## 1. *Obligationes* as a branch of logic



As a branch of logic, obligations came to flourish with European
medieval logic from the thirteenth century onwards, and they slowly
vanished when logic as a discipline, a.k.a. 'scholastic
logic' retreated in the early modern times. During the peak
period in the fourteenth century, many or perhaps even most European
logicians wrote a dedicated text on obligations (*de
obligationibus*). These texts were either stand-alone treatises or
sections in overall presentations of logic. Of the former kind,
treatises by Walter Burley ("Treatise on Obligations",
written c. 1302; Latin text in Green [1963], English translation in
Kretzmann and Stump [1988]), Roger Swyneshed (written 1330-35;
Spade [1977]) and Robert
 Eland[1]
 (written 1335-1370; Spade [1980]) have been particularly
important in shaping the modern understanding of the topic. Of the
latter kind, for example William Ockham's *Summa logicae*
(written c. 1323; Boehner, Gal, & Brown [1974]) and Paul of
Venice's *Logica magna* (written 1397-98; Ashworth
[1988]) can be mentioned as outstanding examples.



The technique of obligations assumes a dialectical setting or even
something like a game between two persons. This setting is usually
very succinctly described in the texts themselves. In a nutshell, what
happens in an obligational disputation is that one person, the
opponent, is putting forward propositions
 (*propositiones*)[2]
 and another person, the respondent, is expected to give direct
evaluations of these propositions as they come. There are two main
answers, granting the proposition or denying it. Only rarely, the
texts allow other answers like doubting the proposition (see e.g.,
Uckelman, Maat and Rybalko 2018). The sequence of propositions put
forward and answered is taken to be a temporal sequence, but most
authors thought that the truth values of the propositions are not to
be taken to change during the disputation.



The sequence of propositions evaluated this way is launched by the
opponent giving the respondent a specified obligation, in the paradigm
case that of granting an obviously false proposition. This specified
obligation must be respected throughout the sequence, regardless of
how complex the sequence gets. The underlying idea is that although
the specified obligation relieves the respondent from always following
truth in the evaluations, the requirement of some kind of logical
coherence is not suspended.



These special obligations come in different varieties. Most medieval
treatments of the branch of logic are divided by discussions of
different varieties. The most common is called
'*positio*', or 'laying down'. In
*'positio*' a false proposition is laid down at the
beginning of the disputation, and the respondent remains obligated to
grant it whenever the opponent puts it forward. Modern attention has
been mainly on *positio*. (On '*sit verum*',
see Spade 1994-97 and Uckelman 2015.)



Walter Burley, for example, discusses six kinds of obligations (Green
1963, Kretzmann and Stump 1988):



(1)
'*Positio*', where a specified proposition
needs to be always granted.
(2)
'*Depositio*' where a specified proposition
needs to be always denied.
(3)
'*Dubitatio*', where a specified proposition
needs to be always doubted.
(4)
'*Institutio*' or
'*impositio*' where a new meaning is given to some
linguistic item.
(5)
'*Sit verum*', where a specified proposition is
to be taken as true.
(6)
'*Petitio*', where some other requirement is
put on the respondent.


Of these kinds only (1), (2), and (4) are commonly addressed in later
discussions of the genre, but all of them can be found in texts other
than Burley's treatise. However, for a general understanding of
the technique it does not seem necessary to account for varieties
other than '*positio*'.



Almost all the medieval dedicated treatments of obligations proceed
through giving rules on how the respondent ought to give the
evaluations, or what is to be granted and what is to be denied. The
essential rules are based on the idea that each proposition put
forward must be recognized as 'relevant' or
'irrelevant' in relation to the obligation given at the
beginning. Thus, if the respondent is given as a
'*positio*' the proposition 'Plato is
running', the proposition 'Plato is moving', and
indeed any proposition following from 'Plato is running'
is to be deemed sequentially relevant. Similarly, any proposition
incompatible with 'Plato is running' is to be deemed
incompatibly relevant: 'Plato is sitting' will have to be
denied. On the other hand, 'Socrates is running' is to be
deemed irrelevant at least if it is put forward immediately after the
'*positio*', since there is no logical connection
between Plato and Socrates running.



The core two rules of '*positio*' require the
respondent to grant any sequentially relevant proposition, and to deny
any incompatibly relevant proposition. These two rules are accepted by
all authors, but under different interpretations. Since obligational
disputations are technically constituted by sequences of propositions
with their respective responses, what is relevant and what is not at a
particular step need not be determined solely by the
'*positio*'. Also, within the context of
obligations, it cannot be taken as univocally given how
'follows' (*sequitur*), or the logical consequence
relation, is to be understood.



The core terminology of obligations centers around 'has to be
granted' (*est concedendum*) and 'has to be
denied' (*est negandum*). These expressions were widely
used in medieval European academic texts even outside logic, but
normally with the implication that things that have to be granted are
true, and vice versa, what is true must be granted. In obligations
treatises, both the connection and the contrast of these expressions
to truth is of special importance. What is true and what has to be
granted are often not the same thing, because propositions are granted
as following from the false *positio*, or on the basis of some
other special obligation. That the terms 'has to be
granted' and 'has to be denied' are used in general
academic contexts outside obligations can presumably be attributed to
the fact that medieval academia predominantly thought about arguments
in a dialectical context. Obligations as such would not always be at
issue when such terms are used. But it does seem that at least in the
fourteenth century, obligations as a branch of logic was taken to
address core aspects of what it means when something has to be granted
in an argumentative setting.



The medieval logical genre of sophismata (see the entry on
 sophismata
 in this encyclopedia) appears especially relevant to obligations. A
typical sophism in this sense consists of a case (*casus*), a
sophisma sentence, arguments for and against the sophisma sentence
based on the case, and a solution. The case is often imagined rather
than real or even genuinely possible. One relatively natural way to
think of a sophism would thus be to take the case (*casus*) as
a *positio* in the sense of obligational disputations. Indeed,
Richard Kilvington's comments of the obligational rules in his
*Sophismata* show him thinking that obligational rules regulate
how one ought to evaluate sentences on the basis of the case
(*casus*) in a sophisma. (See Kretzmann & Kretzmann 1990a
and 1990b.)


## 2. Background in Aristotelian logic



The core structure of an obligational disputation is closely similar
to what we find from Plato's dialogues. In a so-called Socratic
*elenchus* Socrates is putting forward questions, and most of
them can be and are answered by 'yes' or by
'no'. The sequence of questions in effect builds up an
argument leading to a situation where the person responding to
Socrates' questions is often forced to admit the opposite of
what was put as a starting point in the dialectical encounter. As a
core background assumption, the person answering has to grant to
Socrates some propositions because of other propositions that he has
already granted. In a somewhat implicit form, the obligational
principle that whatever follows needs to be granted is thus present
already in the Socratic *elenchus*. (See e.g., Robinson
1953)



There are however very central differences between the medieval
obligational disputations and the Socratic *elenchus*. Indeed,
it seems possible to sum up the similarities as two core ideas:


1. Both are dialectical encounters constituted by one party putting
forward propositions and the other evaluating them.
2. Both are taken to proceed in a form that is logical, although in
the case of Plato the term 'logical' needs to be taken
with caution for anachronism.



The main difference between Socratic *elenchus* and medieval
obligational disputations is in what happens at the beginning. While
Socrates normally challenges his discussion partner on a substantial
philosophical thesis whose truth would be important to know, there is
no real issue under scrutiny in obligational disputations. The thesis
studied in a Socratic *elenchus* is never known to be obviously
false, but this is the standard way to start an obligational
disputation.



As is well known, Aristotle developed dialectical encounters in a
strictly theoretical manner in his *Topics*. One question that
is still open in the modern scholarship on obligations is to account
for the role of Aristotle's *Topics* in the development
of the obligations genre in the middle ages. Medieval commentaries on
*Topics* do not regularly take up to discuss obligations
systematically (see Green-Pedersen 1984). One interesting exception is
Boethius of Dacia's commentary (written in the 1270s;
Breen-Pedersen & Pinborg 1976), but the rules that he proposes for
obligational disputations appear quite different from what we find
from the mainstream authors of the genre itself (e.g., Burley,
Swyneshed, and Eland). In any case, even judging on the basis of
Boethius' treatment of the technique in the commentary, the
technique does not seem a direct descendant of the so called
'topical' tradition in logic.



There is another medieval descendant of the structure of the Socratic
*elenchus* arising more directly from the Aristotelian approach
to the technique in his *Topics*. During the thirteenth
century, a number of treatises "On the Modes of Opposing and
Responding" were written. (Cf. De Rijk 1980,
Perez-Ilzarbe 2011). These treatises discuss and develop
techniques of winning in a disputation, or in other words ways in
which the opponent can force or trick the respondent into granting the
opposite of the thesis under dispute, or ways in which the respondent
can avoid such contradictions. These treatises form an interesting
parallel to the obligations technique, but even there the core feature
of obligations - discussing an obviously false thesis - is
not to be found.



Aristotle discussed various times (e.g., *Prior Analytics* and
*Metaphysics*) what may happen in reasoning that starts with an
impossible hypothesis, and these discussions have to be taken into
account as a second aspect in which obligational disputations have
roots in Aristotle's works. When reasoning from an impossible
hypothesis one is not to worry about truth but about what follows
logically. This is one of the central technical issues in obligational
disputations. Discussions concerning an impossible hypothesis have a
long history that can be traced all the way from Aristotle to the
Latin medieval period in which obligational disputations properly so
called were developed. And it can also be pointed out that at the
earliest stages of the Latin medieval development of obligational
technique, assuming an impossibility (*positio impossibilis*)
was one, or perhaps even the standard kind of obligation to be
considered. (See Martin 1993; 1997; 2007; Kukkonen 2002; 2005.)



Judging from texts known to modern scholars, the obligational
technique seems to have achieved a structured format during the
thirteenth century, perhaps already in its first half. The oldest
currently known dedicated treatises on the obligational technique seem
to be a pair of anonymous treatises edited by L. M. de Rijk as
*The* *Emmeran Treatise on False Positio* and *The
Emmeran Treatise on Impossible Positio*. (De Rijk 1974.) De Rijk
dated the treatises in his edition to the early thirteenth century,
and there seems to be no reason to doubt this rough date. The rules of
the technique are in these treatises almost the same as in Walter
Burley's *Treatise* *on* *Obligations*,
which might be taken as the standard bearing treatise, written around
1302 (Green 1963; Kretzmann and Stump 1988). In terms of historical
development, the obligational technique belongs to what is often
called *logica moderna*, which built on Aristotelian logic but
took clear distance from the Aristotelian texts themselves.


## 3. Walter Burley's standard theory



According to Walter Burley's presentation of the obligational
theory in the species of *positio*, the respondent faces a
sequence of propositions that he must either grant or deny, one by
one. The sequence starts with a *positum* that is normally a
false sentence which the respondent has to grant. Then the following
propositions are evaluated in accordance with what Burley calls the
essential rules of obligations. Three of these rules can be found from
practically all medieval discussions of the technique. According to
them, the *positum* must be granted if it is put forward,
anything following from the *positum* must be granted, and
anything incompatible with the *positum* must be denied. (Cf.
Green 1963, Kretzmann and Stump 1988. Modern discussions of
Burley's rules can be found in e.g., Ashworth 1981; Keffer 2001;
Spade 1982; Spade and Stump 1983; Stump 1985; Yrjonsuuri 1994 and
2001.)



In Burley's formulations the core rules are as follows:



1.
Everything that is posited and put forward in the form of the
*positum* during the time of the *positio* must be
granted.
2.
Everything that follows from the *positum* must be
granted.
3.
Everything incompatible with the *positum* must be
denied.


Already these rules show that the technique (in Burley's
version) is aiming at consistency in the answers. He seems to have
this insight in mind when he gives later in the treatise a
"useful rule" that "all responses must be directed
to the same instant", as if implying that any description of one
instant of time needs to be consistent.



The above three basic rules are not sufficient, as they cover only
what follows from the *positum* alone or is incompatible with
it alone. An important part of the obligational technique is to
understand all propositions put forward in an obligational disputation
as either relevant (*pertinens*) or irrelevant
(*impertinens*). A proposition that follows or is incompatible,
is relevant, and others are irrelevant. Relevant propositions are to
be answered according to their logical status, or in other words
according to rules 2. and 3. On how to answer irrelevant propositions,
Burley gives the following rule:



4.
If it is irrelevant, it must be responded to on the basis of its
own quality.


As Burley carefully explains, "its own quality" means what
we take its truth value to be. Thus, a proposition known to be true
must be granted, and one known to be false must be denied. Answering
with doubt is also possible, according to Burley.



The rule concerning irrelevant propositions is generally accepted by
all medieval authors writing on obligations, but in importantly
different exact versions. We will come back to some of these versions
in the following, but first we must make clear about Burley's
version. He puts special attention to the following question: a
proposition is relevant if it follows or is incompatible, but follows
from exactly what, or is incompatible with exactly what? As Burley
sees it, this relevance must be determined on the basis of all answers
that have been given in the disputation so far. On the first step
immediately after the *positio*, relevance is based on
following from the *positum*, or being incompatible with it.
But suppose the respondent must evaluate some salient irrelevant
proposition on the first step. Then this answer must be taken into
account in determining whether a later proposition follows or is
incompatible in the disputation. That is, the respondent must take
care not only that he does not deny his *positum* but that all
his answers remain consistent as a set.



Burley formulated these ideas as further clarifications of rules 2.
and 3., but let us here give them new numbers:



5.
Everything that follows from the *positum* either together
with an already granted proposition (or propositions), or together
with the opposite of a proposition (or the opposites of propositions)
already correctly denied and known to be such, must be granted.
6.
Everything that is incompatible with the *positum* together
with an already granted proposition (or propositions), or together
with the opposite of a proposition (or the opposites of propositions)
already correctly denied and known to be such, must be denied.


A crucial effect of these rules is that at every step where an
irrelevant proposition is answered, the class of relevant propositions
expands. Burley makes this explicit through a "useful
rule" that "one must pay special attention to the order
[of the propositions]". Something that is irrelevant at the
first step might not be so anymore later in the disputation.



Now, consider the following disputation as an example:





|  |  |  |
| --- | --- | --- |
| *Positum* | You are in Rome. | Accepted as possible. |
| Proposition 1 | You are not in Rome or you are a bishop | Granted, true and irrelevant (rule 4). |
| Proposition 2 | You are a bishop. | Granted, follows (rule 5). |



Disputation D1




According to Burley's approach, proposition 1 has to be
evaluated as irrelevant, because it does not follow from the
*positum* alone, and because of the first disjunct it is taken
to be true even if the respondent is not a bishop (apologies to any
reader actually in Rome). It must therefore be granted. However, this
answer makes the other disjunct relevant, because the first disjunct
happens to be the negation of the *positum*. Thus, Proposition
2 must be granted. Indeed, by suitable selection of propositions,
almost anything at all may have to be granted at the later steps of a
disputation with a false *positum*. Burley gives a useful rule
to make this clear: "when a false contingent proposition is
posited, one can prove any false proposition that is compossible with
it." (Kretzmann and Stump 1988, 391.)



Burley uses the word "compossible"
(*compossibile*). It is important to notice that at the time
Burley was writing, the concept of consistency was not a standard part
of technical logical language. He and his contemporaries did not speak
about consistent sets of sentences, because they did not have the
vocabulary. This seems like a simple reason why Burley never says
directly and explicitly that all answers must stick to consistency.
Instead, it seems more probable that building the concept was a part
of the development in the obligational technique.



In any case, the above example and the principle that almost anything
may have to be granted are not problematic in Burley's eyes.
This reflects a clear-eyed understanding that obligational
disputations were not about what is true and what is not, they were
about logical relationships between sentences.


## 4. Revisions of the rules



Walter Burley's rules of obligational disputations can be called
the standard approach, but they were also challenged in the fourteenth
century discussions. All these challenges may be taken to be directed
at understanding in what sense the respondent must avoid contradicting
himself in an obligational disputation.



*Time and modality*. When discussing modalities in his
*Ordinatio* (I, d. 38, 2), John Duns Scotus rejects a principle
which he calls an obligational rule, namely that with a false
*positio* one must "deny the present instant". And
in fact, Walter Burley (writing roughly at the same time) mentions
this as a valid, useful rule. The reasoning behind the rule is the
necessity of the present. For example, suppose "you are
running" is the false *positum* when you are in fact
standing in a university hall and participating in an obligational
disputation. Now, if one considers the possibility that you are
running, this possibility must according to Burley be referred to some
other time, because of the necessity of the present. You cannot be
running now since you are standing now, but afterwards you can of
course run. (Kretzmann and Stump 1988, 394.)



Duns Scotus denied the necessity of the present and thought that it
need not be incoherent to consider actually false propositions as
possible at the present instant. That you are standing does not
exclude that you are running is a logical possibility, it only
excludes actual running. This kind of thinking has been found to
reflect a major change in how alethic modalities are conceived. After
Scotus, many thinkers thought of modalities as related to
conceivability and logic, and not so much as based on being in
someone's power, or true at some time or other. That is, the
possibility of a proposition became taken to be an issue unrelated to
the flow of time.



This change is visible in obligations treatises. When Burley mentions
denying the present instant as a useful rule, he takes up a topic that
is found in the earlier treatises, but in fact not often in later
treatises. It looks as if Scotus' rejection of the rule became
accepted.



*Counterfactual thinking*. Richard Kilvington comments on
obligations in his *Sophismata* (S47; Kretzmann & Kretzmann
1990a; 1990b). In his view, it should not be possible to force the
respondent to grant any falsities whatsoever, and the order in which
propositions are put forward ought not to have an effect on their
evaluation. As Kilvington sees it, the respondent should grant only
those falsities which would be true if the *positum* were true.
Kilvington gives the following example (S47, (q); Kretzmann &
Kretzmann 1990b, 127-8):





|  |  |  |
| --- | --- | --- |
| *Positum* | You are in Rome. | Accepted as possible. |
| Proposition 1 | 'You are in Rome' and 'you are a bishop'
are similar in truth value. | Denied, since you are not a bishop. |
| Proposition 2 | You are a bishop. | Denied, does not follow. |



Disputation D2




Assuming that the respondent is not in Rome or a bishop both of the
propositions mentioned in Proposition 1 are false. According
Burley's rules, Proposition 1 would have to be granted as
irrelevant and true. But after having granted Proposition 1, it may
appear that Proposition 2 follows and must be granted. According to
Kilvington, that should not be so. Indeed, he goes directly on to
generally denying Burley's useful rule that anything false that
is compossible with the *positum* may have to be granted. By
Kilvington's comparison, the respondent is bound or obligated by
the *positum* "you are in Rome" to answer as he
would if he were in Rome. Thus, he ought not to grant "you are a
bishop" unless he is a bishop.



When concluding his discussion of the example, Kilvington describes
Proposition 1 as one "that is true now and that would not be
true in virtue of its being in fact as is signified by the
*positum*" (S47, (cc); Kretzmann & Kretzmann 1990b,
131). Kilvington is thus inviting the respondent to think
counterfactually when defending a false *positum*, and as he
continues, he applies this to how one ought to answer when evaluating
a sophisma. One ought to grant what would be true and deny what would
be false, given the *positum*.



An anonymous treatise *De arte obligatoria* probably from the
1330s (edited and translated from a Merton College manuscript in
Kretzmann & Stump 1985) seems to develop a central aspect of
Kilvington's suggestion into a complete set of rules for
obligational disputations. Just like Kilvington, the anonymous author
thought that in an obligational disputation the order of the
propositions ought not influence the answers. The core change to
Burley's rules is that rules 5. and 6. above are revised so that
instead of referring to earlier answers, they must refer to
propositions which would be true if the *positum* was true.
Thus, a proposition must be granted as sequentially relevant, if it
follows from the *positum* together with a proposition that
would be true if the *positum* were true. Similarly, it has to
denied as incompatibly relevant if it is incompatible with the
*positum* together with a proposition that would be true if the
*positum* were true. The result is that all answers must be
given "in accordance with how it is possible for things to
be", or "on the basis of the facts then", as the
anonymous author puts it.



This approach was not followed in later theories of
*obligationes*. It may be that obligations were thought of as
concerned with relations between propositions rather than building a
description of a possible situation that could be imagined. The
anonymous author's approach requires the respondent to imagine
what the situation would be like if the *positum* were true and
give evaluations on the basis of that situation. Thus, the evaluations
would not be based on the logical relations between the various
propositions of the disputation itself. If obligational disputations
were conceived as a tool for examining logical relations between
propositions, such counterfactual thinking may not be the best way to
proceed.



*Accepting contradictions*. There is also another direction
into which Kilvington's comments can be developed. In another
comment on the example D2 above, Kilvington says that the respondent
could grant Proposition 1 and deny Proposition 2 (S47, (bb); Kretzmann
& Kretzmann 1990b, 131). His point is that when the propositions
'you are in Rome' and 'you are a bishop' are
claimed to be similar in truth value, they are mentioned rather than
used. Given that the meanings of these strings of words is not
logically necessary, it might be that Proposition 1 (claiming that the
two mentioned propositions are similar in truth value) has nothing to
do with the respondent being in Rome or being a bishop. Thus,
Proposition 2 does not follow from the *positum* and
Proposition 1 together.



This approach is even stronger in a related discussion where
Kilvington rejects what is nowadays called Tarskian biconditionals, or
the equivalence between a proposition and the claim that the
proposition is true. Logically, inferring that you are in Rome from
the claim that the sentence 'you are in Rome' is true is
valid only if 'you are in Rome' means that you are in
Rome. (S47, (t); Kretzmann & Kretzmann 1990b, 128.)



It seems that in his treatise on *obligationes*, Roger
Swyneshed developed his rules for obligational disputations with an
approach like this (the text is edited in Spade 1977). Especially
interesting in this respect is his discussion on *impositio*,
or the kind of obligational disputation where a new meaning is given
to some linguistic entity. According to Swyneshed, even if the
respondent were obligated by the *impositio* that 'you
are a donkey' means that God exists, the respondent must deny
being a donkey. The proposition "'you are a donkey'
is true" must however be granted. According to Swyneshed's
principle, "because of an *impositio* of some proposition
the answer to be given to it is not to be changed" (Spade 1977,
254). It seems thus that the *impositio* influences only
situations where the proposition at issue is mentioned rather than
used.



Recent scholarship has found Swyneshed's rules for
*positio* more puzzling (see e.g., Spade 1982; Stump 1981;
Yrjonsuuri 1994 and 2001). They are very clear in terms of how to
answer according to them, but it seems very problematic to give a
sensible interpretation to the set of evaluations to be given
according to them. Swyneshed simply rejects Burley's rules 5 and
6. Just like Kilvington and the above-mentioned anonymous author of
*De arte obligatoria*, Swyneshed does not allow the order of
the propositions to have any effect on the answers. However, he does
not allow counterfactual considerations either, but rather thinks that
the respondent must rely on logical relations between the propositions
in giving the evaluations.



As Swyneshed recognizes, his rules may lead the respondent into
granting triads or larger sets of propositions containing
contradictions. In his formulation, "the conclusion must be
granted that three incompatible propositions must be granted, and
four, and so on" (Spade 1977, 274). The sequence of propositions
in D1 above is an example of this. The respondent cannot according to
Swyneshed rely on Burley's rule 5 to grant Proposition 2.
Instead, the respondent should grant a disjunction but deny both of
its parts. Also, it may happen that a conjunction must be denied
despite granting both of its parts. These structural combinations are
explicitly accepted by Swyneshed (Spade 1977, 257).



It must be kept in mind that in each case of granting a contradictory
set by Swyneshed's rules, at least one evaluation is given on
the basis of one proposition being relevant, and at least one other
answer on the basis of that proposition being irrelevant. Thus, if the
evaluations of irrelevant propositions were not allowed to be mixed
with evaluations of relevant ones, these contradictions would not
arise. However, it seems that this is not how Swyneshed thought about
the rules, since he explicitly allows the contradictions without
discussing the idea of two domains of interpretation.



In the context of obligations, Swyneshed does not say that it would be
acceptable to grant or deny a contradictory pair of propositions in
the same disputation: he discusses contradictory triads, not pairs.
However, in his discussion on the Liar's paradox, he goes as far
as explicitly accepting that in some cases both contradictories are
false (cf. Read 2020). This shows that Swyneshed's understanding
of what contradictions are and how they behave in logic is somewhat
outside the logical mainstream. Thus, it may be that further study may
shed more light on how exactly he thought about allowing
contradictions in obligational evaluations of sets of
propositions.



Historically, it seems that counterfactual thinking as an approach to
obligations was mainly rejected, but Swyneshed's approach
garnered some following. Both of these approaches appear to take issue
with Burley's principle that the order of propositions has
crucial significance in how they are to be evaluated. Since
Swyneshed's way of rejecting this principle found some
followers, this fact may signal that his approach was taken to be
closer to what really is at issue in obligations as a genre of logic.
However, generally speaking Burley's approach became the
mainstream towards the end of the fourteenth century. Perhaps, taking
the order of propositions into account was taken to be less
problematic than Swyneshed's acceptance of contradictory
triads.


## 5. Winning the game



If we look at obligational disputations from the modern logical point
of view, it may appear hard to understand how the sequence of answers
should be understood. One tempting alternative would be to think that
the idea would be to describe a counterfactual situation. As pointed
out above, this may be what Kilvington and the anonymous author were
aiming at. (See also Spade 1982, 1992, and 1993.) However, the
mainstream of medieval authors writing on obligations did not accept
their approach.



The mainstream understanding of the current research into the
obligations treatises seems to emphasize in different ways that
according to the standard approach (exemplified here by Burley's
rules) the set of all answers will be consistent. Thus, it may seem
reasonable to understand obligational disputations as a kind of
consistency maintenance game. Given that in modern logic the concept
of a possible world is often understood in terms of what a potentially
infinite consistent set of sentences describes, the sequence of
answers in an obligational disputation can perhaps be interpreted as
an expanding description of a possible world.



The problem with this interpretation is that it does not fit
Swyneshed's rules, which were accepted by some authors as a
viable alternative to Burley's approach. As the rules stand, the
set of all answers cannot be understood as a description of a possible
world, because it need not be consistent. The set of answers to
relevant propositions is consistent even according to
Swyneshed's rules, but understanding that set as describing a
possible world runs into the difficulty that that set does not expand
from the logical closure of the *positum*, and thus it does not
describe any possible world more fully than the *positum*
alone.



A crucial feature of the Socratic *elenchus* is that the
answerer is led to deny what has already been granted. Avoiding it to
happen that the respondent denies the *positum* is a central
objective in an obligational disputation, and Burley's rules
appear to require that this objective is achieved through a consistent
sequence of answers. Swyneshed allows contradictory triads. In other
words, the set of answers may become inconsistent but the respondent
is not allowed to grant any "contradictory incompatible with the
*positum*" (Spade 1977, 274). It appears that even in
Swyneshed's case a crucial interest is to explicate logical
relations between the propositions put forward by the opponent, and
especially as they relate to the *positum*. As a general
description of what obligational disputations were about, consistency
may not be exactly the right word. Perhaps it would be more exact to
describe obligations as aiming at a dynamic understanding of logical
relations within a set of propositions.



The square of opposition was an important tool in Aristotelian logic
for understanding contradiction and other relations between pairs of
assertoric predicative propositions formed with the same categorematic
terms. As has been recognized, John Buridan and other logicians
expanded this theory to the octagon of opposition to cover also modal
propositions. Obligational disputations can be understood as an
expansion into another direction. An obligational disputation
abstracts from the predication as the standard form of proposition and
turns from the binary logical relations into examining logical
relations within a dynamically constructed sequence of propositions.
(Cf. especially Dutilh Novaes 2005; 2007; 2011; Dutilh Novaes and
Uckelman 2016; Strobino 2011; Uckelman 2013; Yrjonsuuri 2000;
2015.)



In his discussion of *obligationes* from the very end of the
fourteenth century, Paul of Venice compares obligations with another
genre of logic, saying that "the matter of obligations is only
the matter of consequences in a more subtle style" (Paul of
Venice in Ashworth 1988, 32). One aspect of what Paul of Venice means
by this statement is presumably that both obligations and consequences
are concerned with non-syllogistic logic. To this extent, this
description of the theory of *obligationes* seems correct even
from the modern vantage point. However, it does not seem quite right
to think that an obligational disputation would try to spell out the
logical relation between an antecedent and a consequent, or even a set
of premises and a conclusion. The logical relations considered in an
obligational disputation are not distinctively binary. It seems that
they are often approached as logical relations becoming explicit only
in a dynamic sequence of three or more propositions.