A Set Theoretic Interpretation of Aristotle’s Syllogistic
Kenneth Boyd
PHIL 540
April 11, 2006
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A Set Theoretic Interpretation of Aristotle’s Syllogistic
Aristotle’s syllogistic has, by various scholars, been regarded as prototypical to
and foundational of modern logic. Interpreting the syllogistic, however, is not an
immediately straightforward process, and there has been considerable debate as to its
intended function (i.e. as representing ontological facts or simply supplying a template
for proper argumentation) and structure (i.e. the related debate as to whether the
syllogistic is a system of inferences or implications). In this paper I propose that viewing
Aristotle’s syllogistic via a set theoretic paradigm can provide insight into not only
interpretations of the syllogistic and consequences thereof, but also by providing a tie to
Aristotle’s ontology. I will begin by outlining the possible forms of propositions in the
syllogistic, and their analogues in set theory. I will then focus mainly on the structure of
the set in terms of the relationship between members of the set and the set as a whole, and
how Aristotle’s treatment of parts and wholes (and totalities and heaps) and their role in
the syllogistic permits a set-theoretic interpretation. My intentions are not to definitively
settle the debates mentioned above; rather, by showing that the ontological commitments
of set theory are not in opposition with Aristotle’s own ontology, set theory becomes a
viable interpretive paradigm for the syllogistic.
The Syllogistic
In the Prior Analytics, Aristotle introduces his concepts of premises, terms and
the syllogism itself: “A premiss…is a sentence affirming or denying one thing of
another. This is either universal or particular or indefinite. By universal I mean the
statement that something belongs to all or none of something else; by particular that it
belongs to some or not to some or not to all; by indefinite that it does or does not belong,
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without any mark to show whether it is universal or particular, e.g. ‘contraries are
subjects of the same science’, or ‘pleasure is not good’” (Prior Analytics I.1,24a16-21).
Here Aristotle dictates that all syllogisms are to be constructed with premises existing in
one of four relations: universal attribution (B belongs to every A), universal negation (B
belongs to no A), particular attribution (B belongs to some A) and particular negation (B
does not belong to some A). He also introduces his rules of conversion, necessary
consequences and substitutable propositions of the aforementioned four possible
relations. When considering the convertibility of universals, negations must be entirely
convertible, i.e. that “if no pleasure is good, then no good will be pleasure” (Prior
Analytics, 25a7), while universal attributions must be convertible in part, i.e. that “if
every pleasure is good, some good must be pleasure” (Prior Analytics, 25a7).
Convertibility rules also apply to particulars, the particular attribution again in part, i.e.
“if some pleasure is good, then some good will be pleasure” (Prior Analytics, 25a10,
emphasis added). The only syllogistic relation of premises that need not necessarily
convert is the particular negation, such that “if some animal is not man, it does not follow
that some man is not animal” (Prior Analytics, 25a11-12). These four relations and their
convertibility rules form the backbone of Aristotelian syllogisms: for instance, the
syllogism “Barbara” follows the form that “If A is predicated of all B, and B of all C, A
must be predicated of all C” (Prior Analytics, 25b39-26a1).
A Set Theoretic Representation
All of the above propositions can be represented accurately and intuitively using
the formalizations of naïve set theory1. It is, I believe, intuitive to understand the
For the purposes of the discussion, the difference between “naïve” set theory and “modern” set theory are
not important.
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propositions and the conversions diagrammatically. I illustrate these relations below, and
in anticipation of discussion in the next section, indicate their analogues and
formalizations in set theory:
Proposition: B belongs to no A.
A
B
Convertibility Rule: A belongs to no B.
Set Theory Analogue: A  B = 0
(~x)(x  A & x  B)
Proposition: B belongs to every A.
B
Convertibility Rule: Some A belongs to B.
A
Set Theory Analogue: B  A
(x)(x  B  x  A)
Proposition: B belongs to some A.
A
B
x
Convertibility Rule: A belongs to some B.
Set Theory Analogue: A  B  0
(x)(x  A & x  B)
Proposition: B does not belong to some A.
A
B
Convertibility Rule: Some A may belong to B.
Set Theory Analogue: A  B
(x)(x  A  x  B)2
Note that this formalization captures the limiting case as put forth by Aristotle: that “B does not belong to
some A” may be formalized by any of the three previous cases, given the specific relation between A and
B; in this instance, the case where some A does belong to B is shown.
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The formulations above treat Aristotle’s terms (A and B, in this case) as sets. A
number of properties of sets are used in conveying Aristotle’s conditions. First, the
concepts of membership and non-membership (, ), that sets contain elements that are
members of that set, and there are elements that are not members of that set (for example,
the number 2 is a member of the set of even numbers, whereas 3 is not). In Aristotelian
language, there is an analogue with predication: that “man” is predicated of “Socrates” is
interpreted in this formalization as saying that “Socrates is a member of the set of men”.
Notice also that membership need not apply solely to individuals, but that sets can be
members of other sets. This again is analogously representative of the notion that
“animal” can be predicated of “man”, as well as any individual animal; thus the set of
men could be seen as a member of the set of animals.
Second, sets are defined as consisting of elements that all share a certain property.
This is known as “set building”: we define a set S such that S = {x: x}, where  is a
predicate, and S is read as “the set of all x’s such that  of x”. For example, we can
define a set R such that R = {x: x is red}, and thus R represents the set of all red things.
Similarly if we say M = {x: x is a man}, then M can be seen as representing the set of all
men, the set theory equivalent of the Aristotelian universal “man” (universals will be
discussed in further detail in the next section). Third, the above formalizations make use
of the notion of a subset, that is where every member of one set is a member of another,
but not necessarily vice versa. Aristotle captures this notion by noting the limiting case
of non-convertibility of particular negation, to repeat that all men must be animals but it
does not follow that all animals must be men.
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Although set notation seems to convey Aristotle’s intended relations between
terms in syllogisms, and the use of diagrams is convenient in order to convey the essence
of the consequences of the four types of propositions and their convertibility rules, there
remain certain underlying assumptions about the structure of sets, namely the relation
between the set as an entity and its members. I will start with an examination of how
Aristotle defines “terms”.
Aristotle’s Terms
As Jan Łukasiewicz notes in “Aristotle’s Syllogistic”, “nothing in the Prior
Analytics is said about the terms” (Łukasiewicz, 4). Nevertheless, he concludes based on
Aristotle’s definitions in De Interpretatione, that “it is evident that the terms of universal
and particular premisses must be universal” (Łukasiewicz, 4). His argument proceeds as
follows: since individuals cannot be predicated of anything else, and because Aristotle’s
convertibility rules demand that each term be able to be predicated of the other, then the
terms used in the syllogistic must be universals. Łukasiewicz also notes Aristotle’s
mention of a gradient of predicability: on the one end exist the individuals, those things
that cannot be predicated of but other things can be predicated of them, on the other that
which can predicate of but cannot “demonstrate another predicate, save as a matter of
opinion” (Prior Analytics, I.27, 43a30-40). It is what exists between these two extremes
that concerns us when talking about the syllogistic: “Whatever lies between these limits
can be spoken of in both ways: they may be stated of others, and others stated of them.
And as a rule arguments and inquiries are concerned with these things” (Prior Analytics,
I.27, 43a41-44). These things are universals, Aristotle’s terms.
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There is, however, debate surrounding interpretation of Aristotle’s terms, namely
as to whether or not they exist as non-worldly abstractions (what we would consider to be
variables in modern logic) or if they exist rather as “place holders” for objects in the real
world; this particular debate underlies the debate concerning whether the syllogistic
makes ontological or empirical arguments. The former view is held by Łukasiewicz, the
latter by John Corcoran. In his essay, “The Founding of Logic”, Corcoran states his view
that “the ‘terms’ are concrete substantives such as ‘human,’ ‘animal,’ ‘plant,’ etc.”
(Corcoran, 12-13). Further, he states “[t]here is no evidence whatever to support the
widespread belief that Aristotle postulated abstract forms over and above concrete
propositions, concrete arguments, or concrete deductions. Indeed, such a postulation
would have been characteristically platonistic and nonaristotelian” (Corcoran, 13). It
seems that under a set theoretic interpretation, terms as sets may be interpreted as either
abstract variables or concrete substantives. Taking the syllogism “Barbara” again as an
example, if we consider Aristotelian terms as sets, the conclusion “A must be predicated
of all C” follows validly from the premises “if A is predicated of all B, and B of all C”
regardless of the contents of those sets. That sets exhibit the property of transitivity of
inclusion (or predication) is a general feature of sets, and thus a set theoretic
interpretation does not prevent interpreting Aristotle’s terms as variables. If, however,
we are to view sets as Aristotelian universals, we may indeed also view these terms as
concrete substantives without loss in interpretive power. Thus, it seems that a set
theoretic interpretation, at least at this point, allows Łukasiewicz and Corcoran to argue
amongst themselves.
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That we should introduce set notation into the syllogistic does not come without
its ontological consequences. First, it is not clear whether Aristotle’s ontology would
allow for an analogous concept of a set, and thus we run the risk of making an entirely
anachronistic comparison: specifically, it is not immediately apparent if Aristotle would
treat the notion of a set as a “whole”, a “totality”, or a “heap”. Second, set theoretic
principles establish a certain relationship between a set and its elements. It is then
important to examine how Aristotle himself treats the relationship between parts and
wholes, and what his view implies for the plausibility of allowing the concept of a set
into his ontology. These questions will now be addressed in turn.
Wholes, and their Parts
In Metaphysics, Aristotle defines “whole” in the following way (it will be prudent
to quote Aristotle’s definition at length):
(1) that from which is absent none of the parts of which it is said to be naturally a
whole, and (2) that which so contains the things it contains that they form a unity;
and this in two senses – either as being each severally one thing, or as making up
the unity between them. For (a) that which is true of a whole class and is said to
hold good as a whole (which implies that it is a kind of whole) is true of a whole
in the sense that it contains many things by being predicated of each, and by all of
them, e.g. man, horse, god, being severally one single thing, because all are living
things. But (b) the continuous and limited is a whole, when it is a unity consisting
of several parts, especially if they are present only potentially, but, failing this,
even if they are present actually. Of these things themselves, those which are so
by nature are wholes in a higher degree than those which are so by art, as we said
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in the case of unity also, wholeness being in fact a sort of oneness. (Metaphysics
V.26, 1024a26-36)
The definition as it stands, at least superficially, makes it seem as though sets would be
viable candidates for being wholes under Aristotle’s definition. His condition that “that
which is true of a whole class and is said to hold good as a whole” in fact “implies that it
is a kind of whole”. Thus Aristotle’s examples of wholes, “man, horse, god” exist as
universals, wholes inasmuch as every part of each whole is predicated of the same thing,
i.e. that “man” is a whole consists of each member of the whole being a man. In addition,
Aristotle offers the comparison of the individual whole, that is what is “continuous and
limited”, whose parts can exist in either actuality or potentiality. He briefly mentions a
“higher degree” of wholeness: presumably, the whole whose parts are potential is a
higher degree of whole than that whose parts are actual, and that natural wholes exist in a
higher degree than those “which are so by art”. The “degree” of wholeness of a set is not
immediately important. Rather, what concerns us now is, since we have committed
ourselves to treating sets as universals in the syllogistic, whether they should be
interpreted as universal wholes or individual wholes.
Before facing this problem, however, that sets are comprised of members
encourages examination of Aristotle’s definition of “part”:
(1) that into which a quantum [“that which is divisible into two or more
constituent parts of which each is by nature a ‘one’ and a ‘this’” (Metaphysics
V.12, 1020a7-8)] can in any way be divided…e.g. two is called in a sense a part of
three. It means (b), of the parts in the first sense, only those which measure is the
whole; this is why two, though in one sense it is, in another is not, called a part of
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three. – (2) The elements into which a kind might be divided apart from the
quantity are also called parts of it; for which reason we say the species are parts of
the genus. – (3) The elements into which a whole is divided… (4) The elements in
the definition which explains a thing are also parts of the whole; this is why the
genus is called a part of the species, though in another sense the species is part of
the genus (Metaphysics V.25, 1023b12-25).
Here there is a certain amount of tension in calling a set a whole, inasmuch as a whole
consists of parts. In this definition Aristotle seems to want to define “part” in terms of
that into which a whole can be divided. This seems to imply that wholes are, in a sense,
ontologically prior to their parts: the tension exists inasmuch as sets are considered to be
comprised of their parts, and as the familiar set theory maxim states, a set is defined by
its elements. Thus when constructing sets we “build them up” out of their parts, a
procedure resulting in a relationship between parts and compound that Aristotle might not
immediately endorse as whole-worthy. Aristotle does, however, define the concepts of
totalities and heaps. Would a set be better characterized as one of these?
Heaps, Totals, and “Something Else”
Aristotle distinguishes between two types of compounds: those that are
composed of elements such that they create a whole, and those that result in heaps. To
distinguish the two, he gives the example of a syllable:
now the syllable is not its elements, ba is not the same as b and a, nor is flesh fire
and earth (for when these are separated the wholes, i.e. the flesh and the syllable,
no longer exist, but the elements of the syllable exist, and so do fire and earth);
the syllable, then, is something – not only its elements (the vowel and consonant)
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but also something else, and the flesh is not only fire and earth or the hot and the
cold, but also something else… (Metaphysics VII.17, 1041b12-18)
Thus for Aristotle the distinguishing feature between a compound as a heap and a
compound as a whole is this “something else”, something such that the whole is
something greater than, or is in addition to, the sum of its parts, whereas the heap is
simply the sum of its parts (and nothing else). Aristotle reaches a conclusion as to the
nature of this “something else” in the following way: it cannot be another element of the
whole, for if it were then there would be a further “something else” that would
encapsulate this additional element, and the process would repeat ad infinitum; a similar
argument applies if the “something else” is thought to be a compound (Metaphysics
VII.17, 1041b20-25). What this “something else” is, concludes Aristotle, is “the cause
which makes this thing flesh and that a syllable…And this is the substance of each thing
(for this is the primary cause of its being)” (Metaphysics VII.17, 1041b25-30). He further
describes this cause as a “kind of ‘nature’, which is not an element but a principle”
(Metaphysics VII.17, 1041b30-31). Again it is tempting here to draw parallels with the
defining characteristic of sets: that sets should be “built up” on the basis of elements
meeting the criterion of a defining predicate is analogous to the “cause” or “principle”
behind Aristotle’s wholes. Thus if we bestow the defining predicate of a set with the
status being the “essence” of that set, again Aristotle’s criteria for wholeness seem to be
satisfied.
There is another peculiarity of wholes that Aristotle mentions, in contrast with the
concept of a “total”:
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Again (3), of quanta that have a beginning and a middle and an end, those to
which the position does not make a difference are called totals, and those to which
it does, wholes. Those which admit of both descriptions are both wholes and
totals. These are the things whose nature remains the same after transposition, but
whose form does not, e.g. wax or a coat; they are called both wholes and totals;
for they have both characteristics (Metaphysics V.26, 1024a1-3, emphasis added).
An important characteristic of sets is that the order or repetition of their elements makes
no difference as to the identity of that set. For example, the set A = {1,2,3,4} is
considered equivalent to the set B = {2,4,3,1} as well as to the set C = {1,1,1,1,2,3,4}.
That the repetition of elements found in C does not, I believe, pose a serious problem, as
this repetition says nothing about the structure or nature of the set, but merely represents
a redundancy3. That sets A and C are equivalent, however, seems to bestow sets with the
properties of being both totals and wholes: it could be argued that sets A and B have
different forms (inasmuch as their parts are differently ordered), but are identical in
nature (that is, are defined by the same predicate).
The Apparent Priority of Wholes over Parts
Recapping what has thus far been accomplished: that set theoretic principles are
able to accurately and intuitively capture the nature and logical implications of the
propositions of the syllogistic tempts us to explore the plausibility of viewing Aristotelian
terms as sets. The nature of Aristotle’s terms would require sets to be treated as
universals, and Aristotle’s view of wholes, totals, heaps and parts implies that sets
defined by a predicate also need to be considered as things that are both wholes and
That the set “men” might be represented as {Socrates, Plato, Aristotle, Socrates, Cicero, Socrates, Zeno,
Socrates, Socrates…} says nothing about the nature of men; rather, it is simply a frivolous formalization.
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totals. What has not been dealt with thus far is the tension that exists between the
structure of sets and the ontological priority of wholes that Aristotle mentions at various
times throughout the Metaphysics and elsewhere. Terence Irwin in “Aristotle’s First
Principles” outlines the problem at hand:
What Aristotle calls second substances species and genera, and says first
substances ‘belong in’ them…we might take him to mean that second substances
are classes of which first substances are members. Classes, however, are defined
extensionally…and if Aristotle thinks second substances are classes, he cannot
plausibly claim that they reveal what first substances are. Since the class of men
is coextensive with the class of grammarians (i.e. things capable of grammatical
knowledge), it follows that if man reveals what particular men are, grammarian
must reveal it also…Aristotle must assume that ‘F reveals the essence of G’ does
not allow the substitution of coreferential terms for ‘F’, and that the species of
man cannot be defined extensionally; hence it cannot be a class” (Irwin, 79).
Irwin’s argument rests on the reductio ad absurdum that were “men” to be a set4, it
would be coextensive with the set of “grammarians”, i.e. that they have the same
members and are therefore the same set. But we obviously feel that there is a difference
between what makes “men” men, and what makes “grammarians” grammarians, and
Irwin’s assumption is that if we treat these secondary substances as sets, then we
necessarily conflate secondary substances with properties of primary substances. I find
this argument unconvincing for a number of reasons: first, it is not clear that if we were to
take “men” and “grammarians” as sets, that they would be coextensive. Surely there
“Set” and “class” are taken here to be synonymous; nothing of importance rests on the distinction in the
present context.
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exist human beings incapable of using grammar; it seems more likely that “grammarian”
would be a subset of “men” instead of an equivalent set. Second, if we were to find that
two sets defined by different predicates had the exact same members, would we not want
to say that these two predicates are in fact synonymous? For instance, if we construct the
set of men defined by the predicate “is a man” and find it coextensive with the set defined
by the predicate “is a homo sapien” we should not find the set theoretic interpretation
flawed; indeed, we are talking about the same set! What is still of interest, however, is
Irwin’s mention of sets being defined extensionally, and the apparent priority of wholes
over their parts. It is to this subject I turn now, and finally.
Aristotle makes mention of the separation of parts and wholes using the example
of a finger severed from the hand: “it is not a finger in any and every state that is the
finger of a living thing, but a dead finger is a finger only in name” (Metaphysics VII.10,
1035b24-26). Here the tension is apparent: if a whole is to be considered, at least in
some ways, prior to its elements, then the characteristic of sets being created out of
elements presents a potential incompatibility. Elsewhere Aristotle makes a similar
statement concerning the wholeness of an animal: “To say, then, that shape and color
constitute the animal is an inadequate statement, and is much the same as if a woodcarver
were to insist that the hand he had cut out was really a hand” (Parts of Animals, 640b3536). Aristotle also makes mention that the whole is explanatorily prior to its parts: “The
circle and the semicircle also are in this relation; for the semicircle is defined by the
circle; and so is the finger by the whole body, for a finger is ‘such and such a part of a
man’” (Metaphysics VII.10, 1035b8-9). A set, however, by nature does not share this
property: an element is not defined by the set in which it is a member, but rather the set
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is defined by elements that share a certain property. Thus, sets seem to be explanatorily
posterior to its elements; a similar argument can be made concerning the ontological
priority of a sets elements over the set itself.
What needs to be distinguished is Aristotle’s mention of parts of matter and parts
of form. He distinguishes between two different types of formulae: “the formula of the
circle does not include that of the segments, but the formula of the syllable includes that
of the letters; for the letters are parts of the formula of the form, and not matter, but the
segments are parts in the sense of matter on which the form supervenes” (Metaphysics
VII.9, 1035a9-13). Thus for Aristotle what is prior are parts of the formula that constitute
the whole, whereas the whole itself is prior to the parts of matter, i.e. in concrete
instances. Thus we are still able to maintain our original attribution that sets exist as
wholes, as long as the elements of that set, i.e. its parts, are considered to be parts of the
set in the same way that the letters of a syllable are parts of the formula of that syllable.
What has been accomplished? By examining Aristotle’s notions of universals,
wholes, and their constituent parts, it seems that if we choose to interpret the syllogistic
in set theoretic terms there will not be an immediate conflict in ontology. Set theory
does, however, carry with it a number of logical and ontological consequences that could
potentially be considered non-Aristotelian. First, there exists the notion of the empty set,
that is the set that has no members. The empty set itself is still an entity, at least within
set theory, but it is not clear whether Aristotle would accept a whole that has no parts at
all. Second, if we are to take sets as existing as universals, that there are an unlimited
number of sets would require an ontology that allows for an unlimited number of
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universals. These two additional problems are outside of the scope of this paper, but
present interesting problems worthy of investigation.
Conclusion
That the principles of set theory can guide our understanding of the syllogistic, its
terms and its propositions would make such a paradigm of interpretation valuable. That
it is unclear, however, whether the syllogistic was intended as a logic making ontological
claims or merely empirical ones means that for set theory to be a viable mode of
interpretation, it should be compatible with both the structure of Aristotle’s logic and his
ontology. Throughout this paper the former has been shown in set theory’s ability to
capture the four different possibilities of propositions and their convertibility rules, and
the latter has been shown to be compatible with worries considering the nature of
universals, parts and wholes (although further problems potentially exist). Thus by
viewing Aristotle’s terms as sets, I do not wish to propose the grandiose claim that
Aristotle had sets specifically in mind when constructing the syllogistic; I merely wish to
make the more modest claim that both Aristotle’s logic and ontology of parts and wholes
supports a set theoretic interpretation. Set theory, then, is useful as a tool for
understanding Aristotle’s syllogistic, not necessarily as a guiding force in the debates
surrounding it.
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Works Cited
Corcoran, John. “The Founding of Logic: Modern Interpretations of Aristotle’s Logic”.
Ancient Philosophy 14: 9-24, 1994.
Harte, Verity. Plato on Parts and Wholes: The Metaphysics of Structure. Oxford:
Clarendon Press, 2002.
Irwin, Terrence. Aristotle’s First Principles. Oxford: Clarendon Press, 1988.
Łukasiewicz, Jan. Aristotle’s Syllogistic. Oxford: Clarendon Press, 1951.
McKeon, Richard, ed. The Basic Works of Aristotle. New York: Modern Library, 2001.
Potter, Michael. Set Theory and its Philosophy. Oxford: Oxford University Press, 2004.