## 1. Life and Works



Richard Kilvington (we know almost seventy different spellings of his
name) was born at the beginning of the fourteenth century in the
village of Kilvington, Yorkshire. He was the son of a priest from the
diocese of York. He studied at Oxford, where he became Master of Arts
(1324/25) then Doctor of Theology (ca. 1335) (for biographical
details, see Kretzmann and Kretzmann 1990b, Jung-Palczewska 2000b).
His academic career was followed by a diplomatic and an ecclesiastical
one, working in the service of Edward III and taking part in
diplomatic missions. His career culminated in his appointment as Dean
of St. Paul's Cathedral in London. Along with Richard Fitzralph,
Kilvington was involved in the battle against the mendicant friars, an
argument that continued almost until his death in 1361.



Other than a few sermons, all of Kilvington's known works stem
from his lectures at Oxford. None is written in the usual commentary
fashion, following the order of books in the respective works of
Aristotle. In accordance with the fourteenth-century Oxford practice,
the number of topics discussed was reduced to certain central issues,
which were fully developed with no more than ten questions in each
set. The reduction in the range of topics is counterbalanced by deeper
analysis in the questions chosen for treatment. Some of
Kilvington's questions cover fifteen folios, which in a modern
edition yield about 120 pages. His philosophical works, the
*Sophismata* and *Quaestiones super libros De generatione et
corruptione*, composed before 1325, came from his lectures as a
Bachelor of Arts; the *Quaestiones super libros Physicorum*
(1325/26) and *Quaestiones super libros Ethicorum* (1326/32)
date from his time as an Arts Master; after he advanced to the Faculty
of Theology, he penned eight questions on Peter Lombard's
*Sentences*, composed before 1334.



The manuscript tradition of Kilvington's works is complex. The
question on the *Ethics* and on the *Sentences* are
often edited in different ways, with many omissions and corrections by
scribes (Michalowska 2011b). The most complicated, however, is
the tradition of his questions on Aristotles' *libri
naturales*, that is, on the *On generation and corruption*
and the *Physics*. It testifies perfectly well to the
dispersion of these works, which probably occurred shortly after the
questions were written. The reconstruction of these commentaries was
possible thanks to cross references to be found in the questions (Jung
& Podkonski 2020: 16-17).


## 2. Method in Science



Like many other English thinkers, Kilvington was a leader in three
main disciplines: terminist logic, mathematical physics, and the new,
"mathematical" theology. Methods and procedures developed
in the first two disciplines were used in the third. The application
of the terminist logic and the refutation of the Aristotelian
prohibition against *metabasis* resulted in Kilvington's
broad use of logic and mathematics in all branches of inquiry to gain
certain knowledge. Four types of measurements were involved in his
research. The predominant form of measurement by limits, i.e., by the
beginning and ending of successive or permanent things
(*incipit/desinit*), by the first and last instants of the
beginning and ending of continuous processes (*de primo et ultimo
instanti*), and by the intrinsic and extrinsic limits of
capacities of passive and active potencies (*de maximo et
minimo*), does not appear to be straightforwardly mathematical,
though it raises mathematical considerations insofar as it prescribes
measure for natural processes, such as changing the elements, heating,
changing the speed of motion, acquiring knowledge, getting better or
forming the habit to perform meritorious deeds. The second type of
measurement, by latitude of forms, describes processes in which
accidental forms are intensified or diminished in terms of the
distribution of natural qualities such as heat or whiteness or moral
qualities such as love, grace, sin, will, or desire. In his
measurement of the intension and remission of forms, Kilvington is
interested in determining how the highest degree of a quality can be
introduced into a subject already possessing the same quality to a
certain degree by undergoing an alteration, and consequently in
establishing the possibility of a most intense or diminished degree
of, e.g., heat and cold, or virtue and vice. The third type of
measurement, the strictly mathematical, employs a new calculus of
compounding proportions to measure speed in local motion or speed in
the distribution of love. Finally, the fourth type of measurement
describes a 'rule' permitting the comparison of
infinities, treated as infinite sets containing infinite subsets, and
determining which of them is equal, lesser, or greater.



Kilvington's concept of infinity is one of his greatest
achievements. He takes for granted Ockham's definition of a
continuum, according to which each continuum is a being that contains
an actually infinite number of smaller and smaller proportional parts.
When we are given any continuum, we are also given its halves, the
halves of these halves and, consequently, Ockham claims, an infinite
number of its parts. The actual infinity is not understood here as an
actually infinite continuum "that is so great that there cannot
be any greater one". Thus, it is impossible for a process of
infinite division to be accomplished, for if it were accomplished, it
would no longer be infinite, but finite. For Ockham, an actually
infinite continuum is a set of actually existing parts which could be
enumerated without end. Kilvington supports Ockham's view in all
his works, starting from *Sophismata* up to the commentary on
the *Sentences*. Ockham, however, never gives any mathematical
proof for this statement and limits himself to exploring Duns
Scotus's geometrical arguments against indivisibility.
Kilvington puts more effort into demonstrating that infinite
proportional parts actually exist in a continuum (S
42[43]: 112-15; Kretzmann 1990b: 307; see also Jung &
Podkonski 2009a; Podkonski 2016). In his philosophical and
theological works, Kilvington provides many examples proving that
infinite sets are not equal, that is, that they are not of the same
density. He also explains how to establish a one-to-one correspondence
of the elements of such infinite sets. Kilvington affirms that all
created infinities, being infinite, are equal 'in
multitude.' This, however, does not imply that infinities are
necessarily equal 'in magnitude', since two created
infinities can be at once equal and unequal with respect to multitude
of their parts or their dimensions. Infinities can also be
'qualitatively' unequal because there is a qualitative
difference between God's infinity and every created
being's infinity. God as the most perfect is the only being
absolutely infinite, whereas creatures as contingent beings can only
be relatively infinite. As Podkonski affirms "Kilvington
must have been convinced that an infinite set can be put in one-to-one
correspondence with infinite subsets. And it is worth noting that this
paradoxical feature of infinite sets nowadays serves as a criterion of
determining them" (see Podkonski 2009: 142).



Kilvington employs all types of measurement to describe different
cases both real and imaginable. Having adopted Ockham's
ontological minimalism, he claims that absolutes, i.e., substances and
qualities, are the only subjects that undergo changes. Therefore, such
notions as 'motion,' 'time,'
'latitude,', and 'degree' do not have any
representation in reality and only serve as tools for the faultless
description of the processes of various changes. Thus, Kilvington
contrasts things that are really distinct with things distinguishable
only in reason, i.e., in the imagination. Imaginary cases are
descriptions of hypothetical situations; to be imaginable means to be
possible, i.e., not such as to generate a contradiction. Everything
imaginable must be logically possible within a natural framework. So
we can imagine the void and formulate the rules of motion in it,
because there is no contradiction either on the part of nature or on
the part of God for the void to exist. Any theory, whether it
describes an imaginary case or an observed phenomenon, must be
coherent. Therefore, logic and mathematics are the best methods to
describe any reality: factual or imaginable.



There are four levels in Kilvington's *secundum
imaginationem* analyses. These levels may be classified according
to their increasing abstraction and decreasing probability. On the
first level, there are imaginary cases which are potentially
observable and which might occur in nature, such as Socrates'
becoming white. On the second level are imaginary cases which cannot
be observed, even though they belong to the natural order. These cases
illustrate the necessary consequences of the application of rules
properly describing natural phenomena--the best example being the
Earth's rectilinear motion, which is caused by its desire to
unite the center of gravity with its own center. On the third level
are cases not observable but theoretically possible, such as reaching
infinite speed in an instant. The fourth level concerns cases that are
only theoretically possible. Kilvington uses the last two groups of
imaginable, i.e., hypothetical cases to reveal inconsistencies in
received theories, especially from Aristotle, demonstrating
mathematically the paradoxes that arise from Aristotle's laws of
motion. If hypothetical cases do not involve contradiction, there is
no reason to reject them or exclude them from the realm of speculation
(see Jung 2016).



Kilvington's *secundum imaginationem* analyses go
together with his *ceteris paribus* method: he assumes that all
circumstances in the case being considered are the same, and that only
one or two chosen factors, which change during the process, cause
changes in the results.


## 3. Logic



Kilvington's *Sophismata*, written before 1325, is his
only logical work. A *sophisma* or sophism is neither a
standard paradox of disputation nor a sophistical argument but a
statement the truth of which is in question. The first sophism
Kilvington discusses typifies the basic structure: a statement of the
sophism sentence followed by a case or hypothesis, arguments for and
against the sophism sentence, the resolution of the sophism sentence
and reply to the arguments on the opposing side, ending with an
introduction to the next sophism sentence.



Kilvington's sophisms are meant to be of logical interest, but
they also pose important questions in physics or natural philosophy.
In constructing his sophisms, Kilvington sometimes makes use of
observable physical motion and at other times appeals to imaginable
cases that have no reference to outside reality. Although the latter
cases are impossible physically, they are theoretically possible,
i.e., they do not involve a formal contradiction. At one point he
writes, "thus even though the hypothesis supposed there is
impossible in fact ... it is nevertheless possible per se; and
for purposes of the sophisma, that is enough [*unde licet casus
idem positus sit impossibilis de facto ... tamen per se
possibilis est; et hoc sufficit pro sophismate*]" (S 29: 69;
tr. Kretzmann and Kretzmann 1990b: 68).



The first eleven sophisms deal with the process of whitening, in which
the motion of alteration is conceived as a successive entity
extrinsically limited at its beginning and end. There is no first
instant of alteration, claims Kilvington, but only a last instant
before the alteration begins; likewise, there is no last instant of
alteration, but only the first instant at which the final degree has
been introduced. There is no minimum degree of whiteness or speed
gained in motion, but rather smaller and smaller degrees ad infinitum
down to zero, since the qualities change continuously. Integers are
potentially infinite because one can always find a higher integer, but
not actually infinite since there is no single infinite number. In
Kilvington's view, since any continuity--e.g., time, space,
motion, heat, whiteness--is infinitely divisible, it can be
spoken of quantitatively and measured in terms of infinite sets of
integers. The subjects of sophisms 29-44 reveal
Kilvington's special interest in local motion with respect to
causes, i.e., active and passive potencies, and effects, i.e., time,
distance traversed, and speed in motion. He considers both uniform and
difform motion caused by voluntary agents and points out the
questionable measure of instantaneous speed through the comparison of
speed in uniform and accelerated motion (see Kretzmann 1982).



The last four sophisms are ostensibly connected with epistemology and
the logic of knowledge, i.e., sentences on knowing and doubting
involving intentional contexts, such as S45: "You know this to
be everything that is this". The most interesting among them is
S47, "You know that the king is seated", where Kilvington
calls some rules of obligational disputation into question (see
Kretzmann and Kretzmann 1990: 330--47; d'Ors 1991). In the
opinion of Stump, "what Kilvington has done in his work on S47,
by his change in the rule for irrelevant propositions, is to shift the
whole purpose of obligations" (Stump 1982: 332).


## 4. Natural Philosophy



Although Kilvington does not enjoy the reputation in natural
philosophy that he does in logic, recent research reveals that his
questions on Aristotle's *De generatione et corruptione*
and *Physics* inspired Thomas Bradwardine's theory of
motion and his famous rule of the proportions of velocities in motion
(see Jung 2022a). Both works stemmed from lectures Kilvington
delivered in the Arts Faculty before 1328, i.e., before
Bradwardine's famous treatise *On the Proportions of
Velocities in Motions*.



Like most medieval natural philosophers, Kilvington accepts
Aristotle's general rules for change to occur, stating that: 1)
there cannot be a change without an active capacity (*virtus
motiva*) and a passive capacity (*virtus resistiva*),
because without resistance, motion would not be temporal; 2) the
proportion of an active capacity to a passive one must be the
proportion of *maioris inequalitatis*, that is, it must be
greater than one. These conditions are sufficient for the occurrence
of natural local motion. Kilvington is primarily interested in the
description of the natural motions of a mixed and simple body both in
a medium and in a vacuum. In his discussion of various motions, such
as the motion of the Earth or the motion of weights, he often relies
on the laws of statics presented in the works of Jordanus de Nemore,
Archimedes, and Euclid (DM: 221, 223, 239-40).



Aristotle's most famous law of motion, "Everything that
moves is moved by something," is only quoted a few times when
Kilvington discusses the processes of transformation of the elements.
In such cases, it is imperative that, for example, hot fire come into
contact with iron and act directly on it, turning its cold into
heat.



Taking substance and quality as the only two absolute things (*res
absolutae*), Kilvington concludes that the reality of motion is
limited to what is in motion: the places, qualities, and quantities
that are successively acquired. Consequently, he is more interested in
measuring local motion in terms of the actions of the causes of
motion, the distance traversed, and the time elapsed, than in the
intensity of speed. In his questions on physics, Kilvington describes
the differences between various types of change, such as generation,
alteration, and augmentation; determines the measurements of active
and passive capacities that cause change; finds rules for the division
of various types of continua; and formulates a mathematically
consistent rule of local motion. He examines the problem of motion
from two points of view: with regard to its causes and with regard to
its effects (see Jung-Palczewska 2002b).



Kilvington's discussion of the measure of motion with respect to
causes, or what we would call his 'dynamic' analysis, has
both a physical aspect involving relations between acting and
resistive potencies, and a mathematical aspect involving concepts of
continuity and limits. He is interested in how a capacity is to be
bounded if it is active or passive potency. Is it subject to weakening
or strengthening? Is it mutable or immutable? How do we determine the
boundaries of an active potency if a body moves in a medium that is
not uniformly resistant?



The mathematical character of Kilvington's theory can be seen in
his use of two kinds of limit for continuous sequences: an intrinsic
boundary (when an element is a member of the sequence of the 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*). Although he did not formulate strict rules
about the different types of division of continua, his study
cases reveal that he approved the following conditions for the
existence of limits:


1. There must be a range in which the capacity can act or be acted
on, and another range in which it cannot act or be acted on.
2. The capacity should be capable of taking on a continuous range of
values between zero and the value which serves as its boundary, and no
other values.



According to Aristotle (*Physics* VIII), motion occurs only if
the ratio of acting capacity (a force F) to passive capacity (a
resistance R) is a ratio of major inequality, i.e., when it is greater
than 1. Kilvington maintains that any small excess of force over
resistance suffices for motion to commence and continue; thus,
whenever force is greater than resistance, motion occurs (DM:
215-25, 252-58). This assumes that force (an active
capacity) is bounded by a minimum upon which it cannot act
(*minimum quod non*), i.e., by the resistance that is equal to
it (DM: 225-29, 258-59). For a passive capacity,
Kilvington accepts the *minimum quod sic* limit "with
respect to circumstances". He accepts Aristotle's claim
that to establish a limit for Socrates' passive capacity of
vision, we should point to the smallest thing he can see. However, it
also happens that we cannot see such a great thing as a cathedral if
we are too close to it. Therefore, a passive capacity cannot be
described by a *minimum quod sic* limit in each case (DM:
229-33, 259-62).



It is clear that Kilvington's belief that mathematics is the
legitimate method of describing natural phenomena also allowed him to
formulate a new rule of local motion. Kilvington agrees that the
proper way of measuring the speed of motion is to describe variations
of proportion of forces to resistances. He was aware that
Aristotle's and Averroes's rules of motion, given in
*Physics* VII, were not universally applicable, and was
convinced that the correct calculus of proportions was formulated by
Euclid in his *Elements*. As a consequence, he noticed that the
proper understanding of Euclid's definition of operations on
proportions necessitates a new interpretation of Aristotle's and
Averroes's theory of motion. On the one hand, Euclid's and
Archimedes's theory of operations on proportions concludes that
doubl~~i~~ng a ratio corresponds to squaring the fraction which we
form from the ratio. On the other hand, Aristotle's and
Averroes's statements clearly indicate that speed is
proportional to the proportion of an active power to resistance, which
is not squared but simply multiplied by two. Having noticed the
inconsistency between these two views, Kilvington first presented two
main arguments against the Aristotelian proposition and finally
concluded that, when talking about a power moving one half of a mobile
(that is, something moveable), Aristotle means precisely a double
ratio between F and R; when talking about a power moving a mobile
twice as heavy, he means taking the square root of the ratio of F : R.
The new mathematical rule describing the speed of motion is in accord
with those of Aristotle only in one case: if the ratio of the power of
the mover to that of its mobile is 2 : 1, the same power will move
half the mobile with exactly twice the speed. Kilvington's
reinterpretation of Aristotle's and Averroes's rules of
motion reveals that the speed cannot be described by multiplication
alone. Kilvington's calculus provides values of the ratio of F
to R greater than 1 : 1 for any speed down to zero, since any root of
a ratio greater than 1 : 1 is always a ratio greater than 1 : 1. And,
with his additional assumption that any excess, however small, of an
active power over resistance is sufficient to initiate and continue
motion, he may have described a very slow motion with a speed greater
than 0 and less than 1 (0 > v < 1). Hence, he avoids a serious
weakness of Aristotle's theory, which cannot explain the
mathematical relation of F and R in very slow motions, at speeds
less than 1 (Jung 2022a: 76-77).



Kilvington applies his new rule of motion to describe the uniformly
difform motion of mixed bodies and the motion of simple bodies both in
a medium and in a vacuum. Explaining Kilvington's theory, we
must keep in mind that temporal motion is possible only if there is
some resistance playing the role of a *virtus impeditiva*. The
motion of a mixed body in a medium is easiest to explain, since then
the acting power has to overcome the external resistance of the medium
as well as the internal resistance of that element which is moved away
from its natural place. The local motion of a simple body in a medium
is not problematic either, since it can be explained by its natural
desire to attain the natural place determined by its heaviness or
lightness. Nor does Kilvington have a problem with explaining the
natural motion of a mixed body in a vacuum. He once again adopts
Ockham's view that if there were a void, it would be a place.
Since place in the Aristotelian sense is something natural that has
essential qualities, it determines the natural motion of elementary
bodies and, moreover, their inclination to remain at rest in their
natural places. Accordingly, says Kilvington, one could imagine a void
in four natural spheres, which despite being empty preserve the proper
qualities characteristic of the natural places of earth, water, air,
and fire. Hence, the temporal motion of a mixed body in a void is the
result of the natural inclination of heavy or light elements to attain
their natural places. The heaviness and lightness play the roles of
force and resistance, respectively. Although there would be no
external resistance in a vacuum, the motion of a mixed body could
occur without any difficulty (see Jung-Palczewska 1997).



The temporal motion of a simple body in a vacuum is the most difficult
to explain. In the opinion of Averroes, a simple body such as a piece
of earth has an elementary form, prime matter, and different
quantitative parts. Because form cannot resist matter, no resistance
can come from its qualitative parts. But there can be resistance
coming from its quantitative parts resisting one another. This time
Kilvington uses Grosseteste's idea, recognizing that the
temporal motion of a simple body in a vacuum is possible due to the
internal resistance that arises when the peripheral parts of a body
that move along oblique lines resist its central parts moving along
straight lines. Such an internal resistance produces motion, and does
not impede; nevertheless, it guarantees temporal motion. Consequently,
if a vacuum existed, the natural motion of a simple body would be
possible. Moreover, the speed of such a motion would be the fastest,
since there is no external resistance to be overcome (Jung-Palczewska
2002a: 169).



In Kilvington's opinion, lower and greater resistance, i.e. the
rarity or density of the medium, is responsible for faster or slower
motion, and the distance of the medium traversed in motion is
responsible for the longer or shorter time consumed in
motion.



According to Kilvington, in order to characterize changes in the speed
of motion, one must analyze the problem of local motion in its
kinematic aspect, which takes into account distance traversed and time
consumed. He correctly recognizes that to measure the speed of motion
that lasts some time, it is enough to establish relations between time
and distance traversed. In his opinion, the same distances traversed
in equal intervals of time characterize uniform motion. Uniformly
difform motions are characterized respectively: accelerated motion by
the same distance traversed in a shorter interval of time; decelerated
motion by the same distance traversed in a longer time; or, according
to modern definitions: the motion is uniformly difforme if the speed
changes by equal amounts in equal intervals. Kilvington provides a
proper description of free fall, noting that speed increases by the
same amount in each time interval, and that speed is equally slower at
the beginning of each interval than at the end of it. Difformly
difform motion is characterized by unequal distances traversed in
unequal intervals of time; the speed of such motion changes unequally
(see Jung-Palczewska 2002b).



Although Kilvington never abandoned Aristotle's physics, he
often goes beyond his theories to resolve paradoxes resulting from his
laws, giving the impression that, behind the facade of
Aristotle's principles and concepts, Kilvington is an Ockhamist.
Despite the fact that Kilvington never explicitly mentions Ockham,
there is no doubt that he not only knew the *Venerable
Inceptor's* opinions, but also accepted them as a natural
way to interpret the works of the Philosopher.


## 5. Ethics



The third Aristotelian work on which Richard Kilvington commented
during his regency in the Arts Faculty was the *Nicomachean
Ethics*. His commentary on the second and tenth books of the
*Ethics* takes a form of ten questions, which deal only with
selected issues that were the subjects of Kilvington's lectures
at Oxford: e.g., creating and destroying moral virtue, free acts of
volition, the behavior of honest people and the delight taken in their
actions (or conversely, the punishment of those who act evilly), and
questions concerning particular virtues such as courage, generosity,
magnanimity, justice, and prudence. As Michalowska has shown,
Kilvington uses terminist logic and mathematical physics to solve
ethical problems (see Michalowska 2011a, 2016).
Michalowska also shows that, just as he did in his questions on
the *Physics*, Kilvington follows Ockham's minimalist
ontology by treating ethical qualities--i.e., vices and virtues,
cognition and wisdom, good and evil--as volition-like objects,
calling them *res*. Being real things and not merely mental
concepts, they can be measured by addition, subtraction, and division
into parts, for they undergo change via increase or decrease and so
have varying degrees of intensity. Such changes--e.g., undergoing
punishment for an evil act--cannot be instantaneous and must
happen in time. Each change is the result of an acting power
overcoming resistance. In the case of moral acts, the changes do not
produce any external effects but internal modifications in terms of
the intensity of virtues and vices. When a vice acts upon a virtue, it
causes its continuous change, and so someone's courage can vary
in intensity. Virtues and vices are opposed in Kilvington's
physical theory, so it is impossible for someone to be vicious and
virtuous at the same time, although it is possible for them to be
generous at one time, miserly at another.



The increase or decrease of a moral quality is either an effect of the
impact of the opposite quality (or a change in the degree of intensity
of the same quality), or else the result of human external acts. For
example, frequent generous actions lead to an increase of generosity.
Performing morally good actions intensifies the virtues, whereas the
constant practice of evil diminishes them. Virtues and vices can be
described in terms of different degrees of intensity, so one can say
that a man can be more or less generous during his lifetime. And just
like physical qualities, Kilvington states that the intensity of a
moral quality has only an extrinsic limit, so that one cannot perfect
one's virtue infinitely.



Virtues and vices have an absolute or relative character, and can be
possessed absolutely (*simpliciter*) or in a certain respect
(*secundum quid*). There are highest--i.e., most
perfect--degrees of intensity of our moral virtues, but there are
no absolutely greatest degrees, like Platonic ideas. In
Kilvington's opinion, people are never absolutely generous or
virtuous. The ultimate perfection, i.e., the highest degree of moral
virtue, is the product of a person's natural dispositions,
socializations, and moral acts. But since people relevantly differ,
each of us is virtuous in our own way. So too, the highest degree of
moral virtue is unique in each of us. In Kilvington's opinion,
if someone is prudent in the highest degree, they must have all the
other virtues in the highest degree as well (Michalowska 2011a:
488-92).



For Kilvington, prudence is one of the primary virtues. It is a habit,
which cooperates with right reason (*recta ratio*) in the
process of making good or bad decisions. Even though Ockham is not
mentioned by name, his theory of the relation between prudence and
moral knowledge is present in Kilvington's discussion. Ockham
distinguishes two types of moral knowledge. The first, which concerns
universal truths, is gained through learning; the second, which
concerns particular statements and particular situations, is gained
through experience. Prudence is understood in two ways: as knowledge
about singular propositions and as universal practical knowledge. In
his opinion, both types of prudence are gained only through
experience, the former concerning singular statements and the latter
universal practical statements. The former is properly called
prudence, whereas the latter is commonly known as prudence. In
Ockham's view, the first kind of knowledge--i.e., of
universal truths--must be distinguished from prudence concerning
singular statements. The second type of knowledge, however, is the
same as prudence, since it is also gained through experience
(*Quaestiones* q.6, a.10). Kilvington identifies two kinds of
moral knowledge. The first is called *scientia necessaria*,
which is composed of general statements and refers to universal truth.
The other is called *scientia ad utrumlibet*, which comprises
particular statements. The *scientia necessaria*, achieved by
means of deduction, is not sufficient to make good moral decisions and
so it must be complemented by a reference to *scientia ad
utrumlibet*, achieved by experience (see Michalowska 2016:
13). Gaining knowledge through experience is an indispensable part
becoming prudent. Kilvington states that someone can err with regard
to a moral choice even though they possess certain and complete
knowledge about universal moral truths; a skilled logician is not
necessarily a moral person. To make good moral decisions, one needs a
fully-developed prudence, which is the same as *scientia ad
utrumlibet*. Kilvington claims that someone who possesses moral
knowledge is not automatically prudent, but a prudent person is always
wise (see Michalowska & Jung 2010: 109-111).



Good choices are possible only when the will is supported by prudence.
The problem of free will and free choice is fully elaborated in
Kilvington's *Ethics*, where he presents his
theory--what Michalowska calls a "dynamic
voluntarism". Kilvington distinguishes three types of human
volitional acts: willing, nilling, and not-willing. Willing always
wills, and can never be passive or in potency. Even when the will
wants nothing (*velle nihil*), it is willing, and so it cannot
rest and is always determined to an act of willing. Here Kilvington
seems to be directly influenced by Scotus, who claims that the will
cannot be suspended (*Ord.* I d.1). The will is absolutely free
in its acts of willing, and the free will of volition is the primary
principle in the genus of contingent propositions. Since the will is
active all the time, it must decide between its three acts of volition
(*velle volitionem*), nolition (*velle nolitionem*), or
not-willing (*non-velle*). With regard to its own internal
acts, the will is absolutely free. With regard to its external acts,
however, it chooses between wanting something (*velle aliquid*)
and not wanting something (*nolle aliquid*). In these cases,
the will is also absolutely free to make such a choice.



For Kilvington it is obvious that prudence plays an essential role in
producing good moral acts. When the habit of prudence is not fully
developed, the will is indecisive. Repeated good moral decisions makes
it hesitate (*non-velle*) less, so that the agent is able to
reach a decision in any context, whether affirmatively *velle*
or negatively *nolle*. Supported by fully developed prudence,
the will makes proper and good moral choices more easily or even
effortlessly (see Michalowska 2016: 14). Kilvington, however, is
of the opinion that most of us rarely make good moral decisions
because we often remain in doubt, stuck in the state of
*non-velle*.


## 6. Theology



In theology, Kilvington applied the new methods of terminist logic and
mathematics to the most popular issues discussed in the
fourteenth-century, such as human and divine love, human and divine
will, God's absolute and ordained power and human free decision,
divine knowledge of future contingents and predestination. Nothing is
considered separately from the Creator; therefore, Kilvington relates
each human action to God. Kilvington's questions on the
*Sentences* testify to his involvement in the lively,
contemporary discussions with semi-Pelagians. Thomas Bradwardine
argued with John Duns Scotus, William Ockham, Adam Wodeham, Robert
Holcot, and Thomas Buckingham, all of whom he regarded as Pelagians.
In his commentary on the *Sentences*, on which he lectured in
1332-1333, Ockham was the theologian he most fiercely attacked.
Kilvington most likely lectured on the *Sentences* in the same
year. It also seems obvious that each took part in other's lectures as
*socii*. In his question *Utrum actus volunatis per se malus
sit per aliquid*, Kilvington debates and criticizes
Bradwardine's views (see Jung & Michalowska 2022: 6).



Kilvington repeatedly cites the opinions of Augustine and Anselm,
which he usually accepts completely, only sometimes reinterpreting
them according to his theory. For Kilvington, however, the greatest
authority is John Duns Scotus, whose views he fully accepts and
successfully develops. First of all, Kilvington adopts Scotus's
distinction (*Ord*. I, d. 44, q. u.) between the absolute and
ordained power of God:



> 
> 
> I say along with Scotus, in *Sentences* I, distinction 44, that
> God's power is called ordained insofar as it is the principle of
> doing something in conformity with right law with regard to the
> established order. God's power is called absolute insofar as it
> exceeds God's ordained power, because by virtue of it God can
> act against the established order. Generally, I say that by virtue of
> absolute power God can do everything that is not contradictory. Thence
> jurists use the terms *de facto* and *de iure* in a
> similar manner, for example, a king can do *de facto* that
> which is not in accordance with law and does not agree with it.
> God's power is called ordained insofar as it is a principle for
> doing something in conformity with a right law with regard to the
> established order (DV: 168).
> 
> 
> 



The established order of nature is the result of God's ordained
power, but God can also act against this order by his absolute power.
In Kilvington's opinion, both of God's powers are
intensively infinite *simpliciter*, but even so, God's
absolute power is infinitely greater than, i.e., infinitely more
powerful than, his ordained power.



Like Scotus, Kilvington is convinced that *potentia dei
absoluta* is a power that really is or can be actualized by God.
Miracles would be examples of God acting against the natural order.
Individual situations also show that God can deviate from laws
established in the natural order, reflecting God's particular
judgment.



Kilvington's theory of *potentia dei absoluta*
and*ordinata* serves to underline the contingency of
creation and the freedom of divine will. Here Kilvington abridges
Scotus's opinions (*Lectura* I, dist. 39) and reorders
his arguments, taking into account only those most useful for his own
theory. Kilvington claims that God's knowledge, existence, and
will are the same as God's essence. However, with regard to
God's absolute knowledge, assertoric statements about the past
and present and contingent statements about the future have the same
certainty since they are absolutely necessary, whereas with regard to
the God's ordained knowledge, they have only ordained
necessity:



> 
> 
> [I]t can be said that God not always knows with the same certainty
> what came about and that which is going to come about. <It is
> so,> because God's knowledge is equally certain with regard
> to future contingents as with regard to necessary events; however, it
> is possible for divine knowledge not to be certain with regard to
> God's creatures, because a creature <i. e., a human> can
> know that (s)he is sitting with as much certainty as that a human is
> an animal. Still, one certainty concerns contingent and the other
> necessary things. Therefore, this consequence is false:
> '<God> knows with equal certainty that which happened and
> that which is going to happen'. Yet, it may happen that
> <God> does not know the things that are going to happen, and
> thus <God> may not know the things that happened. This
> conclusion, however, is true by virtue of God's absolute power.
> And by virtue of God's ordained power it does not follow that
> all events (past, future, and present) are equally known by God (DV:
> 182).
> 
> 
> 



Kilvington's affinity to Scotus may also be seen in his theory
of future contingents. He is in agreement with Scotus
(*Lectura* I, dist. 39, qq. 1-5) in saying that only an
instant in time is present since only the 'now' exists.
Therefore, Aquinas's analogy to God sitting at the center of a
circle and being present to the whole of time fails, whereas
Scotus's concept of a radius sweeping out the circumference of
the circle is correct, since the entire circle does not exist all at
once. He quotes Scotus *verbatim*:



> 
> 
> God is "like the center and all time flowing from the past to
> the future <is> like the circumference, and no matter how the
> circumference moves, it is always equally distant from the center, and
> no matter whether time flows into the past or the future, it is always
> the present time to God" (DV: 183).
> 
> 
> 



Consequently, 'now' moving in one way or another, moves
like a point on the circumference of a circle.



Kilvington, like Scotus, also rejects the view that God knows future
contingents via Ideas because Ideas necessarily represent what they
represent, as in the sentence, "Socrates is white", where
it is said that Socrates is white. Although Kilvington does not
explain his position clearly, it seems that he takes for granted
Scotus's explanation (DV: 178). Scotus says that perhaps Ideas
could represent simple or complexes terms necessarily, although, as
Chris Schabel puts it: "They could not represent contingent
complexes (...), which we can call *X*. If God had the
Idea only, eternally he would know only part of the contradiction, and
there would be no contingency. If He knew both parts, *X* and
~*X* , He would know contradictories to be true simultaneously.
Second, since Ideas represent both futures that are possible but will
not exist, and futures that are possible and will exist, one needs to
posit a way to distinguish between what will exist and what will not
exist" (Schabel 2000: 42).



Kilvington is also in agreement with Scotus when saying that secondary
causes cannot originate any contingency because of the necessity in a
chain of causes. Therefore, a contingency observed in the action of
secondary causes must be routed to the first cause, which is God. To
know contingents, God first has to choose one of two contrary
statements, since otherwise, i.e., when God had an act of knowledge
before his act of will, he would have had only necessary ordained
knowledge about the natural order, which he has already established,
and he would not know contingents. Consequently, God would have only
partial knowledge about one side of a contradiction (i.e., he would
know only one of two contradictory statements, e.g., "Antichrist
will be" or "Antichrist will not be"), and his will
would not be absolutely free. Therefore, contingency must be located
in God's will and not in God's intellect. Kilvington
believes that future contingent events are such because God knows that
they are future contingents and not vice versa. God's accepting
(*beneplacitum*) will, with respect to future contingents, is
naturally prior to God's knowledge, because the following
consequence is true, "God wants *A* to happen; therefore,
God knows it will happen," whereas this is false, "God
knows it will happen (viz., that Socrates will sin); therefore, he
wants him to sin" (DV: 175).



Following Scotus, Kilvington maintains that at the same instant in
which the divine will wills *A*, it is able not to will
*A*, and thus he accepts Scotus's synchronic contingency;
nevertheless, he does accept Scotus' concept of *instantiae
naturae*. To save God's absolute free will and at the same
time to avoid the prospect of mutability in God's
decision-making, Kilvington asserts that by his absolute power God can
make himself not will *A*, while *A* is what God, by his
ordained power, wills in that particular instant, and this happens in
eternity. Such internal acts of God do not cause any changes in
his external acts, since God's will is immutable.
Consequently, by his absolute power, God can undo the past without
causing any change in history, such as, for example, Abraham's
non-existence. God, whose absolute power is infinitely infinite, can
cause the past not to be the past, because there is no past for God.
To say that God cannot undo the past simply means that God has not
revealed a given thing, such as the restoration of one's lost
virginity. It is also not contradictory for God's absolute power
to create the world from eternity, or to annihilate it. The
world's annihilation would not be less just than its continued
existence, since God's justice stems from his essence, which,
like his power, is absolute and ordained. However, God cannot do so by
his ordained power, since, even being infinite, his ordained power is
infinitely less powerful than his absolute power.



Kilvington believes that not only God's will but also
God's knowledge is contingent with regard to future contingents.
This produces a distinction between types of prophecies, with some of
them being absolutely necessary and others being conditionally
necessary and in this way contingent. Everything absolutely and
unconditionally revealed by God necessarily comes about with ordained
necessity. Everything revealed by God's ordained
power--e.g., articles of faith--depends on God's will
and could be changed. Once revealed, however, they would have ordained
necessity, and so they would form a new law. The contingency *ad
utrumlibet* of future events is guaranteed by the conditions under
which such future events come about. Although God absolutely foreknows
that something will come about by absolute and ordained power, he
reveals it absolutely with regard to circumstances, that is,
conditionally. In conditional revelation, the change of status from
being contingent to not being contingent is warranted by the
conditions. When the revealed conditions are fulfilled, the future
event will no longer be contingent. The same concerns future
contingent deeds. Solely those foretold by God are necessary, but the
others are not because they depend on people's voluntary
choices. Nothing that depends on created free will *ad
utrumlibet* can be revealed absolutely relative to
God's ordained power (see Jung & Michalowska 2022:
62-63 )



Like the divine will, human free will is endowed with absolute and
ordained power to perform good and bad deeds. Kilvington argues that
God produces the substance of people's volitional act, that is,
their ability to make choices, and interacts with human free will to
intensify good deeds and to produce certain degrees of these acts. He
argues that if free will were solely responsible for the degree of its
act, then someone could love God whether endowed with grace or not,
and someone without grace could love God above everything else, which
is absurd. God only contributes to the performance of good deeds, and
evil deeds result from the choices of human free will alone, because
"truly we are good because of God, and <we are> evil all
by ourselves."(Jung & Michalowska 2022: 46) It is
obvious, however, that even if humans perform good deeds, God does not
have to take them into account or reward humans for their meritorious
acts. People cannot be the primary cause of their reward, but they are
the primary cause of their punishment because they voluntarily make
wrong choices and perform evil acts. The capacity to sin
testifies to the freedom of people to make free choices (see
Jung & Michalowska 2022: 51-52 )



In his discussion of predestination, Kilvington again appears to be
following Duns Scotus, who maintains that predestination is simply a
choice of salvation and as such has only positive connotations.
Reprobation is solely a consequence of human misdeeds. God is the
primary and sole cause of predestination from eternity, but he also
takes into account the conduct of creatures, that is, their good or
bad use of free will. Kilvington does not accept the concept of double
predestination. The good use of free will, which is a concomitant act
of the intellect, may be a partial cause of predestination when it
comes to the decision on baptism. The bad use of free will is the
primary cause of sin and punishment and the only cause of the
condemnation of adults who are endowed with grace. Although God is the
first cause of people being accepted without any preceding merit or
redemption, God is not the first cause revoking grace without any
preceding demerit. That is because justification and acceptance are
the same as God's goodness and clemency (that is, God's
substance), whereas forsaking and rejection are merely acts of
God's righteousness (that is, God's justice). Therefore,
the evil deeds of grace-filled adults are the only reason why God, by
divine righteousness, abandons people (see
Jung & Michalowska 2022: 56-57)



In Kilvington's *Sentences* commentary, there are also
many places where he appears to follow Ockham, and while Scotus is
often cited by name, Ockham remains in the background. Still,
knowledge of both Scotus and Ockham is crucial to understanding
Kilvington's thought, as his own contributions are often the
result of blending these two strands of fourteenth-century Franciscan
theology.


## 6. Impact and Influence



Besides the particular topics he discussed, Kilvington's
extensive use of *sophisma* argumentation, his mathematization
of ethics and theology, and his frequent use of hypothetical
(*secundum imaginationem*) cases, place his thought in the
mainstream of fourteenth-century English philosophy and theology. His
teachings on logic were influential both in England and on the
Continent. Richard Billingham, Roger Rosetus, William Heytesbury, Adam
Wodeham, and Richard Swineshead were among the English scholars who
benefited from Kilvington's *Sophismata*. His
*Quaestiones super De generatione et corruptione* was quoted by
Richard Fitzralph, Adam Wodeham, and Blasius of Parma, and his
*Quaestiones super Physicam* was familiar to the next
generation of Oxford Calculators, William Heytesbury, John Dumbleton
and Roger Swineshead (who also may have influenced Parisian masters
such as Nicolas Oresme and John Buridan). But Thomas Bradwardine was
perhaps the most famous beneficiary of Kilvington's theory of
motion. In his renowned treatise, *On the Ratios of Velocities in
Motions*, Bradwardine included most of Kilvington's
arguments for a new function describing the relation of motive power
and resistance (see Jung 2022a). Kilvington's views on future
contingents were discussed by masters at the University of Vienna in
the first decade of the fifteenth century such as Nicholas of
Dinkelsbuhl, John Berwart of Villingen, Peter of Pulkau, and the
Carmelite Arnold of Seehausen. His questions on the *Ethics*
and the *Sentences* enjoyed a reputation not only in Oxford but
also Paris and were frequently quoted by Adam Junior, John of
Mirecourt, Johanes de Burgo, and Thomas of Krakow (see
Jung-Palczewska 2000b).