## 1. Aspects of Ancient Modal Paradigms



In speaking about the general features of the universe, ancient
philosophers were inclined to think that all generic possibilities
will be actualized, a habit of thinking called the principle of
plenitude by Arthur O. Lovejoy (1936). Correspondingly, it was natural
for them to think that the invariant structures of reality are
necessary. This line of thought is found, e.g., in Plato's
doctrine of the ideas which are exhaustively imitated in the world by
the Demiurge, in Aristotle's theory of the priority of actuality
over potentiality, in the Stoic doctrine of the rational world-order
and the eternal cosmic cycles, as well as in Plotinus's
metaphysics of emanation (Knuuttila 1993).



In these approaches to the constituents of the universe, modal notions
could be understood in accordance with the so-called
'statistical' or 'temporal frequency' model of
modality where the meaning of modal terms is spelled out extensionally
as follows: what is necessary is always actual, what is impossible is
never actual and what is possible is at least sometimes actual. The
term 'statistical interpretation of modality' was
introduced into the modern discussion by Oscar Becker (1952), and it
has been applied since in descriptions of certain ways of thinking in
the history of philosophy as well, particularly by Jaakko Hintikka
(1973).



Even though Aristotle did not define modal terms with the help of
extensional notions, examples of this way of thinking can be found in
his discussion of eternal beings, the natures of things, the types of
events, and generic statements about such things. (See, for example,
*Metaphysics*. IX.10.) Modal terms refer to the one and only
world of ours and classify the types of things and events on the basis
of their actuality. This paradigm suggests that actualization is the
general criterion of the genuineness of possibilities, but the
deterministic implications of this view compelled Aristotle to seek
ways of speaking about unrealized singular possibilities. Diodorus
Chronus (fl. 300 BCE) was a determinist who found no problem in this
way of thinking. Some commentators have argued that Aristotle's
views showing similarities to the statistical model are based on
special metaphysical and ontological doctrines and not on his
understanding of modal terms themselves. However, it is not clear that
Aristotle had any such distinction in mind. (For different
interpretations and evaluations of the role of this model in
Aristotle, see Hintikka 1973, Sorabji 1980, Seel 1982, Waterlow 1982a,
van Rijen 1989, Gaskin 1995.) In *Posterior Analytics* I.6,
Aristotle states that certain predicates may belong to their subjects
at all times without belonging to them necessarily. Some ancient
commentators took this to mean that Aristotle operated with a
distinction between strong essential *per se* necessities and
weak accidental necessities in the sense of non-essential invariances,
such as inseparable accidents (see also Porphyry, *Isagoge*
3.5-6), and that this distinction played an important role in
his modal syllogistic. See the commentaries on the *Prior
Analytics* 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 of
Averroes and some Latin authors in the Middle Ages. (See below,
section 3.)



Another Aristotelian modal paradigm was that of possibility as a
potency. In *Met*. V.12 and IX.1 potency is said to be the
principle of motion or change either as the activator or as the
receptor of a relevant influence. (For agent and patient in
Aristotle's natural philosophy in general, see Waterlow 1982b.)
The types of potency-based possibilities belonging to a species are
recognized as possibilities because of their actualization -- no
natural potency type remains eternally frustrated. Aristotle says that
when the agent and the patient come together and there is no
hindrance, the one must act and the other must be acted on
(*Met*. IX.5).



In *De Caelo* I.12 Aristotle supposes, *per
impossibile*, that a thing has contrary potencies, one of which is
always actualized. He argues that the alleged eternally unactualized
potencies are not potencies at all because they cannot be assumed to
be realized at any time without contradiction. Aristotle applies here
the model of possibility as non-contradictoriness which is defined in
*Prior Analytics* I.13 as follows: when a possibility is
assumed to be realized, it results in nothing impossible. In speaking
about the assumed non-contradictory actualization of a possibility,
Aristotle thinks that it is realized in real history. Aristotle uses
in *De Caelo* I.12 and in some other places (for example,
*Met* IX.4, *An. pr.* I.15) a *reductio* argument
which consists of assuming the denial of what is to be proved and
showing that this leads to a contradiction. The argument has created
much controversy about how possibilities are supposed to obtain. See
Judson 1983; Rini 2011, 135-156; Rosen and Malink 2012; Smith
2016.) 



Aristotle refers to potencies in criticizing some of his
contemporaries who claimed that only that which takes place is
possible (*Met*. IX.3). The model of possibility as potency
*prima facie* allowed him to speak about all kinds of
unrealized singular possibilities by referring to passive or active
potencies, but taken separately they represent partial possibilities
which do not guarantee that their actualization can take place. More
is required for a real singular possibility, but when the further
requirements are added, such as a contact between the active and
passive factor and the absence of an external hindrance, the potency
model suggests that the potency can really be actualized only when it
is actualized (*Met*. IX.5, *Phys*. VIII.1). It is
possible that this led Aristotle to define motion
(*kinesis*) as the actuality of the potentiality (of the
end) *qua* potentiality (*Phys*. III.1), but this did
not explain the possibility of beginning (Hintikka et al. 1977).



In discussing future contingent statements in In Chapter 9 of *De
interpretatione*, Aristotle says that what is, necessarily is when
it is, but he then qualifies this necessity of the present with the
remark that it does not follow that what is actual is necessary
without qualification. If he meant that the temporal necessity of a
present event does not imply that such an event necessarily takes
place in circumstances of that type, this is an unsatisfactory
'statistical' attempt to avoid the problem that
changeability as a criterion of contingency makes all temporally
definite singular events necessary (Hintikka 1973). Another
interpretation is that Aristotle wanted to show that the necessity of
an event at a certain time does not imply that it would have been
antecedently necessary. Aristotle discusses such singular diachronic
modalities in some places (*Met*. VI.3; *EN* III.5,
1114a17-21; *De int*. 19a13-17) in which he seems
to assume that the conditions which at
*t*1 are necessary for *p* to
obtain at a later time *t*2 are not
necessarily sufficient for this, although they might be sufficient for
the possibility (at *t*1) that
*p* obtains at *t*2. Aristotle
did not elaborate these ideas, which might have been his most
promising attempt to formulate a theory of unrealized singular
possibilities (for example  *De int.* 19a12-14. The
importance 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 various
attempts to characterize the possibilities based on dispositional
properties such as heatable, separable, or countable. Analogous
discussions were not unusual in later ancient philosophy. In
Philo's definition of possibility (ca. 300 BCE), the existence
of a passive potency was regarded as a sufficient ground for speaking
about a singular possibility. The Stoics revised this definition by
adding the condition of the absence of external hindrance, thinking
that otherwise the alleged possibility could not be realized.
According to the deterministic world view of the Stoics, fate as a
kind of active potency necessitates everything, but they did not
accept the Master Argument of Diodorus Cronus for determinism, which
was meant to show that there cannot be possibilities which will not be
realized. The number of passive potencies with respect to a definite
future instant of time is greater than what will be realized. As long
as these possibilities are not prevented from being realized by being
unactualized, they in some sense represent open possibilities.
Alexander of Aphrodisias was influenced by the Stoic ideas of
diachronic modalities, but he thought that it was misleading to speak
about unrealized diachronic possibilities if everything is determined.
He argued for what he took to be Aristotle's view, namely that
there are undetermined prospective alternatives which remain open
options until the moment of time to which they refer. (See Sharples
1983; Bobzien 1993, 1998; Hankinson 1998.) Neither Aristotle nor later
ancient thinkers had any considered conception of simultaneous
alternatives. They thought that what is, necessarily is when it is,
and that the alternative possibilities disappear when the future is
fixed. Alexander's Peripatetic theory of alternative prospective
possibilities could be characterized as the model of diachronic
modalities without simultaneous alternatives: there are transient
singular alternative possibilities, but those which will not be
realized vanish instead of remaining unrealized.



Aristotle often made use of indirect arguments from false or
impossible positions by adding hypotheses which he himself labeled as
impossible. In order to defend Aristotle's procedure against
ancient critics, Alexander of Aphrodisias characterized these
hypotheses as impossibilities that were not nonsensical. (For this
controversy, see Kukkonen 2002.) Some late ancient authors were
interested in impossible hypotheses as tools for conceptual analysis.
In the arguments which were called Eudemian procedures something
impossible was assumed in order to see what followed. The
impossibilities discussed in this way by Philoponus and Boethius show
similarities with Porphyry's characterization of inseparable
accidents as something which cannot occur separately but can be
separated in thought. These hypotheses were not regarded as
formulations of possibilities in the sense of what could be actual;
they were counterpossible and not merely counterfactual (Martin
1999).



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


## 2. Early Medieval Developments



Early medieval thinkers were well acquainted with ancient modal
conceptions through Boethius's works. One of the Aristotelian
modal paradigms occurring in Boethius is that of possibility as
potency (*potestas*, *potentia*). According to Boethius,
when the term 'possibility' (*possibilitas*) is
used in the sense of 'potency', it refers to real powers
or tendencies, the ends of which are either actual or non-actual at
the moment of utterance. Some potencies are never unrealized. They are
said 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 the
potencies of their contraries, for they would remain unrealized
forever and the constitution of nature does not include elements which
would be in vain (*In Periherm*. II, 236). The potencies of
non-necessary features of being do not exclude contrary potencies.
They are not always and universally actualized, but as potency-types
even these potencies are taken to satisfy the actualization criterion
of genuineness (*In Periherm*. I, 120-121; II, 237).



Boethius's view that the types of potencies and potency based
possibilities are sometimes actualized is in agreement with the
Aristotelian frequency interpretation of modality. This is another
Boethian conception of necessity and possibility. He thought that
modal notions can be regarded as tools for expressing temporal or
generic frequencies. According to the temporal version, what always
is, is by necessity, and what never is, is impossible. Possibility is
interpreted as expressing what is at least sometimes actual.
Correspondingly, a generic property of a species is possible only if
it is exemplified at least in one member of that species (*In
Periherm*. I, 120-121, 200-201; II, 237).



Like Aristotle, Boethius often treated statement-making utterances as
temporally indeterminate sentences. The same sentence can be uttered
at different times, and many of these temporally indeterminate
sentences may sometimes be true and sometimes false, depending on the
circumstances at the moment of utterance. If the state of affairs the
actuality of which makes the sentence true is omnitemporally actual,
the sentence is true whenever it is uttered. In this case, it is
necessarily true. If the state of affairs associated with an
assertoric sentence is always non-actual, the sentence is always false
and therefore impossible. A sentence is possible only if what is
asserted is not always non-actual (I, 124-125). Quasi-statistical
ideas are also employed in Ammonius's Greek commentary on
Aristotle's *De interpretatione* which shares some
sources with Boethius's work (88.12-28) and in Alexander
of Aphrodisias's commentary on Aristotle's modal
syllogistic. (See Mueller 1999, 23-31.)



In dealing with Chapter 9 of Aristotle's *De
interpretatione*, Boethius argues (II, 241) that because



(1)
M(*p**t* & ! *p**t*)
(1')
It is possible that *p* obtains at *t* and
not-*p* obtains at *t*


is not acceptable, one should also deny



(2)
*p**t* & M*t* !*p**t*
(2')
*p* obtains at *t* and it is possible at *t* that not-*p* obtains at *t*.


The denial of (2) is equivalent to



(3)
*p**t* -  L*t* *p**t*
(3')
If *p* obtains at *t* then it is necessary at *t* that *p* obtains at *t*.


(2) was generally denied in ancient philosophy and its denial was
taken as an axiom by Boethius as well. Correspondingly, (3) shows how
the necessity of the present was understood in ancient thought.
Boethius thought that the temporal necessity of *p* can be
qualified by shifting attention from temporally definite cases or
statements to their temporally indeterminate counterparts (I,
121-122; II, 242-243; cf. Ammonius 153.24-26). This
was one of Boethius's interpretations of the Aristotelian
distinction between necessity now and necessity without qualification.
But he also made use of the diachronic model according to which the
necessity of *p* at *t* does not imply that, before
*t*, it was necessary that *p* obtains at
*t*.



Boethius developed the diachronic ideas as part of his criticism of
Stoic determinism. If it is not true that everything is causally
necessitated, there must be genuine alternatives in the course of
events. Free choice was the source of contingency in which Boethius
was mainly interested, but he thought in addition that according to
the Peripatetic doctrine there is a real factor of indeterminacy in
the causal nexus of nature. When Boethius refers to chance, free
choice, and possibility in this context, his examples include
temporalized modal notions which refer to diachronic prospective
possibilities at a given moment of time. A temporally determinate
prospective possibility may not be realized at the time to which it
refers, in which case it ceases to be a possibility. Boethius did not
develop the idea of simultaneous alternatives which would remain
intact even when diachronic possibilities had vanished, insisting that
only what is actual at a certain time is at that time possible at that
time (cf. (3) above). But he also thought that there are objective
singular contingencies, so that the result of some prospective
possibilities is indefinite and uncertain 'not only to us who
are ignorant, but to nature' (*In Periherm*. I, 106, 120;
II, 190-192, 197-198, 203, 207). (For Boethius's
modal conceptions, see Kretzmann 1985; Knuuttila 1993,
45-62.)



As for the discussion of future contingent statements in *De
interpretatione* 9, Boethius's view shows similarities to
that of Ammonius, both authors apparently having known some similar
Greek discussions. (Ammonius's Greek commentary on *De
interpretatione* is translated by D. Blank and Boethius's
two Latin commentaries by N. Kretzmann in the same volume, with essays
by R. Sorabji, N. Kretzmann and M. Mignucci, in 1998.) According to
the majority interpretation, Ammonius and Boethius ascribe to
Aristotle the view that the predictions of future contingent events
and their denials differ from other contradictory pairs of
propositions because truth and falsity are not definitely distributed
between them. The propositions are consequently neither true nor
false, but their disjunctions are necessary. This was characterized as
broad bivalence by Kretzmann 1987, 1998. (See also Frede 1985; Craig
1988; Gaskin 1995.) Another interpretation holds that future
contingents are not definitely true or false in Boethius's view
because their truth-makers are not yet determined, but are true or
false in an indeterminate way. No qualification of the principle of
bivalence is involved (Mignucci 1989, 1998; for a related
interpretation of Ammonius, see Seel 2000.) While most medieval
thinkers regarded the latter view as true in theology, many of them
thought that Aristotle's opinion was similar to the Boethian
broad bivalence. Peter Abelard and John Buridan were among those who
read Aristotle as holding that future contingent propositions are true
or false. Peter Auriol argued that these propositions lack a
truth-value; even God is aware of the future in a way which does not
imply 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 only
because the flux of time is present to divine eternity. Some late
medieval thinkers, for example John Duns Scotus and William of Ockham,
found the idea of atemporal presence of history to God problematic and
tried to find other models for foreknowledge. These discussions led to
the so-called middle knowledge theory of the counterfactuals of
freedom (Craig 1988; Freddoso 1988; Dekker 2000).



From the point of view of the history of modal thought, interesting
things took place in theology in the eleventh and twelfth centuries.
Augustine had already criticized the application of the frequency
model of possibility to divine power; for him, God has freely chosen
the actual world and its providential plan from alternatives which he
could have realized but did not will to do (*potuit sed
noluit*). This way of thinking differs from ancient philosophical
modal paradigms because the metaphysical basis is now the eternal
domain of simultaneous alternatives instead of the idea of one
necessary world order. In Augustine, God's eternal ideas of
finite beings represent the possibilities of how the highest being can
be imitated, the possibilities thus having an ontological foundation
in God's essence. This was the dominating conception of
theological modal metaphysics until Duns Scotus departed from it. (See
Knuuttila 2014, 86-89). The discrepancy between the Catholic doctrine
of God's freedom and power and the philosophical modal
conceptions was brought into the scope of discussion in the eleventh
century. Peter Damian argued for the sovereignty of divine omnipotence
in a way that is sometimes mistakenly taken to include the power to
change the past (Holopainen 1996). Anselm of Canterbury tried to
develop a general modal semantics based on the notions of power and
potentiality, with various conceptual distinctions (Knuuttila 2004).
Modal questions in philosophy and theology were developed in a more
sophisticated way in twelfth-century considerations of God's
power and providence and historical contingencies with the idea of
simultaneous alternatives. The idea of divine choice between
alternatives was absent in Avicenna and Averroes, but it was defended
by al-Ghazali, who criticized Avicenna's necessitarian metaphysics.
(See Kukkonen 2000; for metaphysical necessitarianism in Arabic
philosophy, see Belo 2007; De Haan 2020.) 



Even though Abelard (1079-1142) sometimes made use of the
Boethian modal concepts (as Anselm also did), he was interested in the
philosophical significance of the idea of modality as alternativeness.
Assuming that what is actual is temporally necessary at a certain
point of time as no longer avoidable, he adds that unrealized
counterfactual alternatives are possible at the same time in the sense
that they could have happened at that time. There are also merely
imaginable 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 idea
that natural regularities which are called natural necessities are not
absolute, since they are freely chosen by God and can be overridden by
divine power. This basically Augustinian conception was a widespread
theological view in the twelfth century. In explaining Plato's
'Platonitas' Gilbert innovatively argues that this
includes all that Plato was, is and will be as well as what he could
be but never is (*The Commentaries on Boethius*
144.77-78, 274.75-76; Knuuttila 1993, 75-82).



An interesting early thirteenth-century philosophical analysis of
Augustinian modalities was put forward by Robert Grosseteste (d. 1253;
Lewis 1996). Grosseteste taught that while things are primarily called
necessary or possible 'from eternity and without
beginning' with respect to God's eternal knowledge, there
are necessities and impossibilities with a beginning in God's
providence which are eternal contingencies in the sense that God could
have chosen their opposites (*De libero arbitrio*
168.26-170.33, 178.24-29). One of the theses of
twelfth-century authors, later called *nominales*, was that
'What is once true is always true'. It was argued that
while tensed statements about temporally definite singular events have
a changing truth-value, the corresponding non-tensed propositions are
unchangingly true or false, without being necessarily true or false
for this reason (Nuchelmans 1973, 177-189; Iwakuma and Ebbesen
1992). This was in agreement with Abelard's view that future
contingent propositions are true or false. The actuality of a
contingent state of affairs at a specified future time does not
exclude the non-temporalized possibility of simultaneous alternatives,
nor does the truth of a proposition about this state of affairs make
it 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's
remarks in *De interpretatione* 12 and 13, twelfth- and
thirteenth-century logicians often presented the equipollences between
modal terms and opposed relations between modal propositions with the
help of the following diagram:



![square](square.png)

Figure 1.




The square could be taken to refer to modals *de dicto* or
singular modals *de re* (see below.) Abelard tried to define
the opposed relations between quantified *de re* modals as
well, mistakenly thinking that these were the same as those between
singular modal propositions (*Glossae super Perihermeneias*
XII, 468-471, 530-544). This question was not much
discussed before its satisfactory solution in fourteenth-century modal
semantics. (See Hughes 1989 and his description of Buridan's
octagon of modal opposites and equipollences.) While
*possibile* and *contingens* are treated as synonyms in
the figure, it became more usual to associate the former with
one-sided possibility (not impossible) and the latter with two-sided
possibility (neither necessary nor impossible). 



The anonymous *Dialectica Monacensis* (ca. 1200) is one of the
numerous works representing the new terminist approach to logic and
can be used as an example of how modalities were treated in it. (A
collection of late twelfth- and early thirteenth-century logical texts
is 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 or
nominal. The modal adverb qualifies the copula, and the structure of
the 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. One
can apply an adverbial type of reading to them, which is said to be
how Aristotle treats modal sentences in the *Prior Analytics*.
The quality and quantity of such a *de re* modal sentence is
determined by the corresponding non-modal sentence. In a *de
dicto* modal sentence that which is asserted in a non-modal
sentence is considered as the subject about which the mode is
predicated. When modal sentences are understood in this way, they are
always 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 in *De
interpretatione* (De Rijk 1967, II-2, 479.35-480.26). The
idea of the systematic distinction between the readings *de
dicto* (*in sensu composito*) and *de re* (*in
sensu diviso*) of modally qualified statements was employed in
Abelard's investigations of modal statements (*Glossae super
Perihermeneias* XII, 3-106; *Dialectica*
191.1-210.19). Independently of Abelard, the distinction was
often used, as in the *Dialectica Monacensis*, in discussions
of the composition-division ambiguity of sentences. (See also
Maieru 1972, ch. 5; Jacobi 1980, ch. 4.)



Following Boethius, many authors referred to a modal distinction based
on a triple matter of assertoric sentences, namely natural,
contingent, or remote. In a sentence of natural matter, the predicate
belongs essentially to the subject or is its proprium, in contingent
matter, the predicate may belong or not belong to the subject, and in
remote matter, the predicate cannot belong to the subject. See, for
example, Peter of Spain, *Tractatus*, p. 7; *Dialectica
Monacensis* 472.9-473.22. According to Garland the Computist (11th
century), the opposite universal sentences of contingent matter are
both false and the opposite particular sentences are both true
(*Dialectica*, 54.21-30; 82.25-30). The same division of the
sentences of contingent matter based on the frequency interpretation
of 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 the
distinction between modalities *per se* and *per
accidens*, which was based on the idea that the modal status of a
temporally 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' may
not. (See, for example, William of Sherwood, *Introduction to
Logic*, 41). Another distinction between sentences necessary
*per se* and *per accidens* was based on
Aristotle's theory of *per se* predication in
*Posterior Analytics* I.4. A sentence was said to be
accidentally necessary when it was unchangeably true but, as distinct
from *per se* predications, there was no necessary conceptual
connection between subject and predicate. This became an important
part of thirteenth-century interpretations of Aristotle's modal
syllogistics. (See, for example, Robert Kilwardby's *Notule
libri Priorum* 8.133-142; 40.162-174.)



One example of the prevalence of the traditional use of modal notions
can be found in the early medieval *de dicto/de re* analysis of
examples such as 'A standing man can sit'. It was commonly
stated that the composite (*de dicto*) sense is 'It is
possible that a man sits and stands at the same time' and that
on this reading the sentence is false. The divided (*de re*)
sense is 'A man who is now standing can sit' and on this
reading the sentence is true. Many authors formulated the divided
possibility as follows: 'A standing man can sit at another
time'. It was assumed that a possibility refers to an
actualization in the one and only world history and that it cannot
refer to the present moment because of the necessity of the present
understood in the Aristotelian sense formulated in (2) and (3) above.
When authors referred to another time, they thought that the
possibility will be realized at that time or that the divided
possibility refers to the future even though it may remain unrealized.
Those who made use of the (at that time modern) idea of simultaneous
alternatives took the composite reading to refer to one and the same
state of affairs and the divided reading to simultaneous alternative
states of affairs. This analysis was also applied to the question of
whether God's knowledge of things makes them necessary
(Knuuttila 1993, 118-121).



A great deal of Abelard's logical works consisted of discussions
of topics, consequences and conditionals. Like Boethius, Abelard
thought that true conditionals express necessary connections between
the antecedents and the consequents. He argued that inseparability and
entailment between the truth of the antecendent and consequent are
required for the truth of a conditional. Some twelfth-century masters
regarded the principle that the antecedent is not true without the
consequent as a sufficient condition for the truth of a conditional
and accepted the so-called paradoxes of implication. The question of
the nature of conditionals and consequences remained a popular theme
in medieval logic (Martin 1987, 2012).



The principles of propositional modal logic, found in *Prior
Analytics* I.15, were generally expressed as follows: if the
antecedent of a valid consequence is possible/necessary, the
consequent is possible/necessary (Abelard, *Dialectica*
202.6-8). However, the main interest was in modal syllogistic
and modal predicate logic. Avicenna (d. 1037) wrote a brief Arabic
summary of Aristotle's modal syllogistic, but his own theory was
different, being based on the assumptions that the subject terms and
the predicate terms of assertoric and modal propositions stand for all
possible applications and the truth-conditions of assertoric
propositions and corresponding possibility propositions are the same.
It follows, for example, that syllogisms with assertoric premises
coincide with uniform possibility syllogisms (Street 2002, 2005;
Lagerlund 2009). Avicenna was particularly interested in relative
necessities and distinguished between various types of conditional
necessities in terms of temporal determinations. Later Arabic works on
modal theories were much influenced by Avicenna. (See Strobino and
Thom 2016.) While Averroes's commentaries on the *Prior
Analytics* followed the main lines of Aristotle's text, his
separate treatise on modality involved new systematic ideas, mainly
the theory of accidental and *per se* necessary terms and the
interpretation of syllogistic necessity premises as *per se*
necessary predications with *per se* necessary terms. Both
ideas were inspired by Aristotle's remarks in the *Posterior
Analytics* I.4. Since Averroes took modal premises to be of the
divided type, assertoric premises in Aristotelian mixed
necessity-assertoric syllogisms must have a predicate term which is
necessary. The same applies to the subject term of the first premise
in mixed assertoric-necessity syllogisms (*Quaesita octo in librum
Priorum Analyticorum*, IV.3, 84, in *Aristotelis Opera cum
Averrois Commentariis* I.2b; see also Thom 2003, 81-85).
This is a speculative explanation of Aristotle's asymmetric
treatment of mixed necessity-assertoric syllogisms and mixed
assertoric-necessity syllogisms. Gersonides later tried to develop
further Averroes's remarks; see Manekin 1992. Analogous
essentialist ideas were developed in thirteenth-century Latin
discussions. 



The first Latin commentary on the *Prior Analytics* is an
anonymous late twelfth-century treatise ('Anonymus Aurelianensis
III') which involves detailed discussions of modal conversion
and modal syllogisms as well as many problems dealt with in ancient
commentaries. (See Ebbesen 2008; an edition by Thomsen Thornqvist
in 2015.). *Dialectica Monacensis*
involves a brief summary of Aristotle's modal syllogistic the
elements of which were discussed in logic courses in Paris in the
first part of the thirteenth century. Robert Kilwardby's
commentary *Notule libri Priorum* (c. 1240) became an
authoritative work from which the discussions of modal syllogisms in
the commentaries of Albert the Great (ca. 1250) and many others were
largely derived. Abelard, who did not
deal with Aristotle's modal syllogistic, said that the modals in
mixed syllogisms with both modal and assertoric premises should be
understood in a way which he elsewhere characterizes as *de re*
interpretation (*Glossae super Perihermeneias* XII,
189-203). This reading of modal premises was often assumed,
although it was seldom discussed as such. A central problem of
Aristotle's theory is that the structure of the premises is not
analyzed. Even if it is natural to think that the presupposition of
the mixed moods is a *de re* reading of modally qualified
premises, this creates difficulties when applied to the conversion
rules, most of which are unproblematic only if understood as rules for
modals *de dicto*. (According to Aristotle, necessity premises
are 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'. Affirmative
contingency premises are converted to corresponding negative
contingency premises and both these by the conversion of terms to
particular contingency propositions; *Prior Analytics* I.3,
13.)



While many historians think that Aristotle's modal syllogistic
included incompatible elements, this was not the view of
mid-thirteenth century logicians. Many of them discussed the same
alleged counter-examples to the universal convertibility of necessary
propositions, such as



(8)
Everything healthy (or awake) is necessarily an animal.


Robert Kilwardby's explanation is based on the view that
convertible necessity premises are necessity propositions *per
se* and not *per accidens*, like (8), which are not
convertible. (See *Notule libri Priorum* 8.133-146.) In
affirmative necessity propositions *per se*, the subject is
*per se* connected to the predicate. In negative necessity
propositions *per se*, the subject is *per se*
incompatible with the predicate. The terms in *per se*
inherences or incompatibilities are essential and necessarily stand
for the things they signify. The historical background of
Kilwardby's interpretation is not clear; it does show
similarities to Averroes's discussion mentioned above, but there
was no historical link. (See Lagerlund 2000, 25-42; Thom 2007,
19-28.)



As for the conversion of contingency propositions (neither necessary
nor impossible), Kilwardby notes that while the converted propositions
of indefinite (*utrumlibet*) contingency are of the same type
of contingency, the conversion of natural contingency propositions
(true about most cases) results in contingency propositions when
contingency means possibility proper (not impossible). There were
extensive discussions of the kinds of contingency based on various
philosophical ideas in Kilwardby, Albert the Great and their
contemporaries (Knuuttila 2008, 540-541).



Kilwardby follows Aristotle's remark that 'A contingently
belongs to B' may mean either 'A contingently belongs to
that to which B belongs' or 'A contingently belongs to
that to which B contingently belongs', *Notule libri
Priorum* 18.682-697. Kilwardby argues that the subject terms
in contingency syllogisms are read in the second way and ampliated, if
syllogistic relations do not demand restrictions. In explaining the
difference in this respect between necessity propositions and
contingency propositions, he states that since the terms in *per
se* necessity propositions are essential, 'Every A is
necessarily B' and 'Whatever is necessarily A is
necessarily B' behave in the same way in logic. Contingency
propositions which are ampliated do not mean the same as those which
are not ampliated (*Notule libri Priorum* 18.187-207;
18.653-672). For the conversion of both types of contingency
propositions and the role of the distinction between essential and
accidental terms in defining the truth conditions of contingency
propositions, see Thom 2019, 139-149. In the conversion of *de re* readings of unampliated contingency propositions, the
possibility changes into actuality, which is in agreement with the
frequency interpretation of modality.



According to Kilwardby, the modal character of the predication in the
conclusion of the perfect first figure syllogism follows from the
first premise, which involves the whole syllogism in accordance with
the *dici de omni et nullo* (Lagerlund 2000, 41-42). The
premises and the conclusion in uniform necessity syllogisms are
necessary *per se*. In mixed first-figure syllogisms with a
major necessity premise and a minor asertoric premise, the
non-modalized premise should be *simpliciter* assertoric, i.e.,
a necessarily true *per se* predication. Similarly, in mixed
first-figure syllogisms with contingent major and assertoric minor
premises, the assertoric premise must be *simpliciter*
assertoric, but this time the criteria are that the predicate belongs
to the subject *per se*, invariably or by natural contingency
(*Notule libri Priorum* 15.255-301;
20.706-736).



Kilwardby explains that in first-figure mixed necessity-assertoric
syllogisms the necessity premise appropriates to itself a minor which
is necessary *per se*; no such appropriation occurs in
first-figure mixed assertoric-necessity syllogisms. There are similar
appropriation rules for some mixed second-figure and third-figure
moods with assertoric and necessity premises and for various mixed
contingency moods pertaining to the kind of appropriated contingency
premises or assertoric premises (Thom 2007, chs. 5-6).



Kilwardby and his followers regarded Aristotle's modal
syllogistic as the correct theory of modalities, the explication of
which demanded various metaphysical considerations. As exemplified by
the appropriation rules, they assumed that propositions of the same
form had different interpretations, depending on how they were related
to other propositions in a syllogism. From the logical point of view,
these rules have an *ad hoc* character. (For some comparisons
between contemporary philosophical modal logic and thirteenth-century
views, see also Uckelman 2009.)

 Thirteenth-century Aristotelian
authors generally assumed that modalities had a metaphysical
foundation in the hierarchical order of the nature of things that
determined their occurrences in the temporal order accountable by
frequential modalities. This was the background of the popular
Averroist classification of necessary and contingent causes in natural
philosophy (Knuuttila 1993, 129-137). Following a twelfth-century
distinction, theologians divided between God's absolute power
limited by contradiction and God's ordained power as the first
principle of the existing order (Moonan 1994; Gelber 2004, 309-349).


After Kilwardby and Albert the Great, who retailed him, several
thirteenth-century authors wrote treatises on the *Prior
Analytics*. Ebbesen (2010) lists seven literal and six question
commentaries from the 1270s through the 1290s. One of these has been
recently edited, namely the question commentary of Radulphus Brito
(2017). On the basis of the lists of the contents of the question
commentaries included in Ebbesen 2010, one can characterize this as a
Kilwardbian tradition. While Richard Campsall's early
fourteenth-century *Questions on Aristotle's Prior
Analytics* was influenced by this literature, it also involved
some new modal ideas. Campsall thinks that one should discuss *de
dicto* and *de re* modalities separately. In considering
the syllogistic logic of *de re* possibility statements with
terms standing for actual things, he says that an affirmative *de
re* possibility statement as of now implies the corresponding
assertoric statement (5.40) and a negative assertoric statement as of
now implies the corresponding *de re* necessity statement
(5.50). It follows that what is possible now is actualized and things
cannot be otherwise because all true present tense negative statements
are necessarily true. This is Campsall's version of the
traditional doctrine of the necessity of the present. But if the
denial of a present affirmative statement is necessary, how can the
statement itself be non-necessary? Campsall defines a *de re*
contingency statement as a conjunction of an affirmative and
corresponding negative possibility proper statement (7.34-36).
Perhaps he thought that this is applicable to present tense statements
with actual terms and imply the idea of simultaneous alternatives, as
suggested by Lagerlund 2000, 87-90. But Campsall also equates
*de re* necessity with respect to actual things to unchanging
predication and contingency to changing predication. Actual things may
be contingent in the sense that they will be changed in the future
(12.31). For Campshall's confusing formulations, see also
Knuuttila 2018.


## 4. Fourteenth-Century Discussions



John Duns Scotus's theory of simultaneous alternatives was an
exposition of the intensional theory of modality, some elements of
which were put forward in the twelfth century. In criticizing Henry of
Ghent's theory of theological modalities, Scotus sketched a
model of 'divine psychology' in which certain relations
between theological, metaphysical, and modal theoretical notions are
defined. Scotus deviated from the metaphysical tradition in which
possibilities were founded in divine being. According to him, when God
as an omniscient being knows all possibilities, he does not know them
by turning first to his essence. Possibilities can be known in
themselves (*Ord*. I.35, 32). In fact they would be what they
are even if there were no God. Scotus states that if it is assumed
that, *per impossibile*, neither God nor the world exists and
the proposition 'The world is possible' then existed, this
proposition would be true. The actual world is possible as it is, and
this possibility and the possibilities of unrealized things are
primary 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 about
possibilities as such (Knuuttila 1996) or about the independence of
formal possibilities, but not of concrete possibilities, which are
dependent on God (Normore 2003; Hoffmann 2009).



Scotus calls the propositional elements of the modal domain
'logical possibilities'. These express things and states
of affairs to which it is not repugnant to be. Possibilities as such
have no kind of existence of their own nor are they causally
sufficient for the existence of anything, but they form the
precondition for everything that is or can be. God's omniscience
involves an actually infinite number of intelligibles (*De primo
principio* IV.9, 68-69); as an object of divine knowledge, they
receive an intelligible or objective being. As an object of divine
omnipotence, they receive an intentional possible being. Some of these
are included in God's providential plan of creation and will
receive an actual being. The description of how things could be at a
certain moment consists of compossible possibilities. Though
possibilities necessarily are what they are, the actualization of
non-necessary possibilities is contingent. Since all finite things are
contingently actual, their alternatives are possible with respect to
the 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 a
contingent event as follows: 'I do not call something contingent
because it is not always or necessarily the case, but because its
opposite could be actual at the very moment when it occurs'
(*Ord*. I.2.1.1-2, 86). This is a denial of the
traditional thesis of the necessity of the present and the temporal
frequency characterization of contingency. In Scotus's modal
semantics, the meaning of the notion of contingency is spelt out by
considering simultaneous alternatives. What is actual is contingently
so if, instead of being actual, it could be not actual. This
conception of simultaneous contingent alternatives is part of an
argument that the first cause does not act necessarily. According to
Scotus, the eternal creative act of divine will is free only if it
could be other than it is in a real sense (*Lect*.
I.39.1-5, 58). For Scotus's modal theory, see also
Honnefelder 1990; Vos et al. 1994; King 2001; Hoffmann 2002; the
thesis of Scotus's innovativeness is criticized in Pasnau
2020.



Scotus's approach to modalities brought new themes into
philosophical discussion. One of these was the idea of possibility as
a non-existent precondition of all being and thinking. Some of his
followers and critics argued that if there were no God, there would
not be any kind of modality (see Hoffmann 2002, Coombs 2004; for
Bradwardine's criticism, see Frost 2014). Scotus's views
were known in the seventeenth century through the works of
Suarez and some Scotist authors (Honnefelder 1990). In his
discussion of eternal truths, Descartes criticized the classical view
of the ontological foundation of modality as well as the Scotist
theory of modality and conceivability. (There are different
interpretations of Descartes's view of the foundations of
modality and how it is related to late medieval discussions; see
Alanen 1990; Normore 1991, 2006.)



Another influential idea was the distinction between logical and
natural necessities and possibilities. In Scotus's theory,
logically necessary attributes and relations are attached to things in
all those sets of compossibilities in which they occur. Against this
background one could ask which of the natural invariances treated as
necessities in earlier natural philosophy were necessary in this
strong sense of necessity, and which of them were merely empirical
generalizations without being logically necessary. (For a discussion
of logical and natural necessities in the fourteenth century, see
Knuuttila 1993, 155-160, 2001a.) Buridan distinguished between
logical and natural necessities in his classification of four grades
of necessity (*Dialectica* 8.6.3). He used a non-extensional
metaphysical notion of possibility *sencundum imaginationen* in
his questions on Aristotle's *Physics* (Sylla 2001), but
a frequency interpretation in his questions on Aristotle's
*On generation and corruption* I.4 (57). This was the main
modal conception in the Averroist Aristotle commentaries by John of
Jandun (see, for example, his questions on *Metaphysics*, IX.5
114va-b; questions on *On the Heavens* I.34, 21vb) and remained
a popular option until the Renaissance times (Knebel 2003). 



One important branch of medieval logic developed in treatises called
*De obligationibus* dealt, roughly speaking, with how an
increasing set of true and false propositions might remain coherent in
a disputation. According to thirteenth-century rules, a false present
tense statement could be accepted as a starting point only if it was
taken to refer to a moment of time different from the actual one.
Scotus deleted this rule, based on the Aristotelian axiom of the
necessity of the present, and later theories accepted the Scotist
revision. In this new form, obligations logic could be regarded as a
theory of how to analyze possible states of affairs and their mutual
relationships. These discussions influenced the philosophical theory
of counterfactual conditionals (Yrjonsuuri 1994, 2001; Gelber
2004; Dutilh Novaes 2007).



In dealing with counterfactual hypotheses of indirect proofs mentioned
above, Averroes and Thomas Aquinas made use of the idea of abstract
possibilities which did not imply the idea of alternative domains. The
possibilities of a thing can be dealt with at various levels which
correspond to Porphyrian predicables. Something which is possible for
a thing as a member of a genus can be impossible for it as a member of
a species. The same holds of it as a member of a species and an
individuated thing. Thus humans can fly because there are other
animals which can fly. These abstract possibilities are impossible in
the sense that they cannot be actualized. Buridan heavily criticized
this approach from the point of view of his modal theory. He argued
that if a counterfactual state of affairs is possible, it can be
coherently imagined as actual. If something cannot be treated in this
way, calling it possible is based on conceptual confusion. (See
Knuuttila and Kukkonen 2011.) While Scotus, Buridan and many others
understood the basic level of possibility in terms of semantic
consistency, Ockham wanted to preserve the link to the notion of power
in his modal considerations, thinking that necessity is actuality plus
immutability, the past and the present are necessary, and Scotus was
wrong in assuming that things could be different from how they are at
the very moment of their actuality (Normore 2016). 



Influenced by the new ideas about logical modality, William of Ockham
(*Summa logicae*), John Buridan (*Tractatus de
consequentiis*, *Summulae de Dialectica*) and some other
fourteenth-century authors could formulate the principles of modal
logic much more completely and satisfactorily than did their
predecessors. Questions of modal logic were discussed separately with
respect to modal propositions *de dicto* and *de re*;
modal propositions *de re* were further divided into two groups
depending on whether the subject terms refer to actual or possible
beings. It was thought that logicians should also analyze the
relationships between these readings and, furthermore, the
consequences having various types of modal sentences as their parts.
Ockham, Buridan and their followers largely dropped thirteenth-century
essentialist assumptions from modal syllogistic. They regarded the
Aristotelian version as a fragmentary theory in which the distinctions
between different types of fine structures were not explicated and,
consequently, did not try to reconstruct Aristotle's modal
syllogistic as a consistent whole by one unifying analysis of modal
propositions; they believed, like some modern commentators, that such
reconstruction 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 possible
worlds semantics to Buridan's modal system. This is done in
Johnston 2015. In discussing the conception of simultaneous
alternativeness in fourteenth-century logic, many historians have used
the terminology of possible worlds semantics without the metaphysical
and formal details of the modern theory (Yrjonsuuri 1994,
167-174; Dutilh Novaes 2007, 90-91; Read 2021. Ockham and Buridan
state 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*) and
divided (*de re*) readings do not differ at this level, but are
separated in dealing with universal and particular propositions. While
Ockham did not discuss unrestricted divided necessity propositions,
Buridan took the subject terms of all quantified divided modal
propositions as standing for possible beings if they are not
restricted. The truth of these propositions demands the truth of all
or some relevant singular propositions of the type just mentioned; the
demonstrative pronoun is then taken to refer to the possible beings
even though they may not exist. Buridan could have said that the
possible truth of 'This is X' means that it is true in a
possible state of affairs in which the possible being referred to by
'this' occurs and that the necessary truth of 'This
is X' means that it is true in all possible states of affairs in
which the possible being referred to by 'this' occurs
(Hughes 1989). For Buridan's modal logic, see also Read's
introduction to *Treatise on Consequences* (2015); Johnston
2021. 


 The new modal logic was among the most remarkable achievements of
medieval logic. Buridan's modal logic was dominant in late
medieval times, being more systematic than that of Ockham because of
its symmetric treatment of possibility and necessity. It was embraced
by Marsilius of Inghen, Albert of Saxony, Jodocus Trutfetter and
others (Lagerlund 2000, 184-227; for the later influence of
medieval modal theories, see also Coombs 2003; Knebel 2003; Roncaglia
1996, 2003; Schmutz 2006). The rise of the new modal logic was
accompanied by elaborated theories of epistemic logic (Boh 1993) and
deontic logic (Knuuttila and Hallamaa 1995).