The medieval name for paradoxes like the famous Liar Paradox (“This proposition is false”) was “insolubles” or insolubilia,  though besides semantic paradoxes, they included epistemic paradoxes, e.g., “You do not know this proposition”. From the late-twelfth century to the end of the Middle Ages and beyond, such paradoxes were discussed at length by an enormous number of authors. Yet, unlike twentieth century interest in the paradoxes, medieval interest seems not to have been prompted by any sense of theoretical “crisis”.
The history of the medieval discussions can be divided into three main periods: (a) an early stage, from the late-twelfth century to the 1320s; (b) a period of especially intense and original work, during roughly the second quarter of the fourteenth century; (c) a late period, from about 1350 on. The discussion in this article will be organized as follows:
- 1. Origins of the Medieval Discussion
- 2. Early Developments to the 1320s
- 3. The Second Quarter of the Fourteenth Century
- 4. The Late Period
- 5. Observations
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1. Origins of the Medieval Discussion
The Liar Paradox was well known to antiquity. Its discovery is often credited to Eubulides the Megarian (4th century BCE), on the basis of a remark by Diogenes Laertius (Lives of the Philosophers II.108), although in fact Diogenes says only that Eubulides discussed the paradox, not that he discovered it. A little later, the poet and grammarian Philetus (or Philitas) of Cos (c. 330–c. 270 BCE), if we are to believe the story in Athenaeus of Naucratis’s Deipnosophists IX.401e, worried so much over the Liar that he wasted away and died of insomnia, as, according to Athenaeus, his epitaph recorded:
Philetus of Cos am I
’Twas The Liar who made me die,
And the bad nights caused thereby.
Diogenes Laertius also reports (VII.196–98) that, in addition to a huge number of other works on a variety of topics, the Stoic logician Chrysippus (c. 279–206 BCE) wrote:
- Introduction to the Liar;
- Liar Propositions: An Introduction;
- six books on the Liar itself;
- Reply to Those Who Think There Are Propositions That Are Both True and False;
- Reply to Those Who Solve the Liar Proposition by Division;
- On the Solution to the Liar (in three books);
- Reply to Those Who Say The Liar Argument Has False Premises.
Unfortunately, almost none of Chrysippus’ work survives.
1.1 Unlikely Ancient Sources
Nonetheless, it does not appear that medieval interest in insolubles was derived directly from these or any other known ancient sources that discuss the Liar. Many of the relevant works were lost (e.g., the works of Chrysippus), while others were never translated into Latin and so were effectively unavailable to the Latin Middle Ages, though things may be different for the Arabic and Byzantine traditions, which are only starting to be studied (see, e.g., Alwishah & Sanson 2009, Gerogiorgakis 2009, El-Rouayheb 2020 and Zarepour 2020). Indeed, it is not at all clear just what it was that prompted medieval interest. One might have supposed that, even if particular theories about the Liar were not transmitted to the Latin West from antiquity, at least formulations of Liar-type paradoxes must have been known and available to stimulate the medieval discussions. In fact, however, there are strikingly few possibilities.
Seneca (Epistle 45.10), for instance, mentions the Liar paradox by its Greek name pseudomenon, but does not actually formulate it. Again, St Augustine perhaps has the Liar in mind in his Against the Academicians (Contra academicos III.13.29), where he refers to the “most lying calumny, ‘if it is true [it is] false, if it is false it is true’”. But neither passage would likely be sufficient by itself to suggest the special problems of the Liar to anyone not already familiar with them.
Somewhat more explicit is Aulus Gellius’s (2nd century CE) Attic Nights (XVIII.ii.10), “When I lie and say I am lying, am I lying or speaking the truth?” But Gellius was not widely read in the Middle Ages, and no known medieval author cites him in the context of insolubles. Again, Cicero’s Academica priora, II.xxix.95–xxx.97, contains a fairly clear formulation:
If you lie and speak that truth [namely, that you lie], are you lying or speaking the truth? … If you say you lie, and you speak the truth, you lie; but you say you lie, and you speak the truth; therefore, you lie.
But this passage is never cited in the insolubilia-literature. Moreover Cicero, who wrote in Latin and so did not have to be translated to be available to the Middle Ages, calls such paradoxes “inexplicables” (inexplicabilia). If he was the catalyst for the medieval discussions, we would have expected to find that term in the insolubilia-literature, and we do not; the unanimous medieval term is ‘insolubles’.
1.2 St. Paul’s Reference to Epimenides
One initially plausible stimulus for the medieval discussions would appear to be the Epistle to Titus 1:12: “One of themselves, even a prophet of their own, said, The Cretians [= Cretans] are always liars, evil beasts, slow bellies”. The Cretan in question is traditionally said to have been the sixth-century BCE thinker Epimenides. For this reason, the Liar Paradox is nowadays sometimes referred to as the “Epimenides”. Yet, blatant as the paradox is here, and authoritative as the Epistle was taken to be, not a single medieval author is known to have discussed or even acknowledged the logical and semantic problems this text poses. When medieval authors discuss the passage at all, for instance in Scriptural commentaries, they seem to be concerned only with why St. Paul should be quoting pagan sources. It is not known who was the first to link this text with the Liar Paradox.
1.3 Aristotle’s Sophistical Refutations
By contrast with these passages, none of which was cited in the insolubilia-literature, there is a text from Aristotle’s Sophistical Refutations 25, [A-SR]: 180a27–b7, that, almost from the very beginning of the insolubilia-literature to the end of the Middle Ages, served as the framework for discussing insolubles. It occurs in Aristotle’s discussion of the fallacy of confusing things said “in a certain respect” (secundum quid) with things said “absolutely” or “without qualification” (simpliciter). In this context, Aristotle considers someone who takes an oath that he will become an oath-breaker, and then does so. Absolutely or without qualification, Aristotle says, such a man is an oath-breaker, even though with respect to the particular oath to become an oath-breaker he is an oath-keeper. Then Aristotle adds the intriguing remark, “The argument is similar too concerning the same man’s lying and speaking the truth at the same time” ([A-SR]: 180b2–3). It was this sentence that many medieval authors took to be a reference to the Liar Paradox, which therefore, on the authority of Aristotle, could be solved as a fallacy secundum quid et simpliciter.
The widespread appeal to this passage throughout the history of the insolubilia-literature indicates that the text did play some role in prompting medieval interest in insolubles. This suggestion is reinforced by the fact that the earliest known medieval statement of the Liar occurs in 1132, around the time the Sophistical Refutations first began to circulate in Western Europe in Latin translation (see Section 2 below).
Nevertheless, it is not immediately obvious how Aristotle’s remarks can be made to fit the Liar Paradox. The oath-breaker, as the example was generally interpreted, takes two oaths: one, which he keeps, that he will break his oath, and a second (it does not matter what it is) that he breaks, thereby fulfilling the first oath. The man is an oath-breaker and an oath-fulfiller, but with respect to different oaths; by breaking his second oath, rendering it false, he fulfills the first oath, making it true. However, it is possible to interpret the passage as referring to a single oath, when the oath is broken at the same time as it is made. Seen that way, it connects the Liar paradox with the fallacy secundum quid et simpliciter.
In short, it seems clear that the Sophistical Refutations was instrumental in prompting medieval interest in insolubles. But more must have been involved too. Martin (1993) suggests a connection with theories of obligations (cf. Section 3.3 below). Before medieval logicians could formulate genuine Liar-type paradoxes, they first had to go well beyond anything found in Aristotle’s text. At present we cannot say whether they did this on the basis of some still unidentified ancient source or whether it was through their own intellectual power and logical insight.
1.4 The Many Varieties of Insolubles
The medievals discussed many more insolubles than the Liar Paradox, though most can be seen as variants of it. One common variant was what is now called the ‘yes’-‘no’ paradox: Socrates says “What Plato is saying is false”, while Plato says “What Socrates is saying is true” (see, e.g., Buridan [B-SD]: 974). There is also the ‘no’-‘no’ paradox, where Plato instead says “What Socrates is saying is false” (Buridan [B-SD]: 971). There is even a ‘no’-‘no’-‘no’ paradox, where Socrates says that Plato is saying something false, Plato that Cicero is saying something false and Cicero that Socrates is saying something false (Albert [AS-I]: 353). There are conjunctive insolubles, e.g., “God exists and some conjunction is false”, where God has annihilated any other conjunction, and disjunctive insolubles, e.g., “A man is an ass or some disjunction is false”, where God has instead annihilated any other disjunctive proposition (Albert [AS-I]: 357–8). There is also a nice example where a landowner has decreed that only those who speak truly will be allowed across his bridge and those who lie about their business will be thrown in the water (or maybe even hanged on the nearby gallows). When Socrates is challenged on coming to the river, he says “You will throw me in the water” (Bradwardine [B-I]: 135; Buridan [B-SD]: 993; see also Cervantes Don Quixote, vol. II book III ch. XIX, p. 714).
The medievals discovered what is now usually referred to as Curry’s Paradox, in the form “If this conditional is true then a man is an ass” (see Read 2002: section 9), or in its contraposed form, “If God exists then some conditional is false” (where this is the only conditional: Albert [AS-I]: 359); or in the form of an inference (consequentia), nowadays dubbed “V-Curry”, “This inference is valid so you are an ass” (see the discussion in Section 3.9 below). There were also epistemic paradoxes, such as “You do not know this proposition” (Bradwardine [B-I]: 139): you do not know it, for if you did, it would be true and you would not know it. But now you must realise that you do know it. Such insolubles can involve doubt as well as knowledge, e.g., “Socrates knows the proposition written on the wall to be doubtful to him” (see note 32). Further insolubles arise from the medieval theory of (logical) Obligations, e.g., “Something proposed is denied by you” (Bradwardine [B-I]: 125), and “The king is sitting or a disjunctive doubt is proposed” (Bradwardine [B-I]: 151)—in the theory of obligations, the respondent is taken never to know whether the king is sitting.
2. Early Developments to the 1320s
In 1132, Adam of Balsham, the founder of the important logical school of the “Parvipontani” (so called because they gathered at the Petit Pont in Paris), wrote an Art of Discussing (Ars disserendi), in which he treats, among other things, various kinds of yes/no questions, including “whether he speaks truly who says he lies” and “whether he who says nothing but that he lies is speaking the truth”. ([AB]: 107.)
The importance of this passage should not be exaggerated. It is true that it gives us the earliest known explicit medieval formulation of the Liar. But Adam makes no attempt to solve the paradox, does not say it was a current topic of discussion in his day, and in fact does not even indicate he recognized its paradoxicalness. He simply offers it as an example of one kind of yes/no question.
It is not until later in the twelfth century that one finds an explicit statement of the special problems raised by insolubles. In his On the Natures of Things (De naturis rerum), of unknown date but apparently well known by the end of the century, Alexander Neckham ([N-NR]: 289) wrote:
Again, if Socrates says he lies, and says nothing else, he says some proposition. Therefore, either a true one or a false one. Therefore, if Socrates says only that he lies, he says what is true or what is false.
But if (1) Socrates says only the proposition that Socrates lies, and he says what is true, then it is true that Socrates lies. And if it is true that Socrates lies, Socrates says what is false. Therefore, if Socrates says only the proposition that Socrates lies, and he says what is true, he says what is false.
But if (2) Socrates says only the proposition that Socrates lies, and he says what is false, then it is false that Socrates says what is false. And if it is false that Socrates says what is false, Socrates does not say what is false. But if Socrates says only that he lies, he says either what is true or what is false. Therefore, if Socrates says he lies, he says what is true. Therefore, if Socrates says only that he lies, and he says what is false, then he says what is true.
But if Socrates says only that he lies, he says what is true or false. Therefore, if Socrates says only that he lies, he says what is true and says what is false.
Nevertheless, although it is clear that Neckham was fully aware of what is paradoxical about the Liar, he makes no attempt to solve the paradox. He presents it only as an example of the “vanities” logic deals with. This suggests that by his day others were trying to solve the paradox, and in fact in the discussion of the fallacy secundum quid et simpliciter contained in the so-called Munich Logic (= Dialectica Monacensis) from sometime in the second half of the century, we find the remark: “But how this fallacy arises in uttering the insoluble ‘I am saying a falsehood’, that is a matter discussed in the treatise on insolubles”.
The first text we have that actually tries to solve the paradox is an anonymous treatise from the very end of the twelfth or the very early thirteenth century (De Rijk 1966). From then on, there are a great number of surviving treatments (see Spade 1975). In the early 1320s, Thomas Bradwardine, in a preliminary section of his own treatise on insolubles, lists eight views in circulation in his day, in addition to his own (see Bradwardine [B-I], ch. 2; Spade 1987: 43–46). Some of these views can no longer be identified in the texts that survive from the period before Bradwardine, but among the surviving views, we can distinguish five broad approaches to “solving” the paradox. (Sometimes these approaches are combined in a single author.)
2.1 Insolubles as Fallacies secundum quid et simpliciter
As might be expected in view of Section 1.3 above, many of these early theories attempted to analyze insolubles as fallacies secundum quid et simpliciter. Later in the insolubilia-literature, discussions often continued to be cast in terms of this fallacy. It was claimed in (Spade 1987: 32) that their real focus was generally on entirely different theoretical issues, and that the role of the fallacy thus became purely “honorary”, preserving the authority of Aristotle. However, that assessment was challenged in (Dutilh Novaes & Read 2008). Bradwardine, for example, explicitly and repeatedly casts his solution in terms of Aristotle’s discussion, making actual (not merely “honorary”) use of the conceptual framework offered by the fallacy secundum quid et simpliciter.
As remarked above, Aristotle had suggested ([A-SR]: 180b5–7) that insolubles are false simpliciter (absolutely/without qualification), but true secundum quid (in a certain respect). Some authors in the early medieval literature, however, argued that insolubles are without qualification neither true nor false, but only true in a certain respect and false in a certain respect. Others used the terminology of simpliciter and secundum quid, but applied it to reference (suppositio) rather than to truth and falsehood, so that in insolubles certain terms did not refer “without qualification” to their referents, but only “in a certain respect”. This view is in effect a kind of restriction on self-reference.
2.2 The Theory of Transcasus
The theory of transcasus has nothing to do with the fallacy secundum quid et simpliciter, although it too seems to have had its origins in antiquity. The word transcasus is not a common Latin word. It seems to be a literal translation of Greek metaptosis. In Stoic logic, propositions that change their truth value over time were called metapiptonta (from the same root). Walter Burley in fact used the word transcasus exactly this way in 1302 in two short logical works (Spade 1987: 33–34).
Nevertheless, in the particular context of insolubles, while the term transcasus does have an association with time, it does not imply any change of truth value over time. Rather the theory of transcasus held that in the proposition “This statement is false”, the term “false” refers not to the proposition in which it occurs, but rather to some proposition uttered earlier. Thus, when the liar says “I am lying”, what he really means is “What I said just a moment ago was a lie”. If the speaker did not in fact say anything earlier, then his present statement is simply false and no paradox arises.
This view, like the last of those discussed in Section 2.4 below, amounts in practice to a restriction on self-reference. But it is not clear exactly what motivated it. In any event, the theory of transcasus appears to have disappeared as a theory actually held by anyone after the early period, although it continued to be mentioned in later authors’ surveys of earlier views.
2.3 Exercised Act vs. Signified (or Conceived) Act
A third theory from this early period distinguishes the “exercised” act from the “signified” or “conceived” act. The details of this theory are not yet well understood, but the basic strategy is to distinguish what the liar says he is doing (namely, lying) from what he really is doing. The author of the Questions on the Sophistical Refutations attributed to John Duns Scotus, who held a version of this theory (Scotus [DS-Q], questions 52–53, pp. 505–15), thought that what the liar is really doing (the “exercised act”) is speaking the truth. In order to avoid the paradox, this theory would seem to be committed to saying that the exercised act and the signified act are two distinct acts, so that the theory, like the theory of transcasus (Section 2.2 above), is committed to some kind of restriction on self-reference.
2.4 The Theory of Restriction
Even when not combined with transcasus or the theory that distinguishes exercised act from signified act, a very popular approach throughout the insolubilia-literature, in the early period (and for that matter even in our own day), before Bradwardine’s sustained attack on it (see Section 3.1), was to deny or restrict the possibility of self-reference. Such theories had the title “restriction” (restrictio) and their proponents were called “restricters” (restringentes). All such theories maintained that in some or even all cases, terms in propositions could not “supposit for” (stand for, refer to) the propositions in which they occur.
Some theories of restriction went further and also ruled out other patterns of reference. For example:
- Proposition a = ‘b is true’, and b = ‘a is false’ (the ‘yes’-‘no’ paradox). Here a refers to b and b refers back to a. But reference is not a transitive relation, so that there is no real self-reference here. Nevertheless, the situation is paradoxical, and as a result some restricters ruled out all referential “loops”.
- Proposition a is a certain token of the form ‘a is false’, and b is a second token of the same type. Token a is self-referential, but token b is not, since it refers to a, not to itself. Yet some restricters thought the two tokens should be treated semantically alike, so that not only could the subject of a not refer to a itself, neither could the subject of b.
- Proposition a = ‘b is true’, and proposition b = ‘b is false’. Here, b is self-referential, but a is not. Nevertheless, b is the contradictory of a. Hence, by saying its contradictory is true, a is in effect saying that it itself is false. Thus, although it is not self-referential, a is nevertheless paradoxical. Some restricters prevented such cases by maintaining that not only were terms unable to refer to the propositions in which they occurred, they also could not refer to the contradictories of the propositions in which they occurred.
As a general theory, restriction is open to an obvious objection: it rules out innocuous forms of reference along with pathological ones. The sentence “This sentence has five words” is not paradoxical, after all, even though it is self-referential; in fact, it seems obviously true. Yet the general theory of restriction would disallow it.
Medieval authors sometimes raised this objection. As a result, we find two kinds of restriction-theories in the medieval literature: (a) general or strong theories that rule out self-reference, and perhaps other patterns of reference too, in innocuous as well as pathological cases; and (b) more specialized or weaker theories that rule out certain forms of reference only when they result in paradox. Walter Burley and William of Ockham, for example, held the latter form of restriction (Spade 1974).
If general or strong theories of restriction are open to the objection stated above, the weaker theories are open to a different objection: they risk being vacuous if their proponents did not have any independent way of identifying paradoxical cases. Perhaps, their theories amounted to saying “all forms of reference are allowed, except for paradoxical ones, which are not allowed”. This is no doubt true, but it is also a tautology.The restrictivist response largely died out after Bradwardine’s attack on it in chs. 3–4 of his Insolubles. However, in the late 1320s, Walter de Segrave (or Sexgrave) gave a spirited defence of restrictivism against Bradwardine’s arguments (see Section 3.2 and Spade 1975 item LXVIII, pp. 113–6).
Unlike restriction, which remained (and remains) a popular view, the theory of “cassation” disappeared very early (though it has its contemporary advocates). It is maintained in the earliest known treatise on insolubles (De Rijk 1966) and in one other early anonymous text (Spade 1975: 43–44), but seems to have died out after about 1225, although it continued to be mentioned in later authors’ surveys of previous views, no doubt because of its inclusion in Bradwardine’s own survey. It was briefly revived by John Dumbleton in the 1340s: see Section 3.6 and Spade 1975 item XXXVI, pp. 63–5). His essential idea was that signification requires uptake, so any utterance which cannot be understood cannot constitute a proposition—and the insolubles defy understanding, for self-reference generates a regress of deferred intelligibility.
‘Cassation’ is now an archaic word (though it survives in legal documents and other languages), but merely means “making null and void, canceling”. In effect, this theory holds that one who utters an insoluble proposition “isn’t saying anything”. The second of the texts just cited even gives a curious “ordinary language” argument, appealing to the rusticus (the man-in-the-street), who, if you were to say to him “What I am saying is false”, would reply “Nil dicis” (“You are saying nothing”).
The treatise in De Rijk 1966 presents more of a theory. Much of it is obscure to modern scholars, but it seems to appeal to a distinction between a mental act of asserting and a vocal act of uttering a proposition. “Saying” requires both acts; it is “an assertion with utterance”. In the case of the liar who says “What I am saying is false”, the mental act of asserting is present, and for that matter so is the physical act of uttering the words. But somehow (this is the obscure part) there is no “saying”.
It is tempting to interpret this view as an appeal to a kind of fallacy of composition; just as someone who is both good and an author is not necessarily a good author, so too something that is both mentally asserted and vocally uttered is not necessarily “said” (asserted with utterance). It is tempting, yes, but highly speculative. Nevertheless, whatever the correct interpretation, it appears that the distinction between asserting and uttering drawn by this theory escapes the facile “refutation” of it used as early as the mid-thirteenth century, that it “plainly contradicts sensation that is not deceived”.
3. The Second Quarter of the Fourteenth Century
The preceding theories represent the earliest stage of the insolubilia-literature. Although these theories are sometimes mentioned in the later literature, and in the case of “restriction” often accepted in the later literature, much more sophisticated treatments began to emerge in the second quarter of the fourteenth century. The turning point is Thomas Bradwardine, whose own theory was enormously influential on later authors. Shortly after Bradwardine, two other English authors from this middle period are also important: Roger Swyneshed (Section 3.3), and William Heytesbury (Section 3.4). A little later, important contributions were made by Parisian authors, Gregory of Rimini (Section 3.7), John Buridan (Section 3.8) and Albert of Saxony (Section 3.9).
3.1 Thomas Bradwardine
Thomas Bradwardine (c. 1300–1349) wrote his Insolubles at Oxford sometime between 1321 and 1324. It became one of the most important works on the topic in the Middle Ages. In fact, in the late 1350s, Ralph Strode, in his own treatise on the topic, surveys the earlier views (quoting Bradwardine’s own survey and theory almost verbatim), and then says (Spade 1981: 116):
For the opinions mentioned above were those of the old [logicians], who understood little or nothing about insolubles. After them there arose the prince of modern philosophers of nature, namely Master Thomas Bradwardine. He was the first one who discovered something worthwhile about insolubles.
Bradwardine’s theory is built around a distinctive theory of truth, which in turn depends on a conception of signification, described by Spade as an “adverbial” theory of propositional signification (Spade 2007: 180–87 [Other Internet Resources]; cf. Read 2008b: §13.2). By virtue of their constituent terms, propositions signify things; but, in addition, a proposition as a whole signifies that such-and-such is the case. This conception may be related to Walter Burley’s theory of the propositio in re (see, e.g., Cesalli 2001). It is this latter kind of signification that is the basis for Bradwardine’s theory of truth.
For Bradwardine, (D1) a proposition is true if it signifies only as is the case (tantum sicut est), and (D2) it is false if it signifies otherwise than is the case (aliter quam est). Note the absence of the ‘only’ in the criterion for falsehood. Truth therefore, is more demanding than falsehood. In order for a proposition to be true, all of what it signifies to be the case must in fact be the case; if any of what it signifies to be the case fails to be the case, the proposition is false. He will then argue that insolubles signify more than at first appears, and that not everything they signify can be the case. Consequently, they are simply false.
Thus what is most distinctive in Bradwardine’s theory is his “multiple-meanings” theory of signification. For him, propositions signify many things, not in the sense of being ambiguous, but as requiring a multitude of conditions to be satisfied for their truth. For example, ‘Some man is running’ signifies not only that a man is running, but also that there is a man, and a runner. Indeed, Bradwardine claims that a proposition signifies everything which follows from it. This is his famous second postulate, (P2). There is considerable controversy over its correct interpretation (for a careful discussion, see Dutilh Novaes 2009: §1). (P2) is interpreted by Spade (1981:120) as what he calls “Bradwardine’s Principle” (BP):
If p only if q, then P signifies that q,
where the name replacing ‘P’ names the sentence replacing ‘p’. However, he concedes that, when read in this way, the principle does not support the proof Bradwardine gives of his second thesis, (T2), which we will discuss below. Accordingly, Spade attributes a further principle to Bradwardine, the “Converse Bradwardine Principle” (CBP):
Whatever a sentence signifies follows from it. If P signifies that q, then p only if q.
However, he admits that Bradwardine never states or mentions this principle, and that with it, Bradwardine’s solution collapses.
It is claimed in Read 2009 that (P2) should be interpreted more generously, not in terms of how it is actually stated by Bradwardine, but how it is actually used by him. As used, it is a closure principle, that a proposition signifies everything which follows from what it signifies. This arguably has (BP) as a consequence, but is stronger than it, and sufficient for Bradwardine’s proof of (T2).
Bradwardine’s solution to the insolubles is stated in his second thesis, (T2): “Every proposition which signifies that it itself is not true, or is false, also signifies that it is true and is false”. The proof has four stages (Bradwardine B-I §6.6.1–6):
- suppose first that a signifies that a is not true, and nothing else. If a is not true, then by (D1) it does not signify only as is the case, so it is not the case that a is not true (since we are supposing that is all it signifies), that is, a is true. So if a is not true, it is true. But a signifies that a is not true, so by (P2) a signifies that it is true. Thus a does not and cannot signify only that a is not true.
- So suppose that a signifies that a is not true and also that b is c. If a is not true, then by (D1) it does not signify only as is the case, so it is not the case that a is not true and b is c, that is, either a is true or b is not c, by (P4), a statement of the De Morgan Laws. So again by (P2), a signifies either that a is true or b is not c. But a signifies that b is c, so by (P5), Disjunctive Syllogism, and (P2) again, a signifies that a is true.
- suppose that a signifies that a is false. Then by (P1), Bivalence, and (P2), a signifies that a is not true, so by (1) and (2) above, a signifies that a is true.
- so by steps 2 and 3, if a signifies that a is not true or that a is false, a also signifies that a is true. But a cannot be both true and false. So things cannot be only as a signifies, so by (D2) a is false.
In the subsequent chapter, Ch. 7, Bradwardine considers the problem of revenge in various guises. (See, e.g., the entry on the Liar paradox, and especially the section on Expressive power and ‘revenge’.) Take Socrates’ utterance of ‘Socrates utters a falsehood’, where Socrates utters nothing else. Bradwardine’s claim is that Socrates’ utterance is false, that is, that Socrates utters a falsehood. How can Bradwardine’s claim that Socrates utters a falsehood be true, while Socrates’ utterance of the same thing is false? The reason, Bradwardine replies, is that Socrates’ utterance is self-referential, and signifies not only that Socrates’ utterance is false but also, by (T2), that it is true (and so is false), whereas Bradwardine’s utterance is not self-referential (rather, it refers to Socrates’ utterance) and so not subject to (T2).
3.2 Walter Segrave
Walter de Segrave (or de Sexgrave) was at Merton College, Oxford from 1321 until at least 1338, and had obtained his Magister Artium by 1336. From 1340–42 he was Chancellor to Richard Aungerville, that is, Richard de Bury, Bishop of Durham, who famously gathered around him some of the very best minds in the kingdom, including Walter Burley, Thomas Bradwardine and Richard Kilvington. Segrave subsequently became Dean of Chichester, but was dead by 1349. His Insolubles is his only known work, which appears to have been composed at Oxford in the late 1320s or early 1330s, which is consistent with the fact that it is clearly a response to Bradwardine’s own Insolubles. (See Spade 1975 item LXVIII)
Segrave shares many assumptions with Bradwardine, indeed, at a couple of points he appears to endorse Bradwardine’s second postulate (P2), that a proposition signifies everything implied by what it signifies. For the heart of Segrave’s solution is that, since whoever asserts a proposition asserts that it is true, the restriction on supposition that Segrave maintains is that
The extremes of a proposition only supposit for those things for which the whole can mean that it itself is true, assuming that it exists, and those extremes do not supposit for such things for which the whole, assuming that it exists, would mean that it itself is false. And this is my main point.The reason Segrave gives is that
... it is because the extremes take their supposition from the copula, whose significate is that the proposition is true, as was said, so the extreme does not supposit for anything for which the whole would mean that it itself is false or is not true, because this would be inconsistent with the significate of the copula, and so the extremes are restricted by the meaning of the copula.He cites the authority of Aristotle in Metaphysics Δ 7:
Every proposition means things to be in reality as it signifies. This is self-evident and is clear from the Philosopher and the Commentator [that is, Averroes] in comment 14 on the fifth book of the Metaphysics and in the explanation in that comment as a whole: for the copula in the proposition signifies it to be a truth, as is elucidated there.
Burley and Bradwardine agree on one thing: that insolubles commit the fallacy (simpliciter et secundum quid), taking this from Aristotle’s treatment in his De Sophisticis Elenchis of the example of the man who swears that he is forsworn. Segrave says they are mistaken: according to him, insolubles commit the fallacy of accident.
The fallacy of accident is the first of the fallacies described by Aristotle in De Sophisticis Elenchis as those “independent of language,” and discussed at some length in ch.24. The classic example is the Hidden Man puzzle: you know your father (or Coriscus), your father (or Coriscus) is the man approaching, but you don’t know the man approaching (since he is wearing a mask, or too far away to recognise, etc.). Aristotle’s diagnosis was that one or more of the two properties attached to Coriscus (being known by you and being the man approaching) is accidental (or incidental) to him and so there is no essential connection to support inferring the conclusion from the premises.
Aristotle’s discussion of the fallacy of accident is neither clear nor convincing. One medieval attempt to clarify the fallacy of accident is found in Giles of Rome. The fallacy arises, he said, when there is a variation in the supposition of the middle term. Burley extends the idea of variation of the supposition of terms to the extremes (Burley, Tractatus super librum Elenchorum, cited in Ockham 1979, p.231 n.3):
In this fallacy there should be assigned three, namely, the attribute, the accident and the subject thing. And according to Giles, the major extreme is always the attribute and the middle term the subject thing and the minor extreme the accident. But this is not a big worry, for it suffices for there to be this fallacy that some term is not included but is compared to two other terms in the argument. Whence it should be realised that the fallacy of accident sometimes results from a variation of the middle term and sometimes from a variation of the major or minor extreme.Accordingly, Segrave’s analysis appeals to the fallacy of accident:
Insolubles commit the fallacy of accident because by arguing like this:This is said by Socrates and this is a falsehood, so a falsehood is said by Socrates,the term ‘falsehood’ supposits in the minor premise for something it does not supposit for in the conclusion. Similarly, in arguing like this:No falsehood is said by Socrates, this is a falsehood, so this is not said by Socrates,there is a variation in the middle term because the term ‘falsehood’ supposits for one thing in the major premise and another in the minor, according to those advocating this solution. And thus it is clear that they have to solve these kinds of paralogisms according to the fallacy of accident, namely, from a variation of the middle term or of an extreme term.
3.3 Roger Swyneshed
Sometime between roughly 1330 and 1335, the English Benedictine Roger Swyneshed adopted a theory in some respects reminiscent of Bradwardine’s, but with interesting features of its own. Like Bradwardine, Swyneshed held that for a proposition to be true, it is not enough that it “signify as is the case”. But whereas Bradwardine maintained that in addition the proposition must not signify otherwise than is the case (that is, it must signify only as is the case), Swyneshed said that in addition the proposition must not “falsify itself”. Propositions that falsify themselves are said to be those that are “relevant (pertinens) to inferring that they are false”. Thus where Bradwardine amended the account of both signification and consequently of truth, Swyneshed amended only the account of truth. Insolubles do falsify themselves, and so are false for that reason, even though they signify as is the case.
The notion of “relevance” clearly relates to the theory of Obligationes (q.v.). Much of Bradwardine’s, Swyneshed’s and Heytesbury’s treatises uses the terminology of Obligationes-treatises. But the main historical interest of his theory does not lie there. Rather, it lies in three famous and controversial conclusions he drew from his principles:
- Some false propositions signify as is the case. Insolubles do. Thus, where a is the insoluble ‘a is false’, a is self-falsifying and so false. But it signifies as is the case (namely, that it is false).
- In some valid formal inferences, falsehoods follow from truths. For consider the inference “The conclusion of this inference is false; therefore, the conclusion of this inference is false”. The premise and the conclusion of this inference are not only two tokens of the same type, but more importantly say the same thing about the same thing, so, Swyneshed claimed, the inference is a formally valid one. (Bradwardine and Buridan would both disagree.) But while the conclusion is a self-falsifying insoluble, and so is false, the premise is not self-falsifying, and is in fact true. (The conclusion of the inference is false, on Swyneshed’s account, because it falsifies itself.) Here then, a falsehood validly follows from a truth.
- In the case of insolubles, two mutually contradictory propositions are false at the same time. Where a = ‘a is false’, a is insoluble and false. But its contradictory, ‘a is not false’, Swyneshed claims, is not insoluble and is not self-falsifying. Nevertheless, it is false because it signifies otherwise than is the case. The insoluble a really is false.
Many authors found these conclusions ridiculous, especially the second and third ones. But they had their defenders as well.
Swyneshed explicitly considers a situation where a = ‘a does not signify as is the case’, and says that a is neither true nor false in that situation. This is the only known case of a medieval author’s actually allowing failure of bivalence for insolubles, even though several authors refer to (and reject) such theories.
3.4 William Heytesbury
In 1335, the Mertonian logician and philosopher of nature William Heytesbury wrote an important treatise entitled Rules for Solving Sophisms (Regulae solvendi sophismata). The first of its six chapters is on insolubles. The Rules as a whole, and this first chapter in particular, were widely read and commented on, particularly in Italy in the late-fourteenth and fifteenth centuries (see, e.g., Section 4.4 below). Indeed, Heytesbury’s theory is a competitor to Bradwardine’s as the most influential theory of insolubles in the whole of the Middle Ages.
Heytesbury treated insolubles as paradoxical only with respect to certain assumed circumstances (what he calls the casus or “scenario”). For example, the proposition ‘Socrates is uttering a falsehood’ is not paradoxical in the abstract, all by itself, but only in contexts where, say, it is Socrates who utters that proposition, the proposition is the only proposition Socrates utters (it is not an embedded quotation, for instance, part of some larger statement he is making), and where his proposition signifies just as it normally does. Spoken and written language are thoroughly arbitrary, for medieval authors, so that the vocal sequence or inscription ‘Socrates is uttering a falsehood’ could theoretically signify any way you want. It might, for example, signify that 2 + 2 = 4, in which case it would not be insoluble at all but straightforwardly true.
It is the last condition that is the focal point for Heytesbury’s attack. He holds that in the casus where Socrates himself says just ‘Socrates is uttering a falsehood’ and nothing else, his proposition cannot, on pain of contradiction, signify just as it normally does (precise sicut verba communiter pretendunt, as he puts it). If it does signify as it normally does, it must signify some other way as well.
How else might it signify? Heytesbury did not think it was up to him to answer that question, as Bradwardine had. The proposition’s additional signification cannot be predicted, given the arbitrariness of spoken and written language. Depending on what else it signifies, different verdicts about the proposition are appropriate. In short, Heytesbury’s strategy is to say,
You tell me exactly what Socrates’s statement signifies, and I’ll tell you first of all whether the case you describe is possible, and if it is, I’ll tell you whether his statement is true or false.
This “shift the burden” strategy is a consequence of the fact that Heytesbury, even more than Bradwardine and Swyneshed, views the question of insolubles in the context of obligationes, a highly formalized medieval disputation context that is a subject of much recent discussion. But Heytesbury shares one claim at least with Bradwardine: that no proposition, in a suitable scenario, can signify only and precisely that it itself is false. It must signify something more than just this. See Heytesbury [H-OI] §50 and Bradwardine [B-I] § A4.3.
3.5 The Modified Heytesbury Solution
Many later authors felt that Heytesbury had simply sidestepped the real theoretical issue, and went on to stipulate what Heytesbury would not: an insoluble’s “additional” signification. They held that, in circumstances that make it insoluble, a proposition not only signifies as it normally does; it also signifies that it is true. This “adjustment” to Heytesbury’s theory has the effect of combining it with the tradition stemming from Bradwardine. It proved to be an appealing combination. We might call this the modified Heytesbury solution.
That John of Holland subscribes to the modified view is not immediately obvious, but clear enough when his text is examined carefully. In response to a counter-argument, where A is ‘Socrates says a falsehood’, he writes (John of Holland 1985, p.130):
I deny the first inference, namely ‘A is a falsehood, and Socrates says A, therefore Socrates says a falsehood’, because it is a fallacy secundum quid et simpliciter. For the conclusion signifies many things conjunctively, namely that Socrates says a falsehood and something else (according to some people, viz that ‘Socrates says a falsehood’ is true).That parenthetical clause contains the modified Heytesbury solution. At this point, John has mentioned it but hasn’t yet committed himself to it. But in his response to the fourth counter-argument, he writes:
I reply that ‘Socrates does not say a falsehood’ is not the contradictory of the insoluble, because the insoluble is not a singular proposition. Hence it is not necessary that this proposition contradict it. But if one asks, what then is the contradictory of this insoluble, I say that it is ‘Not-(Socrates says a falsehood)’ signifying disjunctively that either Socrates does not say a falsehood or that ‘Socrates says a falsehood’ is not true. Then this disjunction is true if one of its parts is, namely: ‘Socrates says a falsehood’ is not true. And the reason is that from the fact that the insoluble signifies conjunctively that Socrates says a falsehood and that ‘Socrates says a falsehood’ is true (at least as many people say), it is necessary that its contradictory is a proposition in which the negation is preposed to the whole.That strongly suggests that John of Holland accepts the modified view, and stands significantly in contrast to Heytesbury’s view, which refuses to specify what the additional meaning is. But none of those who hold the modified view seems to offer any argument, as Bradwardine had done, to show that this additional signification is that the proposition itself is true.
3.6 John Dumbleton
John Dumbleton became Master of Arts at Oxford and Bachelor of Theology at Paris. He was listed as Fellow of Merton College in 1338 and was also a founding Fellow of Queens College in 1341. He studied theology at Paris in the mid-1340s, returning to Oxford around 1347: he was listed again at Merton in 1348. His Summa Logicae et Philosophiae Naturalis was left unfinished at his death, probably from the Black Death in 1349.
Dumbleton’s treatise on insolubles constitutes a very small part of his long treatise, running to some 400,000 words, entitled Summa Logicae et Philosophiae Naturalis. It is preserved in 21 manuscripts, though two omit the first part, on logic. Part I: Summa Logicae opens with a short treatise on signification. This is essential for underpinning and defending Dumbleton’s approach to the insolubles, for he argues for a version of cassatio: insolubles signify in such a way that they fail to express propositions. The reason for this failure, he says, is that an utterance, spoken or written, is a proposition only if it communicates something from the speaker to the hearer.
Dumbleton restricts the word ‘term’ to spoken sounds and written marks. Given that they don’t signify naturally, he says, we need
... to explain first in what manner written and spoken terms signify; secondly, in what manner they are imposed to signify; thirdly, in what manner a term is significative of any thing whatsoever at our pleasure; fourthly, how we come to know terms; and lastly in what manner any thing can be signified by any term.Dumbleton’s answer is that a term only signifies when and insofar as our understanding performs an act which consists in calling to mind or remembering the true intention of the thing by means of the proper intention of the term:
A term’s signifying to someone is nothing other than that term’s calling to mind the true intention of something else by means of the term’s proper intention.>
This analysis of the notion of signification leads Dumbleton to the first conclusion of his treatise on insolubles:
The first of the conclusions is this: any term signifies only while and insofar as there is actual comprehension by it. This conclusion is clear like this: no term signifies by its own nature, nor by any nature introduced in it by an impositor, but terms only make us comprehend other things; therefore, if a term signifies, it signifies only while and insofar as there is actual apprehension through it. Moreover, an intention only signifies while there is actual apprehension through it, but an intention signifies to us naturally-before and more than a term, and the signification of a term derives from the signification of the intention, therefore all the more so does a term only signify in so far as there is actual apprehension through it.
Applied to insolubles, the idea is that, although there is an utterance involved which signifies in some way, the order of priority results in a regress whereby its signification is not well founded and the utterance fails to constitute a proposition. So the scenario must be rejected in proposing that the utterance does constitute a proposition:
If one supposes that the proposition ‘This proposition is false’, or a similar one, is spoken, call it A, take what is exactly comprehended through its subject—about which the coupling is performed in order to produce the proposition—call that B. B is either proposition A or another proposition. Not another, because then there would not be a composition about the self-same proposition, which contradicts the hypothesis. If what is signified is A, then A is not the proposition ‘This proposition is false’, but this one ‘“This proposition is false” is false’. The inference holds like this: because A is a proposition equivalent to one where the subject refers to the proposition A itself, which signifies in some way about A—e.g., take the proposition: ‘This proposition exists’, call it C, referring to ‘God exists’. C is the same as the proposition ‘“God exists” exists’—so, in the scenario proposed, to the subject of the proposition ‘“This proposition is false” is false’ there would then correspond a further proposition and so on ad infinitum. And because there is successive reference ad infinitum, it follows that it is not fully comprehended.The solution presented by Dumbleton here is highly reminiscent of that put forward in Ryle 1951, sometimes referred to as the “namely-rider” solution. The upshot is that Dumbleton’s solution to the insolubles is part of a broader analysis of linguistic understanding and communication. Indeed, not only do insolubles fail to constitute propositions, so too do many other expressions.
3.7 Gregory of Rimini
Gregory of Rimini’s main writing was done in Paris in the 1340s. Although today we know of no text or passage of his that discusses insolubles, there must have been one, because in 1372 Peter of Ailly cites Gregory’s theory in some detail and uses it in writing his own treatise on insolubles (see Peter of Ailly [P-CI], and Section 4.3 below).
Gregory’s view relies on the traditional medieval notion (going back to Aristotle’s On Interpretation 1, 16a3–5) of “mental language”, the “language of thought” that underlies and is expressed in spoken and written language. Unlike spoken and written languages, where the signification of words and propositions is thoroughly a matter of convention, signification in mental language is fixed by nature once and for all, the same for everyone. Consequently, propositions in mental language can never signify otherwise than they “normally” do. It follows that an analysis like Bradwardine’s or Heytesbury’s, or the modified Heytesbury solution, according to which insolubles do signify otherwise than they normally do, cannot be applied to propositions formed in mental language. Although neither Bradwardine nor Heytesbury seems to have drawn this conclusion, it follows from their theories that insolubles cannot be formulated in mental language.
In the absence of any text by Gregory on the topic, we cannot be sure that he reasoned like this from Heytesbury’s position. But for whatever reason, he apparently did confine insolubles to spoken and written language; for Gregory there are no insolubles in mental language. An insoluble proposition in spoken or written language corresponds to and expresses not the mental proposition one would normally expect on the basis of the usual linguistic conventions, but a complex and non-paradoxical mental proposition.
For example, where a is the spoken or written proposition ‘a is false’, a corresponds to and expresses the conjunction of two mental propositions. The first conjunct signifies that a is false. Note that this mental proposition is not the insoluble a, since that was in spoken or written language whereas this proposition is mental. Unlike a, this proposition is not self-referential; it refers instead to a.
The second conjunct signifies that the first conjunct is false. Since the first conjunct signifies that a is false, this means that the second conjunct amounts to saying that a is not false, but rather true.
One way, therefore, of viewing Gregory’s theory is to say that he adopted the modified Heytesbury view described in Section 3.5 above, the view that combines Heytesbury with Bradwardine, but that Gregory then moved that whole analysis into mental language. As in Heytesbury’s theory, insolubles for Gregory do not signify just as their words normally do—they do not express the mental proposition one would expect from the normal linguistic conventions. As in Bradwardine’s theory, insolubles for Gregory do signify partly that way (through the first conjunct of the mental proposition). But they do not signify precisely that way; they also signify that they are true (through the second conjunct of the mental proposition).
Given our present knowledge of Gregory’s views, this reconstruction must remain speculative.
3.8 John Buridan
John Buridan was another logician of the 14th century holding a theory of the insolubles similar to Bradwardine’s and the modified Heytesbury solution. However, there may have been no direct influence. Like Rimini, Buridan also taught at the University of Paris, but unlike Rimini, rather unusually remained a teaching master in the Arts Faculty for his entire career, from the 1320s until at least 1358. We can trace the development of his approach to the insolubles from his early Quaestiones Elencorum, dating from the 1330s, and Treatise on Consequences, through his commentaries on Aristotle’s Posterior Analytics, Sophistical Refutations, and Metaphysics, to the various treatises of the Summulae de Dialectica, which was repeatedly revised over twenty years. His final view is described in the ninth and last treatise of the Summulae, with the independent title Sophismata, in a version from the mid-1350s (see Pironet 1993).
Buridan’s early view was that every proposition, not just insolubles, signifies its own truth. This idea, which we have already encountered in Segrave and the modified Heytesbury solution, can be found as early as Bonaventure’s Quaestiones disputatae de mysterio Trinitatis (q1 a1), composed in the 1250s. Buridan gives only the briefest of arguments for this claim about signification, not grounding it in any principle like Bradwardine’s (P2). He writes, in his early work Questiones Elencorum (Buridan 1994, p.92):
For every proposition is affirmative or negative. But each of them signifies itself to be true or at least from each it follows that it is true. This is clear first concerning affirmatives, for every affirmative proposition signifies that its subject and predicate supposit for the same, and this is for it to be true ... Secondly, it is clear concerning negatives, for a negative does not signify that the subject and predicate supposit for the same, and this is for the negative proposition to be true.Thus insolubles, which signify their own falsity, signify that they are both true and false, and so are false. Buridan’s view on insolubles may have developed out of Girard of Odo’s, whose Logica was composed in Paris in the early 1320s. Girard’s idea was that the Liar paradox, ‘I say something false’, has four things wrong with it (malitiae), the fourth being that, since it is affirmative, it asserts the unity of its subject with its predicate, but its predicate (‘something false’) denies this (see Giraldus Odonis Logica, 396–8).
However, as early as the Treatise on Consequences, Buridan thinks this final step, concluding that insolubles are false because they signify that they are both true and false, needs qualification. For he rejects the idea that a proposition is true if things are how it signifies, even however it signifies. Things can be, e.g., how ‘No proposition is negative’ signifies, but it cannot be true (since it is itself negative, and thus falsifies itself when formed). Rather, affirmative propositions are true if their terms supposit for the same, negative if they supposit for different things. Indeed, he later rejects the suggestion that propositions signify their own truth. He did so both for ontological reasons, since that would require some kind of propositional meanings (the famous complexe significabilia—see, e.g., Klima 2009: §10.2); and also because that would make every proposition meta-linguistic. Rather, his later theory claims that every proposition virtually implies another proposition asserting the truth of the first. Then an insoluble like ‘Socrates utters a falsehood’, uttered by Socrates, is false not because its terms don’t supposit for the same, but because the terms in the implied proposition can’t also do so. Buridan’s solution has been much discussed in recent decades and has been edited and translated several times (see Buridan [B-S], [B-S2], [B-B], [B-SD], [B-SD2]), but it is deeply problematic (see, e.g., Read 2002: §5; Read 2006: §6; with a response on Buridan’s behalf in Klima 2009, §10.5).
3.9 Albert of Saxony
Much closer to Bradwardine’s theory than to Buridan’s is that of another Paris logician, Albert of Saxony, who arrived in Paris sometime before 1351 and taught there until around 1362. His view of the insolubles is similar to Buridan’s early view, arguing in a similar way that every proposition signifies its own truth. But there is reason to doubt whether Albert was a student or even follower of Buridan, for Buridan belonged to the Picardian Nation at the University, whereas Albert was in the English (or by then “Anglo-German”) Nation, and in general his outlook follows the English logical tradition from earlier in the century. What is perhaps most impressive, and enjoyable, about Albert’s treatise on Insolubles (Albert of Saxony, [AS-I]), the first part of the sixth treatise of his Perutilis Logica (A Really Useful Logic), is the extensive list of insolubles treated, and their variety. For example, we find there a paradox much discussed in the recent literature on paradoxes under the title ‘V-Curry’, closely related to Curry’s paradox. In fact, in its contemporary form it appears in Dumbleton’s discussion of the insolubles (see above, Section 3.6) and in Heytesbury’s Sophismata Asinina ([H-SA}: sophism 18, p. 413): consider the inference with sole premise ‘This inference is valid’ and conclusion ‘A man is an ass’. Albert contraposes it, inferring ‘This inference is invalid’ from the premise ‘God exists’. The text translated in “Insolubles” ([AS-I]: XIV, p. 368) follows the Venice 1522 edition. As given in [AS-L] (p. 1158), following the manuscripts, the text should read:
Let this inference be A, its premise [‘God exists’] B, and its conclusion [‘This inference is invalid’] C, and let ‘this’ indicate the inference itself. Then I put forward inference A, and I ask whether inference A is valid or not.
[i] If one says that it is valid, then, since its premise is true, it follows that its conclusion is true. And if its conclusion is true, then things are as its conclusion signifies. But the conclusion signifies that inference A is not valid. Therefore, inference A is not valid.
[ii] But if one says that inference A is not valid—on the contrary: If inference A is not valid, it is possible that B be true while C is false. But that is false, which I prove as follows. For if A is not valid, then C is true, and so A is valid, since B is not true unless C is true. So if A is not valid, A is valid. The premise [‘If A is not valid then C is true’] is clear, for if A is not valid, things are as C signifies, because C signifies that A is not valid; and, consequently, C is true. The first inference is evident, for in order that A be valid it is enough that B cannot be true without C, if it is formulated. The last inference holds “from the first to the last”.
In a nutshell, Albert’s argument in [ii] is that if A is invalid, it must be possible for B to be true and C false. But if A is invalid, C is true. So it is impossible for B to be true and C false. So even if A is invalid, it is valid, and so A is valid. But by [i], if A is valid it is invalid. So it is both—paradox.
Dumbleton’s response to the paradox was to deny that the premise (of his version, ‘This inference is valid’) constitutes a proposition, given the regressive reference of its subject (‘this inference’—which inference?—‘this inference’, …; see 3.6 John Dumbleton). Albert’s response, however, in line with his general approach to the insolubles, is to agree that his inverted inference is indeed invalid, but to deny that it follows that C (‘This inference is invalid’) is true, for C signifies more than just that A is invalid. Thus the premise (‘God exists’) can be true without the conclusion being true, and so the inference really is invalid.
4. The Late Period
The period of greatest innovation and sophistication in the medieval insolubilia-literature seems to have been the second quarter of the fourteenth century. After about 1350, less original work is known. Insolubles continued to be discussed, but it seems that for the most part the theories adopted were variations or elaborations of the ones already seen. Paul of Venice (1499), writing in about 1396–7, in the final section of his Logica Magna, lists fifteen theories in addition to his own, supplementing Bradwardine’s list of nine with later developments, but most from before 1350 (see Spade 1973: 82–4). This later period is not yet fully researched, however, so it is too early for a clear verdict.
4.1 Marsilius of Inghen
Spade (1980: 6) suggests the Marsilius’s solution to the insolubles derives, like Gregory’s, from Bradwardine’s. It is actually more similar to the modified Heytesbury solution and to Albert of Saxony’s. Marsilius’s treatise was written in Paris, probably in the 1360s, where Marsilius was in the English Nation. (He may even have overlapped with Albert, before Albert left Paris in 1362.) Marsilius’s main idea is that every proposition has two significations, a material signification, which is what we take as its normal or ordinary meaning, about things, and a formal signification, which is meta-linguistic, affirmative propositions signifying that their subject and predicate supposit for the same thing, negatives that they don’t. The consequence is that insolubles falsify themselves. Take ‘This proposition is false’, for example. Its material signification contradicts its formal signification, for its material signification is that it’s false, that is, that its predicate does not apply to its subject (that is, to itself), whereas, being affirmative, its formal signification is that its predicate does apply to its subject. So, Marsilius concludes, every proposition signifying itself to be false falsifies itself—in being self-contradictory, not in Swyneshed’s sense of that phrase.
But Marsilius rejects the rather simple-minded proof that we noted Buridan and Albert give that every proposition asserts or signifies its own truth. For that proof turned on the assumption that every affirmative (resp. negative) proposition is true if its subject and predicate supposit (resp. don’t supposit) for the same thing, by the definition of truth. But insolubles are an immediate counterexample to this. As just noted, ‘This proposition is false’ is false (since it falsifies itself), and so its subject and predicate do supposit for the same thing. Indeed, ‘This is false’ says that ‘This is false’ is false (rather in the way Gregory’s second mental proposition said that the first mental proposition was false) and so, in saying that it’s false that it’s false, its formal signification is that it’s true. It’s implicitly contradictory and so false.
4.2 John Wyclif
Perhaps one of the genuinely new theories to emerge from this late period is that of John Wyclif, who wrote a Summary of Insolubles (Summa insolubilium),  probably in the early 1360s, and included another discussion of insolubles in his Continuation of the Logic (Logicae continuatio), III.8. The theory is essentially the same in the two treatments.
For Wyclif, the key to resolving insolubles is to recognize various senses in which propositions can be true or false. There are three main senses of ‘true’, and accordingly of ‘false’:
- In the transcendental sense, truth is convertible with being, so that any proposition is true in this sense, no matter what it signifies. This sense can be disregarded in discussing insolubles. Nothing (that is, no being) is false in the sense that it fails to be true in this sense.
- In a second sense, a proposition is true if and only if what it “primarily signifies” exists. These “primary significates” are neither substances nor accidents, but rather “beings of reason”. It is perhaps plausible to interpret an existing primary significate as analogous to a “fact” in the modern philosophical sense. A proposition is false in this second sense if and only if its primary significate fails to exist.
- In a third sense, a proposition is true if and only if what it primarily signifies exists and is independent of the proposition itself. It is false in this third sense if and only if its primary significate either fails to exist or else exists but depends on the proposition itself.
The “independence” required by the third kind of truth is an obscure and difficult matter. But here is how it applies to insolubles:
Where a = ‘a is false’, its primary significate either exists or does not. If it does, then in any event it is not independent of a in the sense required by the third kind of truth. In either case, then, a will be false in the third sense. If the word ‘false’ in a is taken in the second sense, therefore, a’s primary significate does exist, since it is a fact that a is false in the third sense. In short, the insoluble is true in the second sense, but false in the third sense.
Our present understanding of Wyclif’s theory does not go much beyond this. Many questions and problems remain. For instance, if the word ‘false’ in a is not taken in the third sense but in the second, the paradox seems to emerge all over again in a form that cannot be handled by this theory.
Whatever its virtues or defects, Wyclif’s theory had some influence on later authors. Robert Alyngton’s own Insolubilia, for instance, from around 1380, explicitly appeals to Wyclif’s theory. Its influence can also be seen in Peter of Mantua’s account (see Section 4.4 below) and in an anonymous late treatise preserved in a Prague manuscript (see Wyclif [W-SI]: xxiv–xxv.)
4.3 Peter of Ailly
As already mentioned (Section 3.7 above), in 1372 the Frenchman Peter of Ailly (Petrus de Alliaco) wrote an Insolubilia that preserves all we know of Gregory of Rimini’s theory. Peter’s theory looks much like Gregory’s. Nevertheless, he did not accept Gregory’s view entirely. Whereas for Gregory, an insoluble in spoken or written language corresponds to or expresses a conjunction of two propositions in mental language, for Peter it corresponds to or expresses two distinct mental propositions, not their conjunction. The two distinct mental propositions are the same two that Gregory had conjoined.
In medieval semantics, propositions that correspond to two distinct mental propositions are ambiguous or equivocal. (Indeed, that is the medieval account of equivocation.) Thus, for Peter, insolubles in spoken or written language are strictly equivocal and do not have a single signification. In one sense (answering to the first of Gregory’s conjuncts), they are true; in another sense (answering to Gregory’s second conjunct), they are false. By contrast, for Gregory, insolubles are just false, not ambiguous at all; they correspond to a single false conjunction, one conjunct of which is true and the other false.
Peter’s theory has the phenomenological advantage that it accounts for the psychological “flip-flop” sense we have when thinking about insolubles. When we look at them one way they seem true; when we look at them another way, they seem false. No other medieval theory seems to account for this psychological fact. For further discussion, see Dutilh Novaes 2008a: §3.8.
4.4 Peter of Mantua
Strobino 2012 contains the first significant discussion of Peter of Mantua’s account of insolubles in modern times. Mantua’s treatise, composed in the early 1390s, shows influences from Albert of Saxony and William Heytesbury (whose opinions he criticises extensively) and of Wyclif’s theory. Mantua’s theory is also mentioned in Paul of Venice’s Logica Magna, in addition to the fifteen theories Paul specifically enumerates. Once again, the theory frames itself in Aristotelian terms, whereby the insolubles are absolutely or unqualifiedly false, but true in a certain respect. Like Wyclif’s, his main notion of truth (Wyclif’s third, Mantua’s second) requires that propositions not be self-referential. If they are self-referential, they are false in this sense. But Mantua’s other notion (similar to but narrower than Wyclif’s second) applies only to self-referential propositions that are true according to their primary signification. For example, ‘This proposition is not true’ is false in the first sense (since it is self-referential), but true in the second sense, because it is self-referential and not true (in the first sense). Strobino (§ 3.4) argues that Mantua’s theory, like Wyclif’s, cannot deal with the problem of revenge, e.g., with such a proposition as ‘This proposition is not true in either sense’.
4.5 Paul of Venice
The authenticity of the Logica Magna is a matter of some contention. Assuming that Paul was the author, we can find discussions by him of the insolubles in at least four works, his Logica Parva (translated in Paul of Venice 1984 and edited in Paul of Venice 2002), his Logica Magna (in the treatise on ‘Insolubles’, in a forthcoming edition and English translation by Barbara Bartocci and Stephen Read from Paul of Venice 1499 and the one extant manuscript), in the Quadratura (in at least four of its 200 chapters, Paul of Venice 1493), and in the final sophism, no.50, in his Sophismata Aurea (Paul of Venice 1483). Paul follows two of the leading solutions to the insolubles in the fourteenth century, the modified Heytesbury solution and Roger Swyneshed’s, in different works: the modified Heytesbury solution in the Logica Parva and the Quadratura, and that of Swyneshed in the Logica Magna and the Sophismata Aurea.[31a]
In the Logica Parva, an elementary pedagogical text, Paul clearly follows Heytesbury in accepting an insoluble scenario only if it is left open that the insoluble proposition has a secondary or additional signification, but (unlike Heytesbury) specifies that additional signification as asserting its own truth. He spells it out towards the end of ch.6:
It should be noted that an insoluble has two significates, one exact (adequatum) and one principal. The exact significate is a subject-predicate significate similar to the insoluble utterance. E.g., the exact significate of ‘Socrates says a falsehood’ is Socrates saying a falsehood or that Socrates says a falsehood. But the principal significate is a compound significate, e.g., that Socrates says a falsehood and that the proposition is true.However, that the solution Paul presents here is not his own, that is, not his preferred solution, is made clear at the end of the chapter, where Paul writes:
Notice that not everything I have said here, or in other treatises, have I said according to my own view, but partly according to the view of others, in order to enable young beginners to progress more easily.Paul’s own view, then, is more likely that which he presents in his major treatment of the insolubles, in his Logica Magna.
Paul devotes the whole of the first chapter of his treatise on insolubles in the Logica Magna to the rejection of fifteen other proposed solutions. The treatment can be seen as falling into four groups, the first three groups corresponding to three sources on which Paul draws: first, he runs rapidly through seven of the eight alternative solutions considered by Bradwardine in his treatise, for the most part summarising almost verbatim Bradwardine’s own criticism. He then turns to Heytesbury’s criticism of alternative solutions, starting with the second of the four solutions considered by Heytesbury, that of John Dumbleton. (The first one Heytesbury rejects is Swyneshed’s, which Paul will himself accept.) The next (third on Heytesbury’s list) is Kilvington’s, and then Paul comes to Bradwardine’s own solution. Thus the first ten solutions considered are all from Oxford, or at least, those discussed at Oxford in the two decades from the early 1320s to the early 1340s. With the eleventh solution, Paul turns to his third source, namely, Peter of Mantua, the eleventh being Albert of Saxony’s solution, presented at Paris in the early 1350s, and possibly the same as John Buridan’s own early solution. (However, Marsilius of Inghen, whose solution is not discussed by Peter or Paul, appears to have distinguished Albert’s solution from Buridan’s early view: see Spade 1975, 79.) Albert’s is the first view discussed by Mantua. Next comes Heytesbury’s (Mantua’s second) solution. Before proceeding to the third solution discussed by Mantua, Paul considers Peter of Ailly’s solution at some length, seemingly drawing directly on Ailly’s own treatise, to the discussion of which Paul appends (without distinguishing it by number) a criticism of Mantua’s solution. Finally, Paul turns to a rejection of restrictivism, the first solution rejected by Bradwardine and the third by Mantua. But Paul deals with the specific form given to it by Walter Segrave (see Section 3.2).
After this extended discussion of alternatives (occupying a quarter of his treatise), Paul sets out to develop his own solution, based firmly on Roger Swyneshed’s proposals from the 1330s. In his second chapter, he systematically lays out his distinctions (divisiones) and assumptions (suppositiones), then draws seven Conclusions and seven Corollaries. The basic idea is Swyneshed’s, namely, to provide a solution which does not depend on postulating tacit or hidden or consequential significates for insoluble propositions beyond what is clearly shown---what they standardly suggest or indicate by the straightforward combination of their parts (in Heytesbury’s phrase, “ sicut termini communiter pretendunt”). Instead, as Swyneshed had proposed, Paul tightens the criterion for truth, to exclude those that falsify themselves, weakening the criterion for falsehood to admit those examples that do falsify themselves even if otherwise impeccable. Roger’s second and third notorious Conclusions reappear as Paul’s fifth and second respectively. Paul will later describe the second Conclusion as a fundamental principle:
I deny the inference: ‘The contradictory of A is false, therefore A is true,’ since in the case of insolubles I uphold as a fundamental principle that two mutually contradictory propositions may both be false.Consequently, Paul defines an insoluble as a self-falsifying proposition:
An insoluble proposition is a proposition having reflection on itself wholly or partially implying its own falsity or that it is not itself true, such as ‘This is false,’ ‘This is not true,’ referring to themselves, and ‘This is true,’ referring to its contradictory. And so on for many other examples which can be identified on the basis of what has been said. Therefore insolubles are self-falsifying propositions.
In the rest of the treatise, Paul considers, in ch.3, the objections which Heytesbury had directed at Swyneshed’s solution, many of which had already been addressed in discussing the Conclusions and Corollaries in ch.2; shows at some length, in ch.4, how his solution deals with the much-discussed example where Socrates only says ‘Socrates says a falsehood’; extends the account, in ch.5, to deal with other examples, such as that where Socrates says that his sole business is to be hung on the gallows, which are not obviously insolubles until the background scenario is added; in ch.6 to examples like ‘A falsehood exists’ which become insoluble in a suitable scenario; and in ch.7 to examples involving exclusive and exceptive propositions, such as ‘Only a false proposition is exclusive’ (assuming it is the only exclusive proposition) and ‘No proposition except A is false’ (where this is A and is the only exceptive proposition).
Several instructive observations can be made about the medieval insolubilia-literature.
First, although this article has focused on Liar-type paradoxes, and although the medieval literature did too, it also included other kinds of puzzles. For example (the ‘no’-‘no’ paradox), where a = ‘b is false’ and b = ‘a is false’, no Liar-type paradox arises; contradiction can be avoided by simply taking one of the two propositions as true and the other as false. But medieval logicians regarded such cases as problematic because they require us to assign different truth values to propositions that are semantically exactly alike; there is no reason to pick a as the true proposition rather than b or conversely (see Read 2006). Cases like this, which violate only a kind of semantic “principle of sufficient reason”, were often included under the heading “insolubles” (for example, Buridan, Sophismata VIII.8). A variety of epistemic and pragmatic puzzles were often included as well. There was rarely any attempt, as is usual in present-day literature on the paradoxes, to ignore all the inessentials and focus in on a single paradigmatic case that gets at the kernel of the issue. For medieval authors, the issue was a broad one. Many did not attempt to give any precise and rigorous characterization of what it takes to be an insoluble. Often, the definitions they did give are quite general and include much more than Liar-type paradoxes. In contrast, Bradwardine’s definition is precise: an insoluble is “a difficult paralogism secundum quid et simpliciter arising from some [speech-] act’s reflection on itself with a privative determination” (Bradwardine [B-I]: §2.1).
Secondly, medieval authors did not have any sense of theoretical “crisis” over insolubles, as modern discussions of the paradoxes often do. The medievals did not regard the paradoxes as threatening the very foundations of reasoning. On the contrary, most authors seem to have regarded them as merely argumentative nuisances, and their main concern was to come up with a way of dealing with them when they arose in disputation. No doubt this difference from the reaction to the logical paradoxes in modern logic is due to the different contexts in which the discussions emerged. Modern logic is a formalized, systematic discipline, closely tied to the foundations of mathematics; medieval logic, by contrast, was much looser and informal (which of course is not to say it lacked insight), much more tied to the give and take of live academic disputation.
Thirdly, and related to the second point, most medieval authors thought it was entirely feasible to find a completely satisfactory “solution” to insolubles. Insolubles were regarded as resting on a straightforward but pernicious fallacy, although authors disagreed over just what the fallacy is. William of Ockham, for instance, writes,
As for insolubles, you should know it is not because they can in no way be solved that some sophisms are called insolubles, but because they are solved with difficulty. (Ockham [O-SL]: III-3, 46)
One medieval author who is known to have departed from this confident view is William Heytesbury, who raises objections against his own view, and then remarks (Heytesbury [H-OI]: 45, emphasis added):
Many objections of this sort can be raised against this view, which it would be difficult or impossible to answer to complete satisfaction.
Again, about his own view he says (p. 21, emphasis added):
I do not claim that it or any [opinion] is altogether satisfactory, because I do not see that this is possible. Nevertheless I rate this one among all of them to be nearer the truth.
Richard Lavenham, an English contemporary of Wyclif, perhaps put the prevailing optimism best (Spade 1975: 93; Heytesbury [H-OI]: 8):
Just as the bond of love is sometimes called insoluble, not because it can in no way be untied (sit solubilis) but because it can be untied [only] with difficulty, so a proposition is sometimes called insoluble, not because it is not solvable but because it is solvable [only] with difficulty.
Primary Literature in Translation
- Albert of Saxony [Albert von Sachsen], [AS-I], “Insolubles”, in The Cambridge Translations of Medieval Philosophical Texts. Volume I: Logic and the Philosophy of Language, Norman Kretzmann and Eleonore Stump (trans.), Cambridge: Cambridge University Press, 1988, pp. 337–68. Translated from the early printed text of Albert’s Perutilis Logica, Venice 1522, reprinted Hildesheim: Olms 1974. doi:10.1017/CBO9781139171557.013
- –––, [AS-L], Logik: Lateinisch-Deutsch, Harald Berger (ed. and trans.), Hamburg: Felix Meiner Verlag, 2010. Critical edition of Albert’s Perutilis Logica from the manuscripts, with German translation.
- Anonymous, Treatise on Insolubles, in Peter of Spain: Tractatus syncategorematum and Selected Anonymous Treatises, (Mediaeval Philosophical Texts in Translation, vol. 13), Joseph P. Mullally (trans.) Milwaukee, WI: Marquette University Press, 1964, pp. 335–9. Among the “anonymous treatises” translated at the end of this volume, this is a late (probably fifteenth century) treatise on insolubles.
- Aristotle, On Interpretation [De Interpretatione], E.M. Edghill (trans.), in The Works of Aristotle, Sir David Ross (ed.), Cambridge: Cambridge University Press, 1928.
- –––, [A-SR], Sophistical Refutations [De Sophisticis Elenchis], W.A. Pickard-Cambridge (trans.), in The Works of Aristotle, Sir David Ross (ed.), Cambridge: Cambridge University Press, 1928.
- Athenaeus of Naucratis, The Deipnosophists, 7 volumes, (Loeb Classical Library), Charles Burton Gulick (trans.), Cambridge, MA: Harvard University Press, 1927–41.
- Bonaventure, “Quaestiones disputatae de mysterio Trinitatis”, in Medieval Philosophy: From St. Augustine to Nicholas of Cusa, (Readings in the history of philosophy), John F. Wippel and Allan B. Wolter (trans.), New York: Free Press, 1969.
- Bradwardine, Thomas, [B-I], Insolubilia, (Dallas Medieval Texts and Translations 10), Stephen Read (ed. and trans.), Leuven: Peeters, 2010.
- Buridan, John, [B-S], Sophisms on Meaning and Truth, Theodore Kermit Scott (trans.), New York: Appleton-Century-Crofts, 1966. A translation of Buridan’s Sophismata, based on the edition published in Buridan [B-S2]. Chap. 8 is on insolubles.
- –––, [B-B], John Buridan on Self-Reference, G.E. Hughes (ed. and trans.), Cambridge: Cambridge University Press, 1982. A translation, with philosophical commentary, of Chap. 8 of Buridan’s Sophismata, on insolubles. [Note: There are two versions of this book, with different pagination. The paperbound publication has the subtitle: Chapter Eight of Buridan’s “Sophismata”, translated with an Introduction and a philosophical Commentary. The hardbound publication includes a Latin edition, and has the slightly different subtitle: Chapter Eight of Buridan’s “Sophismata”, with a Translation, an Introduction, and a philosophical Commentary.]
- –––, [B-SD], Summulae de Dialectica, (Yale Library of Medieval Philosophy), Gyula Klima (trans.), New Haven, CT: Yale University Press, 2001. Treatise 9 of the Summulae de Dialectica is Buridan’s Sophismata. Treatise 9, Chap. 8 is on insolubles—see Buridan [B-S] and [B-B].
- Cervantes Saavandra, Miguel de, The Adventures of Don Quixote de la Mancha, Tobias Smollett (trans.), London: Deutsch, 1986. The original two volumes were published in 1605 and 1615.
- Heytesbury, William, [H-OI], On “Insoluble” Sentences: Chapter One of His Rules for Solving Sophisms, (Mediaeval Sources in Translation, 21), Paul Vincent Spade (trans.), Toronto: Pontifical Institute of Mediaeval Studies, 1979. Translation of chapter 1 of Regulae solvendi sophismata, dated 1335.
- ––– [Guillaume Heytesbury], [H-SA], Sophismata Asinina, présentation, édition critique et analyse par F. Pironet, Paris: Vrin, 1994.
- Paul of Venice, Logica Parva, Alan Perreiah (trans.), Munich: Philosophia Verlag, 1984.
- Peter of Ailly, Concepts and Insolubles: An Annotated Translation, (Synthese Historical Library, 19), Paul Vincent Spade (trans.), Dordrecht: D. Reidel, 1980. Translation of Conceptus et Insolubilia dated 1372.
Primary Literature in the Original Languages
- Adam of Balsham, [AB], Adam Balsamiensis Parvipontani Ars Disserendi (Dialectica Alexandri), in Lorenzo Minio-Paluello (ed.) Twelfth Century Logic: Texts and Studies, vol. 1, Rome: Edizioni di storia e letteratura, 1956.
- Albert von Sachsen [Albert of Saxony], Logik, Harald Berger (ed. and trans.), Hamburg: Felix Meiner Verlag, 2010. (Critical edition of Albert’s Perutilis Logica from the manuscripts, with German translation.) Same as [AS-L] above.
- Anonymous, Insolubilia, in Paul Vincent Spade, “An Anonymous Tract on Insolubilia from Ms Vat. lat. 674: An Edition and Analysis of the Text”, Vivarium, 9: 1–18, 1971. The text comes from 1368. doi:10.1163/156853471X00019
- Augustine, Contra academicos, De beata vita, De ordine, De magistro, De libero arbitrio, (Corpus Christianorum Series Latina, 29), W.M. Green and K.-D. Daur (eds.), Turnholt: Brepols, 1970.
- Bradwardine, Thomas, Insolubilia, (Dallas Medieval Texts and Translations 10), Stephen Read (ed. and trans.), Leuven: Peeters, 2010. Same as [B-I] above.
- Bricot, Thomas, Tractatus insolubilium, (Artistarium, 6), E.J. Ashworth (ed.) Nijmegen: Ingenium, 1986. Bricot was a very late fifteenth century author.
- Buridan, John, Questiones Elencorum, R. van der Lecq and H.A.G. Braakhuis (eds.), Nijmegen: Ingenium, 1994.
- –––, [B-S2], Sophismata, (Grammatica Speculativa, vol. 1), T.K. Scott (ed.), Stutgart-Bad Cannstatt: Frommann, 1977. The Latin edition that formed the basis for Scott’s translation in Buridan [B-S].
- –––, [B-SD2], Summulae: De Practica Sophismatum, Fabienne Pironet (ed.), Turnhout: Brepols, 2004. Critical edition of the Latin text. Translated in Buridan [B-SD], Treatise 9.
- Cicero, Academica, James S. Reid (ed.), Hildesheim: Georg Olms, 1966. (Photoreprint of the London 1885 edition.)
- Duns Scotus, John, [DS-Q], Quaestiones super librum elenchorum, in his Opera philosophica Vol. 2. Quaestiones in Libros Perihermenias Aristotelis , Opus & Opus II; Quaestiones Super Librum Elenchorum Aristotelis; and Theoremata, R. Andrews et al. (eds.), St. Bonaventure, NY: The Franciscan Institute; Washington: The Catholic University of America, 2004.
- Fland, Robert, Insolubilia, in Paul Vincent Spade, 1978, “Robert Fland’s Insolubilia: An Edition, with Comments on the Dating of Fland’s Works”, Mediaeval Studies, 40: 56–80. Read and Thakkar (2017) argue that the author’s real name was Robert Eland. He wrote between 1335 and about 1360. doi:10.1484/J.MS.2.306221
- Gellius, Aulus, Noctes Atticae (Attic Nights), 2 volumes, (Oxford Classical Texts), P.K. Marshall (ed.), Oxford: Clarendon Press, 1968. doi:10.1093/actrade/9780198146520.book.1 doi:10.1093/actrade/9780198146513.book.1
- Giraldus Odonis, Logica, volume 1 of L.M. De Rijk (ed.), Opera Philosophica, Leiden: Brill, 1997.
- Heytesbury, William [Guillaume Heytesbury], Sophismata Asinina, présentation, édition critique et analyse par F. Pironet, Paris: Vrin, 1994. (Same as [H-SA], above)
- John of Holland, Four Tracts on Logic: Suppositiones, Fallacie, Obligationes, Insolubilia, (Artistarium, 5), E.P. Bos (ed.), Nijmegen: Ingenium, 1985. Includes a Latin edition of a widely read text on insolubles.
- Laertius, Diogenes, Vitae philosophorum (Lives of the Philosophers), 2 volumes, (Oxford Classical Texts), H.S. Long (ed.), Oxford: Clarendon Press, 1964.
- Neckham, Alexander, [N-NR], De naturis rerum libri duo, with the Poem of the Same Author, De laudibus divinae sapientiae, (Rerum Britannicarum Medii Aevi Scriptores, or Chronicles and Memorials of Great Britain and Ireland during the Middle Ages [Rolls Series], 34), Thomas Wright (ed.), London: Longman, Green, 1863. Reprint Kraus reprints, 1967.
- Ockham, William of [Guillelmus de Ockham], [O-SL], Summa logicae, Gedeon Gál et al. (ed.), St. Bonaventure, NY: The Franciscan Institute, 1974.
- Ockham, William of [Guillelmus de Ockham], Expositio super libros Elenchorum, Franciscus del Punta (ed.), St. Bonaventure, NY: The Franciscan Institute, 1979.
- Paul of Venice, Sophismata Aurea, Pavia: Nicolaus Girardengus de Novis, 1483.
- Paul of Venice, Quadratura, Venice: Bonetus Locatellus for Octavianus Scotus, 1493.
- Paul of Venice, Logica Magna, Venice: Albertinus Vercellensis for Octavianus Scotus, 1499.
- Paul of Venice, Logica Parva, Alan Perreiah (ed.), Leiden: Brill, 2002.
- Roure, Marie Louise, 1970, “La problématique des propositions insolubles au XIIIe siècle et au début du XIVe, suivie de l’édition des traités de W. Shyreswood, W. Burleigh et Th. Bradwardine”, Archives d’histoire doctrinale et littéraire du moyen age, 37: 205–326. Latin editions of the treatises by Walter Burley and Thomas Bradwardine, as well as of a thirteenth century treatise sometimes attributed to William of Sherwood. Roure’s text of Bradwardine’s treatise is unreliable. See Bradwardine [B-I].
- Seneca, [Epistle], Ad Lucilium Epistulae Morales, 2 volumes, (Oxford Classical Texts), L.D. Reynolds (ed.), Oxford: Clarendon Press, 1965.
- Swyneshed, Roger, Insolubilia, in Paul Vincent Spade, 1979, “Roger Swyneshed’s Insolubilia: Edition and Comments”, Archives d’histoire doctrinale et littéraire du moyen age, 46: 177–220. Reprinted in Spade 1988.
- Wyclif, John, Tractatus de logica, 3 volumes, Michael Henry Dziewicki (ed.), London: Trübner & Co., for the Wyclif Society, 1893–99. Includes his Logicae continuatio.
- –––, [W-SI], Summa insolubilium, (Medieval and Renaissance Texts & Studies, 41), Paul Vincent Spade and Gordon Anthony Wilson (ed.), Binghamton, NY: Medieval & Renaissance Texts & Studies, 1984.
- Alwishah, Ahmed and David Sanson, 2009, “The Early Arabic Liar: the Liar Paradox in the Islamic World from the Mid-Ninth to the Mid-Thirteenth Centuries CE”, Vivarium, 47(1): 97–127. doi:10.1163/156853408X3459090909
- Bochenski, I.M., 1962, A History of Formal Logic, Ivo Thomas (trans.), New York: Chelsea Pub.Co.
- Bottin, Francesco, 1976, Le antinomie semantiche nella logica medievale, Padua: Antonore.
- –––, 1983, “The Mertonians’ Metalinguistic Science and the Insolubilia”, in The Rise of British Logic, (Papers in Mediaeval Studies, 7), P. Osmund Lewry (ed.), Toronto: Pontifical Institute of Mediaeval Studies, pp. 235–48.
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- De Rijk, L.M., 1962–67, Logica Modernorum: A
Contribution to the History of Early Terminist Logic, Assen: Van
Gorcum. Also contains many Latin editions of primary texts.
- Vol. 1: On the Twelfth Century Theories of Fallacy,
- Vol. 2: The Origin and Early Development of the Theory of Supposition.
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- –––, 2008, “William Heytesbury and the treatment of Insolubilia in 14th-century England”, in Rahman, Tulenheimo, & Genot 2008: 251–327.
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- Read, Stephen and Mark Thakkar, 2016, “Robert Fland or Elandus Dialecticus?”, Mediaeval Studies, 78: 167–80.
- Ryle, Gilbert, 1951, “Heterologicality”, Analysis, 11(3): 61–69. doi:10.1093/analys/11.3.61
- Rahman, Shahid, Tero Tulenheimo, and Emmanuel Genot (eds.), 2008, Unity, Truth and the Liar: The Modern Relevance of Medieval Solutions to the Liar Paradox, Berlin: Springer-Verlag.
- Spade, Paul Vincent, 1973, “The Origins of the Mediaeval Insolubilia-Literature”, Franciscan Studies, 33: 292–309. Reprinted in Spade 1988.
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How to cite this entry. Preview the PDF version of this entry at the Friends of the SEP Society. Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers, with links to its database.
Other Internet Resources
- Spade, Paul Vincent, Thoughts, Words and Things: An Introduction to Late Mediaeval Logic and Semantic Theory, Version 1.2 (December 27, 2007), (PDF format).
- Spade, Paul Vincent, “Three Questions by John of Wesel on Obligationes and Insolubilia” (PDF format). Includes a Latin edition of the first of his five questions on insolubles, which discusses Swyneshed’s second conclusion.
- Spade, Paul Vincent, “A Proof Concerning Bradwardine’s Theory”, online manuscript.
- William of Ockham, Summa logicae III-3, 46 (PDF format). An English translation by Paul Vincent Spade of Ockham’s discussion of insolubilia.