## 1. Probability: Now and then



Modern notions of probability are quantitative (Hajek 2011).
They underlie the representation of probabilities by real (initially:
fractional) numbers from the interval between 0 and 1. On this basis,
a probability calculus with its own mathematical rules (the Kolmogorov
axioms) has
 arisen.[1]
 The present mathematics of probability marks the end-point of a
development which began in the middle of the seventeenth century with
the calculation of chances in games of fortune by Pascal, Fermat, and
Huygens. These early calculations were considerably refined in the
eighteenth century by the Bernoullis, Montmort, De Moivre, Laplace,
Bayes, and others (Daston 1988; Hacking 2006; Hald 2003). Modern
conceptualizations of probability emerged at the same time. They
evolved around the ideas of a ratio of possibilities, a relative
frequency of events, a degree of belief, or a degree of
confirmation.



Pre-modern notions of probability were very different (Franklin 2016).
*No* pre-modern notion of probability was numerical. A calculus
of probabilities simply did not exist before the seventeenth
 century.[2]
 Pre-modern probability was a qualitative predicate mainly applied to
propositions (e.g., by calling an opinion probable), but extending to
other subject matters as well. The use of the word
'probable' in ordinary language still resembles this older
usage to some extent (Franklin 2001, 126; Franklin 2013; Hacking 2006,
18). Yet, whereas the mathematics of probability was invented in the
early modern era, some continuities in the meaning(s) of probability
exist. These continuities will be discussed in Section 6. Until that
point, it is best to forget the connotations of modern notions of
probability and to approach medieval probability-related terms without
modern preconceptions. For their vocabulary, medieval and Renaissance
authors could draw on a well-stocked ancient Greek and Roman store of
probability-related terms. In consequence, a variety of Latin terms
were in use, which are today translated as 'probable',
'plausible', 'reputable'
(*probabilis*); 'truth-like', 'apparently
true' (*verisimilis*); 'credible',
'worthy of belief' (*credibilis*,
*opinabilis*); or 'most often',
'frequently' (*ut frequenter*). Modern readers may
not always find it obvious that these terms relate to probability;
but, in fact, they are highly relevant for its conceptual evolution.
Up to the nineteenth century, when philosophers spoke about
probability their claims were still often rooted in medieval notions
of probability.



Medieval and Renaissance philosophers used probability-related
vocabulary; but they did not analyze the underlying concepts in depth,
at least in comparison with the thoroughness of their investigations
into the notions of knowledge or science (*scientia*).
Probability-related terms were usually introduced in passing, and the
most differentiated treatments extended to a folio page at best.
Nevertheless, issues of probability were not peripheral to medieval
thought. Probability-related terms and concepts played an enormous
role in the regulation of everyday conduct and in many arts and
sciences. Rhetoric and dialectic were by definition concerned with
persuasion, credibility, and probability. Yet medieval thinkers were
also highly aware that many claims in jurisprudence, theology, and
medicine could not be derived with certainty from indubitable
premises. Hence, arguments in these fields had to rely more often than
not on probable reasoning. The notorious disagreements between
scholastic academics in their fields of inquiry, nourished and
represented by the practice of scholastic disputation, further
increased the demand for probable reasoning. Wherever disagreement was
licit and impossible to extirpate by argument, the claims of all sides
counted as merely probable. Since medieval theologians and lawyers
were no more in agreement concerning the ascription of sins (or of
moral actions) than on other matters, the guidance of human conduct by
confessors was also a matter of probability. In this domain, the
regulation of decisions of conscience relied on a systematic framework
of probable reasoning and decision in conditions of uncertainty. In
short, for a satisfactory understanding of the medieval intellectual
edifice, inspection of the uses of probability seems
indispensable.



This is true for the medieval period, the Renaissance, and in
different ways for early modern scholasticism until the eighteenth
century. We should not assume, however, that the uses and
understanding of probability remained unchanged throughout this long
span of time. For this reason, the period from roughly 1200-1500
is singled out here for discussion. It is the period of mature
medieval scholasticism and covers much of the Renaissance. In the
sixteenth century, the discourse on probability underwent significant
changes, on the scholastic as well as the humanist side. This early
modern discourse needs to be clearly differentiated from its medieval
precursor and is therefore excluded from consideration here (but see
the references in Sections 4.3 and 6).



Within the period under consideration, a distinction between
scholastic and humanist uses of probability can be made. This is not
to say that there was a big divide between them. The same terms were
used by scholastics and humanists with similar meanings. The normative
and regulatory uses of probability, however, were anchored in
scholastic theology and jurisprudence; therefore, they are in the
foreground here. At least some Renaissance humanists used
probability-related terms differently from the scholastic mainstream;
and these differences will be addressed in Section 5.



That the conceptualization of probability was not exclusively a
European affair also deserves attention. Many cultures around the
world tilled the conceptual field of probability, in antiquity and not
least in the medieval era. The question of the extent to which
non-Western notions and uses of probability shaped the treatment of
probability in the Latin Middle Ages (or vice versa) cannot be
answered here; however, some of the scholastic authors discussed below
explicitly referred to Islamic sources in their accounts of
probability.


## 2. Medieval probability-related terminology



Terms, such as the Latin *probabilis* were already in use in
the early Middle Ages and can, for instance, be found in Carolingian
texts on rhetoric (e.g., Alcuin, *Rhetoric*, 113). The ancient
authorities on the subject were Cicero and Boethius, whereas
Aristotle's most seminal statements on probability-related
concepts became known only after the middle of the twelfth century
(see Cox and Ward 2006; Fredborg 1988, 150; Greenberg-Pedersen 1984).
Particularly significant in this respect were the introduction of
Aristotle's *Topics* into academic discourse in the
middle of the twelfth century (but the full impact was not felt before
1200) and the new translations of Aristotle's *Nicomachean
Ethics* and *Rhetoric* in the thirteenth
 century.[3]
 By the end of this century, Aristotle's definitions had largely
(though not entirely) supplanted those of Cicero and Boethius in
scholastic thought. The impact of this change should not be
exaggerated because the notions of probability held by Cicero and, in
particular, Boethius were in significant respects close to those of
Aristotle.



The main probability-related terms in this period were
*probabilis*, *verisimilis*, *credibilis*,
*opinabilis*, and some expressions indicating frequency such as
*ut frequenter*, *ut* *in pluribus*, and
*frequentius*. We need to look carefully at the specific
meanings of these words and phrases before proceeding to concepts of
probability in a narrower sense.


### 2.1 *Probabilis*



*Probabilis* was the most important probability-related
predicate in the Middle Ages (and in the whole scholastic tradition).
It is often possible to translate *probabilis* as
'probable' in an ordinary language sense, but attention
should be paid to meanings arising from a combination with particular
nouns. The predicate *probabilis* mainly pertained to
propositions, sentences, or opinions, but these are not the only
meaningful referents. Sometimes persons were described as probable, as
in the phrase "the more probable philosophers"
(*philosophi probabiliores*). Thomas Deman (1933, 262) renders
this phrase as "the philosophers who deserve most
consideration"; but one could also translate it as "the
philosophers who are more likely to find the truth". Signs were
also sometimes called probable (*signum probabile*), as in
Silvester de Prierio's *Summa* under the heading
'probabile'. Last but not least, probability was
attributed to events, for instance, when Thomas Aquinas, in *Summa
theologiae* II-II, q. 32, 5, 1322, says "according as things
probably and generally occur / according to what probably and
generally occurs" (*secundum ea quae probabiliter et in
pluribus occurrunt*). As we shall see, however, it is conceivable
that all these assertions are elliptic references to the probability
of propositions.



A proposition was probable (*probabilis*) if specific
indicators of truth justified assent (i.e., holding it true) or
acceptance as a premise for
 action.[4]
 The main justifications for judgments of probability will be
discussed in more detail in Section 3, on "Scholastic Concepts
of Probability". For the moment, it is enough to say that
authority, testimony, a sufficient (but not maximal) appearance of
truth, and frequent truth could buttress an ascription of
probability.



The most prolific category of probable propositions were opinions in
the medieval technical sense of this term. An opinion
(*opinio*) was often defined as a proposition held to be true
but which the person holding it feared might be wrong (see Byrne 1968,
Franklin 2001). This was only one of several medieval understandings
of opinion. Robert Grosseteste ([?]1175-1253) reminded his
readers that *opinio* could be understood in three
 ways.[5]
 According to the simplest meaning, opinion stood for any cognition
characterized by assent; and, in this understanding, propositions
expressing knowledge or conviction were also opinions. According to a
more proper definition, opinion denoted assent to a proposition with a
fear that the opposite might be true. Finally, according to the most
precise definitions, opinions were limited to contingent propositions
to which assent was given and the opposite of which were somehow
feared to be true. Grosseteste's distinctions show that
thirteenth-century scholastics were already aware that the term
*opinio* was used far from uniformly by contemporary authors.
That said, it should nevertheless be kept in mind that medieval
discussions of the use of probable opinions predominantly operated on
the basis of the second meaning of *opinio* as a proposition to
which assent was given and which was accompanied by a fear that the
opposite might be true. The fear in question resulted at least in part
from an awareness that one's judgment might be fallible.
According to a widely observed medieval ranking of epistemological
confidence and doxological firmness, the category of opinion stood
below faith or full conviction (*fides*) and knowledge
 (*scientia*).[6]
 All three epistemic attitudes involved assent, but only faith and
knowledge implied full subjective certainty (*certitudo*).



On this basis, "probable opinion" (*opinio
probabilis*) became a standard term for a class of propositions
held to be true, which humans had to be content with in many fields of
investigation. In these fields, human fallibility or the changeable
nature of the subject matter precluded full knowledge
(*scientia*), and a lack of religious relevance or accepted
theological contentiousness prevented the intervention of faith
(*fides*). The scope of this domain was enormous. In
philosophical theory, it was the home-ground for dialectical
argumentation and dialectical syllogisms (see Section 4.1); on a
practical level, it was regulated by counselors of conscience and
lawyers.



For the regulation of consciences, another standard term was
'probable certainty' (*certitudo probabilis*).
Probable certainty forestalled sin by enabling one to base one's
actions on correct probable reasoning. Since reasoning which went
beyond the probable was virtually unattainable in many moral contexts,
an agent satisfied her or his epistemic duties by relying on sound
probable reasoning, thus acquiring a subjective certainty of having
done all that could be expected of an agent to avoid sin (Gardeil
1911; Schuessler 2019, 47). In the opinion of many medieval
scholastics, probable reasoning could not generate epistemological
certainty; the phrase "probable certainty", however,
sprang from compliance with moral demands (which we would call
epistemic duties today) and not from strong conviction based on
evidence. Perhaps because of the risk of misunderstanding it implied,
'probable certainty' was largely supplanted, from the
fifteenth century onward, by the similar concept of 'moral
certainty' (*certitudo moralis*), (Franklin 2001, 69;
Knebel 2000, 55; Schuessler 2009).



In the field of jurisprudence, the term *probabilis* was
typically combined with presumptions (*presumptiones*). A
probable presumption for or against a culprit was one that in itself
did not suffice to incline a reasonable judge to a verdict (only
vehement or violent presumptions could achieve that). Nevertheless,
probable presumptions had some juridical weight (in contrast to
careless presumptions, *presumptiones temerariae*) and could be
aggregated to justify a verdict (Franklin 2001, 43).



In the realm of economics, a distinct lexicon pertaining to
risk-taking, with terms such as risk (*risicum*) and danger of
loss (*periculum sortis*) was in use, and institutions designed
for risk, such as insurance contracts, began to emerge
(Ceccarelli 2001; Ceccarelli 2021; Zwierlein 2016). Economic risk
often found itself associated with the concept of probability (Januard
2023); and moral theological regulations for navigating doubt and
probability held normative implications for the legitimacy of engaging
in economic risk-taking. It is important to note that this does not
imply that the medieval era discouraged risk-taking. Particularly in
the late Middle Ages, engaging in economic risk did not necessarily
entail moral failure or a likelihood of sin.


### 2.2 *Verisimilis*



In medieval usage, *verisimilis* meant what it literally said:
that something was 'truth-like'. This did not, however,
imply that the thing was actually close to truth, because it might
have only the appearance - possibly spurious or superficial
- of truth. Signs, images, and circumstantial evidence were
often called *verisimilis* if they satisfied these conditions.
Medieval lawyers spoke of 'truth-like circumstantial
evidence' (*indicia verisimilia*), and Albert of Cologne
(*Logica,* 241) maintained that what was
'truth-like' resided in 'signs'
(*verisimile est in
 signis*).[7]



Was *verisimilis* used as a synonym for *probabilis* by
medieval scholastics? Albert (ibid.) claims straightforwardly that
"probable things are truth-like" (*probabilia ...
sunt
 verisimilia*).[8]
 Other scholastic uses of these words suggest instead a mere relation
of frequent mutual implication. Probable opinions were usually
truth-like because their probability prima facie justified an
assumption of truth. Truth-like signs often justified an ascription of
probability to an assertion. The widespread combination of
*probabilis* with particular kinds of proposition (Aristotelian
*endoxa*; see Section 3.1) and of *verisimilis* with
signs are nevertheless good reasons to assume that these predicates
were not synonymous. The distinction between the terms may reflect
established differences in ancient Greek terminology, tying
*probabilis* to *pithanon* (persuasive, adoptable) and
*verisimilis* to *eikos* (truth-like). Such differences
arguably already pervaded Cicero's usage of *probabilis*
and *verisimilis*, although the matter is
 contentious.[9]


### 2.3 *Credibilis*, *opinabilis*



A proposition was regarded as *credibilis* or
*opinabilis* if it could legitimately be accepted as
one's opinion. The underlying criteria were often the same as
for *probabilis*. Propositions which did not satisfy the
respective criteria were called *inopinabilis* or
*improbabilis*, and deserved to be rejected by rational and
responsible epistemic agents. This meant that improbable propositions
ought not to be taken for true or accepted as a premise for action,
but not necessarily that they must be regarded as
 false.[10]


### 2.4 Frequency terms



A host of frequency terms such as "frequently"
(*frequenter*), "more frequently" (*ut*
*frequentius*) and "for the most part" (*ut in
pluribus*), or their converses such as "rarely"
(*ut in paucioribus*), often accompanied ascriptions of
probability in medieval scholasticism. What frequently or usually
happened was prima facie considered to be probable. Scholastic usage
of frequency terms followed ancient precedent, most notably
Aristotle's statements of what is true or false "in most
cases" (*hos epi to polu*). Whether this background
in Aristotelian science and methodology justifies the assumption of a
medieval frequency concept of probability will be discussed in Section
3.2.


## 3. Scholastic Concepts of Probability



In some areas of medieval thought, probability-related terms were
important for the regulation of decision-making and conduct. In these
contexts, the term "probable" (*probabilis*) took
on a specific significance. Whether a merchant wanted to follow an
opinion concerning the moral legitimacy of a contract or a lawyer
formulated a presumption, it had to be *probabilis* to possess
moral or juridical weight. It is now necessary to discuss the most
important medieval understandings of *probabilis*, grounding
them in ancient definitions or precedent.



Until the thirteenth century, the definitions of
"probable" by Cicero and Boethius very much shaped the
medieval understanding of probability:



> 
> 
> That is probable which for the most part usually comes to pass, or
> which is a part of the ordinary beliefs of mankind, or which contains
> in itself some resemblance to these qualities, whether such
> resemblance be true or false. (Cicero, *De inventione*,
> I.29.46)
> 
> 
> 
> Something is readily believable (*probabilis*) if it seems true
> to everyone or to the most people or to the wise - and of the
> wise, either to all of them or most of them or to the most famous and
> distinguished - or to an expert in his own field, for example,
> to a doctor in the field of medicine or to a pilot in the navigation
> of ships, or, finally, if it seems true to the person with whom one is
> having the conversation or who is judging it. (Boethius, *De
> topicis*, 1180b28)
> 
> 
> 



For Cicero, probability represents the usual course of events,
commonly held beliefs, or what resembles them. His characterization of
probability thus contains the roots of at least two different concepts
of probability. Boethius repeats Aristotle's definition of an
*endoxon* (see Section 3.1) as a readily believable proposition
almost verbatim and calls it *probabilis*. In this respect, he
prepared the way for the reception of Aristotle's original
definition; but note that Boethius addresses the role of experts more
explicitly than Aristotle, who in medieval translations summarily
refers to *sapientes*, the
 wise.[11]



Aristotle replaced Cicero and Boethius as supreme authority on
probability in the thirteenth century. This is also the time when
medieval probability discourse fully unfolded. For the period from
(roughly) 1200 to 1500, it is possible to identify four different
concepts of probability: the endoxic, proto-frequentist, testimonial
or juridical, and semantic. The number of concepts is, however, a
matter of interpretation since it is possible to split up the
juridical category of probability
 further.[12]



The use of probability-related terms was almost certainly wider than
the concepts of probability listed above suggest. In philosophical and
theological debates, as well as in the regulation of action,
*probabilis* could be used as a predicate for the argumentative
support which a proposition had (translatable as
"plausible" or "rationally tenable"). In the
early modern era, this aspect of medieval probability became
explicitly integrated into the definition of a probable opinion
(Knebel 2000; Maryks 2008; Schuessler 2019; Tutino 2018). It was not,
however, expressly addressed in medieval and, in particular,
scholastic characterizations of probability. Here, the focus will be
on such characterizations in medieval sources.


### 3.1 Endoxic (or topical) probability



The most prominent scholastic understanding of probability derived
from Aristotle's notion of *endoxon*, which shaped the
concept of "probable opinion" (*opinio
probabilis*). Aristotle defines *endoxon* in
*Topics* I, 100b20 as:



> 
> 
> [T]hose opinions are reputable [*endoxa*] which are accepted by
> everyone or by the majority or by the wise - i.e., by all, or by
> the majority, or by the most notable and reputable of them.
> 
> 
> 



Boethius and medieval commentators on the *Topics* translated
*endoxa* as *probabilia* ("probable things")
or *opiniones probabiles* ("probable opinions"). In
light of the rise of Aristotelianism, it is thus unsurprising that
Thomas Aquinas and other scholastics turned to this understanding of
probability when they discussed conclusions from uncertain
 premises:[13]



> 
> 
> [P]ropositions are called probable because they are more known to the
> wise or to the multitude. (Thomas Aquinas, Commentary on the
> *Posterior Analytics*, I.8.35)
> 
> 
> 



Many scholastic characterizations or definitions of probability
contain an explicit reference to *Topics* I, such as this one
by Gregory of Rimini:



> 
> 
> Moreover, since "probable things" mean "what appears
> true to the many or the wise" according to the Philosopher
> [i.e., Aristotle] in *Topics* I, which also agrees with
> Augustine's *Contra Cresconium*, Book III, where he says
> that things are called probable because "they are endorsed, that
> is, approved and believed", [to which] add "to be
> true", it follows that approbation by the authority of the wise
> of this world is a reason which generates probability (*ratio
>  probabilis*).[14]
> 
> 
> 



As this quotation shows, Augustine's understanding of
probability was thought to coincide with Aristotle's, a fact
that must have buttressed the endoxic notion of probability. Yet,
while Augustine is not often referred to in medieval definitions of
probability, a reference to *Topics* *I* became a
commonplace. It seems immediately clear why an endoxic understanding
of *probabilis* led to the modern translations
"reputable", "plausible",
"believable", "approvable", or
"approved". Reputability and approval, however, are not
necessarily indicators of truth, and therefore the role that
*endoxa* played for Aristotle and for the scholastics needs
clarification.



*Endoxa* were important for ancient rhetoric and dialectic.
Rhetoric aims at persuasion, and opinions which are generally accepted
or backed up by the intellectual reputation of their sources tend to
be persuasive. In rhetorical usage, therefore, *endoxon* and
*probabilis* can be equated to a proposition with
"expected persuasiveness". Dialectic is concerned with
disputes and inquiry through the exchange of arguments.
*Endoxa*, in ancient dialectic, were starting, intermediate,
and often also endpoints of such activities; and, to this extent, they
might be seen as instruments for intellectual training. This, however,
would create the wrong impression of the overall role of
*endoxa* in Aristotle's methodology of the sciences
(Haskins 2004; Kraut 2006; Renon 1998). He also takes them to be
reasonable starting points for scientific inquiry, as well as material
for testing its results. The process of inquiry should select the
right *endoxa* and order them systematically. In this respect,
the connection between likely truth and being held true by many or
wise people is presumed by Aristotle. There is no guarantee, however,
that an *endoxon* is true, which makes it an apt object of
inquiry.



Scholastic usage of *opinio probabilis* conformed less to the
rhetorical and more to the dialectical and methodological roles of
*endoxa* (see also Section 4.1) - even if the opinions in
question were applied in oratory and preaching. A scholastic author,
in particular from the thirteenth century onward, who called a
philosophical position or a theological claim probable alluded to an
epistemic quality which justified serious consideration over and above
a proposition's suitability for inducing agreement in hearers.
Sensory perception is sometimes mentioned as a basis for probable
opinions which are held by all people ("Snow is
 white").[15]Apparently,
 the opinions of the uneducated (*rudes*) and laymen
(*idiotae*) did not qualify as probable, unless they were
observation statements, for which uneducated people had as much
competence as the educated. Competence in judging a matter was
therefore the key to scholastic ascriptions of probability. The
category of the "wisest" in medieval definitions of
probability was a stand-in for "expert", including
top-ranking lawyers, theologians, medical doctors and philosophers
- or even architects, engineers and other *virtuosi* of
mechanical arts (as in Boethius's definition of probable
opinion). This expert status was acquired through training in a
university or with a master. Moreover, an expert counted as such only
in his field of
 expertise.[16]
 The *sapientes* to which the definition of probable opinion
refers nevertheless had to possess at least some positive
non-epistemic traits: they had to be decent (*probus*) and
trustworthy people in order to possess weighty opinions.



The role of authority in the endoxic definition of probability has
been widely discussed and
 criticized.[17]
 Although the word *auctoritas* does not occur regularly in
definitions of "the probable", it seems indisputable that
these definitions refer to authority as the basis of probability. The
authority in question is, in principle, intellectual or epistemic
(based on expertise); but the distinction between epistemic and
hierarchical authority was always blurry, and the latter could (and
often did) prevail in practice. Given the dismal training of parish
priests, this was surely often the case because the uneducated were
called on to accept the opinions of their parish priest or confessor
(Thomas Aquinas, *Quodlibetales*, III, q. 4, 2, 47; and
Antoninus, *Summa*, I, 3, 10, 64). Moreover, scholastic
reliance on institutional indicators for competence may have impeded a
clear distinction between epistemic and hierarchical authority, since
epistemic authority was to some extent tied to hierarchical success in
academia. Students, for instance, were prima facie expected to believe
the opinions of their professors (i.e., masters).



The distinction between hierarchical and epistemic authority should
not be confused with the distinction between external and internal
reasons for holding a proposition true. Endoxic probability seems to
be entirely based on the opinions of others and thus on external (or
indirect) reasons for holding a proposition true. (Internal or direct
reasons are those a person herself can assess.) Yet some difficulties
with the view that medieval probability was wholly external and thus
authority-based deserve to be addressed. Endoxic probability was not
the only concept of probability in medieval scholasticism, and others
were (as we shall see) more open to internal reasons. Moreover, an
expert might claim probability for his own judgment - something
not excluded by the definition of *endoxon*. Nevertheless,
agents were often asked to conform to the opinion of a qualified
majority ("the larger and more reasonable part", *maior
et sanior pars*) of competent
 evaluators.[18]
 This requirement shows that the endoxic understanding of probable
opinions was indeed skewed towards external
 authority.[19]


### 3.2 Proto-frequentist probability



Hacking (1975) famously claimed that frequentist conceptions of
probability largely arose as a side-product of the seventeenth century
revolution in probability. Historians of ideas quickly identified the
weaknesses in Hacking's thesis (Brown 1987; Daston 1988; Garber
and Zabell 1979). An understanding of probability as what happens most
of the time or for the most part (*hos epi to polu*) can
already be found in Aristotle and the scholastics. The frequentist
notion of probability therefore seems to possess a long pre-modern
 history.[20]
 Unfortunately, things are far more complicated. Modern frequentism
arose as a complex of ideas in the nineteenth century, perhaps
evolving only in the twentieth century into a full-fledged conception
of probability (Gillies 2000, 88-113). Hence, early modern uses
(such as Jacob Bernoulli's) of the relative frequency of
occurrences in a sequence of events are not yet frequentist in the
modern sense. Medieval (and ancient) uses of *ut frequenter*
considerations differ, in turn, from early modern (proto-)frequentism
(see Schuessler 2019, chap. 12). Having made this point, in what
follows, proto-frequentism will be used without the qualification
"medieval".



As has been shown, medieval readers could have gleaned frequentist
ideas from Cicero even before the rise of Aristotelianism in the
twelfth and thirteenth centuries. Cicero stipulates that
*probabilis* can mean "for the most part", and
medieval authors who followed him therefore held a proto-frequentist
conception of probability. Later on, such a conception was often
derived from the works of Aristotle. Did this make any difference? Let
us look at four examples:


1. It is sufficient that you obtain a probable certainty, which means
that in most cases (*ut in pluribus*) you are right and only in
a few cases (*ut in paucioribus*) are you wrong. (Thomas
Aquinas, *Summa theologiae* II-II, q. 70,
 2)[21]
2. It is not probable that, among the vast number of the faithful,
there would not be many people who would readily supply the needs of
those whom they hold in reverence because of the perfection of their
virtue. (Thomas Aquinas, *Summa contra gentiles*, lib. 3, 135,
19, 187)
3. But, from some pre-existing causes future effects do not follow
necessarily, but usually. For instance, in most cases (*ut in
pluribus*) a perfect human being results from the insemination of
a mother by a man's semen; sometimes, however, monsters are
generated, because of some obstruction which overcomes the operation
of the natural capacity. (Thomas Aquinas, *Summa contra
gentiles,* lib. 3, 154, 11, 243)
4. Hence, in this proposal we have men and women, who at age 25 buy a
life-long annuity for a price which they recover within eight years
and although they can die within these eight years it is more probable
that they live twice the time. In this way what happens more
frequently and is more probable is to the advantage of the buyer.
(Alexander of Alessandria, *Tractatus de usuris*, c. 72, Y f.
 146r)[22]



Passage (i) should not be understood as granting probable certainty
(*certitudo probabilis*) whenever a person is more often right
than wrong. The few exceptions mentioned ought to be really
exceptional in order to allow for a probable certainty, which was
deemed sufficient for blameless moral action. Probable certainty was
thus a threshold rather than a frequency concept.



Passage (ii) refers to the high likelihood that at least some among a
large number of people will possess a desired trait. This is a
proto-statistical statement, but it does not show that probability
*means* frequent occurrence. The claim that, among a large
number of people, some will have a rare trait can easily be believed
- depending on the trait in question - by a multitude or
by "the wise". Consequently, *ut in pluribus*
claims might often have been indicators of endoxic probability rather
than expressions of an independent concept of probability.



Passage (iii) postulates a natural regularity. Sexual conception
usually results in "perfect" (as opposed to deformed or
"monstrous") offspring. This *ut in pluribus* claim
refers to a majority of occurrences, but its background in
Aristotelian natural philosophy is significant. Scholars debate
whether Aristotle's understanding of "for the most
part" judgments is equivalent to a frequentist concept of
probability (Judson 1991; Kraut 2006; Winter 1997). Aristotelian
"for the most part" judgments imply much more in terms of
law-like or natural connections than can be captured by mere reference
to relative frequencies. It is therefore plausible to assume the same
for medieval Aristotelians, in particular for those who wrote about
natural processes.



Passage (iv) is probably closest to implicit statistical reasoning and
refers, not by chance, to an economic context. The passage quoted
deals with fairness requirements between buyers and sellers of life
annuities. A lump sum payment of a buyer and his or her (widows liked
to buy annuities) uncertain stream of income require an equitable
balance. The uncertainty or risk in question pertains to the possible
death of the owner of an annuity; and it is claimed that persons who
are 25 years old will probably live more than eight further years. It
may be apt to speak in this case of a probability of more than 0.5 for
further survival. Medieval considerations concerning insurance or
annuities show that numerical probability was never too far away,
which adds to our amazement that this step did not occur before the
seventeenth century. Passage (iv), however, contains two
probability-related terms: the buyer is favored by what occurs more
often (*frequentius*) and is more probable
(*probabilius*). Kantola assumes that the words
*frequentius* and *probabilius* are synonymous here, as
in several other cases in which they occur
 together.[23]
 Yet why would scholastic authors repeat two synonymous words if one
would do? The combined use of "frequent" and
"probable" often occurs in matter of fact contexts (as in
our example) which require no rhetorical flourishes. There is no
apparent need for emphasis through a repetition of synonyms. A
plausible answer is that the two terms are not, in fact, synonymous
and that frequency justifies an ascription of probability, although
probability *means* something else. It would make sense to
combine both terms precisely because frequency justifies a probability
ascription in the given case, but only probability confers legitimacy
on actions or contracts. In short, frequent occurrence might only have
served as indicator of endoxic probability for medieval scholastics.
The frequency of events may have prima facie justified the assumption
that everyone, a majority or the experts will believe that the events
in question will occur.



Should proto-frequentism therefore be dropped as a genuine concept of
probability? This step would be premature as long as the
interpretation of medieval *ut frequenter* statements is still
being debated. Continuity with ancient notions of probability also
suggests that we should tread carefully. For this reason,
proto-frequentism is included here among the medieval concepts of
probability.


### 3.3 Testimonial and juridical probability



Testimonial probability arises from the testimony of witnesses. It is
one of the two kinds of probability which Silvester de Prierio listed
in his *Summa summarum* (the other is endoxic probability):
"'Probable' is used in two ways. First as the
opposite of hidden, that is, what is proved by witnesses
 ..."[24]
 The rapid growth of jurisprudence after the twelfth century nourished
a great interest in witness
 testimony.[25]
 It was very difficult to rely on circumstantial evidence in medieval
jurisprudence; so the testimony of witnesses was in many ways the key
to a court decision. A complicated system of "half proofs"
and proofs was designed to safe-guard the rationality of juridical
procedures (Evans 2002; Franklin 2001, 15; Rosoni 1995, 89). Moreover,
even outside the narrow confines of jurisprudence, witness testimony
played an important role. Thomas Aquinas, for instance, described a
certainty sufficient for action:



> 
> 
> And yet the fact that in so many it is not possible to have certitude
> without fear of error is no reason why we should reject the certitude
> which can probably be had [*quae probabiliter haberi potest*]
> through two or three witnesses ... (Thomas Aquinas, *Summa
> theologiae*, II-II, q. 70, 2, 1488)
> 
> 
> 



Thomas's remark alludes to the norm that the testimony of one
eye-witness confers probability on an assertion, whereas the
(uncontested) testimony of two or more witnesses creates probable
certainty.



A related but wider field for using the predicate
"probable" in juridical terminology was the classification
of presumptions. A "probable presumption" (*probabilis
presumptio*) had some force in inclining a judge towards or
against a verdict (Hubert 2009; Motzenbacker 1958, 120, 159).
Moreover, witness testimony was just one kind of evidence which could
render a presumption probable. Probability could also result from
"truth-like circumstantial evidence" (*indicia
verisimilia*), as this statement from Huguccio documents:
"Probable is what is accepted because of common human opinion or
truth-like circumstantial evidence
 (*indicia*)."[26]
 Huguccio's reference to common opinion shows that endoxic
probability was also relevant in the law
 courts.[27]
 Nevertheless, the treatment of signs and *indicia* as
probability generating evidence, which was strongly promoted by
scholastic jurists, created an important precedent for connecting the
notion of probability to the direct evidence of an observer.


### 3.4 Semantic probability



Aristotle tied the notion of probability to questions of necessity and
contingency. The probable did not follow with necessity but only
contingently (and for the most part) from the nature of things. In
medieval scholasticism, this connection was often stated in semantic
terms. Boethius of Dacia called probability: "a property which
disposes (*habilitans*) but does not necessitate a subject to
the partaking of a
 predicate".[28]
 Probability is here a degree of participation of a predicate in a
subject. Full participation under all conditions stands for necessity;
participation under most conditions (for the most part) leads to
probability. The semantic notion of probability as conditioned
participation of a predicate in a subject is thus closely related to
the proto-frequentist interpretation discussed above. It is, however,
formally distinct as a notion of probability, not least because some
scholastics addressed it as such. Peter Richeri, *Topica
Aristotelis* (A. 37), I q. 2,
117ra-118ra, referred to Boethius (the
ancient one, not the medieval philosopher of Dacia) in this
context:



> 
> 
> Yet Boethius defines probable things differently in his
> *Topics* I: probable things are those to which the mind readily
> acquiesces, although it does not possess a firmness of truth in them,
> as in this proposition: "If she is a mother, she loves [her
> child]." With respect to this explication, it should moreover be
> known that a proposition is called probable when its subject contains
> a property which disposes, but does not necessitate, it to partake of
> the predicate
>  ....[29]
> 
> 
> 



The semantic relation to which Richeri refers is not explicit in
Boethius's definition. What mainly matters here, however, is
Richeri's use of a semantic notion of probability. Albert of
Cologne displays the same notion in connection with a perception-based
explanation; such propositions are held true by everyone, the most, or
the
 wise.[30]
 The less sense-based and the more intellectual a perception is the
more is it restricted to the wise and the thoughtful, according to
Albert. Interestingly, he acknowledges Arabic sources for this view
(Albert *Logica*, I, 1, 2, 241; Bach 1881; Cortabarria Beita
1953). He does not mention a specific source, but this hint uncovers
an important connection indicating that medieval Islamic philosophy,
theology, and law were no less based on probability than their
Christian counterparts (Black 1990, 108; Daiber 1990, 218; Miller
1984, 55).



Nicolas of Autrecourt offers an alternative perspective on the
contrast between complete and partial participation. Nicolas posits
that perfect knowledge corresponds to the full appearance of an
object, whereas probability represents a degree of approaching this
completeness. Certainty, in turn, arises when there exists a
sufficiently high degree of completeness in the object's manifestation
(Grellard 2005, 93; Salvestrini 2022, 968).


## 4. Further aspects of medieval probability


### 4.1 The dialectical syllogism



Syllogisms are patterns of logical argumentation. A syllogism with
merely probable premises was called dialectical in medieval and
Renaissance philosophy. Since dialectic was the "art of rational
disputation" or "controversial inquiry" (*ars
disserendi*), and disputation was a characteristic activity of
universities in the Middle Ages, argumentation with probable
propositions possessed an enormous significance for medieval thought.
It is also characteristic of logic-fed scholasticism that probable
reasoning was mainly conceived as logical deduction from probable
 premises.[31]
 After the rise of Aristotelianism, these premises were mostly framed
as *endoxa*. Thomas Aquinas writes:



> 
> 
> The dialectician is concerned only with proceeding from propositions
> which are as acceptable as possible. These are propositions which seem
> true to most people and especially to the wise. (Thomas Aquinas,
> *Posterior Analytics*, I, 31, 3, 142)
> 
> 
> 



It is not, however, clear whether authors who summarily refer to
probable premises only intend *endoxa* in their definitions of
dialectical syllogism, or whether they operate, like Albert of
Cologne, with a broader notion of probable premise. Albert
(*Logica*, I, 4, 2, 278) speaks with respect to the dialectical
syllogism about premises which are always or most often (*in
pluribus*) true. There was also some discussion about the
epistemic status of the argumentative scheme of the dialectical
syllogism and the conclusions from probable premises. The common
assumption was that the dialectic syllogism is an indubitably valid
deductive scheme, but that probable premises produce only probable
conclusions or opinions (Buridan, *Quaestiones*, 19; Buridan,
*Summulae*, 347).



Ockham offers some remarkable comments in *Summa logicae*
III.1.1 concerning probability in the dialectical
 syllogism.[32]
 Like many others, Ockham assumes that the dialectical syllogism
starts from merely probable propositions (*probabilia*), which
he characterizes as *endoxa*. Yet he then claims that probable
propositions are true and necessary, even though they are not per se
or derivatively known to be true with certainty (*probabilia sunt
illa, quae cum sint vera et necesseria, non tamen per se nota, nec ex
per se notis syllogizabilia, nec etiam per experientiam evidenter
nota, nec ex talibus sequentia*). Probability, thus, hinges on an
epistemically deficient state of an observer. However, a dialectical
syllogism need not produce uncertainty in an observer since it need
not produce fear of error but can instead engender firm conviction in
its
 conclusion.[33]
 Ockham deviates here from some familiar scholastic assumptions
concerning probability. His *probabilia* cannot be probable
opinions in the third sense of Grosseteste (see above), which are by
definition contingent. The conclusion of a dialectical syllogism is
also not necessarily a probable opinion because then it would be
accompanied by fear of error. This cluster of sentences is
nevertheless all Ockham has to say about probability in the *Summa
logicae* (and Burley, Buridan, and others say even less in their
logical works). No deeper analysis concerning the role of
*probabilia* in his philosophy or theology ensues, and his
uncommon remarks apparently did not spark any discussion by
 others.[34]
 Elsewhere in his work, Ockham judges specific philosophical theses to
be probable or more probable in the same way as other scholastic
authors did. If his short comments on *probabilia* in the
*Summa logicae* offer a glimpse of the deeper modal and
ontological ramifications of probability, they also document that
these ramifications were hardly pursued by medieval
 scholastics.[35]



However important dialectical syllogisms may have been for medieval
disputation, it should also be recognized that action-planning in
conscience was conceived as deduction from (partly) probable premises.
The dialectical syllogism therefore provided a logical basis for moral
decisions and served as a tool for moral theology.



The humanist dialectic of the Renaissance shared many assumptions
concerning the dialectical syllogism with the scholastics, but some
authors developed a different understanding of dialectic. These
developments will be discussed in Section 5.


### 4.2 Order relations between probabilities



Pre-modern probability was not a number or ratio, but mainly a binary
property which a proposition either had or did not have. Yet
pre-modern probability was also an ordinal concept. Some propositions
were regarded as more probable (*probabilior*,
*probabilius*) than others. More probable opinions were a
subclass of probable opinions. Consequently, an opinion had at least
to be probable in order to be more probable than another opinion. In
line with these premises, strict order relations (i.e.
">" and "<") for opinions were common in
medieval and Renaissance philosophy. In the Middle Ages,
attributions of equal probability did exist but seemed to be less
 widespread.[36]
 An equilibrium between opposing reasons led to doubt
(*dubium*). However, it is important to note that the ethical
principles guiding decisions in a state of doubt were distinct from
those applied when choosing between probable opinions.


### 4.3 Both-sided probability



Scholastic authors often ascribed probability at the same time to a
proposition and its negation or to a proposition and a
counter-proposition which was logically incompatible with it. A
proposition could even be regarded as more probable
(*probabilior*) without its negation losing probability. In
fact, calling an opinion more probable than another regularly implied
that the other was also probable, because comparison and choice
between probable alternatives was intended.



For some modern commentators, this impugns an understanding of
"probable" as readily affirmable or adoptable. It seems to
be a minimum requirement of being affirmable that a proposition
*x* is judged to be more likely true than false, that is,
having probability p(*x*) > 0.5. Since a proposition and its
negation cannot both have p > 0.5, they cannot both be affirmable
at the same time. The scholastics, of course, did not operate with
such numerical considerations. For them, "both-sided
probability" (as one might call it) resulted naturally from the
endoxic concept of probability. A proposition and its negation can
both be held true at the same time by different
 experts.[37]



It is not possible, of course, to assume both-sided probability based
on a proto-frequentist or semantic understanding of probability.
Contradictory sentences cannot both be most often true. Yet this
problem did not impugn medieval ascriptions of both-sided probability
in academic debates or in the practice of the confessional, which
usually explicitly or implicitly proceeded on the basis of the endoxic
understanding of probability.



There are two different ways in which an evaluator can consider a
proposition as more probable and a counter-proposition as probable at
the same time. First, the evaluator can believe that more and/or
better experts hold *x* true than *y*, although
*y* is approved by a sufficient number of experts to count as
 probable.[38]
 Second, the evaluator may consider a proposition as more probable
according to her own weighing up of the reasons, while the
counter-proposition is considered probable because it is the opinion
of many or weighty
 experts.[39]
 Both options show that there is no obvious logical problem with the
assumption that two incompatible propositions can both be regarded as
rationally
 affirmable.[40]
 This does not, however, invalidate the claim that only one of the
propositions can be rationally affirmed at a given time and by a given
person from her own standpoint. The person in question can, it seems,
only rationally affirm what she regards as more probable than its
negation. Consequently, regarding a rival opinion as probable means
considering it as affirmable by others - or, in modern
terminology, by one's epistemic
 peers.[41]
 This insight is an important step towards opinion pluralism and the
conceptualization of reasonable disagreement, and it is significant
that it is *in nuce* implied in the both-sided probability of
medieval scholastics (Schuessler 2014).



The epistemological difficulties of both-sided probability (and of
reasonable disagreement between competent reasoners) were not,
however, analyzed in depth by medieval scholastics. Nor did they
investigate how an epistemology of acceptance of probable opinions as
premises for action differs from one which assumes assent or
affirmation of the opinions in question. All this happened much later
in the scholastic tradition, in the seventeenth century; but this is
another story (Deman 1936; Knebel 2000; Schuessler 2019; Schwartz
2019; Tutino 2018). The developments in question were linked to the
rise of a new scholastic doctrine called probabilism in the
seventeenth century. Probabilism, as understood at the time (the
modern understanding of the term is quite different), legitimizes a
specific way of using probable
 opinions.[42]
 It is *not* synonymous with any kind of probabilistic
reasoning. However, we need not deal with scholastic probabilism any
further because the doctrine did not exist in the period now under
consideration. It is sometimes attributed to medieval authors, but
such ascriptions should be contested and, in any case, should not be
accepted without deeper discussion for authors who wrote before
1500.


### 4.4 Subjectivity and objectivity



The question of whether medieval forms of probability were subjective
or objective has long puzzled modern researchers (Kantola 1994). It is
difficult to come up with an answer, not least because medieval
authors did not employ the (characteristically modern) terminology of
subjectivity and objectivity with respect to judgments of
 probability.[43]
 Nevertheless, the discussion above of medieval uses of
probability-related terms sheds some light on matters of subjectivity
and objectivity.



As has been shown, *probabilis* or *verisimilis* were
predicates for the qualitative support a body of evidence gave to the
truth of a proposition or the fittingness of a sign. We have already
seen what kinds of evidence for probability were accepted at the time.
A subjective aspect of probability ascriptions thus consisted in a
belief obtained by the supporting relationship in a particular case.
An objective aspect depended on the actual existence of the supporting
basis.



This kind of objectivity does not make probability a feature of the
world or a theoretical construct based on facts of nature. Such
stronger forms of objectivity underlie, for instance, a true
proto-frequentist understanding of
 probability.[44]
 If probability *means* "what frequently happens",
probability will often derive from objective features of the world. If
frequent occurrence of *x* is only an indicator for the
affirmability of sentences about *x*, medieval
proto-frequentist probability is not strongly objective in this way.
Nevertheless, at least the medieval semantic notion of probability
seems to be objective in the strong sense. A proposition, the
predicate of which mostly inheres in its subject, is at least in some
cases objectively probable on ontological grounds from a scholastic
point of view. Facts of nature determine the degree to which
predicates inhere in a subject, so that it becomes objectively
probable that a mother loves her child.



Moreover, medieval probability ascriptions were in significant
respects inter-subjectively testable - and it is characteristic
of scholasticism to assume that communities are in a position to
assess individual claims of
 rationality.[45]
 Propositions can be identified as semantically probable through the
common use of terms. A communitarian basis also existed for the
endoxic notion of probability, which in principle derives from
observable majorities or a shared ascription of wisdom or expertise.
Endoxic probability could thus become an instrument of social (and
moral) control. Yet, although collective control of endoxic
probability ascriptions may have existed, it was never absolute in the
period under discussion. Often, no consensus concerning the relative
weight of experts existed; and it therefore remained contentious which
side was "larger and more reasonable" (*maior et
sanior*) or even whether a proposition was supported by enough
leading experts to count as probable. On these grounds, some
perspectival variation became possible. Walter Burley wrote that a
thing can be probable to one person and improbable to
 another.[46]
 This statement shows that a limited multi-perspectival variety of
probability ascriptions was acknowledged in the late Middle Ages.


## 5. Probability in the Renaissance



The Renaissance is considered here as a cultural movement which
bridged two historical epochs, the Middle Ages and early modernity.
There is notorious disagreement on the temporal boundaries of the
Renaissance. One familiar view is that it covers the period from the
middle of the fourteenth century to the end of the sixteenth. The
Renaissance is closely linked to the rise of humanism as a style of
thought, and humanism is often conceived of as a rival and antagonist
of scholasticism. Recent scholarship has found less clear-cut
boundaries between humanism and scholasticism than were formerly
assumed, and this is also true for humanist and scholastic uses of
probability-related terms. The whole range of Latin
probability-related words was used by Renaissance humanists and
medieval scholastics alike, with roughly similar meanings based on
ancient precedent. In the period from 1200-1500, there were no
great debates on probability; and what Renaissance authors, like their
contemporaries, had to say about the subject fills a few lines at most
in their works. Nonetheless, the differences between humanist and
scholastic uses of probability should be noted.



Humanist concern with probability was largely stimulated by attempts
to clarify the relationship between rhetoric and dialectic. A second,
but related, field in which probability-terms became important for
humanists was the translation, recovery, and interpretation of ancient
texts, such as Aristotle's *Topics* or Quintilian's
*Institutio oratoria*. Yet the renewed attention to the texts
of Cicero and Quintilian, or those of Aristotle's Greek
commentators, in the fifteenth century does not seem to have produced
an immediate challenge to scholastic usages of probability. In
general, the terms *probabilis* or *probabilitas* appear
to have been less central for humanists than for scholastic (moral)
theology and jurisprudence. Many humanists prolifically used the word
*probabilis*, but some shunned it and spoke instead of
worthiness of belief (*credibilitas*). This may often signal
not much more than a predilection for the Latin of some ancient
author. A contrast to scholastic usage can at best be expected from
authors who intended to make a break with scholastic dialectic such as
Lorenzo Valla and Rudolph Agricola (Mack 1993, 31, 146; Nauta 2009,
233; Spranzi-Zuber 2011, 65).



Lorenzo Valla (*Repastinatio*, 253) attacked scholastic
dialectic, referring to worthiness of belief and belief-worthy things
(*credibilia*) rather than to probability and
*probabilia*. He follows Quintilian's (the complete text
of the *Institutio* *oratoria* was rediscovered in the
Latin West in 1416) division of belief-worthiness into very firm
(*firmissimum*), strongly disposed (*propensius*), and
not ill-disposed (*non
 repugnans*).[47]
 Credibility and truth-likeness are also related to the modal category
of possibility. Valla assumes degrees of possibility. Something can be
very or slightly possible. Only propositions qualified as very
possible were worthy of belief and truth-like (*verisimilis*).
In this context, Valla never once uses the word *probabilis*.
But he approvingly quotes Cicero's connection of argumentation
with probable (*probabilis*) invention in Book 3 of his
*Repastinatio dialecticae*, where he gets down to dialectic and
logic after much previous analysis of language. On the whole,
Valla's probability-related concepts do not seem to differ too
starkly from the semantic and proto-frequentist notions of probability
(or belief-worthiness) which he shared with the scholastics.



Rudolph Agricola, another humanist innovator, put the word
*probabilis* center stage in his path-breaking reinterpretation
of dialectic. For him, dialectic was the art of speaking with
probability on whatever issue. This definition appears traditional
enough, but Agricola (*De inventione dialectica*, 210)
explicitly distanced himself from Aristotle's *endoxon*
as the basis for probability in dialectic. As has been noted by modern
scholars, "speaking with probability" is a feature of a
process for Agricola rather than one grounded in a property of
propositions used in a dialectical syllogism. Probability arises from
the argumentative quality (*argumentosus*), aptness
(*aptus*), and fittingness (*consentaneus*) of a
reasoning process (Agricola, *De inventione dialectica*, 210,
306; Mack 1993, 170; Spranzi-Zuber 2011, 89). This directs dialectic
away from the reliance on authority implied by Aristotle's
definition of reputable opinion. It is also significant that Valla and
Agricola focus on the production of conviction (*fides*)
instead of mere opinion. They thus seem to call for more rhetorical
persuasion or for higher epistemological standards in
dialectic's quest for truth. It is difficult, however, to
develop an uncontentious interpretation of what Agricola exactly meant
by his peculiar, but undoubtedly influential, view of probability. In
sixteenth-century century humanist discourse on probability,
Agricola's dialectic was a key innovation, as were the new
interest in Aristotle's *Topics*, the rise of a
Protestant tradition of dialectic, and much more. Together with
advancements in early modern scholasticism, these developments provide
a justification for ending the present survey around 1500.


## 6. Preview of early modern and modern probability



Medieval and Renaissance notions of probability largely derived from
the same ancient sources and remained related to each other through
interchanges between scholastics and humanists. Yet soon after 1500,
pre-modern probability discourse began to assume a new shape. Although
the many differences between medieval and early modern scholastic and
humanist uses of "the probable" cannot be summarized here,
they should not be underestimated. Above all, at the end of the
sixteenth century, the Aristotelian *endoxon* gave way in moral
theology to a two-tiered definition of probability as either intrinsic
(based on known reasons) or extrinsic (based on the opinions of
others). How much such innovations fostered the rise of numerical
probability in the middle of the seventeenth century is still largely
an open question. Knebel (2000) and Schuessler (2019) show that it is
reasonable to assume a significant contribution by early modern
scholasticism. Maryks (2008) claims that the humanist precedent
prompted the Jesuits to introduce the distinction between internal and
external probability. At present, however, the cross-pollinations
between early modern humanism and scholasticism are scarcely better
understood than the developments leading to the invention of numerical
probability.



In any case, the notions of probability of many philosophers, from
Descartes and Locke to Kant, recognizably rely on a stock of meanings
with roots in Renaissance humanism and medieval scholasticism. The
philosophers in question deviated notably from this stock, but so did
their scholastic contemporaries. The modernization of probability
therefore occurred along a broad front of schools, trends and
traditions in the seventeenth century. It is instructive to relate the
four medieval concepts of probability listed above to their modern
successors. On the modern side, three major groups of interpretations
or concepts of probability have emerged (Hajek 2011):


* quasi-logical approaches: probability as a measure of objective
evidential support;
* degree-of-confidence or degree-of-belief approaches: probability
as a measure of subjective graded belief or confidence;
* feature-of-the-world approaches: probability as a measure for
undetermined features of the world.



The first category mainly contains classical and logical concepts of
 probability.[48]
 Classical concepts define probability through the equal possibility
of states of affairs. Logical concepts focus on evidential support for
the truth of propositions. The most prominent variants of the second
category are Bayesian theories of probability and mathematical
representations of subjective expectation. The last category comprises
frequentist and propensity interpretations of
 probability.[49]
 For frequentists, probability is the limit of a relative frequency in
a series of events; for the propensity view, it is an irreducible
feature of the physical world.



Medieval notions of probability exhibit parallels to all modern groups
of probability concepts. Endoxic and juridical probability are species
of quasi-logical probability because they arise from evidential
support for holding a proposition true. *Endoxa*, that is,
probable opinions of others, are evidence for holding a proposition
true. The witness testimony and *indicia* of juridical notions
of probability are also, of course, kinds of evidence. The second
category of modern interpretations of probability is related to
medieval probability understood as a specific amount of confidence in
the truth of a proposition. It is the confidence that characterizes
opinions as the lowest rank of an epistemological order leading via
*fides* (standing for fully confident belief or faith) to
knowledge. Finally, probability is a feature of the world in modern
propensity views, as well as for the medieval semantic notion of
probability. Whether this also holds for proto-frequentist probability
is, as indicated, a matter of interpretation.



Conspicuously lacking in the period from 1200-1500 is an
ancestor of the "classical" notion of probability,
grounded on the equal possibility of events. This, then, seems to be a
genuinely modern interpretation of
 probability.[50]