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IS THE SYLLOGISTIC A LOGIC?

Phil Corkum

University of California, Los Angeles

Delivered to the American Philosophical Association

Central Division Meeting, April 2004

Much of the last fifty years of scholarship on Aristotle’s syllogistic suggests a conceptual

framework under which the syllogistic is a logic, a system of inferential reasoning, only if

it isn’t a formal ontology, a theory of general facts about the world. I argue that this isn’t

obviously the right interpretative framework. The paper comes in three parts. In the first

part, I’ll present a debate between a theoretical and a logical interpretation of the

syllogistic. In the second part, I’ll argue that the framework of this debate rests on just

one of two distinct conceptions of logic. And in the third part of the paper, I’ll argue that

this conception isn’t obviously Aristotle’s.

In this first part of the paper, I’ll present a debate between a theoretical and a logical

interpretation of the syllogistic. The debate centers on the interpretation of syllogisms as

either implications or inferences. But the significance of this question has been taken to

concern the nature and subject-matter of the syllogistic, and how it ought to be

represented by modern techniques.

Is the Syllogistic a Logic?

I’ll begin by reminding readers of the broad outlines of the syllogistic. It’s

controversial how to describe what’s going on in the Prior Analytics; indeed, this

controversy is part of the subject matter of this paper. So I’ll begin with a fairly full

description under one interpretation and then go on to flag some of what’s controversial.

Syllogisms or ‘moods’ are two-premise arguments with categorical propositions

as the premises and conclusion. The assertoric categorical propositions have the forms: B

belongs to every A; B belongs to no A; B belongs to some A; and B doesn’t belong to

some A. The syllogisms are classified into three ‘figures’, which have the following

format. The premises contain the two terms of the conclusion respectively and a common

or ‘middle’ term: in the first figure, the middle term is in the predicate position of the first

premise and in the subject position of the second premise; in the second and third figures,

the middle is the predicate or the subject, respectively, of both premises. So, for example,

one of the moods of the first figure, called by its medieval mnemonic, ‘Barbara’, has this

form:

(Barbara)

A belongs to every B
B belongs to every C
So A belongs to every C.

In chapters A4-7 of the Prior Analytics, Aristotle considers various combinations for

these three figures and shows which are valid and which invalid. The valid moods of the

first figure are taken to be evidently valid; the validity of the valid moods of the higher-

order figures is established by showing that these moods stand in a certain relation to one

of the moods of the first figure; this derivation process is called perfection. The valid

syllogisms, then, form a bipartite structure; following John Corcoran 1994, let’s call this

Is the Syllogistic a Logic?

an initial-vs.-derivative structure. Finally, the invalidity of the invalid combinations is

established by counterexample.

The outline just drawn presents syllogisms as inferences. And this is part of what

was controversial. Lukasiewicz (1957: 1-3, 20-30)

noted that Aristotle generally presents

syllogisms in conditional form. For example, Barbara is stated as: “if A is said of every B

and B of every C, then it is necessary for A to be predicated of every C.” This suggests

that syllogisms aren’t inferences but implications.

I’ll turn to the critical attention this view received in a minute. But first I’ll note

the apparent consequences of our position on whether syllogisms are inferences or

implications for the interpretation of the syllogistic. For, if syllogisms are implications,

propositions with factual content, then it seems that the syllogistic, insofar as it’s partly a

systematic taxonomy of syllogisms, is an ontology, a system of general facts. And

furthermore, the most natural modern representation of the syllogistic then would be as an

axiomatic system. But, if syllogisms are inferences, arguments proceeding from premises

to a conclusion, then it seems that the syllogistic is a logic or system of inferential

reasoning. And the most natural modern representation of the syllogistic then would be as

a natural deduction system, as in Corcoran 1974b and Smilely 1973.

The contrast here is partly between the derivation of theorems and the derivation

of arguments. Theorems are established as true by deriving them from other propositions,

axioms or theorems, whose truth has already been established or, in the case of axioms,

accepted without derivation. Arguments, on the other hand, are established as valid by

Is the Syllogistic a Logic?

assuming the truth of the premises and deriving the conclusion using accepted rules of

inference.

So, to summarize, Lukasiewicz held all of the following:

1) Syllogisms express implications; specifically, true universalized conditionals;

2) The syllogistic is an axiomatic system; moods of the first figure are axioms;
moods of the higher-order figures are derived theorems;

3) The syllogistic is a formal ontology: it concerns worldly or extralogical facts;
specifically, such relations among classes as inclusion, exclusion, overlap and
non-inclusion.

I’ll say something later on about the sense in which Lukasiewicz’s interpretation of the

syllogistic concerns facts which might be labelled ‘worldly’.

The aim of the paper is to show that some of these apparent consequences—from

the falsity of (1) to the denial of (3), for example—aren’t obviously right. But it’ll be

useful to first consider a little the question: are syllogisms inferences or implications? So

in the rest of this section, I’ll rehearse the evidence cited in support of, and criticism

levied against, (1), the claim that syllogisms are implications. As I’ve noted, Lukasiewicz

defends the claim by noting that Aristotle generally presents syllogisms in conditional

form. But the textual evidence for the claim is inconclusive: Aristotle often, but not

always, presents syllogisms in conditional form. Moreover, as Austin (1952) and

Corcoran (1972: 278) note, it would be natural in some contexts to express arguments as

conditionals where, if the premises hold, then the conclusion follows.

Recent scholarship has focused on the evidence of indirect proof, one method of

perfection.

For example, the indirect proof of Baroco, from APr A5 27a36-b1, is:

Is the Syllogistic a Logic?

If M belongs to all N, but not to some X, it is necessary that N should not belong
to some X; for if N belongs to all X, and M is predicated also of all N, M must
belong to all X; but is was assumed that M does not belong to some X.

It’s controversial how to describe what happens in Aristotle’s indirect proofs. But

according to one plausible reading, the above passage assumes the premises of Baroco

and shows that its conclusion follows by assuming the negation of one of its premises and

using Barbara to derive a contradiction.

Lukasiewicz noted that an indirect proof of a true universalized conditional must

take as its hypothetical assumption not the negation of the conclusion, as Aristotle does in

converting Baroco, but the negation of the conditional. So either (1) is false, under the

plausible assumption that the only propositions syllogisms could be are conditionals, or

we must ascribe a serious error to Aristotle. Lukasiewicz (1957: 58) opts for the second

disjunct, writing that “Aristotle does not understand the nature of hypothetical

arguments.” This allowed Lukasiewicz to continue to endorse (1).

It’s more tempting to use the evidence as an argument against (1). For suppose

one were persuaded by the evidence from indirect proof to hold the disjunctive

conclusion that either (1) is false or Aristotle makes a blunder. Nonetheless, you adhere to

some such hermeneutic principle as: ascribe errors to Aristotle only as a last resort. So

against Lukasiewicz, you opt for the first disjunct, arguing that (1) is false from this

evidence. This is surely the more attractive line, if indeed we’re forced to make this

decision between the two disjuncts.

However, the evidence from indirect proof fails to support the disjunctive

conclusion and so makes for a poor argument for either disjunct. Lukasiewicz is right to

Is the Syllogistic a Logic?

note that, if syllogisms are implications, propositions with conditional form, then an

indirect proof of a syllogism would begin by assuming the negation of that syllogism. But

the negation of a conditional, of course, can be expressed as a conjunction where the

antecedent obtains and the consequent fails to obtain. And this is just what happens in the

proof of Baroco. Admittedly, the indirect proof doesn’t explicitly make the first move of

assuming the negation of the conditional—along the lines of saying: “Suppose it’s not the

case that if M belongs to all N, but not to some X, it’s necessary that N should not belong

to some X.” But still, it’s open for us to hold that the proof of Baroco starts in medias res,

by explicitly assuming the truth of the two conjuncts of the antecedent and the falsity of

the consequent under the tacit assumption of the negation of the conditional. That is, the

absence of an explicit assumption of the negation of the conditional only shows that the

passage is crabbed, not that either syllogisms aren’t implications or Aristotle was

confused about the nature of indirect proofs.

So the evidence from indirect proof is

inconclusive support for the denial of (1).

So far I’ve broached the issues whether syllogisms are inferences or implications (and, in

the long footnote, whether the syllogistic is better represented as a natural deduction

system or an axiomatic system). And although I hope I’ve given a sense of the complexity

of these questions, I haven’t advanced answers. For my main target in this paper is the

alleged significance of such questions as whether syllogisms are inferences or

Is the Syllogistic a Logic?

implications. That is to say, my target isn’t the truth or falsity of (1) and (2) but the

inference from the falsity of (1) and (2) to the denial of (3). In this third part of the paper,

I’ll argue that this inference is valid under only one of two distinct conceptions of the

nature of logic. This requires a little set-up.

Aristotle is widely and rightly credited as the founder of logic, the formal study of

consequence. That is to say, Aristotle founded a study of what it is for a conclusion to

follow from premises; and the way in which Aristotle conducted this study is formal or

topic-neutral, in



roughly



the following sense. The syllogism, ‘All Greeks are men; all

men are mortal; so all Greeks are mortal’ is a valid inference but it’s validity doesn’t

depend on the meaning of the nonlogical words, ‘Greek’, ‘men’ or ‘mortal’. The

inference would be licensed regardless of what these words meant. The inference from

‘John is a bachelor’ to ‘John is unmarried’, on the other hand, is also a permissible

inference but it’s permissibility depends on the meanings of the nonlogical words. If

‘bachelor’ meant Canadian, then the conclusion wouldn’t follow from the premise.

Under one conception, then, logic is characterized by its indifference to all

worldly facts or its abstraction from all semantic content whatsoever. This conception is

often drawn on in contemporary characterizations of logic; it underlies, for example,

Ernest Nagel’s (1956: 66) claim that logical laws are empty: they don’t tell us anything

about the world. To give just one more example: the conception underlies the view Quine

(1970: 95) ascribes to Carnap: that “it is language that makes logical truths true—purely

language, and nothing to do with the nature of the world.”

Is the Syllogistic a Logic?

The thesis that logical truths hold in abstraction from all facts naturally leads to a

collorary concerning that in virtue of which a logical truth holds: namely, that logical

truths hold solely in virtue of their form. For it’s difficult to imagine what else it may be

in virtue of which a logical truth holds, if not its form, under the conception of logic as

indifferent to worldly facts. So call this the Formal conception of logic.

According to another conception, logic is characterized by its generality or

abstraction from specific content. Such a conception of logic, unlike the Formal

conception, is compatible with the claim that logical truths hold in virtue of highly

general facts about the world. So call this the General conception of logic.

Such a

conception underlies Russell’s famous claim that “logic is concerned with the real world

just as truly as zoology, though with its more abstract and general features.”

I’m thinking

of worldly facts in Quine’s sense when he writes:

A logical truth, staying true as it does under all lexical substituitions, admittedly
depends upon none of those features of the world that are reflected in lexical
distinctions; but may it not depend on other features of the world, features that our
language reflects in its grammatical constructions rather than it’s lexicon?

So, for example, we might hold that the predicate calculus depends on such worldly facts

as the fact that objects bear relations to other objects. Likewise, we might take the

syllogistic to concern worldly or extralogical facts reflected in the categorical

propositions.

Notice, although the stronger Formal conception is common in contemporary

characterizations of logic, the weaker General conception is consistent with the model-

theoretic or Tarski-style treatment of logical consequence pervasive in contemporary

logic. Such approaches view permutation invariance as the distinguishing mark of the

Is the Syllogistic a Logic?

logical; for example, they treat logical truths are those truths which are closed under

permutation of the non-logical constants. But to claim that a truth is invariant under

permutation is not necessarily to claim that the truth holds in abstraction from all factual

content.

I’ll next argue that the inference from the falsity of (1) to the denial of (3) rests on

the Formal conception of logic. Put contrapositively, the inference rests on the move, if

(3) then (1). This move is valid under the Formal construal of logic. For under this

construal of logic, the claim that the syllogistic concerns worldly features entails that

syllogisms are not inferences and the syllogistic, not a logic. But, under the General

construal of logic, (3) doesn’t necessarily entail (1). For it’s consistent to hold, under the

General construal, that the syllogistic concerns worldly features yet that syllogisms are

nonetheless inferences.

Now I’ll argue that Aristotle endorses the General conception of logic but not obviously

the Formal conception of logic.

There’s good reason to think that Aristotle believes that an argument is valid only

if every argument in the same form is valid. This claim is only tacit in the Prior Analytics

but it plays two roles there, as Corcoran (1974) noted. First, to establish validity of all

arguments in the same form as a given argument, he establishes the validity of an

arbitrary argument in the same form



that is to say, leaving its content words

Is the Syllogistic a Logic?

unspecified. As we’ve seen, he uses letters for the terms when stating syllogisms and

when proving the higher-order syllogisms valid by conversion.

Second, to establish invalidity of all arguments in the same form as a given

argument, he produces a specific argument in the required form for which the intended

interpretation is a counter interpretation.

Now, this supports the ascription to Aristotle of the General conception of logic.

But it doesn’t go so far as to support the ascription to Aristotle of the Formal conception.

That is, although arguments in the same form are either all valid or all invalid, this

doesn’t show that the way the world is is a matter of indifference to the question of an

argument’s validity.

And, especially in light of the fact that the Formal conception of logic is a

currently controversial thesis, we need to proceed carefully. Aristotle nowhere expresses

the Formal conception of logic. His methods do not require it. And it’s a substantial and

controversial thesis. So we have no reason to ascribe to Aristotle anything stronger than

the General conception.

I’ll bring the paper to a conclusion. I’ve discussed the inference from the view that

syllogisms are implications to the conclusion that the syllogistic is a theory and so not a

logic. I’ve argued that this inference follows easily from a conception of logic which isn’t

obviously Aristotle’s. Of course, this doesn’t establish that the inference is invalid; but I

hope I’ve shown that the burden of proof is on one who would endorse the inference. At

very least, answering our titular question is more difficult than it may seem.

Is the Syllogistic a Logic?

Cf. Patzig 1968: 3-4, 13.

If p and q are open sentences and Q a string of universal quantifiers, one for each free

variable in (p

⊃

q), then Q(p

⊃

q) is a universalized conditional. So Barbara looks like

this: For all A, B, C: if B holds of every A and C holds of every B, then C holds of every
A.

Another consideration given in favor of treating syllogisms as implications and not

inferences: Lukasiewicz (1957: 21) claims that “no syllogism is formulated by Aristotle
as an inference with the word ‘therefore’ (ara).” Austin (1952: 397) counters that
Aristotle uses ara and h ste occasionally—for example, at APo 2.16 and 1.13
respectively. Of course, little can be made to rest on such evidence.

See Lukasiewicz 1951: 58, Austin 1952: 397-8, Corcoran 1974: 280, Smiley 1973: 137-

Let me note that Corcoran (1974a) doesn’t argue in this way. He agrees with

Lukasiewicz that an explicit reductio is “quite different” from Aristotle’s indirect proofs,
notes that Lukasiewicz doesn’t understand the nature of Aristotelian syllogisms and holds
that syllogisms are inferences but doesn’t go so far as to argue that the evidence from
indirect proof supports the claim that syllogisms are inferences.

Lukasiewicz (1957: 54-7) raises several other considerations. He notes that an inference

allows us to assert the conclusion provided the premises are true; it doesn’t say what
happens when the premises aren’t true. So suppose that the syllogisms are rules of
inference which yield inferences when concrete terms are substituted for the variables.
And now consider the following syllogism:

If all animals are birds
and some owls are not birds
then some owls are not animals.

This follows from the inference rule Baroco when we substitute ‘bird’ for ‘M’, ‘animal
for ‘N’ and ‘owl’ for ‘X’. But, Lukasiewicz urges, we cannot apply the reductio method
Aristotle employs to prove Baroco to this. For one thing, we can hardly admit that the
premises are true since they’re false. Furthermore, we need not suppose that the
conclusion is false: it’s false regardless of whether we suppose its falsity or not. Finally,
the contradictory of the conclusion—all owls are animals—, together with the first
premise yields a true conclusion: all owls are birds. So the reductio isn’t ad impossibile in
this case. This seems confused at every point. A reductio considers the situation,
counterfactual or not, where the premises are true and the conclusion false, and derives a
contradiction within this situation. The question whether the situation is actual is
irrelevant.

Whether syllogisms are presented as conditionals is germane to the question whether

they are propositions only under the assumption that conditional grammatical
constructions in Aristotle refer to propositions. It’s not obvious that this assumption is
right. Thanks to Tad Brennan for pressing me to be clearer on this point.

In turning parenthetically to (2), the claim that the syllogistic is best represented as an

axiomatic system, let me begin by making a disclaimer. Since the locus classicus of

Is the Syllogistic a Logic?

criticism of Lukasiewicz is Corcoran 1974a, it’s worth noting that Corcoran himself
doesn’t infer the denial of (2) from the evidence from indirect proof. There’s a second
consideration brought in favour of this denial and, if I understand Corcoran, this
consideration bears more weight.

I was contrasting before axiomatic and natural deduction systems. The contrast

here, recall, is partly between the derivation of theorems and the derivation of arguments.
Theorems are established as true by deriving them from other propositions, axioms or
theorems, whose truth has already been established or, in the case of axioms, accepted
without derivation. Arguments, on the other hand, are established as valid by assuming
the truth of the premises and deriving the conclusion using accepted rules of inference.

There’s also a relevant difference between an axiomatic system and a natural

deduction system in terms of the logic or reasoning underlying the derivation process
which establishes theorems as true or arguments as valid. In an axiomatic system, the
reasoning underlying the derivation process isn’t explicated within the axiomatic system.
But in a natural deduction system, the initial structures are themselves the basic
inferences used in the deriviation process used to prove the validity of higher order
arguments.

We might follow Corcoran 1994 in holding that in order for a system to be a

logic, it must explicate the very reasoning employed in moving from initial to derivative
structures.

Now, what of the syllogistic? Under the interpretation of the syllogistic as an

axiomatic system, the first figure moods are axioms drawn on in the derivation of the
higher order moods but not themselves embodying the reasoning of this derivation. Under
the interpretation of the syllogistic as a natural deduction system, by contrast, the first
figure moods are the very rules of inference used in the derivation process, along with the
conversion rules.

However, the question whether the syllogisms themselves part of the

inferential reasoning in the derivation process doesn’t decide the issue. Even if we take
the syllogisms to embody part of the reasoning employed in the perfection process, they
are not all that this process needs. Perfection also needs the conversion rules, and these
are not themselves syllogisms. So the syllogistic seems to be at best something of a
hybrid.

It’s not clear to me that the syllogistic couldn’t be as well represented by an

axiomatic system as by a natural deduction system. An axiomatic representation of the
syllogistic could concern rules of inference, but encode these rules as implications. The
system would proceed to prove certain implications true by a proof procedure not itself
expressed by theorems but which is nonetheless explicated within the overall theory.

Such a view of the syllogistic would be problematic, of course, to anyone who has

overheard what the tortoise said to Achilles. But there’s no reason to think that Aristotle
was privy to this conversation.

What hinges on the issue whether the syllogistic is better represented as an

axiomatic system or a natural deduction system is, for Corcoran, the foundation of logic
itself. Corcoran (1974a: 280, italics removed) writes:

Is the Syllogistic a Logic?

if the Lukasiewicz view [that (2) is true] is correct then Aristotle cannot be
regarded as the founder of the science of logic. Indeed Aristotle would merit this
title no more than Euclid, Peano, or Zermelo, regarded as founders, respectively,
of axiomatic geometry, axiomatic arithmetic and axiomatic set theory. Each of
these three men set down axiomatizations of bodies of information without
explicitly developing the underlying logic.

At any rate, even if the syllogistic isn’t fully a logic in Corcoran’s sense, it’s reasonable to
call Aristotle the founder of logic. For suppose that the syllogistic employs an underlying
reasoning which isn’t itself embodied by the syllogisms, and so the syllogistic isn’t a
logic in Corcoran’s sense. Nonetheless, Aristotle shows a logician interest in the
underlying reasoning. Unlike Euclid, Peano and Zermelo, Aristotle is concerned to defend
this reasoning: for example, he proves the validity of the conversion rules.

The issue whether the syllogistic is a theory is germane to the question, is the

syllogistic a logic, only under an assumption about what it is to be a logic. And so it’s to
this assumption to which I’ll now turn in the main body of the paper.

Russell, Introduction to Mathematical Philosophy. London: Allen and Unwin. 1919, p.

169.

Quine, Philosophy of Logic, Prentice-Hall. 1970, p. 95.

For example, we might take the syllogistic to concern such part-whole relations as

containment. I flesh out this interpretation in my “Aristotle on Logical Consequence,” to
be delivered to the Canadian Philosophical Association Annual Congress, Manitoba, May
2004.

The move from the falsity of (2) to the denial of (3) is also easy under the Formal

conception. For if the syllogistic is a theory yielding true implications, then it’s not
something which depends solely on form, and so isn’t a logic. But under the General
conception, (3) doesn’t necessarily entail (2). For it’s consistent to hold, under the
General construal, that the syllogistic is a theory about highly general features of the
world yet is nonetheless a logic. Indeed, under the Formal conception of logic, the
question, is the syllogistic a logic, seems to hinge on the question whether it’s a theory.
Under the General conception, these questions are independent.

Thanks to David Blank, Tyler Burge, Alan Code, Andrew Hsu, Henry Mendell, Calvin

Normore, David Sanson, Terry Parsons and Nick White for discussion. Thanks especially
to John Corcoran and Mary Mulhern for written comments. A version of sections 1-2 was
delivered to the Twenty-Sixth Annual Workshop in Ancient Philosophy at Texas A&M
University, College Station, Texas, March 2003. Thanks to the auditors and especially my
respondent, Robin Smith. Another version was delivered at the International Conference
on Ancient and Medieval Philosophy at Fordham University, New York, November 2003.
Thanks to the auditors and especially my respondent, Tad Brennan.