Endoxic (or topical) probability

The most prominent scholastic understanding of probability derived from Aristotle’s notion ofendoxon, which shaped the concept of “probable opinion” (opinio probabilis). Aristotle definesendoxon in Topics I, 100b20 as:

اقتباس :: [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:[12]

اقتباس :: [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 toTopics I, such as this one by Gregory of Rimini:

اقتباس :: 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’sContra 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).[13]

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 Ibecame a commonplace. It seems immediately clear why an endoxic understanding of probabilisled 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 ofendoxa 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”).[14]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.[15]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.[16] 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.[17] This requirement shows that the endoxic understanding of probable opinions was indeed skewed towards external authority.[18]

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 (hōs 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.[19] 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. 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 thatprobabilis 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:

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)[20]
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)
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)
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)[21]

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 probabilitymeans 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 frequentiusand probabilius are synonymous here, as in several other cases in which they occur together.[22]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.

Endoxic (or topical) probability

3.2 Proto-frequentist probability

Endoxic (or topical) probability :: تعاليق

Endoxic (or topical) probability