## 1. Life and Works



William Heytesbury was most likely born before 1313 in Wiltshire
(Salisbury Diocese). He is first mentioned as a fellow at Merton
College in Oxford in 1330; he is thus among the second generation of
Oxford Calculators (a follower of Thomas Bradwardine and Richard
Kilvington, a contemporary of John Dumbleton and Roger Swyneshed, and
a predecessor of Richard Swineshead). He held the administrative
position of a bursar of Merton in 1338-1339, responsible for
determining dues, auditing accounts, and collecting
 revenues.[1]
 By 1340 he had completed his regency in arts at Merton and, together
with John Dumbleton, was named a foundation fellow at the new
Queen's College in 1340, but was soon mentioned among the
fellows of Merton College again. He is recorded as having become a
Doctor of Theology in 1348, yet none of his theological works are
known. He was chancellor of the University in 1371-72 (and
perhaps also in 1353-54) and died shortly thereafter, between
December 1372 and January
 1373.[2]



Heytesbury's extant writings, which are currently agreed to have
originated in 1330s, typically develop a certain logical or physical
theory through analysing sophisms. For most of these texts, there is a
modern (in some cases critical) edition, some are available in
incunabular editions and early prints, and several modern translations
are available. There is a spectrum of works based on how certain their
authenticity is in the present state of research. These break down to
three categories, such that shifts between them in the future are
possible (albeit unlikely in some cases):



**Works agreed to be authentic:**


* *Regulae solvendi sophismata* [RSS] or *Logica*
* *Sophismata* [Soph]
* *Sophismata asinina* [SophAs]
* *Iuxta hunc textum* [IHT] or *Consequentiae
Heytesbury*
* *De sensu composito et diviso* [SCD]



*Regulae solvendi sophismata* [RSS] or *Logica* (a
collection of six treatises: on paradoxes, on knowing and doubting, on
relative terms, on beginning and ceasing, on maxima and minima, and on
the speed of motions) is concerned with instructions for solving
different types of sophisms in the first year of logical studies. In a
way, [RSS] is a paradigm of authenticity for Heytesbury's works
(mutual coherence being used as the criterion of authenticity).
*Sophismata* [Soph] is a collection of sophisms for advanced
students working on natural philosophy. *Sophismata asinina*
[SophAs] is a collection of sophistical proofs that the reader is a
donkey. *Iuxta hunc textum* [IHT], also known as
*Consequentiae Heytesbury*, is a collection of sophisms
designed for testing formal inference rules. *De sensu composito et
diviso* [SCD] is a manual on the logical analysis of the *de
re*/*de dicto* ambiguity.



**Works of uncertain authenticity:**


* *Probationes conclusionum*
* *Casus obligationis* [CO]
* *De propositionibus multiplicibus* or *De propositionum
multiplicium significatione*
* *Termini naturales*



*Probationes conclusionum* follows [RSS] and analyses sophisms
of various kinds. *Casus obligationis* [CO] is a collection of
epistemic sophisms, related to the second chapter of [RSS]. *De
propositionibus multiplicibus* is a short treatise on the analysis
and inferential roles of various kinds of sentences. *Termini
naturales* is a vocabulary of elementary physics, very popular in
the late-medieval
 period.[3]



**Works which have been erroneously attributed to Heytesbury in
the past:**


* *De veritate et falsitate propositionis* (authored by Henry
Hopton)



*De veritate et falsitate propositionis* is a treatise on
sentential semantics motivated by the analysis of the sophism
"*Every sentence is true or
 false*".[4]



## 2. Logic


### 2.1 Obligationes



Since most of Heytesbury's logical views are spelled out either
in the context or in terms of
 *obligationes*,[5]
 a brief note on these must be the starting point of their
presentation. The most important passages are in [SCD] and [Soph] and
to a certain degree in [RSS]. Three problems will be introduced: 1)
the basic concepts, and in particular, relevance; 2) the distinction
between metaphysical and epistemic possibility; and 3) the distinction
between conceding a sentence and conceding that a sentence is
true.



Heytesbury's *obligationes* are zero-sum and dynamic
consistency games participated in by two players, an
"opponent" and a "respondent". The opponent
posits a "*casus*" (*ponitur casus*), i.e.,
the initial linguistic and extra-linguistic assumptions, and proposes
particular sentences. The respondent admits or denies the
*casus* (*admittitur*, *negatur casus*) depending
on its consistency and concedes (*concedo*), negates
(*nego*), or doubts (*dubito*) the proposed sentences,
the criteria being logical relations to either the *casus* or
to the totality of preceding moves (including the *casus*).
Heytesbury chooses the latter option, thus subscribing to so-called
"*antiqua responsio*" (Read 2013: 20-23). The
respondent wins if he maintains consistency.



Sentences proposed to the respondent split into "relevant"
or "irrelevant" depending on their logical relation to
preceding moves. If the sentence proposed to the respondent or its
negation follows from preceding moves, it is relevant
(*pertinens*) and should be either conceded (if implied by
these moves) or negated (if it is incompatible with them); otherwise,
it is irrelevant (*impertinens*). An irrelevant sentence is
either conceded, negated, or doubted depending on the
respondent's knowledge about the "external" world;
it is conceded if known to be actually true, negated if known to be
actually false and doubted otherwise. Such a game is dynamic in the
sense that the status of irrelevant sentences can change in the course
of the game ([SCD] 1494: fol. 4ra-rb [1988a: 432-433]).
This can happen for two reasons. First, if the external world changes
during the game, the respondent's move would change along with
his new information state since the respondent's moves regarding
irrelevant sentences depend on his knowledge of the external world
([SCD] 1494: fol. 4rb [1988a: 433]). The second (and more interesting)
case is brought about by the course of a particular game.
Heytesbury's example is the following game:





Game 1.| *Opponent's moves* | *Respondent's moves* | *Justification* |
| **[O1] POSITIO**: The king is seated or you are in
Rome. | **[R1]** I admit the *casus*. | The *casus* is possible. |
| **[O2] PROPONITUR:** The king is seated. | **[R2]** I doubt that. | Irrelevant and not known to be true or false in the external
world. |
| **[O3] PROPONITUR:** You are in Rome. | **[R3]** I negate that. | Irrelevant and known to be false. |
| **[O4] PROPONITUR:** The king is seated. | **[R4]** I concede that. | Relevant and implied by **[R1]** and
**[R3]**. |




[R2] and [R4] are correct moves because [R1] and [R3] are. When
"the king is seated" is proposed in [O2], it can only be
relevant with respect to [R1], which is not the case as disjunctions
do not entail their sub-formulas. If the respondent is unaware of the
king's whereabouts, the sentence must be doubted. "You are
in Rome" proposed in [O3] is also irrelevant, but can be negated
as known to be false in the external world. At that point, "the
king is seated" becomes relevant and must be conceded as implied
by "the king is seated or you are in Rome" and the
negation of "you are in Rome" via disjunctive syllogism
([SCD] 1494: fol. 4rb [1988a: 433-434]).



Heytesbury considers three types of *casus*: a) possible; b)
impossible but imaginable; and c) impossible and unimaginable ([Soph]
soph. 18 and 31 [1494: fols. 131va-vb and 162va-vb]). The
criteria of possibility, which probably pertain to metaphysics, are
not spelled out clearly. The criteria of imaginability are logical and
epistemological:



> 
> 
> Briefly, every *casus* which is not explicitly inconsistent or
> impossible in a way not easily imaginable (such as a man being a
> donkey etc.) can be admitted for the sake of disputation. But first,
> the respondent has to make the impossibility of such *casus*
> explicit and emphasize that he has not admitted the *casus* as
> possible but merely as imaginable, to defend its implications (as
> imaginable, not as possible) and to negate what is incompatible with
> it in accord with his obligation. ([Soph] soph. 31 [1494: fol.
> 162va-vb])
> 
> 
> 



The realm of the imaginable includes whatever is epistemically
admissible for either logical or epistemic reasons. Neither is purely
objective: what can and cannot be imagined seems player-relative, as
is being "explicitly" inconsistent (for seeing or
overlooking a contradiction is a matter of individual skill). Examples
of the imaginable include the existence of a vacuum, infinite
velocities, expansions of physical objects, and men being immortal,
but not men being
 donkeys.[6]
 The criteria may be practical: certain hypotheses are not productive.
Since the respondent's goal is to maintain consistency,
accepting overtly inconsistent assumptions is a
 nonstarter,[7]
 and certain impossibilities are such that nobody would agree to
defend them even for the sake of
 argument.[8]
 Defining admissibility in terms of imaginability makes
*obligationes* into a framework for discussing
"impossibilities" with applications in
 physics[9]
 as a science based on conceptual analysis and
 thought-experiments,[10]
 where the "*secundum imaginationem*" assumptions
play the same role as the modern day idealizations such as mass points
or frictionless
 movement.[11]



A principle utilized in Heytesbury's solution to various
paradoxes states that a proposed sentence is evaluated according to
its standard meaning, even if it is assumed to signify something else
in the *casus*. For instance, the following moves of the
respondent would be correct:





Game 2.| *Opponent's moves* | *Respondent's moves* | *Justification* |
| **[O1] POSITIO**: "All mice are grey"
signifies that all cats are blue. All cats are white. | **[R1]** I concede the *casus*. | The *casus* is possible. |
| **[O2] PROPONITUR:** All mice are grey. | **[R2]** I concede that. | Irrelevant and known to be true in the external world. |
| **[O3] PROPONITUR:** The sentence "all mice
are grey" is true. | **[R3]** I negate that. | Relevant and inconsistent with the *casus*. |




By [R2], the respondent agrees that all mice are grey, while by [R3]
he denies that all cats are blue. To concede a sentence is to agree
that what the sentence signifies according to its standard meaning is
the case, but to concede that a certain sentence is true is to agree
that what the sentence signifies in the *casus* is the case.
These may or may not be the same depending on the *casus.*
([SophAs]
 412)[12]


### 2.2 Proprietates terminorum



The genre of "*proprietates
 terminorum*" developed in the twelfth century alongside the
reception of Aristotle's *Sophistical refutations*, and
the logico-semantical analysis of fallacies seems to have been one of
its primary
 applications.[13]
 It is an umbrella genre for, among others, theories of meaning and
reference, the use-mention distinction, quantification, and tense.
These are utilized in several of Heytesbury's works with
terminological variations. Three of these issues will be presented
here: the theories of "*suppositio*",
"*mobilitatio*", and
"*ampliatio*".



Supposition is a semantic relation between terms and the object(s)
they stand for in a sentential context. The modes of supposition are
connected to the way sentences can be analyzed (*expositio*),
in particular, to the way general terms can be replaced by a
structured collection of subordinate terms, the so-called
"*descensus*". The following classifications of
suppositions can be reconstructed from Heytesbury's
writings:


* (1) material supposition (*s. materialis*)
* (2) personal supposition (*s. personalis*)
	+ (2.1) discrete supposition (*s. discreta*)
	+ (2.2) common supposition (*s. communis*)
		- (2.21) determinate supposition (*s. determinata*)
		- (2.22) confused supposition (*s. confusa*)
			* (2.221) merely confused or non-distributive supposition (*s.
			confusa tantum*, *confusa non distributiva*)
			* (2.222) confused and distributive supposition (*s. confusa
			distributiva*)



Material supposition ("only you are a donkey" in
"only you are a donkey is false") consists in mentioning
an expression ([SophAs] 403), whereas personal (or significative)
supposition consists in using it ([SophAs] 404). Discrete supposition
is a personal supposition of a singular term ([SophAs]
 390);[14]
 its complement is usually called "common" supposition and
is further divided into determinate and confused suppositions.
Determinate supposition is exemplified by the supposition of
"human" in "some human is an animal" ([SCD]
1494: fol. 3va [1988a: 427]), connected to the *descensus*,
whereby it can be paraphrased as "this human is an animal or
that human is an animal etc." ([SophAs] 387). Merely confused or
non-distributive supposition is exemplified by the supposition of
"animal" in "every human is an animal" and
connected to "disjunctive" *descensus* ([SCD] 1494,
fol. 2va-vb [1988a: 420-421]; [Soph] soph. 7 [1494: fol.
106rb] and soph. 16 (1494: fol. 127vb)), whereby it can be paraphrased
as "every man is this animal or that animal etc." Confused
and distributive suppositions are most typically exemplified by the
supposition of "human" in "every human is an
animal", connecting the *descensus* to every instance of
the common term, whereby it can be paraphrased as "this human is
an animal and that human is an animal etc." ([SCD] 1494: fol.
2va-vb [1988a: 420-421]; [Soph] soph. 7 [1494: fol. 106rb]
and soph. 16 [1494: fol. 127vb]).



The theory of *suppositio* is extended by the theory of
"mobility" and "immobility". If a particular
mode of *descensus* is admissible, that type of supposition is
called "mobile" (*mobilis*); otherwise it is called
"immobile"
 (*immobilis*).[15]
 Heytesbury's example of immobile supposition is the function of
"penny" in "I promise you a penny" or
"human" in "by necessity, every man is an
animal" ([PC] 1494: fol. 188vb; [IHT] arg. 19; [SophAs]
398).



Three points should be noted here. First, the so-called
"simple" ("human" in "human is a
species") supposition is
 omitted.[16]
 Second, Heytesbury's analysis of predicates in terms of the
merely confused supposition is nominalist, as opposed to the realist
analysis of these examples in terms of standing for an abstract
 object.[17]
 Third, Heytesbury endorses the Aristotelian *sensus
compositus*-*sensus divisus* distinction, rather than the
terminist framework in cases where that would be
 appropriate.[18]



Lastly, Heytesbury introduces the so-called
"*ampliatio*", which consists in extending the
range of a term to non-actual entities, such as entities in the past,
future, or mere possibilia. Thus, for instance, the proper paraphrase
of the sentence "a white thing can be black" is
"something that now is white or can be white can be black"
([SCD] 1494: fol. 2va [1988a:
 418]).[19]
 The complementary property is called
"*restrictio*" ([Soph] soph. 4 [1494: fol.
89vb]).


### 2.3 Deductive Validity



The basic textual evidence for
 Heytesbury's theory of deductive validity includes [IHT],
[Soph] and [RSS].



[IHT] is a collection of arguments for both the validity and
invalidity of spurious inferences. Typically, the pro-arguments
construe them as instances of formal inference rules (*bona et
formalis*), the contra-arguments argue that they are not valid
(*non valet*), and the resolutions show that the inference
under scrutiny is not a genuine instance of such a rule. This method
presupposes that deductive validity is ultimately truth-preservation
(an inference is valid if its consequent cannot be false while its
antecedent is true or if its antecedent is incompatible with the
negation of the consequent) ([IHT] arg. 6). The criterion is that it
is not possible to present a counterexample, i.e., a possible
(obligational) situation (*casu possibili posito*) verifying
the antecedent and falsifying the consequent ([IHT] arg. 2). The
method also presupposes that the underlying inference rules are
assumed to be legitimate but misapplied in such sophisms as to require
elucidation, and not revision. The treatise is mostly concerned with
the analysis of logical form (*expositio*) of sentences
containing operators (*officialia*) such as "only",
"nothing but", "begins", "ceases",
etc., with a few applications of modal and epistemic logic.



[IHT] introduces the distinction between the "universal"
(*bona et formalis generaliter*) and "restricted"
(*bona et formalis specialiter*) formal validity:



> 
> 
> An inference is formally valid in the universal sense if the opposite
> of its consequent is formally incompatible with its antecedent and a
> similar form [of argument] holds in all applications (*consimilis
> forma valet in omni materia*) (...) An inference is
> [formally] valid in the restricted sense if the opposite of its
> consequent is formally incompatible with its antecedent, but not every
> such application is valid. ([IHT] arg. 11)
> 
> 
> 



The concept of formal compatibility is not explicitly introduced in
this context.



[Soph] relates the relevance view of formality as a conceptual
connection between premises and conclusions to the substitutional view
of formality as validity under any substitution of extra-logical terms
or in any domain of application (*in omni materia*). If an
inference is valid only in a restricted domain of application, it is
valid based on content or in a particular domain of application
(*de materia*, *gratia materiae*) or "simply
valid" (*bona simpliciter*) ([IHT] arg. 11; [Soph] soph.
3 [1494: fol. 86ra]). The concept of formal incompatibility is
clarified as part of the explication of validity based upon the form
(*de forma*):



> 
> 
> It is commonly agreed that such an inference is not valid based upon
> the form, because the opposite of its consequent and its antecedent do
> not formally imply a contradiction, where "to imply a
> contradiction formally" means that these two cannot be
> distinguished or consistently and distinctively imagined as holding
> simultaneously. ([Soph] soph. 2 [1494: fol. 83rb])
> 
> 
> 



As opposed to testing simple validity against possible counter-models,
formal validity is tested against impossible but imaginable
counter-models to eliminate extra-linguistic influences. Imaginable
*casus* introduced in *obligationes* are thereby
utilized as part of general logic.



Finally, a "necessary" inference is one which can be
reduced to a categorical or hypothetical syllogism via a necessary
auxiliary assumption or "middle" (*per medium
necessarium*) ([Soph] soph. 2 [1494: fol. 83rb]). Similarly, two
sentences are assumed to be logically equivalent
(*convertantur*) if they imply each other formally via a
necessary middle ([RSS] 1494: fol. 15rb-va [1988b:
 461-462]).[20]
 This represents the view of validity as reducibility to
paradigmatically valid
 syllogisms.[21]



In general, Heytesbury introduces three levels of
 validity.[22]
 First, validity as relevance secured by logical form in every domain
of application, explicated as truth-preservation of all inferences
sharing the same logical form (*forma* or *modus
arguendi*) with respect to every imaginable world. Second,
validity as relevance secured by conceptual content with a restricted
domain of application, explicated as truth-preservation with respect
to all imaginable worlds, which is not substitution-invariant. Third,
validity as simple truth-preservation with respect to all possible
worlds. Heytesbury's parallel use of different concepts of
formality makes him a representative of British logic in the
transitional period, containing different stages of development: the
substitutional view (validity preserved in all substitution-instances)
typical for the "Continental tradition", the relevance
view (conceptual connection) typical for the "British
tradition" and later Paul of Venice and Paul of Pergula, and
formality as reducibility to a syllogism typical for the early
fourteenth-century
 tradition.[23]


### 2.4 Logic of Epistemic Statements



Heytesbury's analysis of epistemic statements encompasses the
logical analysis of epistemic statements, inference rules for
sentences containing epistemic operators, and rules for obligational
games with epistemic statements, most of which are applied to
solutions to
 sophisms[24]
 and instructions for obligational games. The underlying principles
pertaining to the logical analysis of language and inference rules for
epistemic statements will be addressed
 below.[25]


#### 2.4.1 *De re*/*de dicto* Ambiguity and Epistemic Statements



Let us consider the following two sentences:



**(dr)**
Socrates I know to be running in Beaumont.
**(dd)**
I know that Socrates is running in Beaumont.


(dr), called "*sensus divisus*" by Heytesbury,
denotes that there is a person (who happens to be Socrates) of whom I
correctly know that he is running in Beaumont, but about whom I may or
may not know that he is Socrates; his kinematics, but not his
identity, is part of my belief. On the other hand, according to (dd),
called "*sensus compositus*" by Heytesbury, I know
that it is Socrates who is running in Beaumont. Hence if I believed
that Socrates died years ago, (dr) could be true but (dd) would be
false ([SCD] 1494: fol. 3va [1988a: 427] and [RSS] 1494, fol. 13rb and
[1988b:
 444]).[26]
 By strictly endorsing the Aristotelian framework and analyzing
epistemic contexts in terms of the *de re*/*de dicto*
ambiguity, Heytesbury does not participate in the fourteenth-century
developments in the logical analysis of
 language.[27]
 Also, his criteria for distinguishing between the two readings are
purely syntactic, whereby scholastic Latin is viewed as an artificial
language with strict formation
 rules.[28]


#### 2.4.2 Inference Rules



Heytesbury's remarks on the "common interpretation of
knowledge" emphasize that a firm and unhesitant belief does not
constitute knowledge unless it is veridical, which results in the
formulation "to know is nothing other than unhesitatingly to
apprehend the truth--i.e., to believe unhesitatingly that it is
so when it is so in reality" ([RSS] 1494: fol. 13vb [1988b:
 447]).[29]
 This implies the veracity of knowledge:



(T)
if x knows that *X*, then *X* is the
 case.[30]


The introspection axioms govern iterated epistemic modalities (Boh
1984 and 1985, Hanke 2018a). While denying the possibility of doubting
one's own knowledge, Heytesbury proposes what is an equivalent
to positive introspection:



(4)
if the agent x perceives that *X* and x is
considering whether x perceives that *X*, then x
perceives that he perceives that *X*.


As perception is in this context replaceable with knowledge, the
principle is equivalent to its modern counterpart, save the emphasis
on active consideration of the problem (i.e., attention), which
appears to correct the principle for the limitations of human agents.
The principles for both first and second-order knowledge are based on
the same evidence ([RSS] 1494: fol. 13vb [1988b: 447-448]).
Heytesbury also holds the contraposition of (4) ([RSS] 1494: fol. 13vb
[1988b: 448]).



In the context of (T), Heytesbury discusses the *casus* where
an agent sees a person who looks exactly like a king, but is not one.
The agent can believe the man to be a king beyond any doubt and even
believe himself to know that. But, by (T), he knows neither that the
man is a king (because that is not true), nor that the man is not a
king (because he does not believe that) ([RSS] 1494: fol. 13vb [1988b:
447]). Even though Heytesbury does not explicitly say so, it seems
natural to assume that this agent does not know that he failed to know
that man is a king. Hence, he fails to know something, but also,
presumably, fails to know that he fails to know it, which is a
counterexample to negative introspection. Accepting the positive
introspection while denying the negative introspection is based on
Heytesbury's interpretation of knowledge as an evidence-based
act of considering something and apprehending its factuality: the
absence of knowledge alone does not constitute conscious
ignorance.

 Heytesbury's account of the closure properties of
knowledge encompasses the closure of meaning under entailment, the
distribution of knowledge over implication, and the distribution of
knowledge over signification.


First, Heytesbury introduces the concept of implied meaning
(*significare ex consequenti*) and believes that sentential
meaning is closed under entailment. As he explains the different
senses of "signifying precisely" (*significare
precise*), he claims that a sentence cannot signify something
precisely in the sense of not signifying anything else. Instead,
"to signify precisely" means that a sentence has a
"primary meaning" ("*significat primo et
principaliter*" or "*significatio primaria et
adequata*") plus whatever its primary meaning entails
(*quod sequitur ipsam significantem quod ita sit*), but nothing
else ([RSS] 1494: fol. 15ra-rb [1988b:
 459-460]).[31]
 To allow that, Heytesbury's semantics must validate some
version of the Bradwardinian sentential
 semantics.[32]



Second, in the possibly inauthentic [CO], outright logical
omniscience, i.e., the closure of knowledge under implication, is
denied:



(O)
If *X* implies *Y* and x knows that *X*,
then x knows that *Y*,


but the distribution of knowledge over implication, i.e., that
knowledge is closed under known inferences (the "axiom
K"), is accepted:



(K)
If x knows that *X* implies *Y*, and x knows
that *X*, then x knows that
 *Y*.[33]


Third, if an agent knows that it is completely the case as a sentence
signifies, does he know that everything it signifies is the
 case?[34]
 The problem of conceding a sentence whose meaning is deductively
closed by an agent with a finite cognitive capacity is solved by
introducing a distinction parallel to (K) and (O) ([RSS] 1494: fol.
15rb [1988b: 460]); Heytesbury denies:



(SO)
If x knows that however *x* signifies is the case, and
*x* signifies that *X*, then x knows that
*X*,


but holds:



(SK)
If x knows that however x signifies is the case, and x knows
that *x* signifies that *X*, then x knows that
*X*.


Together with "strong" semantic closure,
Heytesbury's analysis of epistemic operators validates the
weaker distribution principles and rejects the stronger closure
principles, i.e., different forms of logical
 omniscience.[35]


### 2.5. Insolubilia



Heytesbury analyzes *insolubilia* or self-referential paradoxes
most thoroughly in the first chapter of
 [RSS],[36]
 where he dismisses three alternative solutions to paradoxes to settle
for the fourth. The first three positions agree that paradoxical
sentences can have precisely their standard meaning. According to the
first position held by Roger Swyneshed, paradoxical sentences are
false because they imply their own falsity, and a sentence and its
negation can be false at the same time if one of them is paradoxical
([RSS] 1494: fol. 4va and 6rb [1979: 18-19 and 45]). According
to the second position, paradoxical sentences do not make genuine
statements, hence are neither true nor false ([RSS] 1494: fols.
4va-vb [1979: 19]). That is reminiscent of the so-called
"*cassantes*" who claim that paradoxical sentences
fail to make a true or false statement (literally "say
nothing") (de Rijk 1966), available, for instance, through
Burley or Bradwardine (Roure 1970; Bradwardine B-I). Heytesbury also
could have viewed the second position as the interpretation of
Bradwardine's "*mediantes*", who claims that
paradoxical sentences are neither true nor false, but of a
"middle
 value".[37]
 According to the third position, paradoxical sentences are
either-true-or-false but neither true nor false; they make genuine
statements, but fail to have a standard truth-value ([RSS] 1494: fol.
4vb [1979: 19-20]).



According to the fourth position, paradoxical sentences have their
standard meaning, but say more than what their overt meanings
indicate. It is based on the definitions of a paradoxical
*casus* and a paradoxical sentence. There are two versions of
these definitions, differing in the last clause:



> 
> 
> A paradoxical *casus* is one in which mention is made of some
> sentence such that if in the same *casus* that sentence
> signifies precisely as its words commonly pretend, from its being true
> follows that it is false, and vice versa. ([RSS] 1494: fol. 6rb [1979:
> 47])
> 
> 
> 



or:



> 
> 
> ...it follows that it is true and also false. (Pozzi 1987:
> 236)
> 
> 
> 



Furthermore:



> 
> 
> A paradoxical sentence is one of which mention is made in a
> paradoxical *casus*, such that if in the same *casus* it
> signifies precisely as its words commonly pretend, then from its being
> true follows that it is false, and vice versa. ([RSS] 1494: fol. 6rb
> [1979: 47])
> 
> 
> 



or:



> 
> 
> ...it follows that it is true and also false. (Pozzi 1987:
>  236)[38]
> 
> 
> 



The solution to paradoxes is exemplified by the *casus*, where
Socrates only pronounces the sentence "Socrates is saying
something false", henceforth labelled (s). Five elementary
settings differ in assumptions concerning its meaning:


* (1) the meaning of (s) is not posited
* (2) the meaning of (s) is posited, such that:
	+ (2.1) (s) has precisely its standard meaning
	+ (2.2) (s) has its standard meaning, but not precisely its standard
	meaning, and the additional meaning is not certified
	+ (2.3) (s) has its standard meaning and its additional meaning is
	certified, such that:
		- (2.31) (s) has a logical form of a conjunction
		- (2.32) (s) has a logical form of a disjunction



For Heytesbury, "to solve a paradox" is to instruct the
respondent in the corresponding *obligatio*-game. His
instructions are as follows:



**(R1)**
If someone constructs a paradoxical *casus*, either he
posits how that paradoxical sentence should signify or not. If not,
then: when the paradoxical sentence is proposed, one should respond to
it exactly as one would respond when the *casus* is not
assumed. ([RSS] 1494: fol. 6va [1979: 47])


The corresponding moves are:





Game 3.| *Opponent's moves* | *Respondent's moves* | *Justification* |
| **[O1] POSITIO**: Socrates only pronounces
(s). | **[R1]** I admit the *casus*. | The *casus* is possible. |
| **[O2] PROPONITUR:** (s) is false. | **[R2]** ??? | Irrelevant and ??? |




Heytesbury applies his view of *obligationes*: in the absence
of any other assumptions, (s) is initially irrelevant. As such, if
proposed as the first sentence in the game, the respondent's
move cannot be affected by either the *casus* or the previous
moves, but must follow the respondent's knowledge about the
external world. Heytesbury's emphasis on the stage of the game
suggests that this can change due to the dynamic character of
*obligationes*.



**(R2.1)**
If a paradoxical *casus* is posited and together with that
it is assumed that the paradoxical sentence precisely signifies as its
terms commonly pretend, the *casus* may in no way be admitted.
([RSS] 1494: fol. 6va [1979: 48])


The corresponding moves are:





Game 4.| *Opponent's moves* | *Respondent's moves* | *Justification* |
| **[O1] POSITIO**: Socrates only pronounces (s) and
(s) signifies precisely that (s) is false | **[R1]** I do not admit the *casus*. | The *casus* is inconsistent. |




Since *obligationes* are consistency-maintenance games, the
only non-losing strategy for a respondent is not to start such a game,
for otherwise he would be dragged into inconsistency via the
well-known paradoxical reasoning.



**(R2.2)**
If someone constructs a paradoxical *casus* and if it is
also assumed that the paradoxical sentence signifies as its terms
pretend (but not precisely so), then when this *casus* is
admitted, the insoluble sentence has to be conceded as implied at
whatever stage of the game it is proposed, but that it is true has to
be denied as incompatible. ([RSS] 1494: fol. 6va [1979: 49])


The corresponding moves are:





Game 5.| *Opponent's moves* | *Respondent's moves* | *Justification* |
| **[O1] POSITIO**: Socrates only pronounces (s) and
(s) signifies that (s) is false | **[R1]** I admit the *casus*. | The *casus* is consistent. |
| **[O2] PROPONITUR:** (s) is false. | **[R2]** I concede that. | Implied by the *casus*. |
| **[O3] PROPONITUR:** "(s) is false" is
true. | **[R3]** I deny that. | Incompatible with the *casus*. |




The justification of [R2] is not clearly explained in [RSS]. A
minimalist interpretation is that the proposed sentence is implied by
[R1] via *reductio*: If (s) is false, then it is false
(trivially); if (s) is not false, it is true, and then it is the case
as it signifies, hence it is false. Either way, it is false, hence it
is
 false.[39]
 An obligational explanation is that (s) is regarded as relevant in
this *casus*, and as such can only be conceded or denied. Since
its denial implies an inconsistency, such a move must be avoided in a
consistency-maintenance game; therefore it should be conceded. Since
the *casus* has been admitted as consistent, it cannot validate
an inconsistent move; hence, it must validate its opposite. By the
same reasoning, it is possible to deny that (s) is true. Furthermore,
Heytesbury refuses to certify the additional meaning of (s) ([RSS]
1494: fol. 6va [1979: 49-50]). Unsatisfactory as this may seem,
it is legitimate from the point of view of *obligationes*:
meaning certification is not a move available to the respondent, and
since the instructions are respondent-oriented, the actual meaning of
(s) is an external problem.



(2.31) and (2.32) are dealt with along the same lines.



For Heytesbury, paradoxes are obligational games with self-referential
 sentences.[40]
 However, his criticism of alternative solutions is part of the
fourteenth-century developments relating paradoxes to general
logico-semantical
 issues.[41]
 He views his approach as relatively defensible and probable, but
admits that he cannot solve all objections and considers a completely
satisfactory solution to paradoxes impossible (unfortunately without a
justification). Without too much self-confidence, he closes his
treatise by saying that young students should move past the topic and
do something useful. ([RSS] 1494: fol. 4vb and 7rb [1979: 21 and
58])


## 3. Speculative Physics



Heytesbury's RSS may be taken as representative of Oxford
Calculators' speculative/mathematical physics. It links
interests in logic, mathematics, and physics. As Curtis Wilson says:
"Heytesbury employs the concept of "limit" and
"infinite aggregate" with considerable precision and
subtlety; and in treating of variable physics quantities, he
approaches the ideal of a purely mathematical description"
(Wilson 1960: 3). Not only does Heytesbury analyze physical and
mathematical problems by means of the techniques of the *logica
moderna*, but he also employs physical and mathematical rules in
the analyses of logical or semantic problems of denomination.
Heytesbury is mostly interested in the problem of determining under
what conditions a subject can be denominated such-and-such, e.g.,
"white" or "running". The issue of
denomination arises because the attributes of subjects vary in degrees
of intensity or completeness. He establishes sets of rules of everyday
speech, determining how we denominate a subject under all imaginable
circumstances of change. For example, in the sophism *Omnis homo
qui est albus currit*, he concludes that a man is called to be
"white" if and only if the external surface of the upper
half of him is white (in everyday speech, a man is white if the skin
on his face is white) (Wilson 1960: 22-23).



Like one of the founders of the School of Oxford
Calculators--Richard Kilvington- Heytesbury takes
Ockham's nominalist position and affirms that the real physical
world consists only of *res absolutae*, i.e., substances and
qualities. They both acknowledge that terms such as
"point", "line", "latitude", and
"degree" do not have any representation in reality, but
are useful for describing different types of change. They also hold
that time is not real since time is not distinguished
*realiter* from the motion of celestial spheres, and motion is
not distinguished *realiter* from the body which moves. Thus,
they both contrast things that are really distinct with things
distinguishable only in reason, i.e., in the imagination. Heytesbury
follows Kilvington's frequent use of the phrase *secundum
imaginationem* to describe hypothetical situations in which the
elements of the description, and not the situation itself, are of
primary concern. The only requirement for an imaginable case is that
it should not involve a formal logical contradiction; whether it is
physically possible or not is a matter of indifference. As Wilson
says,



> 
> 
> a distinction is drawn between *realiter* or
> *naturaliter* or *physice loquendo* and *logice*
> or *sophistice loquendo*: *physice* we follow experience
> and the principles laid down in the natural philosophy of Aristotle;
> *logice* or *sophistice* we are free to introduce
> whatever distinctions and cases are convenient and imaginable. (Wilson
> 1960: 25)
> 
> 
> 



Heytesbury, however, remains in the framework of Aristotelian
philosophy of nature, and the problems he discusses are to be found in
Aristotle's *Physica* and *De caelo*.
Nevertheless, Heytesbury is captivated by the mathematical approach es
the earlier Oxford Calculators. Like Thomas Bradwardine and Richard
Kilvington, he examines physical problems using mathematical
apparatus. In Chapters 4-6 of the *Regulae*, Heytesbury
develops measurement by limit, i.e., by the first and last instants of
beginning and ending, and by the intrinsic and extrinsic limits of
passive and active capacities. Although these types of
"measuring" do not appear to be straightforwardly
mathematical, they provide the basis for mathematical considerations
and give excellent results in analyses of the instantaneous motion, as
well as continua. As Wilson states:



> 
> 
> Heytesbury admits "according to the common mode of speech"
> that everything which is, whether time or motion or instant, is in an
> instant, in the sense that it is instantaneously measured by an
> instant [...] the instant in time and the instantaneous position
> in movement are always to be regarded as "limits". (Wilson
> 1960: 41)
> 
> 
> 



The second type of measurement employs a new calculus of compounding
ratios. The third type of measurement, by latitude of forms, describes
motion, in which an accidental form, such as speed, is intensified or
diminished. Heytesbury focuses on establishing the rules for the
"measurement" of speed in every type of the continuous
changes enumerated by Aristotle, i.e., in local motion, in
augmentation, and in alteration.


### 3.1 Beginning and Ceasing



In chapter IV of the *Regulae* (*De incipit et desinit*)
Heytesbury poses and analyzes in great detail cases in which any
process, thing, or state may be said to begin or cease to be. He
begins with possible expositions of the terms "beginning"
(*incipere*) and "ceasing" (*desinere*) in a
way that uncovers certain paradoxical properties of the temporal
continuum. "Beginning" may be expounded by
*positio* of the present (i.e., by the positing of existence in
the present instant) and *remotio* of the past (i.e., by denial
of existence in the past), meaning that in the present instant a thing
or a process is, and immediately prior to the present instant it was
not. "Beginning" may be also expounded by *remotio*
of the present and *positio* of the future, meaning that in the
present instant a thing or a process is not, and immediately after the
present instant it will be. Likewise, "ceasing" may be
expounded in two ways, either by *remotio* of the present and
*positio* of the past, meaning that in the present instant a
thing or a process is not, and immediately prior to the present
instant it was, or by *positio* of the present and
*remotio* of the future, meaning that in the present instant a
thing or a process is, and immediately after the present instant it
will not be. Heytesbury offers a number of sophisms involving the
limit-decision. One of them is particularly interesting since it
clearly exhibits Heytesbury's remarkable skill in solving
problems of instantaneous motion and time without the symbolic
techniques of calculus. Heytesbury poses a case in which both Plato
and Socrates begin to move from rest at the same instant, but Plato
starts to move with a constant acceleration and Socrates with an
acceleration which starts at zero degree and increases uniformly with
time. He concludes that "both Socrates and Plato infinitely
slowly begin to be moved, and yet Socrates infinitely more slowly
begins to be moved than Plato" (Wilson 1960: 54, [RSS] 1494:
fol. 26vb). As Wilson proves, the conclusion is an effect of
comparison of two infinitesimals of different order.



> 
> 
> In the case under consideration both the velocity of Plato (\(v\_p\))
> and the velocity of Socrates (\(v\_s\)) are infinitesimal for time
> *t* approaching zero. Two infinitesimals are said to be of
> different order if the limit of their quotient is either zero or
> infinite. In the case under consideration the limit of the quotient
> \(v\_p/v\_s\) as \(t\rightarrow 0\) is infinite. (Wilson 1960: 55)
> 
> 
> 


### 3.2 Maxima and Minima



In chapter V of the *Regulae*, Heytesbury considers the limits
of capacities, and he applies the limit-concept to the bounding of the
ranges of variables and aggregates. All capacities are sorted as
active or passive. An active capacity (force) is measured against the
passive capacity (resistance) which it can overcome. If an active
capacity can overcome some resistance, it can overcome less. For
example, if Socrates can lift one hundred pounds, he can lift fifty
pounds. Conversely, if a passive capacity can be affected by less, it
can be affected by greater. For example, if Socrates can see a grain
of millet from the distance of a mile, he will be able to see the
Church at the same distance. The boundaries of capacities are of two
kinds: an intrinsic boundary (when an element is a member of the
sequence of elements it bounds: *maximum quod sic, minimum quod
sic*) and an extrinsic boundary (when an element which serves as a
boundary stands outside the range of elements which it bounds:
*maximum quod non, minimum quod non*). From the definitions of
active and passive capacities, it follows that the boundary of an
active capacity must be assigned either by the affirmation of the
maximum (*maximum quod sic*) or by the negation of the minimum
(*minimum quod non*); the boundary of a passive capacity is
assigned either by the affirmation of the minimum (*minimum quod
sic*) or by the negation of the maximum (*maximum quod
non*).



Heytesbury sets the following conditions which must be obtained for
limits to exist: (1) there must be a range in which the capacity can
act or be acted upon, and another range in which it cannot act or be
acted upon, and not both; (2) each capacity should only be able to
take a value in the range on which it is measured from zero and the
value which serves as its boundary. Thus, if an active capacity is
capable of acting upon a given passive capacity in the range, it must
be capable of acting upon any less, and if it is not capable of acting
upon a given passive capacity, it is not capable of acting upon any
greater. Conversely, if a passive capacity is capable of being acted
upon by a certain agent, it must be capable of being acted upon by any
agent of greater strength, and if it is not capable of being acted
upon by a certain agent, it is not capable of being acted upon by a
weaker agent; and (3) infinite capacities like the infinite power of
God are excluded from the discussion, since no termination can be
assigned for them ([RSS] 1494: fol. 29va-vb).



Heytesbury, like Kilvington, classifies all capacities as mutable or
immutable. A mutable capacity is capable of more and less, and both
affirmative and negative parts of the division might be assigned. For
example, with respect to Socrates, a mutable power of lifting (an
active capacity) a *minimum quod non* is assigned. For a
passive capacity the *maximum quod non* is assigned, e.g., a
maximum visible body which cannot be seen at a given distance. An
immutable capacity acts in a determinate way, so it produces only the
one effect it actually produces. Therefore, the affirmative part of
the division is always assigned as a boundary for immutable capacity.
Thus, e.g., there is a maximum distance which, *ceteris
paribus*, Socrates can traverse in an hour by a speed which
increases uniformly from A to B degree ([RSS] 1494: fol. 29vb; Wilson
1960: 73).


### 3.3 Three Categories of Change



Chapter VI (*De tribus praedicamentis*) of the *Regulae*
deals with motion in three categories: place, quantity, and quality.
The first part (*de motu locali*) deals with the speed or
slowness of uniformly or difformly accelerated and decelerated motion
considered in its kinematical aspect. The second part (*de
velocitate motus augmentationis*) deals with augmentation
understood as pure increase in size, and more specifically with the
speed of rarefaction as measured by the effect produced, i.e., by
space traversed in time. The third part (*de velocitate motus
alterationis*) is concentrated on mathematical description of
variation of intensity in space and time. The prime purpose to be
achieved in this chapter is to establish definitions of speed in the
three categories of motion. All debated cases are posed *secundum
imaginationem*.


#### 3.3.1 Local Motion



According to Heytesbury's own words: "local motion by
nature precedes other kinds [of change] just as the first"
(*De motu locali* [DML] 2020: SS 2, 270). He focuses on the
possible measure of the speed of these changes by the effect produced
(i.e., the space traversed) during an equal length of time.
Consequently, he is interested in the kinematic aspect of motion (DML
2020: SS 63,
 296).[42]
 Local motions are grouped into two classes: uniform and difform. A
uniform motion is motion in which equal spaces are traversed
continually in equal parts of time . Difform motions can vary in
infinite ways, both with respect to the magnitude or the subject
moved, and with respect to time (DML 2020: SSSS 3-5,
270). Difform motion with respect to the subject moved is motion in
which different points of the body move with unequal speeds; for
instance, a rolling wheel moves with difform motion since different
points on the wheel vary in regard to distance from the axis of
rotation (DML 2020: SSSS 9-16, 271-273). Difform
motion with respect to time is motion in which unequal spaces are
traversed in equal times. Motion can also be difform with respect to
both time and the subject moved (DML 2020: SSSS 17-20,
273-274). Difform motions are subdivided into two classes: the
uniformly difform (*uniformiter difformis*) motion and the
difformly difform (*difformiter difformis*) motion (DML 2020:
SS 21, 275). Uniformly difform motion is motion in which the speed
either increases or decreases uniformly, that is, a motion in which in
any equal parts of time, equal latitudes of speed are either acquired
or lost (DML 2020: SS 22, 275). A difformly difform motion is
motion in which a greater latitude of velocity is gained or lost in
one part of time than in another equal to it (DML 2020: SS 40,
284).



The most gripping example of uniformly difform motion is accelerated
motion, like a motion of a body moving towards the Earth. Heytesbury
gives a general rule, called the Middle Degree Theorem, by which we
may calculate the distance traversed from the latitude of speed
uniformly acquired. He makes use of this theorem with respect to both
accelerated and decelerated motions. In his own formulation it reads
as follows:



> 
> 
> Certainly, every latitude [of speed] that is acquired or lost
> uniformly, whether commencing at a non-degree or at any degree, as
> long as it is limited to some finite degree, corresponds precisely to
> its own middle degree in such a way that a moving body that uniformly
> acquires or loses its speed in any given time traverses the distance
> perfectly equal to the one it would traverse during the same time when
> moving with [a speed corresponding to] this middle
>  degree.[43]
>  (DML 2020: SS 26, 276; see also Wilson 1960: 122-23)
> 
> 
> 



According to this rule, the distance traversed by a uniformly
accelerated body in a given time is equal to the distance which would
be traversed in the same time in a uniform motion with its middle
degree of speed (half of the sum of the initial and final speed). A
number of conclusions follow from this theorem:


1. Since the medium degree of uniformly acquired latitude of
speed [in uniformly difform motion] which commences at no-degree
and ends at some finite degree is precisely half the degree of that
terminating the motion, it follows that a body which moves with a
uniformly difform motion commencing at no-degree and terminating at
some finite degree [of speed] traverses less than half [the distance]
which would be traversed by a body moving uniformly during the same
time with a speed equal to a given degree terminating the
 motion.[44]
 (DML 2020: SS 27, 272; see also Wilson 1960: 123)
2. [Given that] the medium degree of uniformly difform latitude of
speed which commences at some degree and ends at another is greater
than half the degree terminating the latitude of its more intense
extreme, it follows that a body which moves with uniformly difform
motion commencing at some degree of speed and terminating at another
traverses more than half the distance which would be traversed by a
body moving uniformly during the same time with a speed equal to the
most intense speed in the uniformly difform
 motion.[45]
 (DML 2020: SS 28, 272; see also Wilson 1960: 123)
3. In a uniformly difform motion commencing with no-degree speed and
ending with some finite degree, the distance traversed in the first
half of the time is one third of that traversed in the second half.
And conversely, in a motion in which the speed decreases uniformly
from some degree to no-degree the distance traversed in the first half
of the time is three times that traversed in the second
 half.[46]
 (DML 2020: SS 35, 276; see Wilson 1960: 124).



By the end of the chapter, Heytesbury states that an infinite number
of sophisms can be made concerning speed in local motion, as well as
the intension and remission of such speed (DML 2020: SS 61,
 294).[47]


#### 3.3.2 Alteration



In the subsequent chapter dealing with qualitative changes, Heytesbury
expresses the same concerns. He takes part in the debate over one of
the most widely discussed problems in the fourteenth-century
philosophy of nature focused on the possible "measure" of
the intension and remission of forms. Like most of the Oxford
Calculators, Heytesbury accepts Scotus's and Ockham's
addition theory and assumes that the intension of form is the result
of adding qualitative parts. As an Ockhamist, Heytesbury emphasizes
that the term "motion" in alteration connotes nothing else
than the degrees or latitude of quality that are successively
acquired. A basic assumption is that qualitative intensity is a
certain kind of linearly ordered and additive quantity. Thus, as in
the case of local motion, Heytesbury divides the latitudes of motion
of alteration into the latitude of motion of intension and the
latitude of motion of remission; and the latitude of the latitude of
motion into the latitude of the acquisition of the latitude of motion
and the latitude of the loss of latitude of motion. In modern
terminology, these latitudes correspond to latitudes of speed and of
acceleration and deceleration, respectively. Since there is no limit
to speeds of alteration nor to rates of acceleration or deceleration
of that speed, those latitudes are all infinite.



Heytesbury presents three different ways of "measuring"
the speed of alteration in which the intensity of a quality varies
form one point to another, or from instant to instant of time, or from
point to point of the moving body: (1) by the degree (*gradus*)
induced (a subject is considered to be altered faster when a higher
degree is induced); (2) by the latitude of form acquired in comparison
to the size of the subject (the speed of alteration is proportionally
greater in a larger body); and (3) by the latitude acquired in a given
time, regardless of the size of the subject undergoing alteration.
Heytesbury holds the third position and claims that as in the case of
local motion, the speed of alteration is to be measured at the point
of the body where the rate of change is most rapid, that is, where the
maximum latitude of form is acquired ([RSS] 1494: fol. 51ra).


#### 3.3.3 Augmentation



Heytesbury deals with augmentation as with pure increase in size,
which is the same process as rarefaction; more precisely, he deals
with the speed of rarefaction as measured by the effect produced.
There are three ways of "measuring" the speed of
augmentation acquired in time: (1) by the maximum quantity acquired;
(2) by the latitude of rarity or rareness; and (3) by the ratio of new
size to previous size ([RSS] 1494: fol. 60rb). He holds the third
position, according to which the ratios correspond to different
degrees of augmentation. Augmentation, like the two other motions, can
be uniform or difform. An augmentation is uniform if in equal
intervals of time, the body undergoing augmentation increases in size
by the same ratio. While discussing the above mentioned positions,
Heytesbury makes broad use of the new calculus of ratios invented by
Kilvington and Bradwardine.


## 4. Influence



As a logician, Heytesbury strongly influenced logic in fourteenth and
fifteenth-century Britain, and there is a notable fifteenth and early
sixteenth-century reception in Italy where multiple editions of his
commentaries on [RSS] and [SCD] were printed (Ashworth & Spade
1992; de Rijk 1975, 1977). Despite his dismissive attitude toward this
genre, Heytesbury's treatise is one of the most influential late
medieval texts. These influences include the British logical tradition
(see Pironet 2008), the Italian logical
 tradition,[48]
 and John Mair's
 circle,[49]
 and the treatise is referred to even as late as 1688 (De Benedictis
1688: 580-590; for other influence, see Spade 1989: 273).
Heytesbury's catalogue of solutions to the paradoxes, where the
positions are (correctly or not) attributed to fourteenth-century
British logicians, is further developed and sometimes defended against
Heytesbury's harsh and sarcastic dismissal. (R2.2) is developed
by extending the proof and supplemented by saying that paradoxical
sentences (or all sentences, for that matter) signify their own truth.
Furthermore, Heytesbury's contribution to epistemic logic in the
second chapter of [RSS] appears to have sparked further debate in the
upcoming centuries, as documented by the fourteenth and
fifteenth-century development of the British and Italian treatises
*de scire et dubitare* which typically revolved around the
sophisms formulated by Heytesbury and where Heytesbury's
position was among those which were discussed, sometimes with an
explicit attribution to Heytesbury (Boh 1993 and Hanke 2018a and
2018b). As a philosopher, Heytesbury significantly influenced later
Oxford Calculators: John Dumbleton, anonymous author of the
*Tractatus de sex inconvenientibus*, Richard Swineshead, author
of one of the most famous treatises *Liber Calculationum*, and
continental fourteenth and fifteenth-century philosophers, e.g., John
Casali, John of Holland, Petrus of Mantua, Cajetan of Thiene, Giovanni
Marliani, and Paul of Venice (see Wilson 1956: 25-28, Rommevaux
2017: 154-160, Jung, Podkonski 2020: 130-31,
148-149, 151). The Middle Degree Theorem was widely discussed
during the fourteenth and fifteenth centuries, and later played an
important role in the formulation of the law of free fall (see Damerow
*et al.* 1992: 161-200, esp. 232-236).