Offprint from
OXFO RD STUDI ES
IN ANCI ENT
PHI LOSOPHY
E DIT O R : V I C T O R C A ST O N
VO LU ME LIV
3
CAUSALITY AND
COEXTENSIVENESS IN ARISTOTLE’S
POSTERIOR ANALYTICS 1. 13
LUCAS ANGIONI
. Introduction
I this paper I discuss an important feature of the notion of cause in
Post. An. . , b–. Some scholars have taken the passage as
introducing a false principle about explanation (or even a fallacy of
conversion). I claim that Aristotle is introducing a logical requirement for being the strictly appropriate cause in a scientific demonstration, namely: an appropriate cause must be coextensive with what it
appropriately explains. Some interpretations tend in this direction,
but do not account for the intricacies of the text and, furthermore,
do not explain how Aristotle goes beyond negative causes expressed
in the second-figure syllogisms. My interpretation provides a careful and full discussion of the intricate steps by which Aristotle presents the requirement. Furthermore, I argue that the requirement
is completely consistent with an important feature of Aristotle’s
© Lucas Angioni
Earlier drafts of this paper were presented in Coimbra (‘III International Aristotle
Seminar: Causation in Greek and Medieval Philosophy’, March ), Campinas
(conference ‘Questions on Causality in Aristotle’s Posterior Analytics’, October
), and Helsinki (workshop ‘Aristotelian Themes in Dependence, Modality and
Essence’, January ). On those occasions I benefited from comments, criticisms,
and suggestions from Antonio Martins, David Charles, Nathanael Stein, Carlo
Natali, David Bronstein, Breno Zuppolini, Rodrigo Guerizoli, Michail Peramatzis,
Miira Tuominen, Tuomas Tahko, Benjamin Schneider, Mika Perälä, and others.
I am grateful to Alan Code, Adam Crager, Klaus Corcilius, Laura Castelli, Edgar
Marques, Raphael Zillig, Pieter Sjoerd Hasper, and Henry Mendell for informal
discussions about the main point of this paper. I also thank the anonymous referees
for helpful comments, and Victor Caston for suggestions that improved the style.
See J. Barnes, Aristotle’s Posterior Analytics [Posterior Analytics], Clarendon
Aristotle Series, nd edn. (Oxford, ), ; also the bewilderment expressed by
M. Mignucci, Aristotele: Analitici secondi [Analitici] (Rome and Bari, ), .
Philoponus, In An. Post. . –. Wallies, exonerates Aristotle of such a
charge. His diagnosis is correct: Aristotle is focusing on the relationships between
causes and effects (cf. . –).
Lucas Angioni
notion of scientific explanation, namely, his insistence on katholou
predications as understood in Post. An. . , b–a.
The underlying subject of this paper is the notion of cause understood as one of the key notions involved in Aristotle’s conception of
scientific knowledge in the Posterior Analytics. For ease of reference, I shall use the expression ‘primary cause’ to refer to it. I shall
not examine here Aristotle’s general view on causes, or how it differs from other conceptions, such as the Humean one. As for what
‘cause’ signifies for him, I shall assume that a cause for Aristotle is
a real-world item (a substance’s attribute, or a state of affairs, or a
thing’s essence, or an event, etc.) that grounds another real-world
item—that makes it what it is. Perhaps ‘ground’ would be better
than ‘cause’ as a translation of aition, but for simplicity’s sake I shall
retain the word ‘cause’.
The notion of causality in Aristotle’s theory of demonstration is
cast within the triadic framework of syllogisms: a cause is expressed
as a middle term (B) which explains why a given attribute (A) is present in a given subject (C). Aristotle’s talk about causes can be very
misleading, for sometimes (as in Physics . or Metaphysics Δ ) he
does not make explicit the triadic structure of causal relations and,
For more general accounts of Aristotle’s notion of cause, which address many of
those issues, see J. M. Moravcsik, ‘Aristotle on Adequate Explanations’, Synthese,
(), –; id., ‘What Makes Reality Intelligible? Reflections on Aristotle’s Theory of Aitia’ [‘Aitia’], in L. Judson (ed.), Aristotle’s Physics (Oxford, ), –;
M. Hocutt, ‘Aristotle’s Four Becauses’, Philosophy, (), –; N. Stein,
‘Causation and Explanation in Aristotle’, Philosophy Compass, (), –;
id., ‘Aristotle’s Causal Pluralism’ [‘Pluralism’], Archiv für Geschichte der Philosophie,
(), –.
See R. McKirahan, Principles and Proofs: Aristotle’s Theory of Demonstrative
Science [Principles] (Princeton, ), –, , –, for ‘ground’ as a translation
of aitia or aition (my inspiration is not the current fashion regarding grounding in
contemporary metaphysics). Many prefer ‘explanation’ as a translation of aitia (see
Barnes, Posterior Analytics, ). I stick with the traditional ‘cause’ just for the sake of
simplifying my discussion. Another option would be to transliterate aitia and aition
(see Moravcsik, ‘Aitia’). In this paper I cannot address the question of what is involved or implied in each translation.
The notion of cause as middle term is officially presented in Post. An. . , a–
and is found throughout Aristotle’s discussion of demonstration (especially Post.
An. . and . –, but also . , a–; . , b–; . , b–; . ,
a; . , a ff.; . , a–; . , a–). Aristotle is also concerned
with the triadic framework for causes in Metaphysics Ζ . Cf. D. Charles, Aristotle
on Meaning and Essence (Oxford, ), , –; id., ‘Definition and Explanation in Posterior Analytics and Metaphysics’, in id., Definition in Greek Philosophy
(Oxford, ), –; S. Williams and D. Charles, ‘Essence, Modality and the
Master Craftsman’, in E. Feser (ed.), Aristotle on Method and Metaphysics (New
York, ), – at .
Causality and Coextensiveness in Post. An. .
more importantly for the purposes of this paper, he seems to give
different criteria for sorting out what counts as a relevant explanandum: sometimes the explanandum is explicitly introduced as the
relation between a subject and a predicate, but sometimes Aristotle
seems to take the predicate as the explanandum. I cannot address
this complicated question here. But, as it is important for my purposes to focus on the logical relations between the B-term and the
A-term as attributes of a given subject, I shall prefer to take A as
the ‘effect’ or that of which the cause is cause.
I cannot go into details about Aristotle’s notion of scientific
demonstration, including whether it has an axiomatic model, or how
we should understand the role of necessity, principles, and per se
predications, except to the extent that they figure in my interpretation of the passage that constitutes my main focus. I shall briefly
indicate some assumptions that are indispensable for my points.
First of all, it is important to get clear about what exactly is the
object of the definition Aristotle is giving at Post. An. . , b–
. I take the definiendum to be a higher-level form of knowledge
that can be labelled scientific knowledge. Thus, Aristotle is not
concerned with giving an analysis of the general concept of knowledge. I side with those who take Aristotle’s object as the notion
of scientific understanding. But, more particularly, Aristotle is not
The more articulated story which prevails in Aristotle’s analysis is the following: the B-term being the cause as a middle term, the A-term is treated as ‘that of
which the cause is cause’ in contrast with the C-term as ‘that for which the cause is
cause’ (. , a–). Aristotle sometimes says or suggests that the conclusion of
a syllogism (as a predicative fact AC) is that of which the cause is cause (. , b;
Pr. An. . , b; . , a), and sometimes he uses the word pragma as equivalent to the A-term (. , b), but sometimes pragma refers to an explanandum
with predicative form (. , b). For a different view of pragma see D. Bronstein,
Aristotle on Knowledge and Learning [Learning] (Oxford, ), –.
I use ‘effect’ just for brevity’s sake. What I mean by ‘effect’ is just the A-term
or a predicative explanandum (A–C).
See L. A. Kosman ‘Explanation, Understanding and Insight in Aristotle’s Posterior Analytics’, in H. Lee, A. Mourelatos, and R. Rorty (eds.), Exegesis and Argument: Studies in Greek Philosophy Presented to Gregory Vlastos (Assen, ), –
; J. H. Lesher, ‘Aristotle on ἐπιστήμη as Understanding’, Ancient Philosophy,
(), –; M. F. Burnyeat, ‘Aristotle on Understanding Knowledge’, in E. Berti
(ed.), Aristotle on Science: The Posterior Analytics (Padua, ), –; Bronstein, Learning. My overall view regarding the theory of demonstration in the Posterior Analytics is closer to McKirahan, Principles, –, and M. Ferejohn, Formal
Causes [Formal] (Oxford, ), –. See L. Angioni, ‘Aristotle’s Definition of
Scientific Knowledge (APo b–)’ [‘Definition’], Logical Analysis and History of
Philosophy, (), –, and id., ‘Aristotle on Necessary Principles and on Explaining X through the Essence of X’, Studia Philosophica Estonica, (), –.
Lucas Angioni
ultimately concerned with explicating what a body of full scientific
knowledge amounts to: he is ultimately concerned with explicating what—within such a body of knowledge—is the most important
factor that makes it so special and puts it on a higher level. And this
factor, as Aristotle will develop as he goes on in the Posterior Analytics, is the explanation through the most appropriate cause. Now,
appropriate explanation in this context is not mere justification. Appropriate explanation involves the identification of real-world items
that are primarily responsible for making their explananda what
they are. Aristotle is aware that it is really hard to find explanations that count as the appropriate ones (. , a–), but he is
interested in explicating in what they consist. Now, one important
ingredient of the notion of an appropriate explanation is the notion
of reciprocation between cause and effect: being a primary cause
that delivers the appropriate explanation of its effect involves (but
does not collapse into) being a necessary and sufficient condition for
its effect to obtain. When this reciprocation is cast within the syllogistic framework of demonstrations, the result is (besides other
things) that the middle term encapsulating the cause is coextensive
with the major term it is meant to explain.
Now, on my view, the passage Post. An. . , b–, is advancing just such a point: a primary cause must be coextensive with its
effect. Aristotle’s phrasing in b– is complicated and perhaps
convoluted. The few discussions of it that we find in the literature
are not satisfactory: they either disconnect the passage from the previous discussion about primary causes and so attribute to Aristotle a
false principle about explanation in general, or they see the connection between coextensiveness and primary (or proximate) causes,
but fail to explain how Aristotle’s point is a general one about scientific demonstrations and is not restricted to the negative causes
expressed in second-figure syllogisms. Thus, in offering a careful discussion of each step in the passage b–, my aim is to
show how Aristotle is coherently advancing a general point about
primary causes which is not restricted either to negative cases or to
second-figure syllogisms: primary causes must be coextensive with
their effects. This picture also helps us to understand Aristotle’s
Barnes, Posterior Analytics, .
e.g. Philoponus’ commentary on the Posterior Analytics; W. D. Ross, Aristotle’s
Prior and Posterior Analytics: A Revised Text with Introduction and Commentary
[Prior and Posterior Analytics] (Oxford, ), ; McKirahan, Principles, .
Causality and Coextensiveness in Post. An. .
insistence on katholou predications and hence to attain a more coherent and unified interpretation of the overall theory of scientific
demonstrations presented in the Posterior Analytics.
. Examination of the key passage b–
I shall divide the passage into five sections and refer to it as [T].
Starting with a neutral translation (based on Barnes’s), I discuss
each section and in due course present a different translation:
[T] [] καὶ γὰρ ἐν τούτοις τοῦ ὅτι καὶ οὐ τοῦ διότι ἡ ἀπόδειξις· οὐ
γὰρ λέγεται τὸ αἴτιον. οἷον διὰ τί οὐκ ἀναπνεῖ ὁ τοῖχος;
ὅτι οὐ ζῷον. [] εἰ γὰρ τοῦτο τοῦ μὴ ἀναπνεῖν αἴτιον, ἔδει τὸ
ζῷον εἶναι αἴτιον τοῦ ἀναπνεῖν, [a] οἷον εἰ ἡ ἀπόφασις αἰτία τοῦ
μὴ ὑπάρχειν, ἡ κατάφασις τοῦ ὑπάρχειν, [a′] ὥσπερ εἰ τὸ ἀσύμμετρα εἶναι τὰ θερμὰ καὶ τὰ ψυχρὰ τοῦ μὴ ὑγιαίνειν, τὸ
σύμμετρα εἶναι τοῦ ὑγιαίνειν—[b] ὁμοίως δὲ καὶ εἰ ἡ κατάφασις τοῦ ὑπάρχειν, ἡ ἀπόφασις τοῦ μὴ ὑπάρχειν. [] ἐπὶ δὲ
τῶν οὕτως ἀποδεδομένων οὐ συμβαίνει τὸ λεχθέν· οὐ γὰρ
ἅπαν ἀναπνεῖ ζῷον. [] ὁ δὲ συλλογισμὸς γίνεται τῆς τοιαύτης αἰτίας ἐν τῷ μέσῳ σχήματι. οἷον ἔστω τὸ Α ζῷον, ἐφ᾿
ᾧ Β τὸ ἀναπνεῖν, ἐφ᾿ ᾧ Γ τοῖχος. τῷ μὲν οὖν Β παντὶ
ὑπάρχει τὸ Α (πᾶν γὰρ τὸ ἀναπνέον ζῷον), τῷ δὲ Γ οὐθενί, ὥστε οὐδὲ τὸ Β τῷ Γ οὐθενί· οὐκ ἄρα ἀναπνεῖ ὁ τοῖχος. (b–)
[] For also in these cases the [attempted] demonstration is of the
fact that but not of the reason why, for the cause is not stated. For
instance: why do walls not breathe? Because walls are not animals.
[] If this were the cause of not breathing, being an animal would
have to be the cause of breathing, [a] that is, if the negation is the
cause of something’s not being attributed, then the affirmation is the
cause of its being attributed ([a′] thus if an imbalance of the hot and
cold elements is the cause of not being healthy, their balance is the
cause of being healthy), [b] and, similarly, if the affirmation is the
cause of something’s being attributed, then the negation is the cause
of its not being attributed. [] But what was said does not obtain in
the cases introduced (or explained) in that way: indeed, not every animal breathes. [] The syllogism of this sort of cause comes about in
the middle figure: for example, let A be animal, B breathing, C wall.
A holds of every B (for everything which breathes is an animal), but
of no C: hence B too holds of no C—therefore walls do not breathe.
(trans. Barnes, modified)
The Greek text is from Ross, Prior and Posterior Analytics, here preserving the
lineation.
Lucas Angioni
Aristotle is presenting an attempted demonstration in which the
cause is not stated (step []). Although ‘cause’ does not go with any
adjective there, I submit that Aristotle is talking about the cause
which yields scientific understanding of the fact expressed in the
conclusion of a demonstration: the appropriate or primary cause.
Actually, he has mentioned the notion of a primary cause (τὸ πρῶτον
αἴτιον) a few lines earlier: ‘this syllogism is of the reason why, for the
primary cause [τὸ πρῶτον αἴτιον] has been captured’ (b–). Aristotle refers to the syllogism that explains why planets do not twinkle
through the middle term ‘being near the Earth’, which he has contrasted with the syllogism in which one infers that planets are near
the Earth by their non-twinkling. Thus, I argue that Aristotle’s
use of ‘the cause’ (τὸ αἴτιον) in Post. An. b, , , picks up this
same notion of a primary cause.
In step [] Aristotle is talking about a pattern of explanation
which will be spelt out in step [] as the following Camestres syllogism:
Everything which breathes is an animal.
Walls are not animals.
∴ Walls do not breathe.
Aristotle claims that such a syllogism does not capture the appropriate cause which would ultimately explain why walls do not breathe.
Aristotle’s point has not been understood: he is not interested in a
general principle about explanations which will allow one to infer
positive causes from negative ones (and vice versa), nor is he exclusively concerned with negative causes expressed in second-figure
syllogisms. It might be useful to give a survey of the claims I shall
argue for in what follows:
• In steps [] and [] Aristotle is introducing a requirement which
More on this on sect. .
According to Philoponus, In An. Post. . – Wallies, Alexander has understood Aristotle as referring to the προσεχὴς αἰτία (proximate cause). Aristotle’s use
of τὸ αἴτιον (with no adjective) picks up the stricter notion of primary cause in many
passages (. , a; . , a; . , b–; Post. An. . ).
See Barnes, Posterior Analytics, –; Mignucci, Analitici, . Philoponus,
In An. Post. . –. Wallies (esp. . –, . –), thinks that Aristotle’s
point is that a cause for a negative fact must be expressed in the second figure. Ross,
Prior and Posterior Analytics, , gets the essentials of Aristotle’s point (the same
goes for McKirahan, Principles, –), but is far from disentangling the argument
and understanding its general reach. On the other hand, it is surely a disappointment
that many scholars addressing Aristotle’s theory of explanation avoid this passage.
Causality and Coextensiveness in Post. An. .
a cause must satisfy for being the appropriate and primary cause
to explain a given explanandum; step [] implies the requirement in contrast with the particular counter-example at stake,
but step [] moves to a general introduction of the requirement: Aristotle presents the logical features which a cause must
display in order to be the primary cause capable of providing
strictly scientific understanding about a given explanandum;
(these logical features are only necessary but not sufficient conditions for a cause to be primary).
• In step [] Aristotle explains what was implied in [], namely,
why the Camestres syllogism suggested at step [] and spelt
out in step [] fails at satisfying the requirement. Then, the
Camestres syllogism at [] is employed as a foil for highlighting
the requirement.
Let me develop these points as I proceed to step [] of [T]:
[] εἰ γὰρ τοῦτο τοῦ μὴ ἀναπνεῖν αἴτιον, ἔδει τὸ
ζῷον εἶναι αἴτιον τοῦ ἀναπνεῖν. (b–)
If this [= wall’s not being an animal] were the cause of [wall’s] not breathing, being an animal would have to be the cause of [wall’s] breathing. (my
translation)
It is clear not only from the previous context but also from the next
steps in Aristotle’s discussion that τοῦτο at b means ‘wall’s not
being an animal’: otherwise, the contrast between τοῦτο and τὸ ζῷον
(translated as ‘being an animal’) would be pointless. It is also important to note that τὸ ζῷον in this context works as shorthand for a
predicate attributed to wall (see Barnes’s translation). What is most
important is that Aristotle is using a counterfactual mode in [],
and this fact points to step [], which I translate as follows:
Aristotle is not saying that walls’ not being an animal does not give any explanation of walls’ not breathing. He is saying that our Camestres does not capture
the primary cause (or ‘the precise ground’, as phrased by Ross, Prior and Posterior
Analytics, ) that would give scientific understanding of walls’ not breathing—and
this is compatible with the obvious truth that our Camestres gives at least some explanation of its explanandum. Aristotle is not relying on a ‘general principle about
explanation’ (Barnes, Posterior Analytics, ) that would allow him to pass from
any explanation of negative facts through negations to the explanation of the correlated positive fact through affirmations. His point is more specific: he is stating a
requirement for a cause to be the primary one for its explanandum.
There is no ἄν particle in the apodosis etc., but Philoponus (In An. Post.
. – Wallies), Ross (Prior and Posterior Analytics, , ), and most translators (Barnes, Posterior Analytics, ; Mignucci, Analitici, ; G. R. G. Mure,
Lucas Angioni
[] ἐπὶ δὲ
τῶν οὕτως ἀποδεδομένων οὐ συμβαίνει τὸ λεχθέν· οὐ γὰρ
ἅπαν ἀναπνεῖ ζῷον. (b–)
But what was said does not obtain in the cases introduced (or explained) in
that way: indeed, not every animal breathes. (my translation)
Now, ‘what was said’ refers exactly to the requirement expounded
in steps [] and [], so that Aristotle’s point is that the Camestres
syllogism to be spelt out in [] and already implied in [] and []
(‘the cases explained in that way’) falls short of satisfying the requirement and so works as a foil to it. Thus—coming back to []—
Aristotle’s use of the counterfactual mode in step [] presupposes
the requirement and actually states that the present foil case does
not meet it.
The requirement is then explicated in step [] of [T], which is
the most difficult to unpack. The main difficulties involved in this
step are the following: (i) what ‘the negation’ (ἡ ἀπόφασις) and ‘the
affirmation’ (ἡ κατάφασις) are referring to; (ii) whether Aristotle’s
point is or is not restricted to negative explanations formulated in
the second figure; (iii) why Aristotle needs [b], which seems to be
a redundant repetition of [a].
There are two ways of handling difficulty (i). A first option is to
take the expression ‘the negation’ as an abbreviated reference to the
negative sentence in which ‘animal’ is denied of ‘wall’ and, similarly, to take ‘the affirmation’ as an abbreviated reference to the
affirmative sentence in which ‘animal’ would be universally predicated of ‘wall’. In this way, step [] would remain entirely in the
counterfactual mode: Aristotle would be indirectly introducing the
requirement for being a primary cause by reporting a case in which
the requirement is not fulfilled. The expression οἷον (‘i.e.’) would
be indicating, then, a mere rephrasing of the same point made at
step [].
A second option is to take the expression ‘the negation’ not as
Posterior Analytics, in W. D. Ross (ed.), The Complete Works of Aristotle Translated
into English (Oxford, ); P. Pellegrin, Aristote: Seconds Analytiques (Paris, ),
) are right in translating or paraphrasing the sentence as a counterfactual.
Another issue is (iv) why Aristotle has shifted from αἴτιον to αἰτία. But this shift
is not important to my claims here. Aristotle just moves from the notion of an explanatory factor to the notion of an explanation.
A Greek article can have demonstrative force in many contexts, and it might be
misleading to translate it with a non-committal ‘the’ when the context makes it clear
that the referent is a given one. See e.g. NE . , a (with dunamis).
Causality and Coextensiveness in Post. An. .
an abbreviated reference to the specific negative sentence in which
‘animal’ is denied of ‘wall’, but as an abbreviated reference to a general pattern of sentence in which a term (expressing a cause) is
denied of another term—as if Aristotle has dropped the concrete
terms (‘wall’, ‘animal’, ‘breathe’) and had in mind schematic letters (‘C’, ‘B’, ‘A’) instead. A similar story will apply to ‘the affirmation’. In this option, Aristotle will be extracting a general point
from his previous example. The expression οἷον (‘i.e.’) should be
taken not as introducing a mere rephrasing, but as a sort of generalizing clarification.
In any case, it is clear from the context that Aristotle has a syllogistic framework in mind: ‘ἀπόφασις’ (‘the negation’) and ‘κατάφασις’ (‘the affirmation’) pick up particularly the denial/affirmation of
a middle term (be it the schematic letter B or a concrete term replacing B), and ‘ὑπάρχειν’ (‘being attributed’) and ‘μὴ ὑπάρχειν’ (‘not
being attributed’) refer to the major term’s being or not being attributed to something (either to the schematic letter C or to a concrete term replacing C). Thus, the middle term can be attributed
or not to the minor term, and the major term can be attributed or
not to the minor term: Aristotle is particularly concerned with settling whether the attributive tie between the middle term and the
minor term is a ground for the major’s being attributed (or not)
to the minor term. Aristotle’s issue is whether and how the minor
premiss (B–C) grounds the conclusion (A–C).
Now, according to the first option, the translation of step [] will
be the following (let me call it ‘Version ’ of step []):
[a] οἷον εἰ ἡ ἀπόφασις αἰτία τοῦ
μὴ ὑπάρχειν, ἡ κατάφασις τοῦ ὑπάρχειν, . . .
...
—[b] ὁμοίως δὲ καὶ εἰ ἡ κατάφασις τοῦ ὑπάρχειν, ἡ ἀπόφασις τοῦ μὴ ὑπάρχειν. (b–)
[Version ] [a] i.e. if animal’s being denied of walls were the cause of breathing’s not being attributed to walls, then animal’s being asserted of walls
In favour of taking [a] as a generalization is the fact that [a′] introduces a
different but equivalent example, as if Aristotle were concerned with saying: ‘Don’t
get stuck on the particularity of my single example; I have generalized my point, and
have another concrete example so that you may better understand me.’ The fact that
ἀσύμμετρα has the negation transposed into the predicate is irrelevant. Aristotle relies
on the translatability of ‘hot and cold being incommensurate’ into ‘hot and cold not
being commensurate’. For discussion see M. Ferejohn, The Origins of Aristotelian
Science [Origins] (New Haven, ), –.
Lucas Angioni
would be the cause of breathing’s being attributed to walls . . . [b] and,
similarly, if animal’s being asserted of walls were the cause of breathing’s
being attributed to walls, then animal’s being denied of walls would be the
cause of breathing’s not being attributed to walls. (my translation)
According to the second option for taking step [], its translation
(let it be called ‘Version ’) will be something like this:
[Version ] [a] that is, if B’s being denied of C is the cause of A’s not being
attributed to C, then B’s being asserted of C would be the cause of A being
attributed to C . . . [b] and, similarly, if B’s being asserted of C is the cause
of A’s being attributed to C, then B’s being denied of C would be the cause
of A not being attributed to C. (my translation)
According to Version , steps [a] and [b] describe some logical
features of the general notion of a primary cause. In Version , only
the consequents of the conditionals advanced in [a] and [b] must
be taken in the counterfactual mode, whereas the antecedent must
be formulated in the indicative. I shall explain later why there is
such a difference between the antecedent and the consequent. For
the time being, I remark that in Version , in its turn, explicit reference to the attempted demonstration of why walls do not breathe
requires the antecedent of the conditionals to be in the counterfactual mode too. Cases explained in this way—namely, following the
pattern of the logical relations between the concrete terms of the
Camestres syllogism implied in []—illustrate the requirement by
presenting a situation in which it is not satisfied. On the other hand,
Version attains more generality in expressing the requirement itself by avoiding concrete terms. In what follows I shall argue that
Version should be adopted.
Let me spell out the requirement. Take the Camestres schema,
which is presupposed by Aristotle in step [] and actually used in
step [] of [T]:
Every A is B.
No C is B
∴ No C is A.
‘Its negation’ (in step [a]) might be taken to refer to the minor
premiss of the Camestres schema: this premiss denies that B is
I thank an anonymous referee for improving my formulation here.
Causality and Coextensiveness in Post. An. .
attributed to C. And ‘not being attributed’ (b–) is shorthand
for ‘A’s not being attributed to C’, which is what the conclusion expresses. Thus, when Aristotle says (in the antecedent of [a]) that
‘its negation is an aitia for not being attributed’, he thereby describes a logical property of the primary cause as it can be displayed
in a Camestres schema: if B is the primary cause of A, then the
fact that B is denied of something (as it is of C in the minor premiss
of the Camestres) entails the denial of the attribute it primarily explains; A is not attributed to C either. Let me provisionally call
this logical property the first formal condition for being a primary
cause.
But this, of course, is not the end of the story. The first formal
condition, as depicted in the previous paragraph, is incomplete in
itself: it is only a logical property shared by many other middle
terms which deliver sound Camestres syllogisms but are not
primary causes. This is why the second part of [a] is strictly
needed. It works as a second half of the requirement for primary
causes: ‘its affirmation is an aitia for being attributed’. Now, this
second half of the requiment is precisely what fails in our Camestres
case about walls’ not breathing. If things were contrary to what
is stated in its minor premiss—that is, if walls were animals—this
would not entail that walls would breathe, for, in Aristotle’s view,
there are some species of animals that do not breathe. The inference
from walls’ being animals to walls’ breathing would be a ‘fallacy
of the consequent’ (cf. SE b –) in the second figure. In order
to get a sound inference, the major premiss should be convertible:
‘all animals breathe’ should also be true. But this sentence is false
in Aristotle’s view.
Thus, the conjunction of the first and the second halves of the
requirement implies that a primary cause reciprocates with the
attribute it is meant to explain. What one gets from picking up
together the antecedent and the consequent of the conditional
expressed in [a] is a schema stating that B and A are convertible.
The Camestres schema is useful for displaying the logical property Aristotle is
highlighting, but this does not imply that Aristotle’s point is restricted to secondfigure syllogisms. More on this later.
Cf. Resp. , b–; PA . , b; . , a.
For a similar result see Philop. In An. Post. . – Wallies; McKirahan, Principles, . On reciprocation of causal relations see Post. An. . – and Cat.
(where the causal priority of a cause over its effect is compatible with the reciprocation in their mutual implication of being the case, b–).
Lucas Angioni
But this convertibility is meant as a condition for primary causes,
not as a general rule about any sort of explanation whatsoever.
Let me spell this out carefully. First, take the antecedent itself
(‘C’s not being B is the aitia of C’s not being A’) as a conditional
such as ‘if C is not B, then C is not A’ and translate it into predicate
calculus:
[a] (Antecedent) ∀x (¬Bx → ¬Ax).
Take also the consequent (‘C’s being B is the aitia of C’s being A’)
as a conditional (‘if C is B, then C is A’) and translate it into formal
language:
[a] (Consequent) ∀x (Bx → Ax).
Now, one should be very careful in employing such formalizations,
for Aristotle is definitely not committed to a fallacy of conversion
such as the following (let it be labelled [a*] for future reference):
[a*]: ∀x ((¬Bx → ¬Ax) → (Bx → Ax)).
Aristotle’s point was never meant as an inference from the antecedent to the consequent of [a*]. A reading of [a] in terms of an
intended inference would lead us to attribute a fallacy to Aristotle.
Now, it is clear that Aristotle did not mean [a] as an inference valid
for any proposition whatsoever in general, but something restricted
to explanations. However, even with such a restriction, one might
be tempted to see [a] as stating an inferential rule about explanations in general: a rule that would permit the inference from ‘not
being B is an explanation of not being A’ to ‘being B is an explanation of being A’. This is what Barnes has proposed. His paraphrase for [a] is the following:
() If (∀x) (if not-Fx, then not-Fx because not-Gx) then (∀x) (if
Fx, then Fx because Gx).
Barnes’s idea seems to be that, according to this ‘general principle
about explanation’, once one gets an explanation of not-Fx in terms
of not-Gx, one will be permitted to infer that Gx explains Fx.
Barnes is right in saying that the principle is false. What Barnes has
Pace Barnes, Posterior Analytics, .
See Philop. In An. Post. . –. Wallies for such a charge and his de
fence of Aristotle.
Posterior Analytics, .
Causality and Coextensiveness in Post. An. .
missed is that Aristotle has not meant [a] as a general principle
about any sort of explanation whatsoever. Aristotle is not interested
in how one can infer new explanations from a previous one but is
rather specifying criteria for primary explanations: [a] spells out a
logical condition for being a primary cause—and a logical condition
that is not sufficient for being a primary cause. Therefore, Aristotle’s point should rather be expressed in the following way (let
it be labelled [aPC], where the superscript ‘PC’ means that it is a
condition for being a primary cause):
[aPC] If B is the primary cause of A, then ∀x ((¬Bx → ¬Ax) &
(Bx → Ax)).
And the conjunction in the consequent of [aPC] can be packed into
a biconditional:
[aPC]* If B is the primary cause of A, then ∀x (Bx ↔ Ax).
The antecedent in [aPC], namely, ‘if B is the primary cause of A’, is
supplied from the context. Philoponus (In An. Post. . – Wallies) is on the right track in taking Aristotle to be proposing convertibility between the cause and what it causes. But Aristotle is more
particularly concerned in [T] with showing that ‘cases explained
in this way’ (b–)—i.e. cases following the pattern of the logical relations between ‘animal’, ‘wall’, and ‘breathing’—are not appropriate demonstrations because they fail to capture the primary
cause. He says that ‘the [attempted] demonstration is of the that
but not of the why’ (b). However, just a few lines earlier he
made it clear that capturing the primary cause is the rationale that
See McKirahan, Principles, , for a similar criticism of Barnes: Aristotle’s
point is about demonstration involving immediate principles, not about any sort of explanation. On the other hand, the amendment () which Barnes, Posterior Analytics,
, advances might be true in itself—(= ‘If not-P is among the conditions explanatory of not-Q, then P is among the conditions explanatory of Q’)—but is not a
correct exegesis of [a].
From [b] one can get an equivalent formula matching Aristotle’s phrasing,
namely, [bPC]: ‘If B is the primary cause of A, then ∀x ((Bx → Ax) & (¬Bx → ¬Ax))’.
Now, since the order of the conjuncts does not matter, [bPC] can be taken as equivalent to[aPC]. But then, why has Aristotle explicitly expressed [b]? I deal with this
issue later.
Even if the expression ‘cause of this sort’ (τοιαύτης αἰτίας, b–) refers particularly to negative causes to be expressed in second-figure syllogisms, the relevant
point is that they instantiate the notion of an ‘outside middle term’ (μέσον ἐξω, b),
i.e. causes that are not coextensive with their explananda (‘animal’, for example, is
not coextensive with ‘breathing’), and thereby fail to be primary ones.
Lucas Angioni
makes a syllogism what he calls a demonstration of the why: ‘and
this syllogism is of the why, for the primary cause is captured’ (b–
). He has explicated previously (in a–) how an attempted
demonstration can fail to capture the primary cause in the case of
convertible or coextensive terms. When he comes to b–, he
shows a similar failure involving terms that are not coextensive—
but he does so in such a way that coextensiveness comes out as a
formal requirement for primary causes. What he is trying to describe and characterize in [a] is the logical behaviour of a primary
cause. Therefore, [a] must be understood strictly as [aPC], i.e. as
something restricted to primary causes.
. Is Aristotle’s expression redundant in step [b]?
Another important issue in [T] is why Aristotle was not fully satisfied with step [a], but added the seemingly redundant step [b]. We
can shed light on this by considering the syllogistic framework Aristotle employs. An important point is that the major premiss of the
Camestres schema must be convertible in the case of primary causes:
not only ‘every A is B’, but also ‘every B is A’ must be true. When
the latter sentential schema also yields a true sentence, its truth allows us to construct a sound Barbara showing that the ‘affirmation
of the cause’ (in the minor premiss) is an aitia for the major term A
to be attributed to C:
Every B is A.
Every C is B.
∴ Every C is A.
Now, Aristotle could never be suggesting that both the Barbara
schema and the Camestres schema should deliver sound syllogisms
for the same minor term: the truth of ‘every C is A’ will be incompatible with the truth of ‘no C is A’ for any given interpretation of
‘C’. What Aristotle means is that, B being convertible with A (given
that B is assumed as the primary cause of A), then if a Camestres
See sect. .
Aristotle is not explicit in a– about primary causes—he has just said αἴτιον with no adjective—but since the case in a– is complementary to the case
in a–b and since Aristotle says explicitly that the latter captures the primary
cause, it is fair to infer that the first case (a–) is one in which the terms are
coextensive but the primary cause was not captured.
Causality and Coextensiveness in Post. An. .
syllogism captures the appropriate cause to explain why a given C
is not A, then a correlated Barbara syllogism will correctly express
the counterfactual situation confirming that C’s being B would have
caused C to have the feature A: this is what Aristotle expresses in
[a]; and, conversely, if a Barbara syllogism captures the appropriate cause for a given C’s being A, then a correlated Camestres syllogism will correctly express the counterfactual situation confirming
that, if C were not B, C would not be A: this is what is expressed
in [b]. This is why Aristotle spells out his requirement in a way
that might seem redundant, but is actually needed within his conceptual framework—and we miss the point if we stick to our formal
language with ‘x’ instead of ‘C’.
Let us consider [aPC]S, a syllogistic version of [aPC] (the superscript ‘S’ indicates that the formula is meant only for syllogistic
terms):
[aPC]S If B is the primary cause of A, then ∀C ((BeC → AeC) &
(BaC → AaC)contrary-to-fact).
There can also be obtained a syllogistic version for [b]:
[bPC]S If B is the primary cause of A, then ∀C ((BaC → AaC) &
(BeC → AeC)contrary-to-fact).
Consider [aPC]S: no concrete instance of ‘C’ can satisfy both the
antecedent of the first conditional (‘BeC → AeC’) and the antecedent of the second conditional (‘BaC → AaC’). That is why the
second conditional must be taken as contrary-to-fact, as expressing
a situation that does not actually obtain. And the same sort of
claim holds for [bPC]S. Thus, this syllogistic interpretation with
the second conditional as contrary-to-fact depicts precisely what
Aristotle means in [a] and [b]. If we continue to formulate the
conditionals in our usual language—with ‘x’ referring to individuals of the universal class—[b] will seem to be a redundant
As I shall show, it makes a difference whether we formulate the point with ‘x’,
standing for an individual of the universal class, or with the schematic letter ‘C’,
standing for a term able to be used as argument in a categorical form (this is what I
mean by ‘syllogistic term’). I need not discuss this intricate issue. For a recent approach see M. Malink, Aristotle’s Modal Syllogistic (Cambridge, Mass., ), ff.
I am grateful to an anonymous referee and the editor Victor Caston for greater
clarity on this point, as well as for finding the right expression in the formal notations.
Lucas Angioni
duplication of the point already made in [a]. But a syllogistic
version of [a] with ‘C’ standing for a syllogistic term requires
[b] as its counterpart—if only to dispel the misleading appearance
that Aristotle was exclusively focused on negative primary causes and
second-figure syllogisms. Aristotle’s underlying concern is a general
point about primary causes, not restricted to negative causes. My
interpretation does not hang on how the expression ‘cases in which
the middle term has outside position’ (ἐφ᾿ ὧν τὸ μέσον ἔξω τίθεται,
b) must be understood—whether as a sign that the ensuing
discussion will focus exclusively on second-figure syllogisms or as
pointing to middle terms failing the coextensiveness (or proximateness) requirement. I believe that the sentence at b should
be paraphrased in the following way: ‘furthermore, it happens the
same [namely, one does not reach knowledge of the primary explanation] also in cases where the middle term has more extension
than required’—and this is the way Philoponus, In An. Post. .
– Wallies, takes the expression (following Alexander). However, if one insists that the expression ‘cases in which the middle
term has outside position’ (ἐφ᾿ ὧν τὸ μέσον ἔξω τίθεται) at b
must be taken as pointing to second-figure syllogisms, my answer
is that, even so, Aristotle starts with second-figure syllogisms only
as a useful tool for highlighting the coextensiveness requirement
for primary causes in general. But his point is a general one:
if one starts with a given C that stands to B such that ‘BeC’ is
true—this is [a]—then ‘BaC’ must be counterfactual; but if one
starts with a given C that stands to B such that ‘BaC’ is true—this
Barnes, Posterior Analytics, , takes [b] to be simply the contrapositive of
[a] without explaining why Aristotle has taken the trouble to express it. On the
other hand, Barnes’s formulation (‘If (∀x) (if not-Fx, then not-Fx because not-Gx)
then (∀x) (if Fx, then Fx because Gx)’) cleverly grasps the generality of Aristotle’s
point with ‘if not-Fx’ as the antecedent inside the antecedent of the conditional and
‘if Fx’ as the antecedent inside the consequent of the conditional.
Most intepretations favour the second option: Philop. In An. Post. . –
Wallies; Ross, Prior and Posterior Analytics, ; Barnes, Posterior Analytlics, ;
McKirahan, Principles, –; Ferejohn, Origins, –.
In Pr. An. b the expression points to second-figure syllogisms, but in
a– it points to third-figure syllogisms, so it is not compelling to argue that
Aristotle has used it in b to restrict his remarks to second-figure syllogisms (see
Ross, Prior and Posterior Analytics, ). It is not possible to settle this issue by
appealing to κατὰ τὴν τῶν μέσων θέσιν at b–, since this expression can also be
taken in different ways: as pointing to the syntactical difference in the position of
the middle term in each figure, or as pointing to the choice of concrete middle terms
that have different levels of explanatory relevance (I prefer the latter option).
Causality and Coextensiveness in Post. An. .
is [b]—then ‘BeC’ must be counterfactual. Had Aristotle had a
different language at his disposal—for instance, a language capable of expressing coextensiveness—he would not have needed to
‘duplicate’ his point with [b].
To sum up, Aristotle’s underlying argument in steps []–[]
and [] of [T] might be taken as a modus tollens on the basis of
[aPC]S: the case depicted in the Camestres syllogism with the
terms ‘breathing’, ‘wall’, and ‘animal’ falsifies the consequent of
[aPC]S, since it verifies only one of the conjuncts which constitute
that consequent—even if all walls were animals (counterfactual), it
would not be true that all walls would be breathing things. Aristotle thereby concludes (or suggests) that the middle term ‘animal’
cannot, then, be taken as the primary cause for explaining why
walls do not breathe. And [bPC]S is in order where it is: [bPC]S
stresses that Aristotle’s point is not restricted to negative causes
formulated in the second figure, but ranges over primary causes in
general.
. The coextensiveness requirement for primary causes
Let me focus on the ‘antecedent’ or first half of [a]—which is equivalent to the ‘consequent’ or second half of [b]: ‘its negation is aitia
of not being attributed’. This phrase advances by itself a logical requirement for being a primary cause:
(R) B’s being attributed to a given C must be a necessary (sine
qua non) condition for C’s being A.
Similarly, the ‘consequent’ or second half of [a]—which is equivalent to the ‘antecedent’ or first half of [b]—expresses by itself
Aristotle uses the verb ἀντικατηγορεῖσθαι to mean that two terms are coextensive
with each other, but my point is that the four categorical forms available to express
predicative sentences in his syllogistics do not enable him to express coextensiveness in
any straightforward way. Aristotle has discussed what happens with the valid moods
when terms are coextensive (in Pr. An. . –), but this does not modify my point.
I thank an anonymous referee for greater clarity on this matter.
For a different approach see Ferejohn, Origins, –. On Ferejohn’s view, Aristotle’s worry is with causes that are too remote to explain (as if the ‘Anacharsis case’
at b– presented the central concern), whereas on my view Aristotle is more
specifically saying that only coextensive causes can be primarily explanatory. It is
important to stress that causes can be non-remote without being coextensive: Ferejohn’s view will not cover them.
Lucas Angioni
another logical requirement for being a primary cause: ‘its affirmation is aitia of being attributed’. This implies the following:
(R) B’s being attributed to a given C must be a sufficient condition for C’s being A.
And (R) together with (R) results in the requirement that B must
be coextensive or convertible with A.
[T] is made harder to disentangle by all the circumlocutions
used by Aristotle. He starts with a counterfactual mode to present
his requirement by means of a foil that does not satisfy it (step []).
But the expression of the requirement itself involves a counterfactual mode when interpreted with a triplet of syllogistic terms, since
there could be no interpretation of ‘C’ such that B would be asserted and denied of C at the same time. Such a tangled expression
is due to the limits of his syllogistic language, which (besides other
things) cannot capture coextensiveness as a single categorical form.
When the two conjuncts encapsulated in the consequent of [aPC]
are interpreted in the syllogistic framework as [aPC]S (depicting relations between minor premisses and conclusions), it is not possible
to express both in syllogisms that are simultaneously sound. Thus,
[a] and [b] are strictly needed to cover different triplets of terms
for which either the negation or the affirmation will actually (i.e. not
counterfactually) obtain. It was Aristotle’s concern with a general
point—not restricted to negative causes expressed in second-figure
syllogisms—that led him to [b], which is not a mere or redundant
duplication of [a].
Let me spell out again how the argument goes. Step [a] starts
with generalizing from cases such as the Camestres syllogism implied in step [] and formulated in []. Requirement (R) is satisfied: since walls are not animals and being an animal is a necessary condition for breathing (thus satisfying (R)), it follows that
walls do not breathe. However, requirement (R) is not satisfied in
that case, and this is a sign that the Camestres syllogism formulated in step [] fails at stating the primary cause and thereby falls
short of being an appropriate demonstration. However, there are
Ross, Prior and Posterior Analytics, , saw that coextensiveness is at stake (as
something involved in what is the ‘precise ground’ of the explanandum). See also
McKirahan, Principles, , . But both scholars are far from explicating what is
going on in each step of the text—nor do they explain why Aristotle needs [b] after
[a]. Ferejohn, Formal, is very sympathetic to the coextensiveness requirement for
primary causes, but he does not address b– directly.
Causality and Coextensiveness in Post. An. .
still the cases in which the situation will be the reverse: requirement (R) is satisfied, whereas requirement (R) is not. Such cases
can be depicted as sound Barbara syllogisms. For instance: since
humans are mammals and being a mammal is a sufficient condition for being mortal (thus satisfying (R)), it follows that humans
are mortal. However, (R) will not be satisfied, since humans could
still be mortal even if they were not mammals—in short, being a
mammal is not a necessary condition for being mortal and, consequently, being a mammal cannot be the primary cause of being
mortal. Therefore, Aristotle’s point in [b], far from being a redundant repetition of the point made at [a], is welcome in context:
Aristotle thereby stresses that his requirement regarding the logical features of primary causes is a general one, not restricted to the
negative causes as depicted in second-figure syllogisms. Now, this
also shows that steps [a]–[b] are not a mere rephrasing of the particular point made at step [] about walls’ not being animals, but are
rather a generalization aimed at formulating universal requirements
for primary causes. Therefore, Version of step [] is preferable to
Version .
Once the reasons for the ‘duplication’ in [b] become clear, the
full logical requirement for being a primary cause can be expressed
in the following way:
(= [aPC]*) If B is the primary cause of A, then ∀x (Bx ↔
Ax).
Now, the entailment in the itself does not convert: this means
that the convertibility between B (the cause) and A (that of which it
is the cause) is only a necessary but not a sufficient condition for B
to qualify as the primary cause of A. In order to be a primary cause,
a given middle term B must also present an explanatory appropriateness which, for Aristotle, cannot be reduced to its convertibility
with the effect.
Other intepretations, such as those found in Ross, Prior and Posterior Analytics,
, and McKirahan, Principles, , cannot account for [b] in any interesting way
and, furthermore, restrict Aristotle’s point to negative causes. The same goes for
Philop. In An. Post. – Wallies: although he understands Aristotle’s point about
convertibility between causes and effects (. –. ), as he goes on (. –
. , esp. . –, . –) he takes Aristotle as trying to establish that a cause
for a negative fact must be expressed in the second figure.
As I shall discuss in the next section, Aristotle’s point about irreducibility was
Lucas Angioni
. Beyond the formal features of primary causes
It is of the utmost importance to stress my last remark. I claim that
Aristotle considers (R) and (R) as joint conditions for being a
primary cause. But I do not claim that Aristotle reduces the notion
of primary cause to the combination of (R) and (R). Conditions
(R) and (R) are just formal or logical features of a primary cause.
Now, the first half of Posterior Analytics . (besides many other
texts) makes it clear that these two conditions cannot exhaust the requirements for being a primary cause—since, for instance, not only
not twinkling but also being near the Earth as attributes of planets satisfy both conditions, although one of them is the cause of the other
but not vice versa (see also Post. An. . , especially b–).
Aristotle surely has some additional criteria for sorting out
primary causes, besides these logical conditions. These additional
criteria rest on the notion of explanatory relevance and cannot be
cashed out in formal or logical features of their own. This is not
the place for a full discussion of these additional criteria, but I
shall sketch their main outlines. The first and main point is the
one that has already been implied in the previous paragraphs: the
explanatory appropriateness of a primary cause is not marked by
any logical asymmetry with its effect: as the effect can be soundly
deduced from the cause, in the same way the cause can be soundly
deduced from the effect (see a–b, b–). This means
that explanatory appropriateness cannot be reduced to any logical
property of a cause. Second, the explanatory appropriateness of
a primary cause has a direct link with definitional priority: the
primary cause is an important factor in the definiens account of the
thing it is the cause of, but not the other way round (cf. b–).
established in a–b and in Post. An. . , b–. See R. J. Hankinson,
Cause and Explanation in Ancient Greek Thought (Oxford, ), –; McKirahan,
Principles, –; Ferejohn, Formal, –. For a recent approach to the relation
between coextensiveness and causation see A. M. Leroi, The Lagoon: How Aristotle
Invented Science (London, ), –.
See M. Ferejohn, ‘The Immediate Premises of Aristotelian Demonstration’
[‘Immediate’], Ancient Philosophy, (), – at –; Ferejohn, Formal, –
; McKirahan, Principles, –; Stein, ‘Pluralism’, –; K. Koslicki ‘Essence,
Necessity and Explanation’, in T. Tahko (ed.), Contemporary Aristotelian Metaphysics (Cambridge, ), – at –; O. Goldin, ‘Circular Justification and
Explanation in Aristotle’, Phronesis, (), – at –.
The word αἴτιον is not accompanied by any adjective in b–, but τὸ αἴτιον
Causality and Coextensiveness in Post. An. .
The definitional priority gives expression to the claim that the fully
appropriate explanation of X is tantamount to understanding the
essence of X (see a–, –, a). Third, the explanatory
appropriateness of a primary cause is also marked by the fact that it
makes a series of why questions come to an end: once the primary
cause has been reached, there is no more sense in pursuing why
questions about the explanandum at stake (cf. b–). Fourth,
although Aristotle never offers a conceptual analysis of the notion
of primary cause or the notion of explanatory appropriateness, he
relies on the intuitiveness of some uncontroversial examples to put
us on the right track: it is because planets are near the Earth that
they do not twinkle, but not the other way round (cf. a–); it
is because the Moon is spherical that it waxes and wanes in the
way it does, but not the other way round (b–); the Earth’s
interposition causes the Moon to be deprived of light, but the
Moon’s privation of light does not make the Earth stay in the
middle (b–).
Instead of substantiating these outlines, my main concern in the
remainder of this paper is to highlight the fact that the notion of
primary cause as depicted in b– fits very well with another
feature of Aristotle’s theory of scientific demonstration in the Posterior Analytics.
. Primary cause, coextensiveness, and katholou predication
My interpretation allows us to gain a better understanding of Aristotle’s insistence on coextensiveness between terms in a scientific
explanation. Coextensiveness is the formal feature of the stricter
notion of katholou introduced in Post. An. . , b ff. Of course,
this notion also has intensional features which are mostly relevant
in Aristotle’s theory, but all I need to say now is that coextensiveness between B- and A-terms is indeed an important requirement
for scientific explanations. Aristotle also insists on coextensiveness
in some contexts refers to what I am calling the primary cause, namely: the cause
that delivers the appropriate explanation and thereby scientific understanding of its
explanandum.
I disagree with R. Smith, ‘Immediate Propositions and Aristotle’s Proof Theory’, Ancient Philosophy, (), – at : ‘we have no idea how he finally resolved the problem of fitting convertible terms into sciences: it is possible that he
abandoned them in despair . . . or that he simply lost interest in the whole enterprise
Lucas Angioni
between C- and A-terms at a ff.—which implies coextensiveness between the three terms of the demonstration—but let me keep
this further complication out of my present case. Some scholars
have said or implied that the notion of coextensive or commensurate katholou introduced at Post. An. . , b ff., is a peculiarity
confined to those specific passages with no important role to play
in Aristotle’s theory. Ross says that ‘this strict sense of katholou is,
perhaps, found nowhere else in Aristotle. Usually the word is used
in the sense of kata pantos simply’. Barnes refers to it as a ‘singular sense’. As it stands, this is a remark about terminology, which
I take to be wrong on its own. But the mistake is to assume or
infer that the notion of commensurate katholou has no major use in
Aristotle’s theory of demonstration.
First, coextensiveness between the A- and the B-term is an important feature of the appropriate demonstrations depicted in Post.
An. a–b. One might be tempted to say that coextensiveness
should not be taken as the paradigmatic case. Aristotle has picked
up cases involving coextensive terms in order to stress that the explanatory appropriateness of a cause is not reducible to being a
necessary and sufficient condition for deducing the explanandum.
when its difficulties became too great’. I agree rather with Ferejohn, ‘Immediate’,
–, McKirahan, Principles, , and Ferejohn, Formal, –. Certainly, coextensiveness is not the end of the story (more on this below), since there still might
be intensional gaps in an attempted demonstration with coextensive terms (see P. S.
Hasper, ‘Sources of Delusion in Analytica Posteriora . ’, Phronesis, (), –
at –, and Angioni, ‘Definition’, –). A difficulty stems from the fact that, in
Post. An. . , a–, Aristotle has said that coextensive terms are rare in demonstrations (ἐπειδὴ ὀλίγα τοιαῦτα ἐν ταῖς ἀποδείξεσι). I cannot address this difficulty
here, but I do not believe that a– jeopardizes my interpretation. Many things
depend on how ἀποδείξεις should be taken in a. I take Aristotle to be describing
what his adversaries would have to admit: for them, there is circular and reciprocal
demonstration, but at the same time they take demonstrations as ranging over any
sort of terms (as Aristotle has done with ‘demonstrations’ in his syllogistics), not
only over coextensive terms. I thank an anonymous referee for stressing the need to
mention a–.
Coextensiveness between the three terms seems to be the concern in Post. An.
. –, whereas in Post. An. . – Aristotle considers prominent cases where coextensiveness between the major and the middle is good enough: the subordinate
sciences (. ); application of theorems to particular instances (. ); and cases such
as the lunar eclipse (. ).
Prior and Posterior Analytics, .
Posterior Analytics, .
Katholou has the ‘strict sense’ throughout chapters . –, . , and . of
the Posterior Analytics (as well as in . , b: see L. Angioni, ‘Knowledge and
Opinion about the Same Thing in APo A-’, Dois Pontos, (), –). See
also . , b; . , b, , , a–, ; . , a, .
Causality and Coextensiveness in Post. An. .
Once this thesis has been established, Aristotle might proceed with
other examples in which the terms are not coextensive with each
other. One might pursue this same line of argument concerning
Post. An. . , a chapter in which, examining the logical relations
between cause and effect, Aristotle concludes that ‘also the middle
term in these cases must be equal to that of which it is the cause,
i.e. must convert’ (b–). The expression ‘in these cases’ refers
exactly to cases in which the explanandum at stake is a commensurate feature attributed to its proper subject. One might then argue
that ‘these cases’ are peculiar cases and nothing guarantees that they
must be taken as central cases.
However, this line of argument overlooks the fact that Post. An.
. is a chapter devoted to examining the logical relations between
the cause and that of which the cause is cause. The chapter starts
by asking whether there is a relation of mutual entailment between
cause and effect (a–b) and, having established that there is such
a relation (b–), argues that the asymmetry between cause and
effect is compatible with that logical relation (b–). From this
it clearly follows that the explanatory appropriateness of a primary
cause cannot be reduced to this relation of mutual entailment. It
is not a coincidence that this string of claims chimes with the argument developed at Post. An. . , a–b, which is another
chapter mainly concerned with clarifying what being a cause delivering a proper demonstration amounts to. It does not seem reasonable to say that central examples and central arguments in both
official treatments of causes and their logical relations to their effects in the Posterior Analytics should not be taken as the paradigmatic case for scientific demonstration—especially because further
evidence is furnished by Aristotle’s conspicuous insistence on coextensiveness and katholou predications in chapters – of book .
And Aristotle coherently employs the terminology introduced in
Post. An. . when he characterizes the mutual entailment or the
coextensiveness between causes and explananda at Post. An. . :
[T] ἢ εἰ ἀεὶ καθόλου τὸ πρόβλημά ἐστι, καὶ τὸ αἴτιον ὅλον τι, καὶ οὗ αἴτιον,
καθόλου; οἷον τὸ φυλλορροεῖν ὅλῳ τινὶ ἀφωρισμένον, κἂν εἴδη αὐτοῦ ᾖ, καὶ
τοισδὶ καθόλου, ἢ φυτοῖς ἢ τοιοισδὶ φυτοῖς· ὥστε καὶ τὸ μέσον ἴσον δεῖ
εἶναι ἐπὶ τούτων καὶ οὗ αἴτιον, καὶ ἀντιστρέφειν. (b–)
Whenever the explanandum [problēma] is a commensurate universal [katholou], then the cause is also a whole, and that of which it is
the cause is a commensurate universal. For instance, leaf-shedding is
Lucas Angioni
confined to a given whole, even if there are species of it, and it is a
commensurate universal [katholou] for those—be it plants or plants
of a specific sort. Consequently, also the middle term in these cases
must be equal to that of which it is the cause, i.e. must convert. (my
translation)
There are many controversial intricacies in this passage, but it is
clear that it confirms what Aristotle has established in [T]: the
primary cause delivering the appropriate explanation for a given
explanandum must be (when put in the triadic framework of syllogistic demonstration) a middle term coextensive with the A-term
(and, in most cases, coextensive with the C-term too). And this
means that: (i) the primary cause is a necessary and sufficient condition for its explanandum to obtain (and vice versa), although (ii) its
explanatory appropriateness cannot be reduced to its being a necessary and sufficient condition for its explanandum to obtain. This is
the .
Furthermore, my interpretation of the requirements for being a
primary cause is also suited to Aristotle’s notion of causal priority
as described in Cat. b–. All I need do here is to remark that
the relevant sort of causal priority (which, of course, requires some
asymmetry between cause and effect) is perfectly compatible with
logical convertibility (or reciprocability) between cause and effect,
and such convertibility, when cashed out in the syllogistic framework, amounts to coextensiveness between B- and A-terms and to
mutual entailment between the minor premiss and the conclusion
of Barbara and Camestres syllogisms. This is enough to strengthen
the case for the claim that the notion of a primary cause and its logical feature of being coextensive with that of which it is the cause
is not a sui generis peculiarity confined to a weird paragraph in Post.
An. . , but plays a central role in Aristotle’s overall theory of
demonstration.
Furthermore, coextensiveness (besides other things) is also implied in Post. An.
. , a chapter concerned with establishing that universal demonstrations are superior to partial demonstrations. As a quick survey of the examples is enough to
settle, the contrast in Post. An. . involves two kinds of Barbara demonstration,
one (the ‘universal’) in which the A-term () is a commensurate universal of the Cterm (triangle), and another (the ‘particular’) in which the C-term (isosceles) does
not exhaust the extension of the A-term ().
On this point see M. Peramatzis, Priority in Aristotle’s Metaphysics (Oxford,
), .
Causality and Coextensiveness in Post. An. .
. Conclusion
I have shown that b–, far from being a desperately convoluted passage committed to a false principle about explanation, is a
coherent argument in which Aristotle presents logical requirements
for being a primary cause. I have also shown that the seeming redundancy of Aristotle’s expression is rather his way of making a
general point about primary causes which is not restricted to either
negative causes or second-figure syllogisms. The requirements are
in accordance with another feature of Aristotle’s theory of scientific
demonstration, namely, the notion of a commensurate universal as
developed in Post. An. . . I have not discussed the important intensional features of this notion, because it was enough for my point
to focus on its extensional feature: being a commensurate universal
includes (but does not collapse into) the satisfaction of requirements
(R) and (R) for being the appropriate cause of a given explanandum. I hope to have shown that passage b–, in expounding
(R) and (R) as logical criteria for being a primary cause, makes an
important contribution to Aristotle’s overall project in the Posterior
Analytics.
University of Campinas
B I B LI O G RAPH Y
Angioni, L., ‘Aristotle on Necessary Principles and on Explaining X
through the Essence of X’, Studia Philosophica Estonica, (),
–.
Angioni, L., ‘Aristotle’s Definition of Scientific Knowledge (APo b–
)’ [‘Definition’], Logical Analysis and History of Philosophy, (),
–.
Angioni, L. ‘Knowledge and Opinion about the Same Thing in APo A-’,
Dois Pontos, (), –.
Barnes, J., Aristotle’s Posterior Analytics [Posterior Analytics], Clarendon
Aristotle Series, nd edn. (Oxford, ).
Bronstein, D., Aristotle on Knowledge and Learning [Learning] (Oxford,
).
Burnyeat, M. F., ‘Aristotle on Understanding Knowledge’, in E. Berti
(ed.), Aristotle on Science: The Posterior Analytics (Padua, ),
–.
Lucas Angioni
Charles, D., Aristotle on Meaning and Essence (Oxford, ).
Charles, D., ‘Definition and Explanation in Posterior Analytics and Metaphysics’, in id., Definition in Greek Philosophy (Oxford, ), –.
Ferejohn, M., Formal Causes [Formal] (Oxford, ).
Ferejohn, M., ‘The Immediate Premises of Aristotelian Demonstration’
[‘Immediate’], Ancient Philosophy, (), –.
Ferejohn, M., The Origins of Aristotelian Science [Origins] (New Haven,
).
Goldin, O., ‘Circular Justification and Explanation in Aristotle’, Phronesis,
(), –.
Hankinson, R. J., Cause and Explanation in Ancient Greek Thought (Oxford, ).
Hasper, P. S., ‘Sources of Delusion in Analytica Posteriora . ’, Phronesis,
(), –.
Hocutt, M., ‘Aristotle’s Four Becauses’, Philosophy, (), –.
Koslicki, K., ‘Essence, Necessity and Explanation’, in T. Tahko (ed.),
Contemporary Aristotelian Metaphysics (Cambridge, ), –.
Kosman, L. A., ‘Explanation, Understanding and Insight in Aristotle’s
Posterior Analytics’, in H. Lee, A. Mourelatos, and R. Rorty (eds.), Exegesis and Argument: Studies in Greek Philosophy Presented to Gregory
Vlastos (Assen, ), –.
Leroi. A. M., The Lagoon: How Aristotle Invented Science (London, ).
Lesher, J. H., ‘Aristotle on ἐπιστήμη as Understanding’, Ancient Philosophy, (), –.
McKirahan, R., Principles and Proofs: Aristotle’s Theory of Demonstrative
Science [Principles] (Princeton, ).
Malink, M., Aristotle’s Modal Syllogistic (Cambridge, Mass., ).
Mignucci, M., Aristotele: Analitici secondi [Analitici] (Rome and Bari,
).
Moravcsik, J. M., ‘Aristotle on Adequate Explanations’, Synthese,
(), –.
Moravcsik, J. M., ‘What Makes Reality Intelligible? Reflections on Aristotle’s Theory of Aitia’ [‘Aitia’], in L. Judson (ed.), Aristotle’s Physics
(Oxford, ), –.
Mure, G. R. G., Posterior Analytics, in W. D. Ross (ed.), The Complete
Works of Aristotle Translated into English (Oxford, ).
Pellegrin, P., Aristote: Seconds Analytiques (Paris, ).
Peramatzis, M., Priority in Aristotle’s Metaphysics (Oxford, ).
Ross, W. D., Aristotle’s Prior and Posterior Analytics: A Revised Text with
Introduction and Commentary [Prior and Posterior Analytics] (Oxford,
).
Smith, R., ‘Immediate Propositions and Aristotle’s Proof Theory’, Ancient
Philosophy, (), –.
Causality and Coextensiveness in Post. An. .
Stein, N., ‘Aristotle’s Causal Pluralism’ [‘Pluralism’], Archiv für Geschichte
der Philosophie, (), –.
Stein, N., ‘Causation and Explanation in Aristotle’, Philosophy Compass,
(), –.
Williams, S., and Charles, D., ‘Essence, Modality and the Master Craftsman’, in E. Feser (ed.), Aristotle on Method and Metaphysics (New York,
), –.