## 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.[2]
 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.[3]
> 
> 
> 



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.[4]
 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.[5]
 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*.[6]



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.[7]
 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[8]:



> 
> 
> 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".[9]



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.[10]
 (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*.[11]



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.[12]
 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.[13]


### 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.[14]



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.[15]


### 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.[16]


### 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.[17]


 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).

### 2.5 Cassation



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".[18]


## 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: SS13.2). 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:
SS1). (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):[19]



> 
> 
> 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
SS6.6.1-6):


1. 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.
2. 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.
3. 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.
4. 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* D 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.[21]
 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.[22]
 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.[23]



Many authors found these conclusions ridiculous, especially the second
and third ones. But they had their defenders as
 well.[24]



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.[25]


### 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*).[26]
 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.[27]



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.[28]
 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] SS50 and Bradwardine [B-I] SS
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.[29]
 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.[20]
### 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.[30]
 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:
SS10.2); 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: SS5;
Read 2006: SS6; with a response on Buridan's behalf in Klima
2009, SS10.5).


### 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*),
 [31]
 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: SS3.8.


### 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 (SS 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). 


## 5. Observations



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.[32]
 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]: SS2.1).



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.
> 
> 
>