## 1. Preliminary Considerations


### 1.1 A genealogy of modern conceptions of consequence



In his much-discussed 1936 paper 'On the concept of logical
consequence', Tarski presents two criteria of material adequacy
for formal accounts of logical consequence, which jointly capture the
'common notion' of logical consequence (or so he claims).
They are formulated as the following condition:



> 
> If in the sentences of the class *K* and in the sentence
> *X* we replace the constant terms which are not general-logical
> terms correspondingly by arbitrary other constant terms (where we
> replace equiform constants everywhere by equiform constants) and in
> this way we obtain a new class of sentences *K*' and a
> new sentence *X*, then the sentence *X*' must be
> true if only all sentences of the class *K*' are true.
> (Tarski 2002, SS2.3)
> 



In more mundane terms, the two core aspects that Tarski attributes to
the so-called common notion of logical consequence can be formulated
as:



(TP)
*necessary truth-preservation*: it is impossible for the
antecedent to be true while the consequent is not true;
(ST)
*substitution of terms*: the relation of consequence is
preserved under any (suitable) substitution of the non-logical terms
of the sentences in question; this is now often referred to as the
*formality criterion*.


Different accounts of logical consequence can be (and have been)
formulated on the basis of (TP) and/or (ST): they can be viewed as
both necessary but independent components of the notion of logical
consequence, as Tarski seems to suggest in the passage above; they can
also be viewed as closely related, in particular if (TP) can be
*reduced* to (ST) (i.e. satisfaction of (ST) would entail
satisfaction of (TP) and vice-versa) -- a view that Etchemendy
(1990) attributes to Tarski; or one may hold that the actual core of
the notion of (logical) consequence is (TP), and that (ST) simply
specifies a particular subclass of valid consequences, often referred
to as *formal consequences* (Read 1994).



Tarski correctly identified these two features as key components of
the notion of logical consequence as entertained by philosophers and
mathematicians of his time (and also today). But the question arises:
why these two features and not others? In particular, through which
(historical) processes have they come to constitute the conceptual
core of the notion of logical consequence? These questions are even
more pressing in view of the fact that both features have
been challenged as correct renditions of the conceptual core of
logical consequence -- Etchemendy (1990) challenged the
centrality of formality and (ST); Fields (2008) challenged the
centrality of necessary truth-preservation in view of the semantic
paradoxes.



To make further progress in these debates, an important element is
arguably the historical development of the notion of (logical)
consequence over the centuries, so that we may come to understand
where the informal notion(s) of logical consequence come(s) from.
Engaging in an exercise of 'conceptual
history' may allow for a better grasp of the reasons why this
notion (now widely endorsed) established itself as such in the first
place. If these are compelling reasons, then they may count as
arguments in favor of the centrality of formality and necessary
truth-preservation; but if they rest on disputable, contentious
assumptions, then the analysis may provide elements for a critical
evaluation of each of these two components as truly constitutive of
the concept of logical consequence.



From this point of view, the historical developments in the Latin
Middle Ages, in particular from the 12th to the
14th century, occupy a prominent position. As will be
argued, it is in this period that concepts and ideas inherited from
Greek Antiquity (Aristotle in particular, but also the ancient
commentators) were shaped and consolidated into conceptions of
consequence that bear a remarkable resemblance to the Tarskian
condition of material adequacy presented above. Thus, an analysis of
these historical developments is likely to contribute significantly to
our understanding of the current notion(s) of logical consequence.



Besides variations of (TP) and (ST), criteria of relevance,
containment, causation and explanatoriness also appear frequently in the writings
of medieval authors (as well as earlier authors). As Normore (2015,
357) puts it, there seems to be



> 
> a pervasive ambiguity of terms like 'ergo' and 'igitur' in
> Latin, 'therefore' in English, and similar particles in other
> European languages between a broadly truth-preserving and a broadly
> causal/explanatory sense.
> 



This ambiguity is reflected in the different core conceptions of
consequence throughout the history of logic: is necessary truth
preservation sufficient, or do we require something else, namely a
deeper causal (and/or epistemic) connection between antecedent and
consequent for the consequence to hold? Indeed, in addition to
variations of (TP) and (ST), there is a third recurrent theme in
medieval discussions on consequence:



(Co)
*containment*: in a valid consequence, the conclusion is
contained/understood in the premises.


Different interpretations of this clause run through medieval
discussions on consequence, ranging from the 12th to the 15th century
and beyond. A number of authors seem to treat the notion of containment in metaphysical and/or semantic terms, while others (in particular the British
authors of the second half of the 14th century) lean more heavily
towards what appears to be an epistemic interpretation (see section
3.4). And just as (TP) and (ST) still very much inform how most of us
think about (logical) consequence, variations of the containment
theme, as captured in (Co), remain pervasive among contemporary
authors, motivating for example the development of a number of
relevance logics (see entry on relevance logic in this
encyclopedia). However, notice that (Co) is also interpreted metaphysically by a number of medieval authors; this metaphysical focus is not nearly as present among contemporary philosophers and logicians (with the exception perhaps of some of the recent discussions on the concept of grounding).



Naturally, as with any historical analysis, an investigation of these
developments has intrinsic historical value in and of itself,
independently of its possible contribution to modern debates. Indeed,
medieval theories of consequence are a genuine medieval contribution:
while medieval authors are clearly taking ancient Greek sources and
ideas as their starting point, the emergence of theories of
consequence as such is a Latin medieval innovation. But as it turns out, following the thread provided by the three key notions (TP), (ST) and (Co) as formulated above provides a suitable vantage point to
investigate the development of the notion of consequence in the Latin
Middle Ages. In other words, historical and conceptual analysis can
easily be combined in this case.


### 1.2 What are medieval theories of consequence theories *of*?



At first sight, it is not immediately clear what the object of
analysis of medieval theories of consequence is (Boh 1982). Is it the
semantics of conditional sentences? Is it the validity of inferences
and arguments? Is it the relation of consequence, construed as an
abstract entity? In fact, at times it seems that medieval authors do not properly distinguish between these different notions. After all, these are very different concepts: a conditional
is a sentence, which can be true or false; an argument or inference is
an act, a consecution of assertions, which can be valid or invalid; a
consequence is a relation between sentential/propositional entities,
which can hold or fail to hold (Sundholm 1998).



However, even though the medieval authors may use the same terminology
to refer to these different concepts, this does not mean that they are
not aware of the relevant differences, in particular between a
conditional and a consequence (Normore 2015). As Buridan
(*Tractatus de Consequentiis* (henceforth TC), 21) remarks, it
is for the most part a matter of terminology: he says he will adopt
the definition of consequence as a true hypothetical sentence, but
then throughout his text also uses the terminology of a consequence
being valid or holding rather than simply being true or false. At any
rate, it seems fair to say that, even though analyses of conditionals
are often in the background (as is especially obvious in Boethius and Abelard with examples such as 'if it is a human, it is an animal', and in analyses of the syncategorematic term
'*si*', 'if' (Spruyt 2018)), the main focus of
medieval theories of consequence tends to be the logical relations
between sentential/propositional components (King 2001; Read 2010),
essentially (though not entirely) in the spirit of modern accounts of
the notion of logical consequence (Shapiro 2005) (Caret and Hjortland 2015). Some modern scholars
(e.g. Spade in his translation of Burley's *De Puritate*; Read
2010) prefer to translate the medieval term
'*consequentia*' as 'inference', but
arguably 'consequence' is a more appropriate translation,
both for etymological and for conceptual reasons.



One may also wonder to what extent medieval theories of consequence
really add anything novel to the Aristotelian logical legacy. Kant
(in)famously claimed that Aristotle had discovered everything there
was to know about logic, and insofar as they deal with logical
relations between sentences, it might be thought that theories of
consequence would not have added anything substantially new to
Aristotle's theory of syllogistic in particular. In fact, the
relations between theories of syllogistic and theories of consequence
at different times seem essentially to fall within one of three
categories:


1. Syllogistic and consequence are essentially disjoint concepts,
each having their own foundations and scope. In such cases, the
framework of Aristotle's *Topics* is often (though not always)
summoned to provide the foundations for non-syllogistic
arguments/consequences.
2. All valid arguments, including non-syllogistic ones, are
ultimately to be reduced to syllogistic arguments, as syllogistic
offers the grounds for the validity of every single valid argument. A
proponent of this approach is the 13th century author
Robert Kilwardby.
3. Theories of consequence are seen as an expansion and
generalization of syllogistic; syllogistic is a special case of
consequence. In these cases, syllogistic is absorbed by consequence, which is a more general framework in that it can deal with a much wider range of inferences (Klima 2016). The 14th century author John
Buridan, for example, treats extensively of syllogisms, both
assertoric and modal, in his treatise on consequence.



It is fair to say that approach 3 became predominant in the
14th century, the golden age of medieval theories of
consequence; but the earlier Boethian view that all valid arguments
(including syllogistic arguments) are valid in virtue of topical rules
can also be seen as belonging to category 3. However, given that
medieval theories of syllogistic are treated extensively elsewhere
(see the entry on medieval theories of the syllogism of this
encyclopedia), in what follows we shall focus on non-syllogistic
consequences/arguments, but with the proviso that many of the
interesting developments in syllogistic in the 14th century
are presented in treatises or chapters on consequence.



Another point worth mentioning is the fact that medieval discussions
of the concept of consequence cover both what we would now describe as
'philosophy of logic' and as 'logic proper'.
As for the latter, a number of medieval authors such as Abelard
(Martin 2004), Burley (*De Puritate*), and Buridan (TC)
formulated rules of inference and proved theorems about them (Klima 2016). Many
authors had for the most part understood the logical behavior of what
we now view as the main sentential/propositional operators, such as
'if...then', 'or', negating terms, as
well as meta-level rules such as the transitivity of consequence,
'from the impossible anything follows', or 'the
necessary follows from anything' (the latter two were however
not unanimously endorsed -- see (Martin 1986), (Read 1993,
2010)). (For discussions of the rules formulated by different authors,
see (Pozzi 1978), (Boh 2001), (Dutilh Novaes 2008)). They also offered
sophisticated investigations of the logical behavior of e.g. modal
terms (Buridan, TC) (Johnston 2014).



Besides this more technical layer, medieval authors also
discussed extensively the very nature of the notion of consequence:
what counts as appropriate grounds for a valid consequence, adequate
definitions, subdivisions of kinds of consequence etc. In what
follows, the predominant focus will be on the 'philosophy of
logic' side of medieval theories of consequence, i.e. how they
articulated this very notion, rather than on spelling out the exact
inferential rules endorsed by the different authors. But some medieval
treatises on consequence also contain a high level of technical
sophistication, even though the language used is the regimented
academic Latin of the time - the only symbolic device present is
the use of schematic letters, which in fact dates back to
Aristotle.


## 2. Early Theories of Consequence


### 2.1 Precursors



One of the most important ancient sources for the development of
theories of consequence by medieval authors is, unsurprisingly,
Aristotle. The *Prior Analytics* and the theory of syllogistic
provided the main model for the correctness/validity of arguments for
centuries, and even though theories of consequences can be seen as a
generalization of the rather narrow theory of validity presented in
the *Prior Analytics*, it is clear that syllogistic remains one
of the key elements in the background. Indeed, the famous definition
of a valid deduction (syllogism) at the beginning of the *Prior
Analytics* is already a formulation of the necessary
truth-preservation criterion (TP):



> 
> A deduction is a discourse in which, certain things being stated,
> something other than what is stated follows of necessity from their
> being so. (24b19-20)
> 



Retracing the historical sources for the development of this notion
would take us too far afield, but it seems that the emergence of the
idea of 'following of necessity' is closely related to dialectical practices of debates (Dutilh Novaes 2021), both in philosophy/logic (Castelnerac and Marion 2009) and in mathematics (Netz 1999). But while it is a
necessary condition, necessary truth-preservation is notoriously not a
sufficient condition for syllogistic validity. For example, as is well
known, Aristotle's syllogistic does not validate the principle of
reflexivity, i.e., 'A implies A' for any sentence A, even
though this principle is the most transparent occurrence of necessary
truth-preservation one can think of. Instead, it seems that
syllogistic validity requires a great deal more to hold (Thom 2010) (Thom 2016).
Interestingly, Aristotle himself suggests that the premises in a
syllogism are the material causes for the conclusion
(*Metaphysics* 1013b19-20; *Physics* 195a18-19),
"in the sense of 'that from which'", thus
somewhat in the spirit of the idea of premises containing the
conclusion expressed in (Co), and more generally the idea that the antecedent is the cause of the consequent (which would become widespread in the 13th century (Spruyt 2018)). Indeed, it has been claimed that



> 
> Ancient logics were all in some sense relevance logics. They insisted
> that for an argument to be valid, conditions must be met that
> guaranteed both that it would be impossible for the premises to be
> true and the conclusion false and that there would be connections of
> various kinds between the premises and conclusions. (Normore 1993,
> 448)
> 



Indeed, besides variations of (TP) and (ST), criteria of causation, relevance and containment also appear frequently in the writings of medieval authors.



The extent to which (ST) is present in the *Prior Analytics* is
also a moot point (Thom 2010) (Dutilh Novaes 2021). Aristotle does not apply the concepts
of form and matter anywhere in his logical writings, but his
consistent use of schematic letters and many of his argumentative
strategies in this work suggest that he relies on something resembling
what we now refer to as the 'logical form' of arguments.
What is not clear is whether Aristotle relies on (ST) merely as a
convenient technical device to capture the more fundamental property
of necessary truth-preservation, or whether for him (TP) and (ST) are
independent core components of the concept of a syllogism/deduction
(or perhaps even that (TP) is to be reduced to (ST)).



The *Prior Analytics* is not the only Aristotelian text
providing the historical background for the development of the notion
of consequence. Equally important is one of his (presumably)
'older' logical texts, the *Topics*; this text,
which even more explicitly than the *Analytics* clearly presupposes a dialectical
background, presents various considerations on how to
argue well in dialectical encounters (see the
entry on ancient logic of this encyclopedia, section 2.1). By discussing which moves are allowed in a debate, it naturally ends up addressing the general idea of 'what follows from
what' (Dutilh Novaes 2021). As we will see, the *Topics* became an important
starting point for discussions on the validity of arguments; the
theory of syllogistic only covers a rather limited range of arguments
(two-premised arguments containing only the four kinds of categorical
sentences), and the framework of the *Topics* was often called
upon to fill in the gap between what syllogistic had to offer and the
much larger range of putatively valid arguments one might be
interested in (Klima 2016).



Two other ancient traditions which may have contributed to the
development of medieval theories of consequence are the Stoic
tradition (see the entry on ancient logic of this encyclopedia,
section 5) and the tradition of the ancient commentators (Barnes 1990,
2008; entry on the ancient commentators of this encyclopedia). But in
fact, while a Stoic connection is prima facie plausible -- unlike
Aristotelian term-based logic, Stoic logic is also largely
sentence-based -- historical evidence for direct Stoic influence
remains elusive; for now, no record of actual channels of influence
has been
 identified.[4]
 The ancient commentators, by contrast, had significant (both indirect
and direct) impact on the development of the notion of consequence
-- at first via Boethius, later via the Arabic authors, and as
their commentaries were translated and read by the Latin authors in
the 13th century and onwards.



While the notion of necessary truth-preservation was already quite
mature in the *Prior Analytics*, the conceptual development of
the substitutivity criterion is essentially a later contribution of
the ancient commentators (Barnes 1990, 2008; Dutilh Novaes 2012a).
Recall that Aristotle had not applied the metaphysical notions of form
and matter to logical objects such as sentences and arguments in any
systematic way; this crucial step was undertaken by the ancient
commentators. References to the form and matter of syllogisms are
pervasive in their writings, especially in commentaries on the
*Prior Analytics*, from that of Alexander of Aphrodisias
(2nd century AD) up to that of Ammonius (6th
century AD). The ancient commentators not only distinguished between
the form and the matter of syllogisms: they sometimes also suggested
(though usually rather obliquely) that the form of an argument is
precisely that in virtue of which it is valid and reliable. This would
later pave the way for the distinction between formal and material
consequences and the idea of validity in virtue of form. Here is an
illustrative passage by Alexander of Aphrodisias:



> 
> Combinations are called syllogistic and reliable if they do not alter
> together with differences in the matter -- i.e. if they do not
> deduce and prove different things at different times, but always and
> in every material instance preserve one and the same form in the
> conclusion. Combinations which change and alter configuration together
> with the matter and acquire different and conflicting conclusions at
> different times, are non-syllogistic and unreliable. (Alexander of
> Aphrodisias, *in Apr* 52.20-24, 114)
> 



Alexander also comments on Aristotle's use of schematic letters, and
clearly relates what takes the place of schematic letters to the
*matter* of the argument:



> 
> He uses letters in his exposition in order to indicate to us that the
> conclusions do not depend on the matter but on the figure, on the
> conjunction of the premises and on the moods. For so-and-so is deduced
> syllogistically not because the matter is of such-and-such a kind but
> because the combination is so-and-so. The letters, then, show that the
> conclusion will be such-and-such universally, always, and for every
> assumption. (Alexander of Aphrodisias, *in Apr*
> 53.28-54.2, 116)
> 



In first instance, the Greek logical heritage was almost
single-handedly (though selectively) passed on to the Latin tradition
by one man, the neo-platonic philosopher Boethius. Prior to the late
12th century (Aristotle's and other ancient texts became
widely read again in the Christian parts of Europe only in the
12th century -- see (Dod 1982)), what the medieval
authors had inherited from Greek logic had been almost exclusively
transmitted by Boethius, who had also established the logical
terminology in Latin. His translations of Aristotle's
*Categories* and *De Interpretatione* were widely read,
as well as his textbooks on syllogistic and his two texts *De
hypotheticis syllogismis (**On Hypothetical Syllogisms*
- HS) (dating 516-522) and *De topicis
differentiis* (*On Topical differentiae* -- TD)
(dating 522-523).



Boethius uses the term '*consequentia*' to refer to
that which a hypothetical sentence such as 'If it's day, then
it's light' signifies:



> 
> For it [the sentence] does not propose that it's day and it's light,
> but rather that if it's day, then it's light. Whence it signifies a
> certain consequence (*consequentia*) and not the being [of
> things]. (Boethius, *Commentary on 'On
> Interpretation'* 2, 109-10, translation in Martin
> 2009,
>  67.)[5]
> 



It is from Boethius that later authors inherited the term
'*consequentia*', but the influence of Boethius is
not only terminological. In HS, he focuses on conditionals of the form
"If something's (not) *A*, then it's (not)
*B*" (*si* (*non*) *est* A,
(*non*) *est* B), and lists a number of principles and
rules governing the logical behavior of such sentences (Martin 2009,
66-78). Boethius' considerations are not sufficiently worked-out
as to be viewed as a full-fledged 'theory of consequence',
and indeed there are a number of tensions and inconsistencies in his
doctrines. But HS will prove to be an important source for the later
development of theories of consequence. For example, in this text
Boethius introduces the distinction between natural and accidental
consequences, which then remains the main subdivision of consequences
up until the beginning of 14th century (when it is surpassed by the
distinction between formal and material consequences) (Martin 2018) (Archambault 2018a). For Boethius, both kinds of consequences, natural and accidental, entail
inseparability, roughly meaning that the antecedent cannot be true
while the consequent is false (i.e. a version of (TP)), but natural
consequences entail something more, namely a real causal, metaphysical
connection between the items in question.



The other text mentioned above, *De topicis differentiis*, is
equally significant for the development of later theories of
consequence. It presents a discussion of Cicero's *Topics*,
which in turn claims to have been inspired by Aristotle's
*Topics*. Cicero's work, however, is very different from
Aristotle's, and Boethius in some sense attempts to offer a
unification of both approaches (he also discusses the dialectical
topics of the Greek commentator Themistius). One of the key concepts
introduced by this text is the concept of 'maximal
propositions', which he claims are the general principles
underlying the correctness of topical arguments. As described by C.
Martin,



> 
> Such [maximal] propositions may either appear as a premise in a
> categorical syllogism or, much more importantly for the history of
> logic, as the warrant for an inference. In this second case they are
> the generalizations of the consequential relation which may hold
> between the premises and conclusion of an enthymeme or the antecedent
> and the conclusion of a conditional proposition. (Martin 2009, 79)
> 



Crucially, topical arguments were originally seen as merely probable,
contrasting with the necessary truth-preservation of syllogisms. So
for maximal propositions to serve as grounds for the relation of
consequence, a transformation on the status of topical arguments (from
probable to necessary) had to occur at a later stage (Stump 1982,
290). Moreover, it is important to notice that, although Boethius is
familiar with the work of the Greek ancient commentators and
incorporates some elements from their discussions, he does not
explicitly apply the form vs. matter distinction to syllogisms, as the
earlier authors had done. Barnes (1990) suggests that the logical
hylomorphism (i.e. the application of Aristotle's doctrine of form and
matter to logic) of these authors is nevertheless present in Boethius'
terminology, such as in the opposition between *propositionum
complexio* and *rerum natura* (the structure of a sentence
vs. the nature of things) (e.g. HS II ii 5). But Boethius does not
present the idea of substitution/variation of terms as a property
related to the validity of arguments, as Alexander of Aphrodisias had
suggested. In other words, substitutivity of terms as captured by (ST)
is not a key element of Boethius' account of validity -- neither
terminologically nor conceptually.



It has also been suggested that Boethius was the first to formulate a
version of (Co) with respect to consequences (Green-Pedersen
1984). However, on closer inspection, Boethius' notion of
containment is not really used to define or describe the relation
between the antecedent and the consequent of a consequence (Bosman
2018). Instead, the Boethian relation of containment in question seems
to pertain primarily to metaphysical relations of containment between
genus, species and differentiae. For example, an argument based on the
topic from genus involves a notion of containment: the genus is taken
to contain something that the species is then also inferred to
contain. For example, since all mammals are mortal, we can infer that
human beings are mortal too: the genus 'mammal' contains something,
namely mortality, that the species 'human being' also contains (Bosman
2018, 235). This metaphysical notion of containment then gives rise to
a related semantic notion, which establishes relations of semantic
containment between concepts (Bosman 2018). The upshot is thus that
Boethius does not seem to have had in mind a relation of containment
between antecedent and consequent, but his observations on what the
metaphysics of genus and species entails for certain forms of
inference may well have been the historical precursor of (Co).


### 2.2 Abelard



The period from Boethius in the 6th century to Abelard in the
12th century remains for now quite understudied, in particular as a consequence of the scarcity of extant manuscript evidence. As far as we can tell, the approach to logic in the 8th and 9th centuries appears to have been largely Platonist and metaphysical in character (Wilks 2016). Subsequently, a more linguistic orientation began to emerge, including investigations into inferential relations between categorical and hypothetical statements (Wilks 2016). It seems fair to say, however, that none of these investigations count as fully-fledged theories of consequence.




Indeed, it was only in the
12th century, with Abelard, that a
novel and highly sophisticated theory of consequence/entailment was to
be formulated. Abelard's starting point is the same material inherited
from Boethius which had been available for centuries, and yet what he
does with it is quite extraordinary; in particular, he understood
better than anyone before him the nature of what we now refer to as
propositional operations. His treatment of the conditional and of the disjunction echoes what is now known as connective logic (Martin 2018) (see entry on connective logic of this encyclopedia). And yet, his account is ultimately untenable, as the resulting system is in fact inconsistent (Martin 2004) (Martin 2018).



Tellingly, his theory of consequence is presented in the part of the
*Dialectica* dedicated to the topical framework (the book
*De Locis*), which again illustrates the close historical
connections between theories of consequence and the *Topics*.
Abelard speaks mostly of '*inferentia*' rather than
'*consequentia*', as the latter is for him a
subspecies of the former. He defines the concept of
*inferentia* as follows:



> 
> Therefore, inference consists in the necessity of consecution, that
> is, in that the sense (*sententia*) of the consequent is
> required (*exigitur*) by the sense (*sensus*) of the
> antecedent, as is asserted with a hypothetical proposition ...
> (*Dial.* 253, translation from (Martin 2004, 170))
> 



The phrase 'necessity of consecution' could be viewed as
Abelard's formulation of the criterion of necessary truth-preservation
(TP), but he seems to demand something more than mere
truth-preservation, namely a tighter connection of relevance
(*exigitur*) between the senses of antecedent and consequent
(Martin 2004, section II.5). Indeed, Abelard seems to entertain
different conceptions of the containment relation (Co) between
antecedent and consequent: as a metaphysical (*Dial.* 255),
semantic (*Dial.* 253), and epistemic (*Dial.* 255)
relation.



Abelard then further distinguishes perfect from imperfect
 inferences,[6]
 and this distinction sets him apart from the whole preceding
tradition:



> 
> But inferences are either perfect or imperfect. An inference is
> perfect when, from the structure of the antecedent itself, the truth
> of the consequent is manifest, and the construction of the antecedent
> is so disposed that it contains also the construction of the
> consequent in itself, just as in syllogisms or in conditionals which
> have the form of syllogisms. (*Dial.* 253/4)
> 



He goes on to argue that what warrants a perfect inference, i.e. its
'*vis inferentiae*', is the construction itself:
"'the truth of perfect inferences comes from the structure
(*complexio*), not from the nature of things"
(*Dial*. 255). This is a novel development, as for authors such
as Boethius and those following him, the warrant of *all*
consequences is ultimately to be found in the 'nature of
things', and is captured by means of topical principles.
(Abelard then goes on to provide arguments directed against this
Boethian view, see (MacFarlane 2000, A.4).) What Abelard refers to as
the construction/structure of an inference is indeed roughly what we
now understand as a *schema* (see entry on schemata of this
encyclopedia), as his discussion of examples suggests: it is the
substitution of *terms by other terms* while preserving the
consecution (i.e. a version of (ST)) that is the hallmark of perfect
inferences. "Whatever terms you substitute, whether they are
compatible or incompatible with one another, the consecution can in no
way be broken." (*Dial*. 255, translation from Martin
2004, 171)



Now, while proto-traces of the substitutional conception of validity
could be perceived in Aristotle as well as in some of the ancient
commentators, with Abelard it is (arguably) for the first time
presented as providing the grounds for a certain class of
consequences. And yet, Abelard's conception of consequence is not
*reduced* to (ST), given that imperfect inferences are just as
legitimate/valid as the perfect ones: imperfect inferences are those
that fail the substitution criterion but satisfy the 'necessity
of consecution' criterion. Thus, for Abelard, (ST) defines a
special subclass among valid inferences, but a combination of (TP) and
(Co) remains the true core of his notion of inference/consequence (see
*Dial.* 283-4 for two senses of necessity of consecution).
Indeed, in his subsequent discussion he deals with imperfect
inferences much more extensively than with the perfect ones.



Many of Abelard's logical concepts were tacitly absorbed by later
authors, though not by means of direct influence, and often with no
explicit attribution to Abelard (Martin 2004). Ultimately, the
Abelardian approach to consequences did not prevail (Normore 2015). It
is revealing that we now have only one surviving copy of his
*Dialectica*, a clear sign that it was not widely read.


### 2.3 13th Century



The two main features of 13th century logic are arguably
the emergence of the terminist tradition (authors such as Peter of
Spain, William of Sherwood and Lambert of Auxerre/Lagny) and the
absorption of the newly rediscovered Aristotelian texts and other
Greek sources, resulting in an extensive commentary tradition (Uckelman and Lagerlund 2016) The latter resulted in the distinction between three
groups of logical theories: what became known as the *logica
vetus* (topics emerging from the traditional texts which had
remained available throughout: the *Categories*, *On
Interpretation*, Porphyry's *Isagoge*); the *logica
nova* (covering the material from the newly discovered
Aristotelian texts); and the logica *modernorum* (topics not
directly related to the Aristotelian corpus, such as consequence,
insolubles and obligations).



The terminist authors did not address consequence as an autonomous
topic of investigation; their views on the matter are scattered along
their analyses of sentences, the *Topics*, fallacies and
syncategoremata (the syncategorema '*si*' in
particular) (Spruyt 2018). For 13th century authors, the two main types of consequence were natural and accidental consequences (Martin 2018); another frequently drawn distinction was that between simple and as-of-now consequences (Archambault 2018a). But one cannot really speak of full-fledged theories of consequence among the terminist authors, at least not if compared with what was to come in the 14th century.



Overall, 13th century authors viewed the semantics of conditionals and the relation of consequence predominantly through metaphysical lenses; the main line of thought was that the antecedent had to be somehow causally related to the consequent (Spruyt 2018). These metaphysical relations continued to be conceptualized in terms of the framework from the *Topics*, as they had been by Boethius. Given this metaphysical focus, 13th century authors typically did not recognize (TP) as a sufficient criterion for a relation of consequence to hold; some stronger connection between antecedent and consequent was required, often conceptualized in terms of containment (Co).



Also significant for the overall development of the concept of
consequence is the increasing presence of Aristotelian hylomorphism in
logical contexts. While hylomorphism had not been entirely unknown to
Latin authors prior to the rediscovery of the remaining Aristotelian
texts in the late 12th and 13th centuries, in
this period an explosion of applications of Aristotelian metaphysical
concepts in other areas occurred, especially in logic (Spruyt 2003).
In particular, applications of the form-matter distinction to
arguments (specifically, syllogisms) became frequent again, after a
hiatus of many centuries since the ancient commentators. Such
applications can be found in the only known 12th-century
commentary on the *Prior Analytics*, the *Anonymus
Aurelianensis* III (Ebbesen 1981), in the *Dialectica
Monacensis* (an anonymous text of the early 13th
century, edited in De Rijk 1962/7), and in Robert Kilwardby's
commentary on the Prior Analytics (1230s -- see (Thom 2007) (Thom 2019)),
among other texts. The significance of these applications is that they
paved the way for the consolidation of the notion of formal
consequence in the 14th century (Dutilh Novaes 2012b),
which in turn was to have a huge impact on the rest of the history of
logic.



Indeed, one of the earliest known uses of the phrase 'formal
consequence' can be found in Simon of Faversham's questions on
the *Sophistical Refutations*, written in the 1280s:



> 
> When it is said that "an animal is a substance; therefore a man
> is a substance" is a good consequence I reply that this
> consequence does not hold in virtue of form (*ratione formae*),
> but rather in virtue of matter. Because according to the Commentator
> [Averroes] on the first book of the *Physics*, an argument
> which is valid (*concludens*) in virtue of form must hold in
> all matter. This consequence, however, holds only for features which
> are essential [...] and so this consequence is not formal
> (*formalis*). (Simon of Faversham, *Quaestiones Super Libro
> Elenchorum*, quaestio 36, 200; translation from (Martin 2005)
> 135.)
> 



It is significant that Simon refers to Averroes' commentary on the
*Physics*, thus illustrating the importation of the
Aristotelian (meta)physical framework into logical analyses. We here
have the notion of 'valid in virtue of form' (as with
Abelard's '*complexio*'), and the association of
form and formality with the idea of substitution of terms (ST). Other
authors of the same period, John Duns Scotus for example, also use the
phrase '*consequentia formalis*' and its variants,
but not in the substitutional sense of holding 'in all
matter' (Martin 2005).



This is the historical background for the consolidation of the
distinction between formal and material consequences in the
14th century: a progression towards general theories of
consequence rather than exclusive focus on syllogisms, and the
increasing application of hylomorphism to arguments -- at first
to syllogisms, and later to arguments and consequences in general.


## 3. 14th Century Theories of Consequence


### 3.1 The emergence of treatises on consequence in the 14th century



The precise historical origins of 14th century theories of
consequence are still debated among scholars. What we know is that, at
the beginning of the 14th century, treatises and chapters
bearing the title *De consequentiis* and similar titles began
to appear. Why then, and not before? Naturally, the subject itself,
that is, the logical/inferential relations between sentences, had been
extensively discussed by earlier authors, as we have seen. But no
treatises or chapters were specifically dedicated to the topic or bore
such titles before the 14th century.



According to a once influential hypothesis, medieval theories of
consequences would have emerged from the tradition commenting on and
discussing Aristotle's *Topics* (Bird 1961; Stump 1982). At
first sight, this hypothesis may seem plausible: traditionally, the
role of the topical framework was often to account for the patterns of
(correct) inference and reasoning which did not fit into the
syllogistic system presented in the *Prior Analytics*. So,
conceptually, it would seem quite natural that the tradition on the
*Topics* might represent the historical origins of theories of
consequences. Moreover, as we have seen, most earlier discussions of
the notion of consequence (e.g. Abelard) were conducted explicitly
within the context of the topical framework, following Boethius.



However, on closer scrutiny, this hypothesis fails to receive
historical and textual confirmation. (Green-Pedersen 1984, chapter E
in particular) is (still) the most comprehensive study on this
subject, covering virtually every text known to us that is relevant
for the hypothesis. Green-Pedersen argues (1984, 270) that the late
13th century literature on the *Topics*, that is, the period
immediately preceding the emergence of treatises on consequences,
gives absolutely no indication of what was to come. In other words,
there are no significant similarities between the contents of these
13th century treatises on the *Topics* and the
14th century treatises on consequences. Therefore, we may
conclude that the *Topics* could not have been the main, and in any case
certainly not the only, source for the emergence of 14th
century theories of consequences.



Be that as it may, the importance of the *Topics* for the development of
14th century theories of consequences should not be altogether
dismissed. It is worth noticing that two of the first
authors who presented systematic discussions of consequence in the
14th century, namely Ockham and Burley, are both in some
way or another influenced by the *Topics*. Burley explicitly says that
all valid consequences are based on dialectical topics (*On the
purity*, p. 158 and 162). In contrast, the relation of Ockham's
theory of consequence to the topical framework is more convoluted; Green-Pedersen
argues convincingly that Bird's reconstruction of Ockham's theory
within the topical framework (Bird 1961) is not satisfactory
(Green-Pedersen 1984, 268), but he also confirms that Ockham's
'intrinsic' and 'extrinsic' middles, crucial
concepts for his theory of consequence (to be explained shortly), are
concepts essentially taken (albeit in modified form) from the topical
framework.



In short, although the current availability of texts still does not
allow for definitive conclusions to be drawn, the picture that at this point seems
most plausible is that different strands of traditional logical
theories converged in order to give rise to the 14th
century theories of consequences. It seems that at least four
traditions contributed substantially to these developments: treatises
on *syncategoremata*, especially in connection with the
syncategorema '*si*'; discussions of hypothetical
syllogisms; commentaries on the *Prior Analytics*; and the
tradition of the *Topics*. Different elements of each of these
traditions contributed to the development of different aspects of the
theories of
 consequence.[7]
 Green-Pedersen (1984, 295) argues, for example, that the late
13th-century treatises that most resemble early
14th-century treatises on consequences are "the
treatises on syncategorematic words and a number of
sophism-collections arranged after syncategoremes."



The different 14th century treatises on consequences can be
divided into four main groups:


1. The treatises on consequences from the very beginning of the
14th century: Burley's *De consequentiis* and two
anonymous treatises of roughly the same time (Green-Pedersen 1981).
They are in fact rather unsystematic collections of rules of
consequence/inference; it seems that their purpose was solely to
provide 'rules of thumb' to deal with sophismata related
to some syncategorematic terms. No conceptual or systematic discussion
of the nature of consequence is presented.
2. The second group is represented by Burley's *De Puritate*,
the chapters on consequence in Ockham's *Summa Logicae*
(III-3), a few Pseudo-Ockham treatises, and the *Liber
consequentiarium* (edited in Schupp 1988). In these texts, the
concept of (intrinsic and extrinsic) middles and other topical
concepts occupy a prominent place. They display a much deeper interest
in the very nature of consequence than the previous group, presenting
general definitions and criteria for what is to count as a
consequence, as well as divisions of kinds of consequence.
3. The third group is represented by Buridan's treatise on
consequence and the treatises inspired by it, most notably Albert of
Saxony's (a chapter of his *Perutilis logica*) and Marsilius of
Inghen's treatise on consequence (Ciola 2018). There is also the
interesting commentary on the *Prior Analytics* formerly
attributed to
 Scotus,[8]
 which is thought to have been composed before or in any case
independently of Buridan's treatise (Lagerlund 2000, chapter 6). In
these treatises, topical vestiges such as the doctrine of intrinsic
and extrinsic middles have disappeared completely. What characterizes
them as a group is the definition of formal consequence based on the
substitutivity criterion, in the spirit of (ST) (more on this below).
This tradition is usually referred to as the Parisian/continental
tradition on consequences.
4. The fourth group of treatises is predominantly British, and is
represented by a significantly greater number of surviving treatises
than group (3). It is represented by the treatises by Robert Fland,
John of Holland, Richard Billingham, Richard Lavenham, Ralph Strode,
and the *Logica Oxoniensis,* among others (Ashworth and Spade
1992). What characterizes this group as such is the definition of
formal consequence in terms of containment of the consequent in the
antecedent, in the spirit of (Co), usually interpreted in epistemic
terms.



Chronologically, the development of theories of consequence in the
14th century is thus characterized by an early and rather
'primitive' stage (1), then by a stage of further
development, in which, nevertheless, topical notions still play a
prominent role (2), and then by two further traditions which run more
or less parallel, namely the Parisian/continental tradition (3) and
the British tradition (4). While they differed in particular in the
various definitions given to the formal vs. material consequence
distinction, they all agreed that necessary truth-preservation (TP) is
a necessary (but perhaps not sufficient) condition for something to count as a (valid) consequence
(Dutilh Novaes 2008).



Before moving on, it is important to note that, in the 14th century, rules of
consequence were often discussed against the background of the genre
of oral disputation known as *obligationes* (see entry on
*obligationes* of this encyclopedia). It is common to encounter
formulations of rules of consequence in obligational terms, for
example: if you have conceded the consequence and its antecedent, then
you must concede the consequent. Thus, interesting reflections on
consequence are also to be found in *obligationes* treatises
(and vice-versa).


### 3.2 Burley and Ockham



Walter Burley is the author of the oldest treatise on consequence with
known authorship (edited by Brown in 1980), but it is in his later
work *De Puritate*, longer version, that one finds his fully
developed theory of consequence (Archambault 2018b). The shorter version of *De
Puritate* is thought to have been composed before Ockham's
*Summa Logicae*, and contains only a section on consequence and
a section on *syncategoremata*. The received view is that,
after becoming acquainted with Ockham's *Summa Logicae*, Burley
abandoned the text of what is now known as the shorter version of
*De Puritate* and began to work on a new draft, which was to
become the longer version (Spade 2000). The theory of consequence
presented in the shorter version is based on ten basic principles,
four of which are clearly sentential/propositional, while the other
six take terms as the basic logical unit (Boh 1982). The only
distinction of consequence that Burley discusses is that between
simple and as-of-now consequence, a traditional distinction which
remained popular in the 14th century:



> 
> First therefore I assume a certain distinction, namely this one: One
> kind of [consequence] is simple, another kind is as-of-now (*ut
> nunc*). A simple [consequence] is one that holds for every time.
> For example 'A man runs; an animal runs.' An as-of-now
> [consequence] holds for a determinate time and not always. For example
> 'Every man runs; therefore, Socrates runs'. For that
> [consequence] does not hold always, but only while Socrates is a man.
> (Burley, *De Puritate*, 3)
> 



This temporal understanding of the simple vs. as-of-now distinction is
the one adopted by most authors, both before and after Burley
(however, see Pseudo-Scotus' formulation, discussed in the next
section), and is repeated verbatim in the longer version of *On the
Purity...* (p. 146). Another interesting feature of the
shorter version is the fact that it treats syllogisms under the
concept of consequence, thus illustrating the absorption of
syllogistic by theories of consequences in the 14th
century. Burley's mature theory of consequence, as presented in the
longer version of *De Puritate*, is best discussed against the
background of the theory of consequence presented in Ockham's
*Summa Logicae*, so let us now turn to Ockham first.



Ockham's *Summa Logicae* is thought to date to the first years
of the 1320s; section 3 of Part III is entirely dedicated to
consequences. In chapter 1 of III-3, Ockham presents a somewhat
confusing account of consequences based on nine distinctions,
including the simple vs. as-of-now distinction; the distinction
between formal and material consequence is the last one
 presented.[9]
 It seems that this important distinction was discussed systematically
for the first time in this very text (Martin 2005), but Ockham offers
virtually no justification for his use of the notions of form and
matter with respect to consequences. It is also possible that Ockham
deliberately ignores the well-entrenched distinction between natural
and accidental consequences, given that he mentions *nine*
distinctions, but not this one. Here is how Ockham introduces the
notion of a formal consequence:



> 
> Formal consequences are of two kinds. Some hold in virtue of an
> extrinsic middle, which concerns the form of propositions. For
> example, such rules as 'from an exclusive to a universal, with
> transposition of terms, is a good consequence'; 'if the
> major premise is necessary and the minor premise is assertoric (*de
> inesse*), the conclusion is necessary'. Others hold
> immediately in virtue of an intrinsic middle, and mediately in virtue
> of an extrinsic middle regarding the general conditions of the
> proposition, [...] such as in 'Socrates is not running,
> therefore a man is not running.' (William of Ockham, *Summa
> Logicae* III-3, ch. 1, lines 45-54)
> 



Thus, according to Ockham, formal consequences are those that hold in
virtue of *middles*, be they intrinsic or extrinsic. A
consequence holds immediately in virtue of an intrinsic middle when it
holds in virtue of the truth of a different sentence formed from its
terms. For example, 'Socrates is not running, therefore a man is
not running' holds in virtue of this middle: 'Socrates is
a man', since if 'Socrates is a man' is not true,
the consequence does not hold. These are typically enthymematic
consequences, i.e. consequences with a 'missing premise'
(with the additional premise, they become a valid syllogism). An
extrinsic middle, by contrast, is a sentence not containing the terms
that form the antecedent and the consequent of the putative
consequence, but which is a general rule describing the
'fact' that warrants the passage from the antecedent to
the consequent (reminiscent of Boethius' maximal propositions), and
which concerns the *form* of sentences. Ockham's example of a
consequence holding immediately in virtue of an extrinsic middle is
'Only a man is a donkey, therefore every donkey is a man',
which holds in virtue of this general rule: 'an exclusive and a
universal with transposed terms signify the same and are
convertible'.



Notice however that 'Socrates is not running, therefore a man is
not running' and 'Only a man is a donkey, therefore every
donkey is a man' are both *formal* consequences for
Ockham (since both hold in virtue of middles), whereas the former is
clearly an enthymeme, not valid in all substitutional instances of the
terms 'Socrates', 'man', and
'running', thus not satisfying (ST) (see (Crimi 2018) on the extensional discrepancy between Ockham's notion of formal consequence and Buridan's (ST)-based notion of formal consequence). The latter, on the
other hand, is valid in all substitutional instances of
'donkey' and 'man', and indeed this seems to
be the case of most, if not all, of Ockham's formal consequences
immediately valid in virtue of extrinsic middles; he explicitly says
for example that syllogisms are of the latter kind. Effectively,
formal consequences immediately valid in virtue of extrinsic middles
satisfy the (ST) criterion of being valid 'in all matter',
but the same does not hold of Ockham's (enthymematic) consequences
valid in virtue of an intrinsic middle.



Curiously, while Ockham can be credited with having been the first to
use the terms 'formal consequence' and 'material
consequence' systematically, the *content* of his
distinction did not pass on to later authors. This is arguably because
Ockham's distinction is cast in terms of intrinsic and extrinsic
middles, peculiar concepts belonging to the Boethian framework which
was already losing its influence by Ockham's time (Green-Pedersen
1984). Indeed, one seldom encounters the concept of
'middles' in writings of the post-Ockham period, except
for texts under his direct influence.



As for material consequences, it is not entirely clear how exactly
Ockham intended to define this class of consequences. He says that
material consequences are those that hold in virtue solely of (the
meaning of) their terms (Ockham, *Summa Logicae* III-3, ch. 1,
lines 55-57) but the two examples he gives are of a consequence
with an impossible antecedent and a consequence with a necessary
consequent. This suggests the reading that this category consists
exclusively of consequences of this sort (*ex impossibili* and
*ad necessarium* consequences), but there is no conclusive
evidence supporting this interpretation; in particular, he did not
offer any explicit motivation for his use of the concept of
*matter* to characterize this class of
 consequences.[10]



As mentioned above, the longer version of Burley's *De
Puritate* is thought to be largely a response to Ockham's
*Summa Logicae*, not exclusively but also with respect to
consequence. Burley recovers the natural vs. accidental distinction
which had been deliberately neglected by Ockham (Archambault 2018b), but in fact
formulates it with a terminology similar to that used by Ockham for
formal consequences, namely in terms of the concepts of intrinsic and
extrinsic topics (rather than 'middles', but this appears
to be above all a mere terminological difference):



> 
> Simple [consequence] is of two kinds. One is natural. That happens
> when the antecedent includes the consequent. Such an inference holds
> through an intrinsic topic. An accidental inference is one that holds
> through an extrinsic topic. That happens when the antecedent does not
> include the consequent but the inference holds through a certain
> extrinsic rule. (Burley, *De Puritate*, 146)
> 



Notice again the reference to the idea that the antecedent may
include/contain the consequent (in natural consequences). One may
conjecture that Burley sought to neutralize Ockham's distinction
between formal and material consequences in terms of intrinsic and
extrinsic middles by formulating the traditional distinction between
natural and accidental consequences in terms of containment and
intrinsic/extrinsic topics. Moreover, when discussing the notion of
formal consequence later in the text (pp. 171-173, replying to a
possible objection), Burley seems to be criticizing Ockham's
definition of material consequences as those which hold solely in
virtue of the meaning of terms:



> 
> Thus for a [consequence] to hold by reason of the terms can happen in
> two ways, either because it holds materially by reason of the terms,
> or because it holds formally by reason of the terms -- that is,
> by the *formal* reason of the terms. (Burley, *On the
> Purity*, 173)
> 



In other words, Burley seems to be saying that Ockham's formulation of
the distinction is ineffective and thus inadequate. The exact
formulation of the distinction between formal and material
consequences presented by Ockham was indeed not adopted by later
authors, but it would be excessively speculative to attribute this
outcome to Burley's criticism. In effect, even less of a Burleian
legacy is to be found in later authors specifically with respect to
consequence, in particular as the formal vs. material distinction
became the main subdivision of consequence later on (albeit under
different formulations).


### 3.3 Buridan and the Parisian tradition



John Buridan's treatise on consequence (TC, edited by H. Hubien in
1976, English translation in (Buridan 2015)) quite likely represents
the pinnacle of sophistication for (Latin) medieval discussions of the
concept of consequence. Its modern editor dates it to the 1330s, thus
belonging to the early stages of Buridan's career. We currently know
much less about Buridan's immediate predecessors than about Burley's
or Ockham's, so it is not clear who Buridan takes inspiration from or
is criticizing. The treatise is composed of four books: Book I
presents general considerations on the very notion of consequence;
Book II treats of consequences involving modal sentences; Book III
treats of syllogisms involving assertoric (i.e. non-modal) sentences;
Book IV deals with syllogisms involving modal sentences. Each of them
is remarkable in its own way (Book III for example represents a
radical subversion of Aristotelian orthodoxy, with the suggestion that
third-figure syllogisms are more foundational than first-figure
syllogisms), but here we shall focus on the first two books (Book I in
particular), given the methodological decision of leaving syllogistic
aside in the present analysis.



In Book I, Buridan presents the general definition of a consequence in
the familiar terms of necessary truth-preservation:



> 
> Hence, many say that of two propositions one is antecedent to the
> other if it is impossible for the one to be true without the other
> being true, and one is consequent to the other if it is impossible for
> the one not to be true when the other is true, so that every
> proposition is antecedent to every other proposition for which it is
> impossible for it to be true without the other being true. (Buridan,
> TC, 21; transl. 67)
> 



He then goes on to reformulate the definition for reasons related to
his view that only actually produced sentences (sentence-tokens) can
have a truth-value (Klima 2004; Dutilh Novaes 2005). 'No
sentence is negative, therefore no donkey is running' comes out
as a valid consequence according to the criterion thus formulated,
because 'No sentence is negative' can never be true: its
mere existence falsifies itself whenever it is produced. According to
Buridan, this example should not count as a valid consequence, and one
reason he gives for this is that its contrapositive 'Some donkey
is running, therefore some sentence is negative' is not a valid
consequence. He formulates a definition of consequences in terms of
'howsoever the antecedent/consequent signifies things to
be' in order to accommodate such counterexamples, but adds that
in most cases, the simpler definition is sufficiently accurate.



Commitment to sentence-tokens aside, Buridan's notion of consequence
clearly has necessary truth-preservation (TP) as its fundamental component.
So for him, enthymematic consequences such as 'a man runs,
therefore an animal runs' are just as valid as syllogistic
consequences or other consequences satisfying the criterion of
preservation of validity under term substitution (ST). However,
Buridan does recognize that there is an important distinction between
consequences which do and those which do not satisfy the
substitutional criterion; giving continuation to a tradition which
includes Alexander of Aphrodisias and Simon of Faversham, he
conceptualizes this distinction in hylomorphic terms, more
specifically in terms of the distinction between formal and material
consequence:



> 
> A consequence is called formal if it is valid in all terms retaining a
> similar form. Or if you want to put it explicitly, a formal
> consequence is one where every proposition similar in form that might
> be formed would be a good consequence, e.g., 'That which is A is
> B, so that which is B is A'. (Buridan, TC, 22-23, transl.
> 68; my emphasis)
> 



Material consequences are those which do satisfy the necessary
truth-preservation criterion (TP) but do not satisfy the
substitutional criterion (ST) (Crimi 2018). At first sight, Buridan's distinction
between formal and material consequence seems very similar to, for
example, Abelard's distinction between perfect and imperfect
inferences. There is, however, a fundamental difference; nowhere does
Buridan suggest that formal consequences are valid *in virtue of
their form*, as Abelard had claimed for the
'*complexio*' of perfect inferences (Dutilh Novaes and Bosman 2017). He does say
that the validity of a material consequence is *made evident*
only by means of a reduction to a formal consequence (TC, 1.4), but
this observation pertains to the epistemic level of how the validity
of a consequence is made apparent to us, not to the quasi-metaphysical
level of what grounds
 it.[11]



Buridan also comments explicitly on what is to be understood as the
form and the matter of a consequence:



> 
> I say that when we speak of matter and form, by the matter of a
> proposition or consequence we mean the purely categorematic terms,
> namely, the subject and predicate, setting aside the syncategoremes
> attached to them by which they are conjoined or denied or distributed
> or given a certain kind of supposition; we say all the rest pertains
> to the form (Buridan, TC, 30, transl. 74)
> 



The view that the form of a consequence/argument pertains to its
syncategorematic terms while its matter pertains to its categorematic
terms is presupposed in both earlier and subsequent texts, but here
with Buridan it receives a rare explicit formulation. A modern version
of this idea still survives, in the form of the doctrine of the
logical form of arguments and the modern preoccupation with logical
constants (Read 1994; Dutilh Novaes 2012a; entry on logical constants
of this encyclopedia). However, it is worth emphasizing once again
that drawing the line between the form and the matter of an
argument/consequence in this manner still does not entail the thesis
that the form is that in virtue of which a valid argument is valid;
nor does it entail the thesis that *only* the
arguments/consequences satisfying the substitutional criterion are
indeed valid. Buridan, in particular, does not hold either one of
these theses.



In the final section of Book I, Buridan formulates a series of general
principles which follow from his proposed definition of consequence,
such as that from the impossible anything follows (first conclusion;
see (D'Ors 1993 and Normore 2015) on Buridan and the *ex
impossibili* principle)), the principle of contraposition (third
conclusion), and also many principles pertaining to the semantic
properties of the categorematic terms in a consequence (see entry on
medieval theories of properties of terms). Thus, here again we see
that medieval theories of consequence never completely abandon the
term perspective to adopt an exclusively sentential/propositional
perspective.



Book II of Buridan's treatise presents a sophisticated analysis of the
logical behavior of modal sentences. Modal sentences can be either
composite or divided, depending on where the modal term occurs: if it
is either the subject or the predicate of the sentence, while the
other term is an embedded sentence in nominalized form (in
'dictum' form, in the medieval terminology), then the
sentence is a composite modal sentence. If however the modal term
occurs as an adverb modifying the copula, then it is a divided modal
sentence. Buridan then proves a series of conclusions and equivalences
for each kind of modal sentences, such as that '*B* is
necessarily *A*' is equivalent to '*B* is
not possibly not *A*' (Johnston 2014).



The other treatises on consequence in the Parisian/continental tradition do not seem to depart significantly from Buridan's overall analysis, with one possible exception: the commentary on the
*Prior Analytics* formerly attributed to Scotus (edited in
Yrjonsuuri 2001), and whose authorship remains controversial. The
dating is equally problematic; crucially, it is not clear whether it
was written before or after Buridan's treatise, but some scholars
(Lagerlund 2000, chapter 6) have argued that at any rate Pseudo-Scotus
displays no knowledge of Buridan's treatise (likewise, there is no
obvious evidence that Buridan was familiar with Pseudo-Scotus'
text).



The treatise proceeds very much in the spirit of chapter 3 of Book I
of Buridan's treatise: a putative definition of consequence is
proposed, but then quickly a counterexample is found, namely something
that should not count as a consequence and yet satisfies the
criterion, or the other way round (Boh 1982,
 307-310).[12]
 But while Buridan rests his case after the third proposed definition,
Pseudo-Scotus goes on, and formulates a counterexample to the
definition that Buridan settles on: 'God exists, hence this
argument is invalid'. If this consequence is valid, then it has
a necessary antecedent and a false consequent (since the consequent
says that it is invalid). But then it is invalid. In sum, if it is
valid, it is invalid; thus, by the *consequentia mirabilis*
((*A* - ~*A*) - ~*A*), it is invalid.
But if it is invalid, it is necessarily so, since the premise is a
necessary sentence; therefore, we have a consequence with a necessary
consequent, thus satisfying the necessary truth-preservation
criterion, but which is plainly invalid. This has been described as a
"proto-version" of Curry's
 paradox.[13]



Moreover, the Pseudo-Scotus treatise also offers an interesting formulation of
the simple vs. as-of-now distinction: in contrast with e.g. Burley,
according to Pseudo-Scotus this distinction applies only to material
consequences (recall that for him, a formal consequence is the one
that satisfies the substitutional criterion), and amounts to the modal
value of the missing premise that can be added in order to turn the
(enthymematic) consequence into a formal one. That is, if the missing
premise is a necessary sentence, then the consequence is an
absolute/simple one. But if the missing premise is a contingent truth
(it has to be true with respect to the time indicated by the verbs of
the consequence, otherwise the original material consequence does not
hold), then the original material consequence holds only in some
situations, namely the situations in which the contingent sentence
happens to be true, and is thus an as-of-now consequence. The same
formulation of the simple vs. as-of-now distinction can be found in
Buridan's treatise, Book I chapter 4, which again illustrates the
conceptual connection between the two texts.


### 3.4 The British School



In the British tradition, which is then continued in Italy in the late
14th century and 15th century (Courtenay 1982),
the definition of consequence in terms of necessary truth-preservation
(TP) is also unanimously adopted, as for example in Billingham
(Billingham/Weber 2003, 80), Strode (cited in (Pozzi 1978, 237)) and
Paul of Venice (*Logica Parva*, p. 167). In fact, these authors
present variations of (TP) without much discussion or analysis,
contrary to what is found in Buridan's treatise, for example. More
generally, the treatises in this tradition are characterized by a
lesser degree of conceptual sophistication if compared to the earlier
treatises by Ockham, Burley or Buridan. The goal seems to be mostly
pedagogical, i.e. presenting 'rules of thumb' to argue
correctly, rather than presenting a systematic, conceptual analysis of
the concept of consequence.



Still, what is characteristic of this tradition is a specific
interpretation of the idea of the conclusion being
contained/understood (*intelligitur*) in the premises --
that is, condition (Co) - which these authors rely on
extensively to define the concept of formal consequence. We have seen
that for Abelard, something like (Co) is a necessary condition for all
consequences/inferences, which Martin (2004) spells out in terms of a
criterion of relevance. 13th century authors, such as
Kilwardby (for whom (Co) provided the definition of natural
consequences) and Faversham, also discussed variations of this idea
(Read 2010, 177/8), but prior to the late 14th century, it
was not always formulated in epistemic/psychological terms. The
authors in the 14th century British tradition typically
formulate the definition of formal consequence on the basis of
variations of (Co), but giving it a very explicitly epistemic
twist.



Lavenham, for example, says (as quoted in King 2001, 133): "A
consequence is formal when the consequent necessarily belongs to the
understanding of the antecedent, as it is in the case of syllogistic
consequence, and in many enthymematic consequences". Strode
presents a similar formulation:



> 
> A consequence said to be formally valid is one of which if it is
> understood to be as is adequately signified through the antecedent
> then it is understood to be just as is adequately signified through
> the consequent. For if someone understands you to be a man then he
> understands you to be an animal. (Translation in Normore 1993, 449).
> 



Several other authors formulated similar definitions, such as
Billingham (Weber 2003, 80) and Fland (Fland/Spade 1976). Normore
(1993, 449) argues that a significant transformation occurred in the
14th century British tradition, which "puts in play
the idea that deduction is not an objective relation between abstract
objects or sentences but a mental operation performed on the bases of
what can be understood or imagined." This is noticeable not only
in the definitions of formal (as opposed to material) consequence, but
also in the very definitions of consequence which begin to mention
mental acts, such as: "a consequence is a derivation
(*illatio*) of the consequent from the antecedent"
(Strode, quoted in Normore 1993, 449).



The epistemic/psychological interpretation of these formulations of
formal consequences has not gone uncontested; arguably, they are
equally compatible with a *semantic* interpretation emphasizing
the signification of sentences (Read 2010, 178). But it is clear that,
while the Parisian tradition defined the concept of formal consequence
in terms of (ST), the British tradition formulated the same concept in
terms of (Co). Both were ideas that had been floating around for
centuries, but which yield very different conceptions of what counts
as a formal consequence. These two approaches are not only
intensionally divergent; they also disagree on the extension of the
class of formal consequences. (ST)-formal consequences do not include
enthymematic consequences such as 'Socrates is a man, therefore
Socrates is an animal', but (Co)-formal consequences typically
do. For British authors, the class of material consequences is often
composed exclusively of consequences of the kind 'from the
impossible anything follows' and 'the necessary follows
from anything', which satisfy the truth-preservation criterion
(TP) *a fortiori*, but typically fail relevant/containment
criteria. 'God does not exist, therefore you are a donkey'
counts as a valid consequence according to (TP) ('God does not
exist' is considered to be an impossible sentence), but the
consequent is not contained in the antecedent in the same way as in
'Socrates is a man, therefore Socrates is an animal'. To
mark this distinction, 14th century British authors would
typically view the former as a material consequence and the latter as
a formal consequence (Ashworth & Spade 1992).


## 4. Conclusion



We started by surveying the ancient background for the emergence of
Latin medieval theories of consequence, in particular Aristotle's
*Topics* and *Prior Analytics*, the commentaries by the
ancient commentators, and Boethius' influential logical texts.
Theories of consequence only became an autonomous topic of
investigation in the 14th century, but previous
developments, in particular Abelard's theory of inference/entailment
and the increasing application of hylomorphism to logic in the
13th century, are equally deserving of attention.
Nevertheless, the golden age for theories of consequence was
the 14th century, when different theories were proposed by
Burley, Ockham, Buridan, Billingham, Strode, Paul of Venice, and many
others. As with much of scholastic logic, the topic of consequence
continued to be explored in the 15th century and beyond
(Ashworth 1974, chapter III), providing the background for much of
what was to come in the history of logic, in particular the persistent
association between logic and forms (MacFarlane 2000).