Earlier this week I attended Computability in Europe, where I enjoyed catching up with many logic and computability friends, and answering random “did they do X in the Middle Ages?” questions. One in particular warranted a blog post; Benjamin Rin (Utrecht) asked me whether there was anything like a historical precursor to infinitary logics in the Middle Ages. I gave him a few references, but then decided it would be worth writing up some of the info in them for wider consumption.
What is infinitary logic?
Classical propositional and predicate logic is finitary: Simple atomic sentences can be formed into more complex sentences via boolean combinations (and the addition of quantifiers, in the case of predicate logic), but these combinations can only be iterated a finite number of times.
Consider a classical predicate language with sufficiently many distinct constants: When interpreted on a finite domain, the universal quantifier is eliminable, as every formula universally quantified formula is equivalent to a conjunction of all of the instantiations of the formula. Similarly, the existential quantifier is eliminable in favor of a disjunction of all the instantiations. However, in the presence of infinite domains, this equivalence breaks down because there is no way to list infinitely many distinct instantiations in a single finite conjunction (or disjunction).
Infinitary logic allows formulas of infinite length — generally in the form of infinite conjunctions and disjunctions, but in some cases also allowing infinitely iterated quantifiers. With infinitary logic, the equivalence between universally quantified sentences and conjunctions, and between existentially quantified sentences and disjunctions, is restored even in infinite domains. (For more information on the motivations for infinitary logics and their syntax, see ).
Asking the right questions
I have spent much of my logical career persuading modern logicians of the value of looking to the history of their field. But when asking historical questions about developments in logic, there are better and worse ways to do it. Sometimes a straightforward “Did they do X in the Middle Ages?” or “When did X first develop?” question is perfectly legitimate — for instance, to the question “Did they have propositional logic in the Middle Ages?”, the answer is a straightforward “yes”. Similarly, “When did quantifier/variable symbolic notation first develop?” has a straightforward, determinate answer (and it’s not medieval).
But some modern developments are embedded in a mathematical and symbolic context that was simply not present in the Middle Ages, which can result in those straightforward questions being answered with false negatives. In order for the question “Was there infinitary logic in the Middle Ages?” to be answered positively, there would have to be some notion of recursive definitions of well-formed formulas, the distinction between finite and infinite symbolic formulas, and a well-developed notion of actual infinity (as opposed to Aristotelian potential infinities) — all things which medieval logicians just didn’t have. But rather than accept “no” as the final answer to that question, we can ask a further question, namely: Did medieval logicians have any of the same concerns that motivated the introduction of infinitiary logics, and, if so, how did they address those concerns? That is a much more interesting question to ask, because the answer says something not only about how logic in the Middle Ages worked, but also about how given different technical tools people can come up with different solutions to similar problems. And then we are in a position to ask “are there any aspects of the medieval solutions to these still-occurring modern worries that could be usefully implemented in a modern context?” And that is why studying the history of logic matters — not so that we know the who, the what, and the when, but so that we know the why and the how, because these whys and hows can still be applicable and relevant to today’s concerns.
If the question merely is “Were there infinitary logics in the Middle Ages?” I’d have to answer know. But if the question is “What did medieval logicians have to say about infinitary (or even unbounded) conjunctions and disjunctions?” then we have something to say.
Rules of ascent and descent
And what we have to say starts with the idea of “rules of ascent and descent”. These are rules that tell us what singular statements we can infer from a subject-predicate proposition with a general noun phrase (including quantified noun phrases) as the subject, and when we can infer the generalised form from a collection of singular statements. (A singular statement is one where the subject term applies to exactly one thing, e.g., when it is a proper name (“Socrates”) or a demonstrative pronoun (“that”) or a demonstrate pronoun attached to a noun (“that cat”) or adjective (“that red [thing]”).)
Discussions of the rules of ascent and descent can be found mainly in three different genres of logical texts, in the 13th and 14th centuries: treatises on supposition, syncategoremata, and sophisms.
Treatises on supposition
The 13th century saw the development of a semantic genre called “the properties of terms”, of which the two most important properties were signification and supposition.  The signification of a term, very roughly speaking, the meaning of the term independent of any considerations of its use in a sentence: For instance, the word ‘cat’ signifies all cats, both present, past, and to come. All meaningful/significative words can be divided into two types, those that signify on their own (such as ‘cat’, ‘dog’, ‘man’, and other nouns, verbs, and adjectives), and those that only signify in conjunction with another term (such as ‘and’, ‘not’, ‘every’, ‘only’); these latter words are said to ‘consignify’. But while a term’s signification is fixed, its supposition — the objects signified by the word that are actually under consideration — varies according to syntactic context. For instance, if I speak of ‘every cat’, then, absent any other modifiers in the sentence, ‘cat’ supposits for every presently existing cat. If, however, I speak of ‘that cat’, then ‘cat’ supposits only for the single cat that I am indicating with the demonstrative pronoun. What, then, is the relationship between statements predicating the same predicate of ‘every cat’ and ‘that cat’? Certainly, if something is true of every cat, then it will be true of that cat as well — if every cat is an animal, then that cat is an animal. And that is an application of a rule of descent! But the rule is stronger: Not only is that cat an animal, but this one is too, and this one, and this other one as well, and so on for all the cats. Alternatively, if that cat is an animal, and this one is as well, and this one, and this other one, and so on for all the cats, why, then all cats are animals. (And this would be a rule of ascent.)
“and so on”? Hold that thought…
Treatises on syncategoremata
Just as words can be identified as either significative or consignificative, they can also be identified as categorematic or syncategorematic.  Specific treatises were devoted to the analyses of syncategorematic words, including universal, partial, and indefinite quantifiers, modifications of quantifiers (e.g., exceptives and exclusives), and demonstrative pronouns, all of which are intimately related to rules of ascent and descent.
Treatises on sophisms
Once you define how words work and the rules that govern them, then you can come up with puzzles — and the inferential relationships between the different types of supposition that a word can have, and how that supposition can be modified by modifying where in the sentence the word (or phrase) occurs were a prime source for such puzzles. These puzzles, called sophisms, were sometimes discussed in separate treatises specifically devoted to the topic; and questions of when and how one can descend from a universal or ascend from a partial were discussed in the context of puzzles that question the standard statements of the rules.
What is the problem, and what do people say about it?
The problem is the “and so on”: How does one know when one has exhausted all of the possibilities, whether ascending or descending? Though the problem is most acute in infinite domains, it even arises in finite domains, because there is still an asymmetry between the quantified sentences and the conjunctions, in that every conjunction entails each conjunct, but no single conjunct entails the entire conjunction (except in degenerate cases where one of the conjuncts is an impossibility, or where the conjunction only has one conjunct). This means that descent is simpler than ascent, as we now show:
Suppose that we have four cats, Widget, Slinky, Goldwine, and Nefertari. If we know that ‘Every cat is an animal’, we can descend to ‘This cat is an animal’, pointing to Widget; ‘This cat is an animal’, pointing to Slinky; and so on. If we stop at Goldwine, then we are still expressing truths, just not all of the truths we could possibly express. Suppose, on the other hand, that we know that ‘this cat is an animal’, pointing to Widget; ‘this cat is an animal’, pointing to Slinky; and ‘this cat is an animal’, pointing to Goldwine; this alone does not warrant an ascent up to ‘Every cat is an animal’, because there is a cat that we haven’t yet included in our enumeration, namely, Nefertari. The problem is that adding Nefertari to the list is not enough: Even after we’ve said ‘this cat is an animal’, point to her, we are not yet in a position to say ‘Every cat is an animal’, because we need a further constraint: “And these are all the cats”. When the domain is finite, it is possible to add this further statement; in an infinite or unbounded domain, it is not always straightforward to do so.
How, then, do medieval authors handle the “and so on”? In this post we will look at four authors; and in a future post I’ll hopefully look at others (particularly some 14th-century ones), as well as some of the sophisms that arise from the ascent/descent rules.
William of Sherwood
William of Sherwood first mentions descent when defining the two types of personal supposition (which is when a word stands for a thing bearing the form signified by the word), in his Introduction to Logic :
[Distributive confused personal supposition is] mobile when a descent can be made, as in the term ‘man’ in the example above [‘Every man is running’]. It is immobile when a descent cannot be made, as here: ‘Only every man is running’ (for one cannot infer ‘therefore only Socrates is running’) [pp. 108-109].
According to Kretzmann, “Sherwood’s notion of logical descent is quite stringent…Sherwood’s descent is unquestionably irreversible” [fn. 29].
Immobility arises purely on the part of descent; a few pages later, in Rule V, Sherwood argues that it may still be possible to ascend in the case of immobile supposition:
Sometimes, however, distribution remains immobile, as in ‘not every man is running’, ‘only every man is running’, and other cases of that sort. It is called immobile, however, not because we cannot ascend in the subject but because we cannot descend [p. 119].
That is, Sherwood indicates, obliquely, that from ‘Not Socrates is running’ we can ascend to ‘Not every man is running’: But this is because of the way in which negation interacts with quantification, namely, the fact that to negate a universal a single counterexample is all that is required.
Sherwood does not discuss the case of general ascent from a conjunction to a universal.
Roger Bacon, in his Art and Science of Logic , also invokes the notions of descent and ascent when defining his typology of supposition. However, he is more explicit as to what descent is:
Confused and distributive supposition occurs, therefore, when a common term supposits for all its inferiors and there can be a descent to any one of them, as in ‘Every man runs; therefore this man runs, and that one, etc.’ [∥ 219]
Merely confused supposition occurs when such a descent cannot be made.
Despite the fact that “ascent” appears in the index, Bacon does not explicitly mention ascent at all.
Lambert of Auxerre/Laigny
Like the preceding two authors, Lambert‘s first mention of ascent or descent in the Summa Lamberti  is in his definition of strong mobile supposition:
Strong mobile supposition is what a common term has when it is interpreted necessarily for all its supposita and a descent can be made under it. This happens when a universal affirmative sign is added directly to a common term, as when one says ‘Every man runs’, and likewise when a universal negative sign is added indirectly or directly to a common term, as when one says ‘No man is a stone’…It is called mobile with respect to a term having such supposition because in a term having such supposition a descent can be made on behalf of the supposita contained under it [∥ 1263].
He does not discuss ascent at all.
Peter of Spain
Finally, we look at what Peter of Spain has to say in his Summule logicales . His primary mention of descent is in the context of defining mobile supposition:
[That term ‘human’ supposits] movably because descent is permitted to anything whatever that it supposits for, as in:
therefore, Plato [p. 249]
However, he also, unlike the others, considers sophisms arising from the rules and definitions of supposition, one of which invokes the notion of descent. Due to the great length this post has already gotten to, we will save it for future work.
The four preceding views that we looked at have a number of things in common: First, they are all 13th-century, and second, all of them (with the exception of Peter’s second example that we haven’t discussed here) come from treatises or chapters on the properties of terms. These two features together provide a likely explanation for the third commonality, namely, that most of them discuss the “and so on” problem. This is something that we can expect to find more explicit discussions of in treatises on sophisms (logical puzzles) arising from supposition theory — as we see in Peter’s second discussion of it –, as well as in later authors (we’ll try to visit the 14th century in a future post!) for whom the fact that there is a potential problem had been raised via the sophisms literature.
 Bell, John L., “Infinitary Logic”, Stanford Encyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.).
 For more on both of these, see both Uckelman, Sara L., “William of Sherwood”, Stanford Encyclopedia of Philosophy (Summer 2016 Edition), Edward N. Zalta (ed.) and Read, Stephen, “Medieval Theories: Properties of Terms”, Stanford Encyclopedia of Philosophy (Spring 2019 Edition), Edward N. Zalta (ed.).
 Well, strictly speaking, words are neither categorematic nor syncategorematic, but they are used categorematically or syncategorematically. Cf. Uckelman, Sara L., “The Logic of Categorematic and Syncategorematic Infinity”, Synthese 192, no. 8 (2015): 2361-2377.
 Kretzmann, Norman (ed.), 1966, William of Sherwood’s Introduction to Logic, Minneapolis: University of Minnesota Press.
 Bacon, Roger, 2009, The Art and Science of Logic, Pontifical Institute of Medieval Studies. Trans. by Thomas S. Maloney.
 Lambert of Auxerre, 2015. Logica or Summa Lamberti. University of Notre Dame. Trans. by Thomas S. Maloney.
 Brian P. Copenhaver (ed.), 2014, Peter of Spain: Summaries of Logic, Text, Translation, Introduction, and Notes. Oxford University Press. With Calvin Normore and Terence Parsons.