronicmodalities in some places (Met. VI.3;EN III.5,1114a17–21;De int. 19a13–17) in which he seemsto assume that the conditions which att₁ are necessary forp toobtain at a later timet₂ are notnecessarily sufficient for this, although they might be sufficient forthe possibility (att₁) thatp obtains att₂. Aristotledid not elaborate these ideas, which might have been his mostpromising attempt to formulate a theory of unrealized singularpossibilities (for example De int. 19a12–14. Theimportance of this model is particularly stressed in Waterlow 1982a;see also von Wright 1984; Weidemann 1986; Gaskin 1995.

Aristotle’s conceptual difficulties can be seen from his variousattempts to characterize the possibilities based on dispositionalproperties such as heatable, separable, or countable. Analogousdiscussions were not unusual in later ancient philosophy. InPhilo’s definition of possibility (ca. 300 BCE), the existenceof a passive potency was regarded as a sufficient ground for speakingabout a singular possibility. The Stoics revised this definition byadding the condition of the absence of external hindrance, thinkingthat otherwise the alleged possibility could not be realized.According to the deterministic world view of the Stoics, fate as akind of active potency necessitates everything, but they did notaccept the Master Argument of Diodorus Cronus for determinism, whichwas meant to show that there cannot be possibilities which will not berealized. The number of passive potencies with respect to a definitefuture instant of time is greater than what will be realized. As longas these possibilities are not prevented from being realized by beingunactualized, they in some sense represent open possibilities.Alexander of Aphrodisias was influenced by the Stoic ideas ofdiachronic modalities, but he thought that it was misleading to speakabout unrealized diachronic possibilities if everything is determined.He argued for what he took to be Aristotle’s view, namely thatthere are undetermined prospective alternatives which remain openoptions until the moment of time to which they refer. (See Sharples1983; Bobzien 1993, 1998; Hankinson 1998.) Neither Aristotle nor laterancient thinkers had any considered conception of simultaneousalternatives. They thought that what is, necessarily is when it is,and that the alternative possibilities disappear when the future isfixed. Alexander’s Peripatetic theory of alternative prospectivepossibilities could be characterized as the model of diachronicmodalities without simultaneous alternatives: there are transientsingular alternative possibilities, but those which will not berealized vanish instead of remaining unrealized.

Aristotle often made use of indirect arguments from false orimpossible positions by adding hypotheses which he himself labeled asimpossible. In order to defend Aristotle’s procedure againstancient critics, Alexander of Aphrodisias characterized thesehypotheses as impossibilities that were not nonsensical. (For thiscontroversy, see Kukkonen 2002.) Some late ancient authors wereinterested in impossible hypotheses as tools for conceptual analysis.In the arguments which were called Eudemian procedures somethingimpossible was assumed in order to see what followed. Theimpossibilities discussed in this way by Philoponus and Boethius showsimilarities with Porphyry’s characterization of inseparableaccidents as something which cannot occur separately but can beseparated in thought. These hypotheses were not regarded asformulations of possibilities in the sense of what could be actual;they were counterpossible and not merely counterfactual (Martin1999).

There are several recent works on Aristotle’s modalsyllogistics, but no generally accepted historical reconstructionwhich would make it a coherent theory. It was apparently based onvarious assumptions which were not fully compatible (Hintikka 1973;Smith 1989; Striker 2009). Some commentators have been interested infinding coherent layers of the theory by explicating them in terms ofAristotle’s other views (van Rijen 1989; Patterson 1995). Thereare also several formal reconstructions such as Rini 2011 (modernpredicate logic), Ebert and Nortmann 2007 (possible worlds semantics),and various set-theoretical approaches discussed in Johnson 2004 andMalink 2006, 2013 (mereological semantics).

2. Early Medieval Developments

Early medieval thinkers were well acquainted with ancient modalconceptions through Boethius’s works. One of the Aristotelianmodal paradigms occurring in Boethius is that of possibility aspotency (potestas,potentia). According to Boethius,when the term ‘possibility’ (possibilitas) isused in the sense of ‘potency’, it refers to real powersor tendencies, the ends of which are either actual or non-actual atthe moment of utterance. Some potencies are never unrealized. They aresaid to be necessarily actual. When potencies are not actualized,their ends are said to exist potentially (In Periherm. II,453–455). Necessarily actual potencies leave no room for thepotencies of their contraries, for they would remain unrealizedforever and the constitution of nature does not include elements whichwould be in vain (In Periherm. II, 236). The potencies ofnon-necessary features of being do not exclude contrary potencies.They are not always and universally actualized, but as potency-typeseven these potencies are taken to satisfy the actualization criterionof genuineness (In Periherm. I, 120–121; II, 237).

Boethius’s view that the types of potencies and potency basedpossibilities are sometimes actualized is in agreement with theAristotelian frequency interpretation of modality. This is anotherBoethian conception of necessity and possibility. He thought thatmodal notions can be regarded as tools for expressing temporal orgeneric frequencies. According to the temporal version, what alwaysis, is by necessity, and what never is, is impossible. Possibility isinterpreted as expressing what is at least sometimes actual.Correspondingly, a generic property of a species is possible only ifit is exemplified at least in one member of that species (InPeriherm. I, 120–121, 200–201; II, 237).

Like Aristotle, Boethius often treated statement-making utterances astemporally indeterminate sentences. The same sentence can be utteredat different times, and many of these temporally indeterminatesentences may sometimes be true and sometimes false, depending on thecircumstances at the moment of utterance. If the state of affairs theactuality of which makes the sentence true is omnitemporally actual,the sentence is true whenever it is uttered. In this case, it isnecessarily true. If the state of affairs associated with anassertoric sentence is always non-actual, the sentence is always falseand therefore impossible. A sentence is possible only if what isasserted is not always non-actual (I, 124–125). Quasi-statisticalideas are also employed in Ammonius’s Greek commentary onAristotle’sDe interpretatione which shares somesources with Boethius’s work (88.12–28) and in Alexanderof Aphrodisias’s commentary on Aristotle’s modalsyllogistic. (See Mueller 1999, 23–31.)

In dealing with Chapter 9 of Aristotle’sDeinterpretatione, Boethius argues (II, 241) that because

(1)

M(p_t & ¬p_t)

(1′)

It is possible thatp obtains att andnot-p obtains att

is not acceptable, one should also deny

(2)

p_t & M_t ¬p_t

(2′)

p obtains att and it is possible att that not-p obtains att.

The denial of (2) is equivalent to

(3)

p_t →  L_tp_t

(3′)

Ifp obtains att then it is necessary att thatp obtains att.

(2) was generally denied in ancient philosophy and its denial wastaken as an axiom by Boethius as well. Correspondingly, (3) shows howthe necessity of the present was understood in ancient thought.Boethius thought that the temporal necessity ofp can bequalified by shifting attention from temporally definite cases orstatements to their temporally indeterminate counterparts (I,121–122; II, 242–243; cf. Ammonius 153.24–26). Thiswas one of Boethius’s interpretations of the Aristoteliandistinction between necessity now and necessity without qualification.But he also made use of the diachronic model according to which thenecessity ofp att does not imply that, beforet, it was necessary thatp obtains att.

Boethius developed the diachronic ideas as part of his criticism ofStoic determinism. If it is not true that everything is causallynecessitated, there must be genuine alternatives in the course ofevents. Free choice was the source of contingency in which Boethiuswas mainly interested, but he thought in addition that according tothe Peripatetic doctrine there is a real factor of indeterminacy inthe causal nexus of nature. When Boethius refers to chance, freechoice, and possibility in this context, his examples includetemporalized modal notions which refer to diachronic prospectivepossibilities at a given moment of time. A temporally determinateprospective possibility may not be realized at the time to which itrefers, in which case it ceases to be a possibility. Boethius did notdevelop the idea of simultaneous alternatives which would remainintact even when diachronic possibilities had vanished, insisting thatonly what is actual at a certain time is at that time possible at thattime (cf. (3) above). But he also thought that there are objectivesingular contingencies, so that the result of some prospectivepossibilities is indefinite and uncertain ‘not only to us whoare ignorant, but to nature’ (In Periherm. I, 106, 120;II, 190–192, 197–198, 203, 207). (For Boethius’smodal conceptions, see Kretzmann 1985; Knuuttila 1993,45–62.)

As for the discussion of future contingent statements inDeinterpretatione 9, Boethius’s view shows similarities tothat of Ammonius, both authors apparently having known some similarGreek discussions. (Ammonius’s Greek commentary onDeinterpretatione is translated by D. Blank and Boethius’stwo Latin commentaries by N. Kretzmann in the same volume, with essaysby R. Sorabji, N. Kretzmann and M. Mignucci, in 1998.) According tothe majority interpretation, Ammonius and Boethius ascribe toAristotle the view that the predictions of future contingent eventsand their denials differ from other contradictory pairs ofpropositions because truth and falsity are not definitely distributedbetween them. The propositions are consequently neither true norfalse, but their disjunctions are necessary. This was characterized asbroad bivalence by Kretzmann 1987, 1998. (See also Frede 1985; Craig1988; Gaskin 1995.) Another interpretation holds that futurecontingents are not definitely true or false in Boethius’s viewbecause their truth-makers are not yet determined, but are true orfalse in an indeterminate way. No qualification of the principle ofbivalence is involved (Mignucci 1989, 1998; for a relatedinterpretation of Ammonius, see Seel 2000.) While most medievalthinkers regarded the latter view as true in theology, many of themthought that Aristotle’s opinion was similar to the Boethianbroad bivalence. Peter Abelard and John Buridan were among those whoread Aristotle as holding that future contingent propositions are trueor false. Peter Auriol argued that these propositions lack atruth-value; even God is aware of the future in a way which does notimply bivalence. This was an exceptional view. (See Normore 1982,1993; Lewis 1987; Schabel 2000; Knuuttila 2010.) Boethius, Aquinas,and many others thought that God can know future contingents onlybecause the flux of time is present to divine eternity. Some latemedieval thinkers, for example John Duns Scotus and William of Ockham,found the idea of atemporal presence of history to God problematic andtried to find other models for foreknowledge. These discussions led tothe so-called middle knowledge theory of the counterfactuals offreedom (Craig 1988; Freddoso 1988; Dekker 2000).

From the point of view of the history of modal thought, interestingthings took place in theology in the eleventh and twelfth centuries.Augustine had already criticized the application of the frequencymodel of possibility to divine power; for him, God has freely chosenthe actual world and its providential plan from alternatives which hecould have realized but did not will to do (potuit sednoluit). This way of thinking differs from ancient philosophicalmodal paradigms because the metaphysical basis is now the eternaldomain of simultaneous alternatives instead of the idea of onenecessary world order. In Augustine, God’s eternal ideas offinite beings represent the possibilities of how the highest being canbe imitated, the possibilities thus having an ontological foundationin God’s essence. This was the dominating conception oftheological modal metaphysics until Duns Scotus departed from it. (SeeKnuuttila 2014, 86–89). The discrepancy between the Catholic doctrineof God’s freedom and power and the philosophical modalconceptions was brought into the scope of discussion in the eleventhcentury. Peter Damian argued for the sovereignty of divine omnipotencein a way that is sometimes mistakenly taken to include the power tochange the past (Holopainen 1996). Anselm of Canterbury tried todevelop a general modal semantics based on the notions of power andpotentiality, with various conceptual distinctions (Knuuttila 2004).Modal questions in philosophy and theology were developed in a moresophisticated way in twelfth-century considerations of God’spower and providence and historical contingencies with the idea ofsimultaneous alternatives. The idea of divine choice betweenalternatives was absent in Avicenna and Averroes, but it was defendedby al-Ghazali, who criticized Avicenna’s necessitarian metaphysics.(See Kukkonen 2000; for metaphysical necessitarianism in Arabicphilosophy, see Belo 2007; De Haan 2020.)

Even though Abelard (1079–1142) sometimes made use of theBoethian modal concepts (as Anselm also did), he was interested in thephilosophical significance of the idea of modality as alternativeness.Assuming that what is actual is temporally necessary at a certainpoint of time as no longer avoidable, he adds that unrealizedcounterfactual alternatives are possible at the same time in the sensethat they could have happened at that time. There are also merelyimaginable alternatives, such as Socrates’s being a bishop,which never had a real basis in things. (See Martin 2001, 2003;Marenbon 2007, 156–158, is sceptical about this interpretation;see also Binini 2019.) Gilbert of Poitiers (d. 1154) stressed the ideathat natural regularities which are called natural necessities are notabsolute, since they are freely chosen by God and can be overridden bydivine power. This basically Augustinian conception was a widespreadtheological view in the twelfth century. In explaining Plato’s‘Platonitas’ Gilbert innovatively argues that thisincludes all that Plato was, is and will be as well as what he couldbe but never is (The Commentaries on Boethius144.77–78, 274.75–76; Knuuttila 1993, 75–82).

An interesting early thirteenth-century philosophical analysis ofAugustinian modalities was put forward by Robert Grosseteste (d. 1253;Lewis 1996). Grosseteste taught that while things are primarily callednecessary or possible ‘from eternity and withoutbeginning’ with respect to God’s eternal knowledge, thereare necessities and impossibilities with a beginning in God’sprovidence which are eternal contingencies in the sense that God couldhave chosen their opposites (De libero arbitrio168.26–170.33, 178.24–29). One of the theses oftwelfth-century authors, later callednominales, was that‘What is once true is always true’. It was argued thatwhile tensed statements about temporally definite singular events havea changing truth-value, the corresponding non-tensed propositions areunchangingly true or false, without being necessarily true or falsefor this reason (Nuchelmans 1973, 177–189; Iwakuma and Ebbesen1992). This was in agreement with Abelard’s view that futurecontingent propositions are true or false. The actuality of acontingent state of affairs at a specified future time does notexclude the non-temporalized possibility of simultaneous alternatives,nor does the truth of a proposition about this state of affairs makeit necessary (Abelard,Glossae super Peri hermeneias IX,520–577; see also Peter of Poitiers (d. 1205),Sententiae I.7.133–43, I.12.164–223, I.14,328–353).

3. Modalities in Thirteenth-Century Logical Treatises

Modifying Boethius’s systematization of Aristotle’sremarks inDe interpretatione 12 and 13, twelfth- andthirteenth-century logicians often presented the equipollences betweenmodal terms and opposed relations between modal propositions with thehelp of the following diagram:

Figure 1.

The square could be taken to refer to modalsde dicto orsingular modalsde re (see below.) Abelard tried to definethe opposed relations between quantifiedde re modals aswell, mistakenly thinking that these were the same as those betweensingular modal propositions (Glossae super PerihermeneiasXII, 468–471, 530–544). This question was not muchdiscussed before its satisfactory solution in fourteenth-century modalsemantics. (See Hughes 1989 and his description of Buridan’soctagon of modal opposites and equipollences.) Whilepossibile andcontingens are treated as synonyms inthe figure, it became more usual to associate the former withone-sided possibility (not impossible) and the latter with two-sidedpossibility (neither necessary nor impossible).

The anonymousDialectica Monacensis (ca. 1200) is one of thenumerous works representing the new terminist approach to logic andcan be used as an example of how modalities were treated in it. (Acollection of late twelfth- and early thirteenth-century logical textsis edited in de Rijk 1962–67.) In discussing the quantity(universal, particular, singular) and quality (affirmative, negative)of the modals, the author states that modal terms may be adverbial ornominal. The modal adverb qualifies the copula, and the structure ofthe sentence can be described as follows:

(4)

quantity/subject/modalized copula/predicate (for example: Some A’s are necessarily B)

In this form, the negation can be located in different places,either

(5)

quantity/subject/copula modalized by a negated mode/predicate (for example: Some A’s are-not-necessarily B)

or

(6)

quantity/subject/modalized negative copula/predicate (for example: Some A’s are-necessarily-not B)

The modal sentences with nominal modes can be read in two ways. Onecan apply an adverbial type of reading to them, which is said to behow Aristotle treats modal sentences in thePrior Analytics.The quality and quantity of such ade re modal sentence isdetermined by the corresponding non-modal sentence. In adedicto modal sentence that which is asserted in a non-modalsentence is considered as the subject about which the mode ispredicated. When modal sentences are understood in this way, they arealways singular, their form being:

(7)

subject/copula/mode (for example: That some A’s are B is necessary.)

This reading is said to be the one which Aristotle presented inDeinterpretatione (De Rijk 1967, II-2, 479.35–480.26). Theidea of the systematic distinction between the readingsdedicto (in sensu composito) andde re (insensu diviso) of modally qualified statements was employed inAbelard’s investigations of modal statements (Glossae superPerihermeneias XII, 3–106;Dialectica191.1–210.19). Independently of Abelard, the distinction wasoften used, as in theDialectica Monacensis, in discussionsof the composition-division ambiguity of sentences. (See alsoMaierù 1972, ch. 5; Jacobi 1980, ch. 4.)

Following Boethius, many authors referred to a modal distinction basedon a triple matter of assertoric sentences, namely natural,contingent, or remote. In a sentence of natural matter, the predicatebelongs essentially to the subject or is its proprium, in contingentmatter, the predicate may belong or not belong to the subject, and inremote matter, the predicate cannot belong to the subject. See, forexample, Peter of Spain,Tractatus, p. 7;DialecticaMonacensis 472.9–473.22. According to Garland the Computist (11thcentury), the opposite universal sentences of contingent matter areboth false and the opposite particular sentences are both true(Dialectica, 54.21–30; 82.25–30). The same division of thesentences of contingent matter based on the frequency interpretationof contingency is found in Aquinas (In Periherm. I.13, 168).See also Boethius,In Periherm. II, 177.18–178.8;303.15–306.13; 325.8–15. Another often discussed theme was thedistinction between modalitiesper se andperaccidens, which was based on the idea that the modal status of atemporally indefinite sentence may be changeable or not; for example,‘You have not been in Paris’ may begin to be impossible,whereas ‘You either have or have not been in Paris’ maynot. (See, for example, William of Sherwood,Introduction toLogic, 41). Another distinction between sentences necessaryper se andper accidens was based onAristotle’s theory ofper se predication inPosterior Analytics I.4. A sentence was said to beaccidentally necessary when it was unchangeably true but, as distinctfromper se predications, there was no necessary conceptualconnection between subject and predicate. This became an importantpart of thirteenth-century interpretations of Aristotle’s modalsyllogistics. (See, for example, Robert Kilwardby’sNotulelibri Priorum 8.133–142; 40.162–174.)

One example of the prevalence of the traditional use of modal notionscan be found in the early medievalde dicto/de re analysis ofexamples such as ‘A standing man can sit’. It was commonlystated that the composite (de dicto) sense is ‘It ispossible that a man sits and stands at the same time’ and thaton this reading the sentence is false. The divided (de re)sense is ‘A man who is now standing can sit’ and on thisreading the sentence is true. Many authors formulated the dividedpossibility as follows: ‘A standing man can sit at anothertime’. It was assumed that a possibility refers to anactualization in the one and only world history and that it cannotrefer to the present moment because of the necessity of the presentunderstood in the Aristotelian sense formulated in (2) and (3) above.When authors referred to another time, they thought that thepossibility will be realized at that time or that the dividedpossibility refers to the future even though it may remain unrealized.Those who made use of the (at that time modern) idea of simultaneousalternatives took the composite reading to refer to one and the samestate of affairs and the divided reading to simultaneous alternativestates of affairs. This analysis was also applied to the question ofwhether God’s knowledge of things makes them necessary(Knuuttila 1993, 118–121).

A great deal of Abelard’s logical works consisted of discussionsof topics, consequences and conditionals. Like Boethius, Abelardthought that true conditionals express necessary connections betweenthe antecedents and the consequents. He argued that inseparability andentailment between the truth of the antecendent and consequent arerequired for the truth of a conditional. Some twelfth-century mastersregarded the principle that the antecedent is not true without theconsequent as a sufficient condition for the truth of a conditionaland accepted the so-called paradoxes of implication. The question ofthe nature of conditionals and consequences remained a popular themein medieval logic (Martin 1987, 2012).

The principles of propositional modal logic, found inPriorAnalytics I.15, were generally expressed as follows: if theantecedent of a valid consequence is possible/necessary, theconsequent is possible/necessary (Abelard,Dialectica202.6–8). However, the main interest was in modal syllogisticand modal predicate logic. Avicenna (d. 1037) wrote a brief Arabicsummary of Aristotle’s modal syllogistic, but his own theory wasdifferent, being based on the assumptions that the subject terms andthe predicate terms of assertoric and modal propositions stand for allpossible applications and the truth-conditions of assertoricpropositions and corresponding possibility propositions are the same.It follows, for example, that syllogisms with assertoric premisescoincide with uniform possibility syllogisms (Street 2002, 2005;Lagerlund 2009). Avicenna was particularly interested in relativenecessities and distinguished between various types of conditionalnecessities in terms of temporal determinations. Later Arabic works onmodal theories were much influenced by Avicenna. (See Strobino andThom 2016.) While Averroes’s commentaries on thePriorAnalytics followed the main lines of Aristotle’s text, hisseparate treatise on modality involved new systematic ideas, mainlythe theory of accidental andper se necessary terms and theinterpretation of syllogistic necessity premises asper senecessary predications withper se necessary terms. Bothideas were inspired by Aristotle’s remarks in thePosteriorAnalytics I.4. Since Averroes took modal premises to be of thedivided type, assertoric premises in Aristotelian mixednecessity-assertoric syllogisms must have a predicate term which isnecessary. The same applies to the subject term of the first premisein mixed assertoric-necessity syllogisms (Quaesita octo in librumPriorum Analyticorum, IV.3, 84, inAristotelis Opera cumAverrois Commentariis I.2b; see also Thom 2003, 81–85).This is a speculative explanation of Aristotle’s asymmetrictreatment of mixed necessity-assertoric syllogisms and mixedassertoric-necessity syllogisms. Gersonides later tried to developfurther Averroes’s remarks; see Manekin 1992. Analogousessentialist ideas were developed in thirteenth-century Latindiscussions.

The first Latin commentary on thePrior Analytics is ananonymous late twelfth-century treatise (‘Anonymus AurelianensisIII’) which involves detailed discussions of modal conversionand modal syllogisms as well as many problems dealt with in ancientcommentaries. (See Ebbesen 2008; an edition by Thomsen Thörnqvistin 2015.).Dialectica Monacensisinvolves a brief summary of Aristotle’s modal syllogistic theelements of which were discussed in logic courses in Paris in thefirst part of the thirteenth century. Robert Kilwardby’scommentaryNotule libri Priorum (c. 1240) became anauthoritative work from which the discussions of modal syllogisms inthe commentaries of Albert the Great (ca. 1250) and many others werelargely derived. Abelard, who did notdeal with Aristotle’s modal syllogistic, said that the modals inmixed syllogisms with both modal and assertoric premises should beunderstood in a way which he elsewhere characterizes asde reinterpretation (Glossae super Perihermeneias XII,189–203). This reading of modal premises was often assumed,although it was seldom discussed as such. A central problem ofAristotle’s theory is that the structure of the premises is notanalyzed. Even if it is natural to think that the presupposition ofthe mixed moods is ade re reading of modally qualifiedpremises, this creates difficulties when applied to the conversionrules, most of which are unproblematic only if understood as rules formodalsde dicto. (According to Aristotle, necessity premisesare converted in the same way as assertoric premises:‘Every/some A is B’ implies ‘Some B is A’ and‘No A is B’ implies ‘No B is A’. Affirmativecontingency premises are converted to corresponding negativecontingency premises and both these by the conversion of terms toparticular contingency propositions;Prior Analytics I.3,13.)

While many historians think that Aristotle’s modal syllogisticincluded incompatible elements, this was not the view ofmid-thirteenth century logicians. Many of them discussed the samealleged counter-examples to the universal convertibility of necessarypropositions, such as

(8)

Everything healthy (or awake) is necessarily an animal.

Robert Kilwardby’s explanation is based on the view thatconvertible necessity premises are necessity propositionsperse and notper accidens, like (8), which are notconvertible. (SeeNotule libri Priorum 8.133–146.) Inaffirmative necessity propositionsper se, the subject isper se connected to the predicate. In negative necessitypropositionsper se, the subject isper seincompatible with the predicate. The terms inper seinherences or incompatibilities are essential and necessarily standfor the things they signify. The historical background ofKilwardby’s interpretation is not clear; it does showsimilarities to Averroes’s discussion mentioned above, but therewas no historical link. (See Lagerlund 2000, 25–42; Thom 2007,19–28.)

As for the conversion of contingency propositions (neither necessarynor impossible), Kilwardby notes that while the converted propositionsof indefinite (utrumlibet) contingency are of the same typeof contingency, the conversion of natural contingency propositions(true about most cases) results in contingency propositions whencontingency means possibility proper (not impossible). There wereextensive discussions of the kinds of contingency based on variousphilosophical ideas in Kilwardby, Albert the Great and theircontemporaries (Knuuttila 2008, 540–541).

Kilwardby follows Aristotle’s remark that ‘A contingentlybelongs to B’ may mean either ‘A contingently belongs tothat to which B belongs’ or ‘A contingently belongs tothat to which B contingently belongs’,Notule libriPriorum 18.682–697. Kilwardby argues that the subject termsin contingency syllogisms are read in the second way and ampliated, ifsyllogistic relations do not demand restrictions. In explaining thedifference in this respect between necessity propositions andcontingency propositions, he states that since the terms inperse necessity propositions are essential, ‘Every A isnecessarily B’ and ‘Whatever is necessarily A isnecessarily B’ behave in the same way in logic. Contingencypropositions which are ampliated do not mean the same as those whichare not ampliated (Notule libri Priorum 18.187–207;18.653–672). For the conversion of both types of contingencypropositions and the role of the distinction between essential andaccidental terms in defining the truth conditions of contingencypropositions, see Thom 2019, 139–149. In the conversion ofde re readings of unampliated contingency propositions, thepossibility changes into actuality, which is in agreement with thefrequency interpretation of modality.

According to Kilwardby, the modal character of the predication in theconclusion of the perfect first figure syllogism follows from thefirst premise, which involves the whole syllogism in accordance withthedici de omni et nullo (Lagerlund 2000, 41–42). Thepremises and the conclusion in uniform necessity syllogisms arenecessaryper se. In mixed first-figure syllogisms with amajor necessity premise and a minor asertoric premise, thenon-modalized premise should besimpliciter assertoric, i.e.,a necessarily trueper se predication. Similarly, in mixedfirst-figure syllogisms with contingent major and assertoric minorpremises, the assertoric premise must besimpliciterassertoric, but this time the criteria are that the predicate belongsto the subjectper se, invariably or by natural contingency(Notule libri Priorum 15.255–301;20.706–736).

Kilwardby explains that in first-figure mixed necessity-assertoricsyllogisms the necessity premise appropriates to itself a minor whichis necessaryper se; no such appropriation occurs infirst-figure mixed assertoric-necessity syllogisms. There are similarappropriation rules for some mixed second-figure and third-figuremoods with assertoric and necessity premises and for various mixedcontingency moods pertaining to the kind of appropriated contingencypremises or assertoric premises (Thom 2007, chs. 5–6).

Kilwardby and his followers regarded Aristotle’s modalsyllogistic as the correct theory of modalities, the explication ofwhich demanded various metaphysical considerations. As exemplified bythe appropriation rules, they assumed that propositions of the sameform had different interpretations, depending on how they were relatedto other propositions in a syllogism. From the logical point of view,these rules have anad hoc character. (For some comparisonsbetween contemporary philosophical modal logic and thirteenth-centuryviews, see also Uckelman 2009.)

After Kilwardby and Albert the Great, who retailed him, severalthirteenth-century authors wrote treatises on thePriorAnalytics. Ebbesen (2010) lists seven literal and six questioncommentaries from the 1270s through the 1290s. One of these has beenrecently edited, namely the question commentary of Radulphus Brito(2017). On the basis of the lists of the contents of the questioncommentaries included in Ebbesen 2010, one can characterize this as aKilwardbian tradition. While Richard Campsall’s earlyfourteenth-centuryQuestions on Aristotle’s PriorAnalytics was influenced by this literature, it also involvedsome new modal ideas. Campsall thinks that one should discussdedicto andde re modalities separately. In consideringthe syllogistic logic ofde re possibility statements withterms standing for actual things, he says that an affirmativedere possibility statement as of now implies the correspondingassertoric statement (5.40) and a negative assertoric statement as ofnow implies the correspondingde re necessity statement(5.50). It follows that what is possible now is actualized and thingscannot be otherwise because all true present tense negative statementsare necessarily true. This is Campsall’s version of thetraditional doctrine of the necessity of the present. But if thedenial of a present affirmative statement is necessary, how can thestatement itself be non-necessary? Campsall defines ade recontingency statement as a conjunction of an affirmative andcorresponding negative possibility proper statement (7.34–36).Perhaps he thought that this is applicable to present tense statementswith actual terms and imply the idea of simultaneous alternatives, assuggested by Lagerlund 2000, 87–90. But Campsall also equatesde re necessity with respect to actual things to unchangingpredication and contingency to changing predication. Actual things maybe contingent in the sense that they will be changed in the future(12.31). For Campshall’s confusing formulations, see alsoKnuuttila 2018.

4. Fourteenth-Century Discussions

John Duns Scotus’s theory of simultaneous alternatives was anexposition of the intensional theory of modality, some elements ofwhich were put forward in the twelfth century. In criticizing Henry ofGhent’s theory of theological modalities, Scotus sketched amodel of ‘divine psychology’ in which certain relationsbetween theological, metaphysical, and modal theoretical notions aredefined. Scotus deviated from the metaphysical tradition in whichpossibilities were founded in divine being. According to him, when Godas an omniscient being knows all possibilities, he does not know themby turning first to his essence. Possibilities can be known inthemselves (Ord. I.35, 32). In fact they would be what theyare even if there were no God. Scotus states that if it is assumedthat,per impossibile, neither God nor the world exists andthe proposition ‘The world is possible’ then existed, thisproposition would be true. The actual world is possible as it is, andthis possibility and the possibilities of unrealized things areprimary metaphysical facts which are not dependent on anything else(Ord. I.7.1, 27;Lect. I.7, 32, I.39.1–5, 49).The commentators are divided on whether Scotus speaks here aboutpossibilities as such (Knuuttila 1996) or about the independence offormal possibilities, but not of concrete possibilities, which aredependent on God (Normore 2003; Hoffmann 2009).

Scotus calls the propositional elements of the modal domain‘logical possibilities’. These express things and statesof affairs to which it is not repugnant to be. Possibilities as suchhave no kind of existence of their own nor are they causallysufficient for the existence of anything, but they form theprecondition for everything that is or can be. God’s omniscienceinvolves an actually infinite number of intelligibles (De primoprincipio IV.9, 68–69); as an object of divine knowledge, theyreceive an intelligible or objective being. As an object of divineomnipotence, they receive an intentional possible being. Some of theseare included in God’s providential plan of creation and willreceive an actual being. The description of how things could be at acertain moment consists of compossible possibilities. Thoughpossibilities necessarily are what they are, the actualization ofnon-necessary possibilities is contingent. Since all finite things arecontingently actual, their alternatives are possible with respect tothe same time, though these are not compossible with what is actual.Impossibilities are incompossibilities between possibilities(Ord. I.35, 32, 49–51, I.38, 10, I.43, 14;Lect. I.39.1–5, 62–65).

In criticizing extensional modal theories, Scotus redefined acontingent event as follows: ‘I do not call something contingentbecause it is not always or necessarily the case, but because itsopposite could be actual at the very moment when it occurs’(Ord. I.2.1.1–2, 86). This is a denial of thetraditional thesis of the necessity of the present and the temporalfrequency characterization of contingency. In Scotus’s modalsemantics, the meaning of the notion of contingency is spelt out byconsidering simultaneous alternatives. What is actual is contingentlyso if, instead of being actual, it could be not actual. Thisconception of simultaneous contingent alternatives is part of anargument that the first cause does not act necessarily. According toScotus, the eternal creative act of divine will is free only if itcould be other than it is in a real sense (Lect.I.39.1–5, 58). For Scotus’s modal theory, see alsoHonnefelder 1990; Vos et al. 1994; King 2001; Hoffmann 2002; thethesis of Scotus’s innovativeness is criticized in Pasnau2020.

Scotus’s approach to modalities brought new themes intophilosophical discussion. One of these was the idea of possibility asa non-existent precondition of all being and thinking. Some of hisfollowers and critics argued that if there were no God, there wouldnot be any kind of modality (see Hoffmann 2002, Coombs 2004; forBradwardine’s criticism, see Frost 2014). Scotus’s viewswere known in the seventeenth century through the works ofSuárez and some Scotist authors (Honnefelder 1990). In hisdiscussion of eternal truths, Descartes criticized the classical viewof the ontological foundation of modality as well as the Scotisttheory of modality and conceivability. (There are differentinterpretations of Descartes’s view of the foundations ofmodality and how it is related to late medieval discussions; seeAlanen 1990; Normore 1991, 2006.)

Another influential idea was the distinction between logical andnatural necessities and possibilities. In Scotus’s theory,logically necessary attributes and relations are attached to things inall those sets of compossibilities in which they occur. Against thisbackground one could ask which of the natural invariances treated asnecessities in earlier natural philosophy were necessary in thisstrong sense of necessity, and which of them were merely empiricalgeneralizations without being logically necessary. (For a discussionof logical and natural necessities in the fourteenth century, seeKnuuttila 1993, 155–160, 2001a.) Buridan distinguished betweenlogical and natural necessities in his classification of four gradesof necessity (Dialectica 8.6.3). He used a non-extensionalmetaphysical notion of possibilitysencundum imaginationen inhis questions on Aristotle’sPhysics (Sylla 2001), buta frequency interpretation in his questions on Aristotle’sOn generation and corruption I.4 (57). This was the mainmodal conception in the Averroist Aristotle commentaries by John ofJandun (see, for example, his questions onMetaphysics, IX.5114va-b; questions onOn the Heavens I.34, 21vb) and remaineda popular option until the Renaissance times (Knebel 2003).

One important branch of medieval logic developed in treatises calledDe obligationibus dealt, roughly speaking, with how anincreasing set of true and false propositions might remain coherent ina disputation. According to thirteenth-century rules, a false presenttense statement could be accepted as a starting point only if it wastaken to refer to a moment of time different from the actual one.Scotus deleted this rule, based on the Aristotelian axiom of thenecessity of the present, and later theories accepted the Scotistrevision. In this new form, obligations logic could be regarded as atheory of how to analyze possible states of affairs and their mutualrelationships. These discussions influenced the philosophical theoryof counterfactual conditionals (Yrjönsuuri 1994, 2001; Gelber2004; Dutilh Novaes 2007).

In dealing with counterfactual hypotheses of indirect proofs mentionedabove, Averroes and Thomas Aquinas made use of the idea of abstractpossibilities which did not imply the idea of alternative domains. Thepossibilities of a thing can be dealt with at various levels whichcorrespond to Porphyrian predicables. Something which is possible fora thing as a member of a genus can be impossible for it as a member ofa species. The same holds of it as a member of a species and anindividuated thing. Thus humans can fly because there are otheranimals which can fly. These abstract possibilities are impossible inthe sense that they cannot be actualized. Buridan heavily criticizedthis approach from the point of view of his modal theory. He arguedthat if a counterfactual state of affairs is possible, it can becoherently imagined as actual. If something cannot be treated in thisway, calling it possible is based on conceptual confusion. (SeeKnuuttila and Kukkonen 2011.) While Scotus, Buridan and many othersunderstood the basic level of possibility in terms of semanticconsistency, Ockham wanted to preserve the link to the notion of powerin his modal considerations, thinking that necessity is actuality plusimmutability, the past and the present are necessary, and Scotus waswrong in assuming that things could be different from how they are atthe very moment of their actuality (Normore 2016).

Influenced by the new ideas about logical modality, William of Ockham(Summa logicae), John Buridan (Tractatus deconsequentiis,Summulae de Dialectica) and some otherfourteenth-century authors could formulate the principles of modallogic much more completely and satisfactorily than did theirpredecessors. Questions of modal logic were discussed separately withrespect to modal propositionsde dicto andde re;modal propositionsde re were further divided into two groupsdepending on whether the subject terms refer to actual or possiblebeings. It was thought that logicians should also analyze therelationships between these readings and, furthermore, theconsequences having various types of modal sentences as their parts.Ockham, Buridan and their followers largely dropped thirteenth-centuryessentialist assumptions from modal syllogistic. They regarded theAristotelian version as a fragmentary theory in which the distinctionsbetween different types of fine structures were not explicated and,consequently, did not try to reconstruct Aristotle’s modalsyllogistic as a consistent whole by one unifying analysis of modalpropositions; they believed, like some modern commentators, that suchreconstruction was not possible. (For fourteenth-century modal logic,see King 1985; Lagerlund 2000; Thom 2003; Knuuttila 2008,551–567; Read 2021.)

According to Hughes (1989), one could supply a Kripke-style possibleworlds semantics to Buridan’s modal system. This is done inJohnston 2015. In discussing the conception of simultaneousalternativeness in fourteenth-century logic, many historians have usedthe terminology of possible worlds semantics without the metaphysicaland formal details of the modern theory (Yrjönsuuri 1994,167–174; Dutilh Novaes 2007, 90–91; Read 2021. Ockham and Buridanstate that the truth of ‘A white thing can be black’demands the truth of ‘This can be black’ and that‘This can be black’ and ‘‘This is black’is possible’ mean the same. Compound (de dicto) anddivided (de re) readings do not differ at this level, but areseparated in dealing with universal and particular propositions. WhileOckham did not discuss unrestricted divided necessity propositions,Buridan took the subject terms of all quantified divided modalpropositions as standing for possible beings if they are notrestricted. The truth of these propositions demands the truth of allor some relevant singular propositions of the type just mentioned; thedemonstrative pronoun is then taken to refer to the possible beingseven though they may not exist. Buridan could have said that thepossible truth of ‘This is X’ means that it is true in apossible state of affairs in which the possible being referred to by‘this’ occurs and that the necessary truth of ‘Thisis X’ means that it is true in all possible states of affairs inwhich the possible being referred to by ‘this’ occurs(Hughes 1989). For Buridan’s modal logic, see also Read’sintroduction toTreatise on Consequences (2015); Johnston2021.

The new modal logic was among the most remarkable achievements ofmedieval logic. Buridan’s modal logic was dominant in latemedieval times, being more systematic than that of Ockham because ofits symmetric treatment of possibility and necessity. It was embracedby Marsilius of Inghen, Albert of Saxony, Jodocus Trutfetter andothers (Lagerlund 2000, 184–227; for the later influence ofmedieval modal theories, see also Coombs 2003; Knebel 2003; Roncaglia1996, 2003; Schmutz 2006). The rise of the new modal logic wasaccompanied by elaborated theories of epistemic logic (Boh 1993) anddeontic logic (Knuuttila and Hallamaa 1995).

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