Two Accounts of Aristotle's Logic - Nicolas Fillion

the University of Western Ontario 

Nicolas Fillion 

250 387 509 

Two Accounts of Aristotle’s Logic 

A Comparison of ̷Lukasiewicz’s and Corcoran-Smiley’s Reconstructions 

Seminar on Aristotelian Logic 

Ph.D. in Philosophy 

April 16, 2007 

Faculty of Arts 

the University of Western Ontario

CONTENTS 2 

Contents 

1 Introductory Remarks 4 

2 Critical Presentation of ̷Lukasiewicz’s View 6 

2.1 The Nature of Syllogism . . . . . . . . . . . . . . . . . . . . . . . 7 

2.2 Perfect Syllogisms, Reduction and Rejection Proofs . . . . . . . . 12 

2.3 Logic of Proposition as Preceding the Logic of Terms . . . . . . . 21 

2.4 ̷Lukasiewicz’s Symbolic System for Aristotle’s Logic . . . . . . . 23 

2.5 Summary of the Discussion . . . . . . . . . . . . . . . . . . . . . 25 

3 Presentation of the Corcoran-Smiley View 26 

3.1 Explanation of the Approach . . . . . . . . . . . . . . . . . . . . 27 

3.2 Formal Natural Deduction System . . . . . . . . . . . . . . . . . 29 

3.3 Objections to the View of Syllogistic as a Natural Deduction System 31 

4 Conclusion: Two Traditions in Logic 32

CONTENTS 3 

Abstract 

It is widely accepted, and quite rightly, that Frege’s publication of the 

Begriffsschrift in 1879 begat the most prolific era of the history of logic. 

However, related to this is the highly dubious view, that “mathematical 

logic differs from the traditional formal logic so markedly in method, and 

so far surpasses it in power and subtlety, as to be generally and not unjustifiably 

regarded as a new science.” (Quine, 1965) The obvious ideological 

correlate of this claim is, that traditional formal logic no longer retains 

any theoretical interest, except maybe for archaic philosophers. The effect 

of this view is that contemporary logicians have looked upon Aristotle’s 

logic with jaundiced eyes, identifying as many mistakes as possible and 

exclusively emphasizing discontinuities. 

In this paper, I will argue that this view is wrong insofar as Aristotle’s 

logic can be reformulated in a way that satisfies the standards of modern 

mathematical logic without compromising its underlying philosophy. The 

argument will consist of a compact presentation and a critical examination 

of the two most famous attempts to produce such a model, namely 

̷Lukasiewicz’ reconstruction of syllogistic as an extension of propositional 

calculus (1957) and Corcoran-Smiley’s natural deduction system (1973-4). 

̷Lukasiewicz’s central interpretive claims are (1) that syllogistic is not a 

theory of logically valid arguments but a system of tautologies and (2) 

that it is not a self-sufficient system of logic, since it depends on propositional 

logic for its correctness. This view will be shown to be inadequate, 

since it fails to do justice to Aristotle’s methods of proof. Therefore, despite 

its authoritative position in the literature, ̷Lukasiewicz’s view does 

not render its lettres de noblesse to traditional logic. 

Corcoran and Smiley have developed a much more promising alternative. 

Their two central theses are opposed to those of ̷Lukasiewicz: (1) 

Aristotelian syllogistic is a theory of logically valid arguments and (2) it 

is a self-sufficient theory of logic that presupposes no other system. It will 

be shown that by giving an account of syllogistic as a system of rules of inference, 

it is possible to make sense of most of Aristotle’s claims regarding 

the constitutive elements of logic and his methods of proof (conversion, 

ecthesis, per impossibile). As a reconstruction of Aristotle’s syllogistic 

according to our current standards, this model shows that Aristotelian 

logic still has a theoretical interest, and still is an active field of research.

1 INTRODUCTORY REMARKS 4 

1 Introductory Remarks 

A very important event for the historical development of logic took place in 

1879, when Frege published the Begriffsschrift. Many subsequent logicians had 

been for ever praising for this seminal work. This very typical attitude is, e.g., 

adopted by Quine (1951, p. 1): 

[. . . ] mathematical logic differs from the traditional formal logic so 

markedly in method, and so far surpasses it in power and subtlety, 

as to be generally and not unjustifiably regarded as a new science. 

Such claims may be very useful as a piece of propaganda for one’s own philosophical 

agenda (in this case, promoting logical atomism over and above what 

reason tolerates), but it seems to us to be entirely wrong. Firstly, it is factually 

wrong to state that the methods developed by Frege are constitutive of 

contemporary logic. There is a trend working in the footsteps of Frege, but 

it is by no means the dominant trend. 1 Secondly, this attitude is misleading 

because the alleged discontinuity is clearly an overstatement. There are important 

differences between traditional formal logic and contemporary formal logic; 

there is no debating that point. The point of a Quine-like claim, however, is 

to suggest that since a radical break happened in 1879, the previous works are 

theoretically worthless. The consequence of this attitude regarding Aristotle’s 

logic have been inauspicuous: 

Modern writers tended to look upon Aristotle’s logic with jaundiced 

eyes, finding fault wherever possible and emphasising differences between 

what they took to be Aristotle’s logic and what they took to 

be modern logic. (Corcoran & Scanlan, 1982, p. 76) 

In this paper, my primary motivation is to give grounds for attenuating this 

unfortunate prejudice. In order to do so, I will examine some contemporary 

contributions to the Aristotelian scholarship on categorical logic. More precisely, 

I will present and critically assess two models of Aristotelian logic built with 

the tools of contemporary mathematical logic. The notion of model relevant 

here is not the so-called “model-theoretic” notion of model. Rather, following 

Corcoran (1974b, 124-5), the notion of ‘model’ that I will use “is the ordinary 

one used in discussion, e.g., wooden models of airplanes, plastic model of boats, 

etc. Here the adjective ‘mathematical’ indicates the kind of material employed 

in the model.” The claim will thus not be that this or this model is actually 

Aristotelian logic. Rather, I will try to identify which model best succeeds in 

reproducing most logically relevant features of Aristotle’s sayings and practices. 

I have two main motivations for undertaking this study. The first reason, 

a “theoretical” one, is that I agree with Frege (1895, p. 33) that “where a line 

1 See, e.g., Hintikka (1988), or our own study of this question in Fillion (2006).

1 INTRODUCTORY REMARKS 5 

of thought can be perfectly expressed in symbols, it will appear briefer and 

more perspicuous in this form than in words.” The second reason, a “pragmatic 

one”, is that – as Aristotle used to say – we must begin with what is familiar 

to us. Since most philosophy students are nowadays taught logic by means of 

symbolic logic textbooks, it seems reasonable to use the medium to approach 

Aristotelian logic. I am confident that such an exercise may pave the way for a 

better understanding of Aristotle’s logic. Also, and this has not to be neglected, 

this kind of study of Aristotelian logic by means of contemporary logic may well 

shed some light on contemporary logic itself! 

Various approaches have been explored in view of modelizing categorical 

syllogistic. Novak (1980) recorded and compared four attempts, each using 

different mathematical tools: (1) first-order predicate calculus, (2) extensions of 

propositional calculus, (3) theory of classes and (4) Leśniewski’s ontology. The 

most famous use of (2) is ̷Lukasiewicz’s reconstruction of the syllogistic as an 

axiomatized extension of propositional calculus. It should also be noticed that 

(5) natural deduction system and (6) diagrammatic logic (see Raymond, 2006) 

have been used. 

The first to become autoritative is ̷Lukasiewicz’s reconstruction: 

In some ways this approach has come to be considered standard, 

probably (and justly) due to the wealth of detailed and critical observations 

contained in ̷Lukasiewicz’ work which makes numerous 

references to the texts of Aristotle himself. (Novak, 1980, p. 233) 

Later, in the seventies, the alternative view (5) have been developed and is now 

considered the standard view. My attention in this paper will be devoted to 

these two models. Here is an overview of the “battlefield” that we will examine: 

• ̷Lukasiewicz. This view has initially been presented in ̷Lukasiewicz (1929, 

chap. V), and more thoroughly in ̷Lukasiewicz (1957). Some of the most 

famous works along this line are those of Bocheński (1948, 1961, 1963) 

and Patzig (1968). 

• Corcoran-Smiley. This view has been developed independently and presented 

simultaneously in Corcoran (1972, 1974a,b) and Smiley (1973). 

Along this line, we also find Lear (1980), Corcoran & Scanlan (1982) and 

Corcoran (1994). 

The ̷Lukasiewiczian view takes Aristotle’s logic to be an axiomatized system presupposing 

the propositional calculus, whereas the Corcoran-Smiley view takes 

Aristotle’s logic to be an axiomless logic, i.e. a natural deduction system. 2 For 

each view, I will examine if they succeed in giving a logically correct account of 

Aristotle. As criteria of adequacy, I will assume (1) that the model recovering 

2 We also note en passant the existence of intermediary approaches, such as Thom (1979).

2 CRITICAL PRESENTATION OF ̷LUKASIEWICZ’S VIEW 6 

the largest part of Aristotle’s explicit doctrine will be preferred and (2) that the 

model recovering Aristotle’s results via methods essentially similar to Aristotle’s 

will be preferred. 3 

These evaluation criteria may give rise to this concern, that “the perhaps 

rather obvious and irritating criticism must be made, that it exaggerates the 

extent to which Aristotle himself was, consistently and self-consciously, on the 

side of the angels.” (Austin, 1952, p. 395-6) To my defense, I will simply say 

that the principle of charity must be applied here, since my goal is to show 

that there is a reliable reconstruction of Aristotle’s logic such that the alleged 

gap between traditional logic and contemporary logic is largely filled. As Smith 

(1989, p. vii) observed, “Aristotle is a major part of our philosophical ancestry, 

and we come to understand ourselves better by studying our ideal origins.” 

2 Critical Presentation of ̷Lukasiewicz’s View 

̷Lukasiewicz (1957) made an extremely influential reconstruction of Aristotelian 

logic along the lines of modern mathematical logic. This work is the result of at 

least forty years of work. It probably contributed more than any other publication 

on the subject to what could be called the “received view” on Aristotle’s 

logic among mathematical logicians. Such comments are typical: 

[̷Lukasiewicz] performed his role [as an analyst and interpreter of 

Organon and Metaphysics] in a creative manner by clarifying the author’s 

ideas while preserving his meaning unchanged. (Kotarbiński, 

1958, p. 58) 

He was lately considered – which seems right – the highest authority 

on Aristotle’s logic [. . . ]. (Kotarbiński, 1958, p. 59) 

A reviewer of ̷Lukasiewicz’s book even proclaimed that: 

In any case, the reviewer firmly believes that this interpretation and 

exposition of Aristotelian syllogistic is definitive, as far as the simple 

categorical syllogism is concerned. (Boehner, 1952, p. 210) 

However, it also received it just part of criticisms. Since my objective is mainly 

to show why it is not an adequate reconstruction, I will focus of the latter. 

3 An impediment to the achievement of these goals is my ignorance of ancient Greek. It 

seems to me that Barnes’ translation is tainted by ̷Lukasiewicz’s view, while Smith’s seems to 

be tainted by the Corcoran-Smiley view (see Gasser, 1991, for a critical review of the translation). 

To overcome this difficulty, I have used and compared the translations of Jenkinson, 

Smith, Barnes and Tricot for difficult passages. It greatly helped me while I was trying to 

make sense in an unbiased way of “the intention of the text”. I have also received precious 

help from Gilles Plante and François Chassé (both specialists of Aristotelian philosophy), for 

which a am very thankful.


Each part of this section will on the one hand present ̷Lukasiewicz’s view on 

a central aspect of Aristotle’s categorical syllogistic. On the other hand, critical 

remarks will be made via a comparison with the relevant excerpts from Aristotle’s 

text and, when appropriate, via presentation of some secondary litetarure. 

As we will see, what represents a real challenge in such a modelizing enterprise is 

to give a correct account of the methods of deduction. I will therefore organize 

the content of this section in order to be able to capture this aspect as efficiently 

as possible. For this reason, I will set aside what concerns the importance of 

the number of figures, what concerns the definition of minor, major and middle 

terms, the order of the premises, the number of premises, the generalization of 

decision problem, etc. 

2.1 The Nature of Syllogism 

The following syllogism is often given as an example of Aristotelian syllogism: 

All men are mortal, 

Socrates is a man, 

therefore, Socrates is mortal. 

(“False Form”) 

̷Lukasiewicz (1957, p. 1) considers that it is a False Form of Aristotelian syllogism, 

and he gives two main reasons to support the claim: 

1. It introduces singular terms or premises in the syllogistic (e.g. “Socrates” 

or “Socrates is a man”), whereas Aristotle does not; 

2. It is an inference from two premisses accepted as true to the conclusion, 

which does not correspond to Aristotle’s sayings. 4 

The central interpretive thesis characterizing ̷Lukasiewicz’s view is the following: 

Syllogisms are implication with the conjunction of premises as antecedent 

and a conclusion. 5 

He also notices two minor differences between Aristotle’s syllogisms and the 

False Form. Firstly, Aristotle always put the predicate first. Thus, instead of 

having a sentence of the form “All A is B”, we must have “B is predicated of 

all A”. 6 Moreover, an exact translation of the conclusion would also include a 

mark of necessity, i.e. “A must be predicated of all B”: 

4 Formulated with the currently used symbolism (e.g. Kleene, 1967), the reason advanced 

here is that it takes the syllogism to be of the form A, B ⊢ C instead of the correct ((A∧B) → 

C). 

5 “All Aristotelian syllogisms are implications of the type ‘If α and β, then γ’, where α and 

β are the two premisses and γ is the conclusion. The conjunction of the premisses ‘α and β’ 

is the antecedent, the conclusion γ is the consequent. [. . . ] The implication is also regarded 

as true for all values of the variables A, B, and C.” (̷Lukasiewicz, 1957, p. 20) 

6 This form is supposed to be as odd in Greek as in English. “Thus, ‘A belongs to all B’ 

becomes ‘All B is A’. In the former, ‘all’ is part of an expressed relation between the subject 

and the predicate. In the latter, ‘all’ qualifies the subject.” (Raymond, 2006, pp. 3-4)


This word ‘must’ (ἀνάγκη) is the sign of the so-called ‘syllogistic 

necessity’. It is used by Aristotle in almost all implications which 

contain variables and represent logical laws, i.e. laws of conversion 

or syllogisms. (̷Lukasiewicz, 1957, p. 10) 

̷Lukasiewicz gives as an example the passage where Aristotle lays down the laws 7 

of conversion: 

[. . . ] if A belongs to some of the Bs, then necessarily B belongs 

to some of the As. (For if it belongs to none, then neither will A 

belong to any of the Bs.) But if A does not belong to some B, it is 

not necessary for B also not to belong to some A. (Pr. An., A.2, 

25a20-24) 

We notice, however, that it is somewhat exaggerated to say that Aristotle uses 

it in almost all conversion laws and syllogisms. For example, Aristotle does not 

use this word “necessary” when he introduces the conversion of A-premises and 

E-premises some five lines before the excerpt referred to by ̷Lukasiewicz (Pr. 

An., A.2,25a14-19). It is used when describing Barbara (and the two invalid 

first-figure moods with premises AE and EE), but not Celarent (Pr. An., 

A.4, 25b38-40 & 26a1-2). Aristotle likely means it, but ̷Lukasiewicz’s appeal to 

textual support is, in this case, at best “borderline”. 

From this alleged systematical use of necessitation, ̷Lukasiewicz understands 

that Aristotle uses the sign of necessity in the consequent of a true implication 

to emphasize that the implication is true of all variables occurring in the 

implication (whether a law of conversion or a syllogism). In contemporary 

terms, we would uses quantifiers instead of the sign of necessity: we would write 

“∀a, b(Iab → Iba)” for conversion laws and “∀a, b, c[(Abc ∧ Aab) → Aac]” for 

syllogistic laws. 8 

Taking the two major and the two minor differences between the False Form 

and Aristotle’s form into account, we obtain the following, which is “the true 

form of the Aristotelian syllogism” (̷Lukasiewicz, 1957, p. 3): 

If A is predicated of all B 

and B is predicated of all C 

(True form) 

then (it follows of necessity that) A is predicated of all C. 

7 Whether these are laws or rules will be subject to discussion later. For now, we just adopt 

̷Lukasiewicz’ terminology. 

8 This must be taken with caution. In the classical presentations of propositional logic, 

necessity is assimilated to generality (in a truth-functional way); however, there is nothing 

that indicates that Aristotle did so: “Rather, Aristotle is working the presemantic idea of 

interpretation by replacement: a statement-form is seen to have various instances. One obtains 

an interpretation of a syllogistic formula by substituting specific terms, of the appropriate 

logical category, for the schematic letters.” (Lear p. 8) The reduction ̷Lukasiewicz makes 

treats necessity as a truth-functional rather than a modal notion. To say the least, it may be 

doubted that it is appropriate.


Material Implication. As we have seen, the strong claim constituting ̷Lukasiewicz’s 

view is that Aristotle’s theory of syllogism is a system of true propositions. 

Following Leśniewski, he calls all true propositions (whether axioms or 

theorems) “theses”. There are four basic constants or operators (represented 

by A, E, I, and O), each characterizing a type of proposition of this system of 

truths. Each of these operator takes two arguments. The arguments of these 

operators are universal terms. The following may thus be taken as a short technical 

description of Aristotle’s syllogistic: “The logic of Aristotle is a theory 

of the relations A, E, I, and O in the field of universal terms.” (̷Lukasiewicz, 

1957, p. 14) Universal terms correspond to the grammatical category of common 

nouns. Aristotle’s logic contain two kinds of implication. The conversion laws 

contain two propositions (A, E, I, O) that are connected by a material conditional. 

Syllogisms contain three propositions (A, E, I, O), one of them being the 

consequent of a material implication and two being put in conjunction to form 

the antecedent of the implication. 

Two kinds of argument are proposed to support this claim. The first argument 

is the textual support: 

It must be said emphatically that no syllogism is formulated by 

Aristotle as an inference with the word ‘therefore’ (ἄρα), as is done 

in traditional logic. (̷Lukasiewicz, 1957, p. 21) 

He claims that the transition from the implicational form to the inferential one 

has only taken place with Alexander. For this reason, he makes the distinction 

between the Aristotelian and the Traditional syllogism. The differentiation he 

makes follows the distinction between proposition and rule of inference. Syllogisms 

as implications are propositions, and as propositions must be either true 

or false. The traditional syllogism (False Form) is not a proposition, but a set of 

propositions forming not a single proposition but an inference. Since inferences 

are not propositions, they are neither true nor false. Rather, they are said to be 

valid or not. More precisely, the traditional syllogism is said to be an inference 

when it is stated in concrete terms. When it is stated with variables, it is a rule 

of inference. As rules of inference, they mean that for A, B, and C such that 

the premises ‘A belongs to all B’ and ‘B belongs to all C’ are true, therefore 

the conclusion ‘A belongs to all C’ must also be accepted as true. Whether 

syllogisms should be taken as implication or as inference connects to the use of 

concrete terms as follows: 

But if the syllogism is formulated in the [traditional form] with the 

variables S, M, P , this cannot mean that one considers the premises 

with the variables to be true: no one will say that the expression 

“All M is P ” is true. That expression is neither true nor false 

before definite terms are substituted for the variables M and P . If 

we want to impart sense to [the traditional form], formulated with 

variables, we must treat it as a scheme of inference which expresses


the following rule: whoever accepts sentences of the type “All M is 

P ” and “All S is M”, he is also entitled to accept a sentence of the 

type “All S is P ”. In this rule we refer not to the acceptance of 

the sentence “All M is P ”, but to the acceptance of a sentence of 

the type of “All M is P ”. And by sentences of the types of “All M 

is P ” we understand those sentences which are obtained from the 

expression “All M is P ” by the substitution for the variables of some 

definite terms.” (̷Lukasiewicz, 1963, p. 11) 

That Aristotle held the view vindicated by ̷Lukasiewicz is supported by the fact, 

at least ̷Lukasiewicz takes it to be a fact, that concrete terms are not part of logic 

(see next subsection), and by his alleged systematical use of “if–then” versus 

“therefore”. Regarding this second argument, two rather strong criticisms must 

be made. The first concerns ̷Lukasiewicz’s exaggeration of Aristotle’s strict 

adherence to “if–then” which, as Austin noticed, can hardly pass: 

[Aristotle] does sometimes formulate syllogisms with the fatal ἄρα 

[. . . ]. And what of the formulation (e.g. in Post. An. i,13 ) which 

has a conclusion beginning with ὥστε Moreover, ̷Lukasiewicz himself 

shows that Aristotle’s proofs by reductio ad impossibile (pp. 54 

ff.) involve treating the syllogism as an inference. (Austin, 1952, 

p. 397) 

The second strong criticism that has to be addressed concerns the seemingly 

inconsistency of ̷Lukasiewicz’s historical claims. In the Elements, he affirms the 

following: 

[Frege] was the first, as it seems, to notice the difference between 

the premises on which a reasoning is based, and the rules of inference, 

that is the rules which determine how we are to proceed in 

order to prove a given thesis on the strength of certain premises. 

(̷Lukasiewicz, 1963, p. 5) 

In Aristotle’s Syllogistic, he however seems to suggest that this distinction had 

first been done by the Stoics, some fifty years after Aristotle (1957, p. 48). The 

tricky question is thus: if Frege or the Stoics have first made the distinction 

between implication and inference after Aristotle’s writing of the Organon, is 

it really reasonable to suppose that Aristotle strictly applied it, with the right 

terminology and all that follows Clearly, ̷Lukasiewicz’s suggestion is misleading. 

Elimination of Concrete Terms. As we have seen, one of the two major 

reasons given by ̷Lukasiewicz for rejecting the False Form is that it contains 

concrete terms whereas Aristotle does not use them.


In Aristotle’s systematic exposition of his syllogistic no examples are 

given of syllogisms with concrete terms. (̷Lukasiewicz, 1957, p. 7) 

̷Lukasiewicz understands by this that concrete terms are not part of logic. 9 

Rather, he says that Aristotle used variables to express all valid syllogisms in 

their generality. On the other hand, Aristotle does exemplify invalid combinations 

of premises by concrete terms. This, however, was by no mean necessary 

and Aristotle could have simply proceeded by identifying variables. But 

it “seems that the straight way in this problem, i.e. the way by identifying 

variables, was not seen by Aristotle.” (̷Lukasiewicz, 1957, p. 9) 

The argument that supports this claim – i.e. that concrete terms are no part 

of logic – is based on the following excerpt: 

Now, of all the things that are, some are such as to be predicated 

of nothing else truly universally (for example, Kleon and Kallias 

and what is individual and perceptible) but to have other things 

predicated of them (for each of these is both a man and an animal); 

some are themselves predicated of others but do not have 

other things predicated previously of them; and some are both predicated 

of others and have others predicated of them (for example, 

man is predicated of Kallias and animal of man). 

[. . . Some other things may not] be predicated (except as a matter of 

opinion), but they are predicated of others. Neither can individuals 

be predicated of other things, but instead other things are predicated 

of them. But is is clear that for those in between, predication is 

possible in both ways (for they can be said of other things, and 

other things can also be said of them). And arguments and inquiries 

are almost always chiefly concerned with these things.” [Pr. An., 

A.1, 27, 43a25-43] 

The laws of conversion require that a term may act as a subject and as a 

predicate. This is the real reason, for ̷Lukasiewicz, that explains why Aristotle 

is occupied solely with those terms that can be predicated and can predicate: 

It is essential for the Aristotelian syllogistic that the same term may 

be used as a subject and as a predicate without any restriction. 

[. . . ] Syllogistic as conceived by Aristotle requires terms to be homogeneous 

with respect to their possible positions as subjects and 

predicates. This seems to be the true reason why singular terms 

were omitted by Aristotle. (̷Lukasiewicz, 1957, p. 7) 

9 (Bocheński, 1948, p. 17) formulates it in a very transparent way: “[. . . ] our syllogistic 

proposition is evidently a frame. But as full propositions never occur in formal logic (except 

with bound variables), there is no fear of confusion.”


But he does not think that Aristotle was right in doing that. He claims that the 

last sentence of Aristotle’s excerpt – that we chiefly deal with terms that predicate 

both ways – is just false: “It is plain that individual terms are as important 

as universal, not only in everyday life but also in scientific reasearches. This 

is the greatest defect of Aristotelian logic, that singular terms and propositions 

have no place in it.” 

There are some criticisms that should be addressed here: 

Doubtless ̷Lukasiewicz is right in saying that syllogistic “requires 

terms to be homogeneous with respect to their possible positions 

as subjects and predicates” (p. 7), if that means (as it is better 

put on p. 130) that syllogistic has no application to terms which 

are not such. Yet it is scarcely the case that Aristotle omitted, or 

actually ‘eliminated’, singular terms from his system so explicitly 

as ̷Lukasiewicz suggests. [. . . ] Aristotle is said to be “inexact” in 

this chapter [. . . where he talks about Callias]. While this may be 

perfectly correct, the argument cannot really begin until the meaning 

of Aristotle’s terms, and other passages bearing on the subject, have 

been discussed, and until the meaning of ̷Lukasiewicz’s terms also 

has been explained. If Aristotle is inexact here, it is an inexactitude 

in which he persevered. (Austin, 1952, p. 396) 

From what we have already seen, there seems to exist some ground for doubting 

that ̷Lukasiewicz’s interpretation in entirely correct. Let us continue our 

analysis. 

2.2 Perfect Syllogisms, Reduction and Rejection Proofs 

Perfect and Imperfect Syllogisms Prior Analytics is about demonstration 

(or proof), and demonstration is a kind of deduction 10 

Deduction [syllogism] should be discussed before demonstration because 

deduction is more universal: a demonstration is a kind of deduction 

[syllogism], but not every deduction [syllogism] is a demonstration. 

(Pr. An., A.4, 25b28-31) 

Aristotle distinguishes two kinds of syllogisms: 

I call a deduction [syllogism] complete if it stands in need of nothing 

else besides the things taken in order for the necessity to be evident; 

10 A strange fact, considering that the very first line of Prior Analyticslays down what is 

the object of logic, is ̷Lukasiewicz’s claim that Aristotle did not answered to that question: 

“What is therefore, according to Aristotle, the object of logic, and why is his logic called 

formal The answer to this question is not given by Aristotle himself but by his followers, the 

Peripatetics.” (̷Lukasiewicz, 1957, p. 13)


I call it incomplete if it still needs either one or several additional 

things which are necessary because of the terms assumed, but yet 

were not taken by means of premises. (Pr. An., A.1, 24b22-26) 

By this distinction, it seems that Aristotle’s strategy consists in isolating a 

handful of obviously valid inferences, and to justify the remaining ones by a sort 

of demonstration. In others words, it would be possible to make do without 

them and still move from the premises to the conclusion (Lear, 1980, p. 6). 

However, this explication does not satisfy ̷Lukasiewicz and he thinks that a 

“translation into logical terminology” is needed (̷Lukasiewicz, 1957, p. 43). As 

we have seen, he takes syllogism to be true material implications. On this basis, 

the distinction between perfect and imperfect syllogisms reduces to being or not 

being an axiom: 

What Aristotle says means, therefore, that in a perfect syllogism 

the connexion between the antecedent and the consequent is evident 

of itself without an additional proposition. Perfect syllogisms 

are self-evident statements which do not possess and do not need 

a demonstration; they are indemonstrable, ἀναπόδεικτοι. Indemonstrable 

true statements of a deductive system are now called axioms. 

The perfect syllogisms, therefore, are the axioms of the syllogistic. 

On the other hand, the imperfect syllogisms are not self-evident; 

they must be proved by means of one or more propositions which 

result from the premisses, but are different from them. (̷Lukasiewicz, 

1957, p. 43) 

Clearly, under ̷Lukasiewicz’s account, a syllogism is not (correctly) described 

as valid or as cogent, but only as true or as logically true or tautological. A 

syllogism has nothing to do with proving theorems, but is itself a proposition 

to be proved. From this point of view, imperfect syllogisms are not axioms and 

need to be proved, i.e. established as theorems. To do that, Aristotle uses a few 

methods of proof, namely proofs by conversion, proofs by ecthesis and proofs by 

reductio ad impossibile. As we will see, ̷Lukasiewicz thinks that these proofs are 

deficient, and rather propose to replace them by the laws of a more primitive 

logical theory (namely propositional logic). He thinks that such an alteration is 

necessary, because the proofs of imperfect syllogisms is a crucial component of 

Aristotelian logic: 

[Aristotle] does not speak of ‘axioms’ or ‘basic truths’ but of ‘perfect 

syllogisms’, and does not ‘demonstrate’ or ‘prove’ the imperfect 

syllogisms but ‘reduces’ them (ἀνάγει or ἀναλύει) to the perfect. 

[. . . The Aristotelian theory of syllogism] is an axiomatized deductive 

system, and the reduction of the other syllogistic moods to those of 

the first figure, i.e. their proofs as theorems by means of the axioms, 

is an indispensable part of the system. (̷Lukasiewicz, 1957, p. 44)


Thus, as we see, ̷Lukasiewicz admittedly says that he made some alterations. 

What he does not say, however, is to which extent these alien notions affect the 

very nature of the system. 

These alterations are source of embarassement for ̷Lukasiewicz’s view, especially 

with respect to Aristotle’s actual methods of proofs: “the price of accepting 

̷Lukasiewicz’s account of syllogisms is [. . . ] wholesale rejection of Aristotle’s 

account of their reduction.” (Smiley, 1973, p. 138) But it is also embarrassing 

with respect to another fundamental aspect: 

[. . . ] one cannot make sense of the claim that a proof [demonstration] 

is a type of syllogism (25b28ff) if one treats a syllogism as a 

conditional. A proof is an argument, with a definite structure, from 

several sentences functioning as premises to a conclusion. It is not 

a single sentence. (Lear, 1980, p. 9) 

Taking syllogisms as conditional propositions, ̷Lukasiewicz is forced to conclude 

that Aristotle was mistaken, i.e. that Aristotle’s statement that demonstration 

is a type of syllogism, is false. Under ̷Lukasiewicz’s reading, Aristotle would 

be making the absurd claim that every proof is a proposition, and that some 

propositions are proofs. The point is that a proof, e.g. the proof of the incommensurability 

of the diagonal, is not a single proposition. The only way to get 

out of this situation is to claim that Aristotle was confused, and to “translate 

into logical terminology”. 

[. . . ] While ̷Lukasiewicz writes as if Aristotle were concerned with 

nothing more than ‘the right answers’, it has long been clear [. . . ] 

that in Prior Analytics Aristotle is most concerned with the question 

of how conclusions are derived from premises. An obvious example 

of this interest is Aristotle’s lengthy discussion of the difference between 

direct and indirect deductions, i.e. sullogismoi. (Gasser, 1991, 

p. 238) 

This inability to handle correctly this very basic feature may suggest that 

̷Lukasiewicz’s claim that Aristotelian syllogisms are material implication is erroneous. 

In what follows, we will examine his account and critique of the methods 

of proofs. 

Proofs by Conversion Those proofs are what is often called “direct proofs” 

in the secondary literature. These are plausibly the most easy to account for. 

However, ̷Lukasiewicz judges that it is important to make some “enhancements” 

to the actual procedure used, because “Aristotle performs the proof intuitively” 

(̷Lukasiewicz, 1957, p. 53). He claims that Aristotle is right when he says that 

only two syllogisms of the first figure are needed to build up the whole theory 

of syllogisms. He also claims the following strange thing:


He forgets, however, that the laws of conversion, which he uses to reduce 

the imperfect moods to the perfect ones, also belong to his theory 

and cannot be proved by means of the syllogisms. (̷Lukasiewicz, 

1957, p. 45) 

To say that Aristotle forgot that conversion rules are part of his theory is somewhat 

awkward, since he uses them explicitly. What is ̷Lukasiewicz’s point, 

then 

Aristotle’s system of logic includes three laws (or rules) of conversion: the 

conversion of the E-premise, of the A-premise, and of the I-premise. Aristotle 

proves the conversion of the E-premise by what he calls ecthesis (Pr. An., A.2, 

25a14-26). But, as we shall see, ̷Lukasiewicz takes it to be “a logical process 

lying outside the limits of the syllogistic” (̷Lukasiewicz, 1957, p. 45). For him, 

as it cannot be proved otherwise, it must be stated as a new axiom of the 

system. Likewise, he claims that the conversion of the A-premise is proved by 

a thesis belonging to the square of opposition of which there is no mention in 

the Prior Analytics. For this reason, we must accept as fourth axiom either 

this law of conversion or the thesis of the square of opposition, from which this 

law follows. 11 ̷Lukasiewicz does not think he makes important modifications 

here, but rather that he is “analysing Aristotle’s intuitions” (̷Lukasiewicz, 1957, 

p. 51). The result is as follows: 

It follows from this analysis that if we add to the perfect syllogisms 

of the first figure and to the laws of conversion three laws of the logic 

of propositions, viz. the law of the hypothetical syllogism and two 

laws of the factor, we get strictly formalized proofs of all imperfect 

syllogisms except Baroco and Bocardo. These two moods require 

other theses of the propositional logic. (̷Lukasiewicz, 1957, p. 54) 

There is again some doubts concerning the truthfulness of ̷Lukasiewicz’s reconstruction. 

Did he only make explicit what was already there, or did he make 

important modifications Let us continue our examination before concluding. 

Proofs by Reductio ad impossibile The moods Baroco and Bocardo cannot 

be reduced to the first figure by conversion (under ̷Lukasiewicz’s reconstruction, 

it includes the propositional add-ons). This is why “Aristotle tries to 

prove these two moods by a reductio ad impossibile, ἀπαγωγὴ εἰς τὸ ἀδύνατον.” 

(̷Lukasiewicz, 1957, p. 54) Aristotle’s proof of the mood Baroco goes as follow: 

Next, if M belongs to every N but does not belong to some X, it is 

necessary for N not to belong to some X. (For if it belongs to every 

11 He affirms that the law of conversion of the I-premise can be proved without a new axiom. 

However, under his account, it is so because he adds laws of identity that Aristotle nowhere 

explicited.


X and M is also predicated of every N, then it is necessary for M to 

belong to every X: but it was assumed not to belong to some.) And 

if M belongs to every N but not to every X, then there will be a 

deduction that N does not belong to every X. (The demonstration 

is the same.) (Pr. An., 1.5, 27a36-b3) 

This is a very concise proof that is traditionally explained along those lines: (1) 

it is admitted that the premises ‘M belongs to every N’ and ‘M does not belong 

to some X’ are true, and (2) we conclude from this that the conclusion ‘N does 

not belong to some X’. For if it were false, its contradictory, ‘N belongs to all 

X’, would be true. This proposition is the starting point of the reduction. As 

it is admitted that the premise ‘M belongs to every N’ is true, we get from this 

premise and the proposition ‘N belongs to every X’ the conclusion ‘M belongs 

to all X’ by the mood Barbara. But this conclusion is false, for it is admitted 

that its contradictory ‘M does not belong to some X’ is true. Therefore, the 

starting point of our reduction, ‘N belongs to every X’, which leads to a false 

conclusion, must be false, and its contradictory, ‘N does not belong to some X’, 

must be true. Aristotle himself explains this method of proof in this way: 

A demonstration [leading] into an impossibility differs from a probative 

demonstration in that it puts as a premise what it wants 

to reject by leading away into an agreed falsehood, while a probative 

demonstration begins from agreed positions. More precisely, 

both demonstration take two agreed premises, but one takes the 

premises which the deduction is from, while the other takes one of 

these premises and, as the other premise, the contradictory of the 

conclusion. (Pr. An., B.14, 62b29-34) 

For ̷Lukasiewicz, such an explication is anathema, for it relies on the conception 

of syllogisms as (rules of) inference. This is why this “argument is only apparently 

convincing; in fact, it does not prove the above syllogism” (̷Lukasiewicz, 

1957, p. 54). Not, at least, if syllogisms are taken to be material implications. 

As such, a syllogism is an implication which is true for all values of the variables 

M, N, and X, and not only for those values that verify the premises. This is 

why ̷Lukasiewicz claims that the “proof given by Aristotle is neither sufficient 

nor a proof by reductio ad impossibile” (̷Lukasiewicz, 1957, p. 55). The proof of 

a syllogistic mood qua implication by reductio ad impossibile can not suppose 

the negation of the conclusion only: 

The indirect proof of the mood Baroco should start from the negation 

of this mood, and not from the negation of its conclusion, and 

this negation should lead to an unconditionally false statement, and 

not to a proposition that is admitted to be false under certain conditions. 

(̷Lukasiewicz, 1957, p. 56)


̷Lukasiewicz proposes an alternative to these “insufficient” proofs by using the 

following propositional law: ((p∧q) → r) → ((p∧¬r)¬q). It is obvious that provided 

such an axiom, ̷Lukasiewicz’s suggestion works in a straightforward way. 

However, as ̷Lukasiewicz willingly confess, it “can easily be seen that this genuine 

proof of the mood Baroco by reductio ad impossibile is quite different from 

that given by Aristotle” (̷Lukasiewicz, 1957, p. 56). 12 This is however supposed 

to be an advantage of his presentation over Aristotle’s, for the insufficiency of 

Aristotle’s approach is merely a sign of his being short-sighted: 

In the systematic exposition of the syllogistic these valid proofs are 

replaced by insufficient demonstrations per impossibile. The reason 

is, I suppose, that Aristotle does not recognize arguments ἐξ 

ὑποθέσεως as instruments of genuine proof. All demonstration is 

for him proof by categorical syllogisms; he is anxious to show that 

the proof per impossibile is a genuine proof in so far as it contains 

at least a part that is a categorical syllogism. [. . . ] All this is, of 

course, not true; Aristotle does not understand the nature of hypothetical 

arguments. The proof of Baroco and Bocardo by the law of 

transposition is not reached by an admission or some other hypothesis, 

but performed by an evident logical law; besides, it is certainly 

a proof of one categorical syllogism on the ground of another, but 

it is not performed by a categorical syllogism. (̷Lukasiewicz, 1957, 

pp. 57-8) 13 

This, of course, is only the case if ̷Lukasiewicz’s interpretive thesis that syllogisms 

are material implications holds. By now, it should be clear that there are 

serious doubts that it is the case. Another aspect of Aristotle’s saying about 

proofs by reductio ad impossibile that can not be accounted for by ̷Lukasiewicz 

is Aristotle’s claims that any conclusion that can be derived by per impossibile 

can be proved by a direct syllogism from the same premises. As noticed by 

Lear (1980, p. 9), this “distinction therefore requires that one attributes to the 

syllogism an argumentative structure which a conditional lacks.” The relevant 

excerpt is the following: 

Deductions [syllogisms] which lead into an impossibility are also in 

the same condition as probative ones: for they too come about by 

means of what each term follows or is followed by, and there is 

the same inquiry in both cases. For whatever is proved probatively 

can also be deduced through an impossibility by means of the same 

terms, and whatever is proved through an impossibility can also be 

deduced probatively [. . . ]. (Pr. An., A.29, 45a22-28) 

12 Among other things, it admits the possibility of syllogism ridiculosi (see Thom, 1979, 

p. 757). 

13 This has supposedly been corrected by the Stoics, who used the aforementioned propositional 

law: “This is one of the neatest argument we owe to the Stoics. We see that competent 

logicians reasoned 2,000 years ago in the same way as we are doing today.” (̷Lukasiewicz, 

1957, p. 59) It seems to suggest that Aristotle is not competent!


The point is that arguing validly requires more than making valid inferences. 

It also requires that the inferences done be known to be valid. By showing the 

equivalence of those methods of proof, Aristotle addresses the latter requirement: 

“[. . . ] the problem of one’s knowledge of the validity of a per impossibile 

syllogism is reduced to that of our knowledge of the validity of a direct inference.” 

(Lear, 1980, p. 53) Thus, it is guaranteed that the only difference between 

per impossibile and direct syllogisms, within the confines of syllogistic, is one of 

argumentative structure. It is impossible to account for this part of Aristotle’s 

theory if syllogisms are material implications. 

Proofs by Ecthesis Apart from the proofs by conversion and the proofs 

per impossibile, there is still a third kind of proof used by Aristotle, namely 

the proofs by exposition or ἔκθεσις. According to ̷Lukasiewicz, there are three 

passages where Aristotle uses that kind of proof in Prior Analytics 14 : (1) proof 

of the conversion of the E-premise, (2) the proof of the mood Darapti and (3) 

the proof of the mood Bocardo. The word ‘ecthesis’, however, appears only in 

(2). ̷Lukasiewicz thinks that this logical process lies outside the limits of the 

syllogistic, and that new axioms are needed in their place (̷Lukasiewicz, 1957, 

p. 45) Let us examine the proof of conversion: 

First, let premise AB be universally privative. Now, if A belongs to 

none of the Bs, then neither will B belong to any of the As. For if it 

does not belong to some (for instance to C), it will not be true that 

A belong to none of the Bs, since C is one of the Bs. (Pr. An., A.2, 

25a14-17) 

The proof by exposition requires the introduction of a new term, called the 

‘exposed term’; here it is C. In his commentary, Alexander says that C is a 

singular term given by perception. For ̷Lukasiewicz, this would not do since “a 

proof by perception is not a logical proof” (̷Lukasiewicz, 1957, p. 60). On the 

other hand, if C is a universal term, then the proof would require existential 

quantifiers (since it produces an instantiation). For this, however, an extension 

of syllogistic to quantification theory would be needed. 

This was perhaps the reason why he dropped this kind of proof 

in his final chapter (7) of Book I of the Prior Analytics, where he 

sums up his systematic investigation of syllogistic. Nobody after him 

understood these proofs. It was reserved for modern formal logic to 

explain them by the idea of the existential quantifier. (̷Lukasiewicz, 

1957, p. 67) 

This issue will be discussed in more details in the next subsection about rejection. 

14 See (Raymond, 2006, p. 33ff) and Smith (1982) for a detailed discussion.


Rejection In Prior Analytics, Aristotle does not only give proofs of correct 

syllogisms, but also shows that the others must be rejected. ̷Lukasiewicz claims 

that Aristotle uses two different methods of rejection. The first method is the 

most common. Consider the following syllogistic form: If A belongs to all B and 

B belongs to no C, then A does not belong to some C. To show that this form 

is not correct, we must show that there exists values of the variables A, B and 

C such that the premises are verified but the conclusion is not. Obviously, the 

“easiest way of showing this is to find concrete terms verifying the premisses 

[. . . ] but not the conclusion” (̷Lukasiewicz, 1957, p. 69). Let us examine an 

instance of this way of proceeding: 

For it is possible for the first extreme to belong to all as well as to 

none of the last. Consequently, neither a particular nor a universal 

conclusion becomes necessary; and, since nothing is necessary because 

of these, there will not be a deduction [syllogism]. Terms for 

belonging to every are animal, man, horse; for belonging to none, 

animal, man, stone. (Pr. An., A.4, 26a5-10) 

Here, Aristotle rejects a form by means of the concrete terms ‘man’, ‘animal’, 

‘horse’, and ‘stone’. This procedure necessitates the use of concrete terms, and 

̷Lukasiewicz criticizes this method of rejection as follows: 

This procedure is correct, but it introduces into logic terms and 

propositions not germane to it. ‘Man’ and ‘animal’ are not logical 

terms, and the proposition ‘All men are animals’ is not a logical 

thesis. Logic cannot depend on concrete terms and statements. If 

we want to avoid this difficulty, we must reject some forms axiomatically. 


According to ̷Lukasiewicz, the second method that Aristotle used is much more 

“logical” in nature. 15 In examining second-figure forms, Aristotle rejects a form 

by reducing it to another form already known to be false (Pr. An., A.5, 27b9- 

23). 

Aristotle, however, does not finish his proof [by finding concrete 

terms], because he sees another possibility: if the form with universal 

negative premisses: (6) If M belongs to no N and M belongs to no 

X, then N does not belong to some X, is rejected, (5) [If M belongs 

to no N and M does not belong to some X, then N does not belong 

to some X,] must be rejected too. For if (5) stands, (6), having 

a stronger premiss than (5), must also stand. (̷Lukasiewicz, 1957, 

p. 71) 

15 Whether it has a real importance in Aristotle’s theory is not that clear, because there is 

no primitive rejection law stated. Raymond (2006, p. 209) even considers that it is not part 

of Aristotle’s methodology for rejecting forms: “In fact, Aristotle never relies upon a rule to 

establish invalidity. Nor does he limit the semantics to a certain class of assertions.”


The rejection strategy in such a case is to show that the falsity of a form follows 

of necessity from the falsity of another form. Thus, on the basis of some 

axiomatically rejected forms, it is possible to prove the rejection of other forms. 

̷Lukasiewicz sees this procedure as literally mirroring Aristotle’s method for perfecting 

syllogisms. Since he takes syllogisms to be material implications, and 

that it is crucial to his conception that the perfection of syllogisms is the proof 

of a theorem from axioms, this second method of rejection is supposed to be an 

operation opposed to Frege’s assertion (see ̷Lukasiewicz, 1957, p. 71ff). Since 

the concrete terms necessary for the first kind of rejection are not part of logic, 

this second method based on axiomatic rejection should be considered as the 

true Aristotelian method for the rejection of false forms. 

Many commentators rightly noticed that it is patently unfaithful to Aristotle’s 

sayings 16 : 

There seems to be some special pleading here. The sense in which 

Aristotle “uses rejection” is far-fetched: he is shown to have used 

once, not very self-consciously, the modus tollens, but not to have 

alluded even once to axioms of rejection, the more crucial matter. 

(Austin, 1952, p. 401) 

Before closing this subsection, a few more words on the use of concrete terms 

for rejection are in order. A proof of rejection aims to show that nothing follows 

of necessity from a set of premises: 

Since a syllogistic consequence follows of necessity from the premisses, 

to establish sterility one must prove a certain possibility. 

That is, one must show that it is possible for the premisses of the 

form in question to be true without any syllogistic conclusion, linking 

the major and minor terms in the prescribed order, being true. 

(Lear, 1980, p. 54) 

The easiest way to show that is to show instances where it does not work, such 

as Aristotle’s terms ‘animal’, ‘man’, and ‘stone’. But these terms are supposedly 

not germane to logic, and that made ̷Lukasiewicz say that producing an instance 

is not a proper logical proof. Geach (1972) agreed with ̷Lukasiewicz on that 

point, with the reserve that it is logically correct if it is based on the exposition 

of a logical object. 17 He therefore proposes to use Caroll’s diagrams to make 

these proofs in an appropriate logical way (Geach, 1972, p. 299ff). Likewise, 

Copi (1961, p. 166ff) proposed to use Venn diagrams. These two tools consist 

in appealing to our spatial intuition, i.e. to some kind of graphs, to expose 

the validity or invalidity of a syllogistic form. Their objection to Aristotle’s 

examples is therefore not about the procedure of exposition per se, but rather 

16 See also Corcoran (1974b, 1994). 

17 What is a logical object is unclear, and there are serious doubts that it may be characterized 

in logical terms (Fillion, 2006, chap. 5).


to the kind of object or term that Aristotle used as counterexamples. They 

think that if the objects or terms exposed are not logical in nature, the proof 

would be contingent on the constitution of the world. Aristotle took as example 

swans, supposing that all swans are white; however, we now know that there are 

black swans. Diagrams thus seem more appropriate (whether Carroll’s, Venn’s 

or Euler’s). But is it really the case 

If we draw an Euler diagram on a sheet of paper, it is convincing 

because we assume that the surface on which it is drawn has the 

same topology as a Euclidean plane. (Lear, 1980, p. 62) 

Now, in the same way that we have discovered black swans, we have discovered 

non-Euclidean topological structures. Actually, Euler diagrams fail to exhibit 

validity and invalidity of our canonically accepted inference if they are represented 

on a torus (which is a perfectly logically acceptable topological structure). 

In response, it may be suggested that they can only be taken as heuristic devices. 

But in this case, we are brought back, with Aristotle, to the use of actual 

counterexamples that have nothing of the logical objects that Geach was after. 

Therefore, both ̷Lukasiewicz’s and Geach’s suggestions have no significant 

advantage over Aristotle’s in explaining the ‘why’ of the rejection. 

2.3 Logic of Proposition as Preceding the Logic of Terms 

We have seen many important differences between what Aristotle actually does, 

and what ̷Lukasiewicz says that Aristotle does. ̷Lukasiewicz takes these differences 

to be mere corrections of a technical nature, not making any deep 

alteration to Aristotle’s system. In other words, he thinks he just “filled the 

gaps”. These gaps are all related to the fundamental flaw of Aristotle’s logic: 

he did not understand that some problems can not be expressed by the four 

syllogistic premises. Most importantly, Aristotle did not understand that the 

theory of syllogisms he himself created was an instance against the doctrine that 

all problems may be resolved by syllogisms. Here is the main grief: 

The syllogistic moods, being implications, are propositions of another 

kind than the syllogistic premisses, but nevertheless they are 

true propositions, and if any of them is not self-evident and indemonstrable 

it requires a proof to establish its truth. The proof, however, 

cannot be done by means of a categorical syllogism, because an implication 

does not have either a subject or a predicate, and it would 

be useless to look for a middle term between non-existent extremes. 


Aristotle cultivated many ambiguities on that very point, so that “no one can 

fully understand Aristotle’s proofs who does not know that there exists besides 

the Aristotelian system another system of logic more fundamental than


the theory of syllogism” (̷Lukasiewicz, 1957, p. 47). Concerning those proofs, 

̷Lukasiewicz emphasized (1) that Aristotle’s proofs by conversion are insufficient 

and need to be completed, (2) that proofs by reductio ad impossibile and by ecthesis 

must replaced by axiomatic acceptation or rejection of some propositions 

and (3) that laws of conversions must be accepted axiomatically. Basically, he 

has “shown” that all methods of proofs can only be understood if we replace 

them by a formalistic deduction in propositional logic. This is why, according 

to ̷Lukasiewicz, it is only possible to understand Aristotle’s proofs if it is 

understood that they necessitate propositional calculus. 

Propositional logic was completely ignored by Aristotle. ̷Lukasiewicz affirms 

that the first system of propositional logic was invented about half a century 

after Aristotle, by the Stoics. However, the understanding of the more fundamental 

character of this logical system has only been acquired in the end of the 

nineteenth century, as a consequence of the 1879 event, due to the great German 

logician Gottlob Frege. This theory has been later brought to perfection by the 

authors of Principia Mathematica, Whitehead and Russell, who developed it so 

profoundly that they made it the core of all mathematics. 18 To observe the 

difference between propositional logic and syllogistic, it is sufficient to compare 

Aab and p → q: 

They differ in their constants, which I call functors: in the first formula 

the functor reads ‘all–is’, in the second ‘if–then’. Both are 

functors of two arguments which are here identical. But the main 

difference lies in the arguments. In both formulae the arguments are 

variables, but of a different kind: the values which may be substituted 

for the variable A are terms [. . . ]. The value of the variable p 

are not terms but propositions [. . . ]. This difference between termvariables 

and proposition-variables is the primary difference between 

the two formulae and consequently between the two systems of logic, 

and, as propositions and terms belong to different semantical categories, 

the difference is a fundamental one. (̷Lukasiewicz, 1957, 

p. 47-8) 

For ̷Lukasiewicz, in contrast with the syllogistic, which is a set of true material 

implications, propositional logic is a system of rules of inference, and that makes 

it the primitive logic underlying all logical reasoning. 

[. . . ] Aristotle’s logic is not a primitive logical theory, and for its 

foundation requires a certain earlier theory that investigates sentences 

in general, without engaging in the study of the details of 

their structure. (̷Lukasiewicz, 1963, p. 106) 

That it is the case, however, depends on his being justified in making the propositional 

additions to Aristotle’s doctrine. We must first notice that ̷Lukasiewicz 

18 See ̷Lukasiewicz (1957, p. 48) for the literal text of the propaganda.


avowedly says that it finds no direct support in Prior Analytics: 

It seems that Aristotle did not suspect the existence of another system 

of logic besides his theory of the syllogism. Yet he uses intuitively 

the laws of propositional logic in his proofs of imperfect 

syllogisms, and even sets forth explicitly three statements belonging 

to this logic in Book II of the Prior Analytics. (̷Lukasiewicz, 1957, 

p. 49) 

The question naturally arises how exactly this system corresponds with the 

syllogistic as developed by Aristotle. We will report our final verdict to the 

section 2.5, and will first present the formalistic reconstruction. 

2.4 ̷Lukasiewicz’s Symbolic System for Aristotle’s Logic 

What often causes difficulty for contemporary logicians who want to understand 

Aristotelian logic is that it is a formal logic without being a formalistic logic. 

The goal of ̷Lukasiewicz was to provided a formalistic system for Aristotelian 

logic. In a compact formulation, the system is the following: 

Formal Axiomatized Theory for ̷Lukasiewicz’ Syllogistic (LS): 

1. The alphabet of LS contains: 

(a) propositional connectives ‘¬, ⇒, ∧’; 

(b) propositional variables ‘p, q, r, s, . . .’; 

(c) term variables ‘a, b, c, d, . . .’; 

(d) syllogistic operators ‘A, I’. 

2. The atomic formulas of LS are the following: 

(a) for any a, b that are term variables, Aab and Iab are atomic formulas; 

(b) nothing else is an atomic formula. 

3. The well-formed formulas of LS are the following: 

(a) all atomic propositions are wffs; 

(b) all propositional variables are wffs; 

(c) for any p, q that are propositional variables, ¬p, p ⇒ q are wffs; 

(d) nothing else is a wff. 

4. The axioms of LS are: 

(a) Aaa (Identity Law) 

(b) Iaa (Identity Law)


(c) (Abc ∧ Aab) ⇒ Aac (Barbara) 

(d) (Abc ∧ Iba) ⇒ Iac (Datisi) 

(e) p ⇒ (q ⇒ p) (Law of Simplification) 

(f) (p ⇒ q) ⇒ ((q ⇒ r) ⇒ (p ⇒)) (Law of Hypothetical Syllogism) 

(g) (p ⇒ (q ⇒ r)) ⇒ (q ⇒ (p ⇒ r)) (Law of Commutation) 

(h) p ⇒ (¬p ⇒ q) (Law of Duns Scotus) 

(i) (¬p ⇒ p) ⇒ p (Law of Clavius) 

(j) (p ⇒ q) ⇒ (¬q ⇒ ¬p) (Law of Transposition) 

(k) ((p ∧ q) ⇒ r) ⇒ (p ⇒ (q ⇒ r)) (Law of Exportation) 

(l) p ⇒ (((p ∧ q) ⇒ r) ⇒ (q ⇒ r)) 

(m) (s ⇒ p) ⇒ (((p ∧ q) ⇒ r) ⇒ ((s ∧ q) ⇒ r)) 

(n) ((p ∧ q) ⇒ r) ⇒ ((s ⇒ q) ⇒ ((p ∧ s) ⇒ r)) 

(o) (r ⇒ s) ⇒ (((p ∧ q) ⇒ r) ⇒ ((q ∧ p) ⇒ s)) 

(p) ((p ∧ q) ⇒ r) ⇒ ((p ∧ ¬r) ⇒ ¬q) 

(q) ((p ∧ q) ⇒ r) ⇒ ((¬r ∧ q) ⇒ ¬p) 

(r) ((p ∧ ¬q) ⇒ ¬r) ⇒ ((p ∧ r) ⇒ q) 

5. The following propositions are axiomatically rejected: 

(a) (Acb ∧ Aab) ⇒ Iac 

(b) (Ecb ∧ Eab) ⇒ Iac 

6. The rules of inference of LS are: 

(a) (Substitution) If p is an asserted expression of LS, then any expression 

produced from p by a valid substitution is also an asserted 

expression. The only valid substitution is to put for term variables 

a, b, c other term-variables, e.g. b for a. 

(b) (Detachment) If p ⇒ q and p are asserted expressions of LS, then q 

is an asserted expression. 

(c) (Rejection by detachment) If the implication p ⇒ q is asserted, but 

the consequent q is rejected, then the antecedent p must be rejected 

too. 

(d) (Rejection by substitution) If q is a substitution of p, and q is rejected, 

then p must be rejected too. 

The semantics of the system is more or less clear, so we will not present it here.


2.5 Summary of the Discussion 

̷Lukasiewicz (1957, p. 130) reconstructed Aristotle’s syllogistic in a mathematical 

framework and, doing so, he has “tried to set forth this system free from foreign 

elements.” He does not think that he made serious alterations to the system, 

but rather that he just used formal proofs where Aristotle used propositional 

logic intuitively. 

In our critical presentation, however, we have seen that “the price of accepting 

̷Lukasiewicz’s account of syllogisms is [. . . ] wholesale rejection of Aristotle’s 

account of their reduction” (Smiley, 1973, p. 138). More precisely, we have 

seen that ̷Lukasiewicz’s reconstruction fails (1) to make sense of the distinction 

between perfect and imperfect syllogisms, (2) to make sense of the claim that 

some demonstrations are syllogisms (but not all syllogisms are demonstrations), 

(3) to make sense of direct proofs (by conversion), (4) to make sense of reductio 

ad impossibile, (5) to make sense of ecthesis and (6) to make sense of Aristotle’s 

method for the rejection of invalid syllogisms. These elements – should I 

emphasize it – are fundamental parts or Aristotle’s categorical logic. 

The question now arises how exact a correspondence this system has with 

Aristotle’s theory of categorical syllogisms. Among the arguments provided by 

̷Lukasiewicz to justify his point of view, we find textual and theoretical reasons. 

We have seen that ̷Lukasiewicz’s appeals to textual support are to a large 

extent shakier than he claimed. The theoretical arguments also lead to complications. 

Notwithstanding the unfortunate consequence that Aristotle’s methods 

of proof are all erroneous, there is also embarrassing discrepancies regarding the 

nature of logic. The chief concern is related to the recourse to propositional 

logic. There doubtlessly are many important differences between Aristotelian 

and propositional logic; it is also doubtless that they diverge regarding what 

pertains to logic. This point has a special importance here, since ̷Lukasiewicz’s 

refers to “what pertains to logic” to justify that Aristotle could not have used 

concrete terms except in a heuristic way. This claim is central for ̷Lukasiewicz’s 

analysis since it decides that syllogisms are axioms, and not inferences or rules 

of inference. As we have seen, ̷Lukasiewicz takes syllogisms to be true (truthfunctionally 

true, or tautological) implications. However, to my knowledge, 

there is not a single passage in the Prior Analytics where Aristotle discussed or 

even mentioned tautological propositions, as being the subject matter of logic. 

Is it an unimportant detail, or is it due to the more likely fact that for Aristotle, 

the role of logic was not just to catalogue tautologies 19 If the latter is the case, 

it would explain why Aristotle did not use ̷Lukasiewicz’s laws of identity and 

why he did not use axiomatic rejection. It would also explain Aristotle’s methods 

of proof, all of which make no sense if we take syllogisms to be axioms, but 

make perfect sense if we take them to be rules of inference (see next section). 

19 For Corcoran (1972, p. 696), this is not a coincidence: “The fact that each sentence 

contains two distinct nonlogical constants reflects a systematic avoidance by Aristotle of “sentences” 

such as Axx or Sxx.”

3 PRESENTATION OF THE CORCORAN-SMILEY VIEW 26 

The influence of ̷Lukasiewicz’s reconstruction was and still is great. Firstly, 

it had the fortunate consequence that mathematical logicians were to have a 

familiar medium through which Aristotelian logic could be learned. The unfortunate 

part of the story, however, is that his critique of Aristotle’s methods 

of proofs contributed more than anything to reinforced the idea that the “real 

thing” is propositional logic, and that Aristotelian logic presents no more theoretical 

interest. 

The last question that arises here is the following: Despite the discrepancies 

of ̷Lukasiewicz’s reconstruction, should we accept it as the best we can do in 

reconstructing Aristotle’s logic by means of mathematical models If it were 

the case, then ̷Lukasiewicz would be, in a sense, justified. If, however, there 

is a more faithful and rigorous reconstruction, then we would have no reason 

whatsoever not to reject ̷Lukasiewicz’s reconstruction – however influential it 

may be. In the next section, we will present such a reconstruction. 

3 Presentation of the Corcoran-Smiley View 

As we have seen, ̷Lukasiewicz rejected the traditional account of syllogisms as 

inference and contended that the “true” form of Aristotelian syllogisms is that 

of a material implication. However, if Aristotle’s methods of proofs are to make 

any sense, it seems necessary to treat syllogisms as being “essentially something 

with a deductive structure as well as premisses and conclusions” (Smiley, 1973, 

p. 136), i.e. an inference. 

Why is it that the ̷Lukasiewiczian view treated syllogisms as if they were 

material implications To answer that is to emphasize a difference of perspective 

regarding what is the nature of logic. ̷Lukasiewicz took logic to be what has 

been developed in the previous decades by Frege and Russell. 20 According to 

them, the central concept of logic is the monadic property “logical truth” rather 

than the traditional dyadic relation “logical consequence”. This is what explains 

̷Lukasiewicz’s desire to reformulate the syllogistic not as a system of deductions 

of consequences from arbitrary (and therefore sometimes false) premises, but 

rather a system of proofs of logical truths (tautologies) based on logical axioms 

and rules (Corcoran & Scanlan, 1982, p. 78). This difference is crucial, since it 

determines what is the philosophical role of logic. 

The influence of the Frege-Russell view of logic started to decline in the 

1930s. Two important technical developments in logic did accelerate this change 

of attitude: (1) Tarki’s rigorization of the notion of logical consequence and 

(2) the construction of rigorous systems of natural deduction by Gentzen and 

Jaśkowski (Pelletier, 1999). The Corcoran-Smiley view can be seen as a byproduct 

of these developments. Taking benefit of this more traditional but 

mathematically rigorous approach, they found themselves in a position to argue, 

20 We may add Wittgenstein, Quine, and the Polish school in general.


as opposed to ̷Lukasiewicz, that Aristotelian logic is logically sound in every 

detail: 

[. . . ] Aristotle’s Prior Analytics contained a logical theory involving 

a well-developed logical system (in the modern sense) which is 

at once self-contained (not presupposing a prior underlying logic) 

and comparable with standard systems of propositional logic in simplicity, 

precision, correctness, and comprehensiveness. (Corcoran & 

Scanlan, 1982, p. 76) 

In this section, we will examine their approach. 

3.1 Explanation of the Approach 

The approach adopted by the proponents of the Corcoran-Smiley view is (to my 

knowledge) best explained in Corcoran & Scanlan (1982). In this subsection, I 

will summarize their methodological discussion. 

In their framework, the object of logic in general – and a fortiori of the 

syllogistic – is the study of valid arguments. It is thus completely natural for 

them to see Prior Analytics as being a treatise about deductive reasoning and, 

more generally, about the methods of determining the validity and invalidity of 

premise-conclusion arguments. An argument has two parts: (1) a set of propositions, 

called the premises and (2) a single proposition, called the conclusion. 

An argument is valid if and only if the conclusion is a logical consequence of 

the premises. The argument is otherwise said to be invalid. Any argument is in 

itself either valid or invalid, and this without regard to whether anyone knows 

which one is the case. The epistemological function of logic is thus to develop 

methods providing thinking humans with the knowledge of the validity of arguments. 

For some very simple arguments, mostly having one or two premises, 

the knowledge of their validity is obvious, in the sense that each of them can be 

seen to be valid without considering any other arguments already known to be 

valid. Of course, most arguments are not obviously valid, and this is why we 

need logic as a discipline. A central topic studied in logic is how we can validly 

chain together obviously valid arguments to gain knowledge of the validity of 

more complex arguments. 

The basic procedure is very neatly described as follow: 

The procedure for establishing validity contains a familiar “Kernel+ 

P rocess” structure. There is a Kernel of valid arguments whose validity 

is seen ab initio and there is a Process which produces knowledge 

of validity from knowledge of validity. (Corcoran & Scanlan, 

1982, p. 79) 

Aristotle, and logicians in general, focus on two epistemic processes:


1. the process of establishing knowledge that a conclusion follows necessarily 

from a set of premises; 

2. the process of establishing knowledge that a conclusion does not follow 

necessarily from a set of premises. 

Basically, an argument is known to be invalid when the premises are known to 

be true, and the conclusion is known to be false. Such a basic case is called a 

counter-example. This method applies trivially to simple cases, but it applies 

only to a very limited class of arguments. More generally, we obtain knowledge 

of the invalidity of arguments through a process involving arguments already 

known to be invalid. For this reason, it may be said that the procedure of 

establishing invalidity also contains a “Kernel + P rocess” structure: “There 

is a Kernel of invalid arguments whose invalidity is established ab initio and 

there is a Process which produces knowledge of invalidity from knowledge of 

invalidity” (Corcoran & Scanlan, 1982, p. 79). 

On this basis, we characterize what is a deduction (syllogism) and a proof. 

A deduction is the process of correctly (validly) inferring a conclusion from 

premises, and this process provides us with knowledge of the validity of an 

argument. In other words, it is by means of a deduction that one comes to see 

(for oneself) or to show (to another) that the argument is valid. 21 A deduction 

is a proof of its conclusion if the premises of the deduction are known to be 

true. In this framework, we naturally have the result, with Aristotle, that every 

proof is a deduction but not every deduction is a proof. 

Under this conception of logic, deduction in general (and syllogisms in particular) 

have an argumentative structure. This view allows to efficiently make 

sense of Aristotle’s distinction between perfect and imperfect syllogisms. Perfect 

syllogisms are those arguments whose validity is seen ab initio. Imperfect 

syllogisms are those in need of a deduction to be known as valid. In other words, 

imperfect syllogisms are “perfected” by chaining together (simple) perfect deductions 

(syllogisms and/or conversions). For this reason, Corcoran & Scanlan 

(1982, p. 80) claim that “Aristotle’s process of perfecting imperfect syllogisms 

is identified with our process of deducing.” Likewise, this conception allows to 

easily account for Aristotle’s distinction between direct and indirect deductions. 

This distinction “is entirely a matter of how their conclusions are derived” (Smiley, 

1973, p. 136). When Aristotle says that every argument that can be proved 

indirectly can also be proved directly, and vice-versa, he just means that we 

can gain knowledge of the validity of an argument by two different (types of) 

deductions. 

Now that we are acquainted with the general character of their approach, 

21 It is important to distinguish argument and deduction: “A deduction is often also called 

an argument but there should be no confusion because the deduction includes a chain of 

reasoning over and above the premises and the conclusion and it is by means of a deduction 

that ones comes to see that the (included) argument is valid.” (Corcoran & Scanlan, 1982, 

p. 79)


let us see their technical reconstruction of Aristotle’s syllogistic as a natural 

deduction system. 

3.2 Formal Natural Deduction System 

Deductive system for Corcoran-Smiley’s Syllogistic (CS) 22 : 

1. The alphabet for CS contains: 

(a) A set V of term constants ‘a, b, c, d, . . .’; 

(b) syllogistic operators ‘A, E, I, O’. 

2. The well-formed formulas L for CS are the following: 

(a) for any two distinct term constants a and b, Aab, Eab, Iab and Oab 

are wffs (or sentences); 

(b) nothing else is a wff. 

3. The rules of inference for CS. For any x, y, z ∈ V , 

C1 Exy ⊢ Eyx (conversion) 

C2 Axy ⊢ Iyx (conversion) 

C3 Ixy ⊢ Iyx (conversion) 

PS1 Azy, Axz ⊢ Axy (Barbara) 

PS2 Ezy, Axz ⊢ Exy (Celarent) 

PS3 Azy, Ixz ⊢ Ixy (Darii) 

PS4 Ezy, Ixz ⊢ Oxy (Ferio) 

CTR If P, C(d) ⊢ e and P, C(d) ⊢ C(e), then P ⊢ d (Reductio) 23 

Definition 1 (Argument) An argument is an ordered pair (P, d) where P is 

a set of sentences (called the premises) and d is a single sentence (called the 

conclusion). 

Definition 2 (Direct Deduction) A direct deduction in CS of d from P is 

defined to be a finite list of sentences ending with d, beginning with all or some 

of the sentences in P and such that each subsequent line (after those in P ) is 

either 

• a repetition of a previous line; 

• a CS-conversion of a previous line; 

22 See (Corcoran, 1972) and (Smiley, 1973). 

23 Axy and Exy are defined to be contradictories of Oxy and Ixy respectively (and vice 

versa) and C(d) indicates the contradictory of d.


• a CS-inference from two previous lines. 

Definition 3 (Indirect Deduction) An indirect deduction in CS of d from 

P is defined to be a finite list of sentences ending in a pair of contradictories [e 

and C(e)], beginning with a list of all or some of the sentences in P followed by 

the contradictory of d, and such that each subsequent additional line (after the 

contradictory of d) is either 

• a repetition of a previous line; 

• a CS-conversion of a previous line; 

• a CS-inference from two previous lines. 

As usual in natural deduction systems, we also define a system of annotation 

for deductions: 

+ Axy := assume Axy as a premise 

Axy := we want to show why Axy follows (we interject the conclusion Axy as 

Axy after the premisses to indicate the goal) 

h Axy := suppose Axy for the purpose of reasoning (for indirect deductions) 

a Axy := we have already accepted Axy 

c Axy := by conversion Axy is entered 

s Axy := by syllogistic inference Axy is entered 

Ba Axy := at the last line of an indirect deduction ‘B’ is prefixed to its other 

annotation so that ‘Ba Axy’ is read ‘but we have already accepted Axy’ 

To illustrate how the system works, consider the annotation of Aristotle’s 

deduction of the mood Cesare (Pr. An., A.5, 27a5-8): 

Let M be predicated of no N [+ Enm] and of all X [+ Axm] [omitted: 

Exn]. Then since the negative premise converts, N belong 

to no M [c Emn]. But it was supposed that M belongs to all X 

[Ba Axm]. Therefore, N will belong to no X [s Exn]. 

In the classical representation by means of a Fitch Diagram 24 , we obtain: 

1 Nnm Premise 

2 Axm Premise 

Deduction of Cesare 

3 Nmn Conversion, 1 

4 Axm Reiteration, 2 

7 Nxn Celarent, 3–4 

24 Neither Corcoran nor Smiley use Fitch diagrams, but it seems to me to be an improvement, 

from a didactic point of view, on their presentation.


Here is an illustration of an indirect deduction, namely Aristotle’s deduction 

of the mood Baroco (Pr. An., A.6, 28b8-12): 

For if R belongs to all S [+ Asr], but P does not belong to some S 

[+ Osp], it is necessary that P does not belong to some R [ Orp]. 

For if P belongs to all R [h Arp], and R belongs to all S [Ba Asr], 

the P will belong to all S [s Asp]. But we assumed that is did not 

[Ba Osp]. 

The same example represented in a Fitch Diagram: 

1 Asr Premise 

2 Osp Premise 

3 Arp Hypothesis 

Deduction of Baroco 

4 Asr Reiteration, 1 

5 Asp Barbara, 3, 4 

6 Osp Reiteration, 2 

7 Orp Reductio, 3–6 

3.3 Objections to the View of Syllogistic as a Natural Deduction 

System 

Without having to compare ̷Lukasiewicz’s view to Corcoran-Smiley’s point by 

point, it appears clear that the latter provides a much better reconstruction 

of Aristotle’s categorical logic. The fact that they succeed to explain and justify 

Aristotle’s methods of proof is entirely sufficient to make us prefer their 

approach. It is more sensitive than ̷Lukasiewicz’s approach both from the historical 

and from the theoretical point of view. 

However, despite the merits of this reconstruction, there is always place for 

some improvement. Raymond (2006) formulated some criticisms of this reconstruction. 

The idea underlying all of them is that their notation for premiseconclusion 

arguments – the ordered pair (P, d) – is not explicitly found in Aristotle’s 

writings. Whether it is important or not is not entirely clear. However, 

he contends that the choice of a notation has a strong influence on our understanding 

of the basic intuition of a system and, for this reason, should not be 

neglected: 

Of the three components to an underlying logic – notation, system of 

deduction and a system of semantics – notation is the one component 

that is presupposed by the other two: a system of deduction and a

4 CONCLUSION: TWO TRADITIONS IN LOGIC 32 

system of semantics both presuppose and exploit the features of a 

system of syntax. (Raymond, 2006, pp. 12-3) 

He considers that many hermeneutic problems remain unexplained due to the 

inadequacy of their linear system of notation for premise-conclusion arguments. 

Here are nine of those problems (Raymond, 2006, p. 114): 

1. why, in a treatment that appears to be systematic and complete, is the 

fourth figure missing 

2. what makes the first figure obvious, when compared to the second and the 

third 

3. what explains the changing account of the major, minor and middle terms 

in a syllogism 

4. how can Aristotle say that probative proofs of imperfect syllogisms produce 

the first figure 

5. how can Aristotle say that indirect proofs (per impossibile) come about 

by means of the first figure 

6. how to account in a satisfying way for Aristotle’s claim that what is ‘no 

syllogism’ can become ‘a syllogism’ by conversion 

7. what is to be done with the incompatibility of the linear view (where 

there is one and only one conclusion) with Aristotle’s claim about the 

relationship between the number of terms and the number of conclusions 

in a syllogism 

8. why is it said that every syllogism is through three terms 

9. why is it that Aristotle’s immediate successor, Theophrastus, added a 

indirect first figure but no fourth figure 

To answer these questions, Raymond develops an alternative notation making 

use of diagrams similar to the commutative diagrams used in category theory 

(see Goldblatt, 1979). However, I will not examine it here. Instead, I will end 

this paper by a discussion of the underlying stakes of the discussion I have 

presented. 

4 Conclusion: Two Traditions in Logic 

Understanding Aristotle’s theory of syllogism is in itself important. However, 

in the context of our discussion, it is especially important, for it illustrates 

fundamental issues about what the nature of logic is (or ought to be). An interesting 

aspect of the comparison between ̷Lukasiewicz’s and Corcoran-Smiley’s


reconstructions is to clarify a plurivocity in the discourse of contemporary logicians, 

concerning what they take logic to be. It has long been noticed that 

there are two qualitatively distinct traditions in contemporary logic. In his famous 

paper, van Heijenoort (1967) calls them “logic as calculus” and “logic as 

language”. Shapiro (2005) calls the former “the algebraic perspective”, while 

Hintikka (1997) calls the latter “logic as universal medium”. Peckhaus (2004) 

calls them, respectively, “logic as lingua characterica” and “logic as calculus 

rationcinator”, after Leibniz famous distinction. However, I think Corcoran 

(1994) makes the most precise characterization of the two traditions, by calling 

them, respectively, “formal ontology” and “formal epistemology”. If both 

groups of logicians call themselves “formal logicians”, it should be emphasized 

that it is for very different reasons. The formal ontologists justify their use of 

the adjective ‘formal’ by the contention that the propositions they deal with are 

expressed “exclusively in general logical terms, without the use of names denoting 

particular objects, particular properties, etc.” (Corcoran, 1994, p. 19) This 

point appears to be ̷Lukasiewicz’s real motivation for the claim that concrete 

terms do not pertain to logic. On the other hand, the formal epistemologists 

justify their use of the adjective ‘formal’ by the contention that they deal not 

with the content of the scientific discourse, but with its form. 

Not surprisingly, ̷Lukasiewicz (1957, pp. 4-5) himself distinguishes two trends 

in mathematical logic. The first, he calls “the algebraic approach”, and it 

includes logicians such as Boole, de Morgan, Peirce, Schröder, Venn, Hilbert, 

etc. The second, which he calls “ontology” (̷Lukasiewicz, 1957, p. 15), has 

been founded by Frege in 1879, and it has been further developed by Russell 

& Whitehead, Lesniewski, the Polish School in general, etc. The question that 

seems to animate the debate between the proponents of the ̷Lukasiewiczian view 

and the Corcoran-Smiley view concerns where Aristotle stands. If ̷Lukasiewicz 

is right to claim that Aristotle’s logic is a formal ontology, then we can take 

Aristotle to be presenting a system of propositions organized deductively. On 

the other hand, if Corcoran and Smiley are right, then we can take Aristotle to 

be presenting a system of deductions, organized epistemically. 

Let us examine more closely the two alternatives. ̷Lukasiewicz clearly takes 

Aristotle to be doing formal ontology: 

In the light of investigations by mathematical logic, Aristotle’s syllogistic 

is a small fragment of a more general theory founded by 

Professor S. Leśniewski and called by him ontology. (̷Lukasiewicz, 

1957, p. 15) 

From this point of view, Aristotle (and logicians in general) aims not to understand 

the processes of determining the validity of premise-conclusion arguments, 

but he aims to establish as true or as false, as the case may be, of certain universally 

quantified material implications. This is why it is not an option for the 

proponents of ̷Lukasiewicz’s view to take syllogisms as inferences. Logic does 

not examine the logical soundness of arguments; it investigates “certain general


aspects of ‘reality’, of ‘being as such’, in itself and without regard to how (or 

even whether) it may be known by thinking agents” (Corcoran, 1994, p. 17). 

The best excerpt we can find to illustrate this view is probably one of Russell 

himself: 

[. . . ] logic is concerned with the real world just as zoology, though 

with its more abstract and general features. (Russell, 1919, p. 169) 

The emphasis is put on (structural) ontology, i.e. on the most general characterization 

of reality itself, and not on the epistemic methods for obtaining 

knowledge of the validity or invalidity of arguments. Of course, the proponents 

of this conception need to refer to an epistemic dimension in a way or another. 

The point is only that this dimension is not what logic is about. Logic is about 

deducing the truth of propositions that can be expressed using only generic 

terms (individual, property, relation, etc.) and other logical expressions. Those 

expressions include only variables and non-logical constants (i.e. symbols for 

logical operators). In this framework, this generality of expression is the only 

difference between formal logic and science (since science have to use concrete 

terms). 

The ontological view of logic does not emphasize the epistemic role of logic 

in any way. The motivation for this attitude goes back to Frege (1994), more 

precisely his celebrated charge of psychologism against Husserl. This is a fact 

that the proponents of the view of logic as formal ontology generally think 

that the study of reasoning is not a part of logic, but a part of psychology. 

̷Lukasiewicz, too, agrees on that point: 

It is not true, however, that logic is the science of the laws of thought. 

[. . . ] Logic has no more to do with thinking than mathematics has. 

[. . . ] But the laws of logic do not concern your thoughts in a greater 

degree than do those of mathematics. What is called ‘psychologism’ 

in logic is a mark of the decay of logic in modern philosophy. For 

this decay Aristotle is by no means responsible. (̷Lukasiewicz, 1957, 

p. 12-3) 

For ̷Lukasiewicz, it was absolutely essential to defend Aristotle against the possible 

charge of psychologism that could have resulted from treating his logic as 

formal epistemology. From this standpoint, the only kind of epistemology that 

is acceptable, as Popper (1972) later said, is an “epistemology without knowing 

subject”. 25 ̷Lukasiewicz saw Aristotle as making an application of the informal 

axiomatic method to logic (i.e. formal ontology). This is why he did not notice, 

or did not want to notice, that Aristotle was more plausibly describing methods 

for the study of deductive reasoning. 

Logic as formal epistemology adopts a radically opposed standpoint. Aristotle’s 

syllogistic – and logic in general – is concerned with a methodological episte- 

25 See Tournier (1987) for a discussion of what is at stake – namely the third world.


mological problem, namely that of developing methods for obtaining knowledge 

of the logical validity of arguments. Typically, we gain this knowledge by making 

a deduction following logically valid rules of inference. This is precisely why 

natural deduction systems are superior to axiomatic systems to treat this epistemological 

problem. However, one may argue, is it not the case that logical 

validity is defined in terms of truth-conditions In other words, is it not the 

case that those inferences that are logically acceptable are the truth-preserving 

ones If it were the case, the distinction between formal ontology and formal 

epistemology would be, at best, artificial. This widely spread opinion is not 

exact 26 , for what really matters is the consequence-conservative property. An 

example of inference that is truth-preserving but not consequence-conservative 

is mathematical induction. As a rule of inference, it is logically erroneous, since 

there is more information in the conclusion than in the premises. However, as a 

constitutive law for the “mathematical realm”, it is totally acceptable. When we 

think of logical consequence merely in truth-functional terms, the importance 

of the epistemological part may seem to vanish. 

For the proponents of the Corcoran-Smiley view, Aristotle was concerned 

with how human beings deduce a conclusion from premises that logically imply 

it. In other words, Aristotle was concerned with the methodology of deduction 

and invalidation. This aspect of his philosophy naturally took place as a part 

of his general epistemology. For Aristotle, science is axiomatic. Traditionally, 

in an axiom system, we prove theorems from axioms that are known to be true 

of their objects. However, doing that, we make use of an underlying logic, i.e. 

a system of rules of deduction that justifies the passage from a proposition to 

another. This underlying logic is not itself a part of any axiom system, but is 

rather presupposed by the axiomatic method itself. From this point of view, 

logic governs the passage from proposition to proposition and has no ontological 

import whatsoever. Such a theory naturally took place in Aristotle’s Organon in 

the Prior Analytics, the preamble to his examination of the axiomatic science in 

the Posterior Analytics. Thus, under the Corcoran-Smiley view, Aristotle is not 

making use of an informal axiomatic method that would need to be “translated 

in logical terminology” (as ̷Lukasiewicz said), for the simple reason that he 

did not employ the axiomatic method for his theory of syllogisms. Rather, his 

syllogistic is meta-axiomatical or metascientific – metascience being a discourse 

taking science for object, i.e. what we nowadays call epistemology. 

The contemporary discipline bearing the name logic is a far cry from what 

it was for Aristotle. Logic today includes things going from critical thinking 

26 “To say that deduction is truth-preserving is an understatement at best; it is usually an 

insensitive and misleading half-truth; and it often betrays ignorance of logical insights already 

achieved by Aristotle. [. . . ] Deduction is not merely truth-preserving, it is informationconservative, 

i.e. consequence-conservative in the sense that every consequence of a proposition 

deduced from a given set of propositions is already a consequence of the given set; there is 

no information in the deduced proposition not already in the set of propositions from which it 

was deduced. Not every truth-preserving transformation is consequence-conservative, but, of 

course, every consequence-conservative transformation is truth-preserving.” (Corcoran, 1994, 

p. 15)


to axiomatic set theory. The canonical view is that we have various first-order 

axiomatized systems of logic whose metalogic is presented in set-theoretical 

terms. This metalogic, which is nothing but a more or less informal version of 

axiomatic set theory, is itself justified by its metatheory. Most of the time, this 

metametatheory is presented in a non-axiomatized way, most of all consisting 

in appeals to the meaning of the basic logic terms (‘and’, ‘or’, ‘belongs to’, 

‘necessarily’, etc.) We can separate all these parts of the discipline logic in two 

categories: (1) axiomatized formal theories and (2) underlying logic (that makes 

the axiomatic exercise possible). If we take for account this stratification of logic, 

absent in Aristotle’s time, we would say, with Lear (1980, p. 13), that “Aristotle 

shares with contemporary logicians a primary interest in metalogic.” Banks 

(1948) even proposes strong arguments to sustain that Aristotle’s conception of 

logic is nothing but our current conception of metalogic. 

If we grant that logic, in its most genuine philosophical sense, is an underlying 

logic, the debate between ̷Lukasiewicz and Corcoran-Smiley would have the 

following crucial historical stake: 

[. . . ] if the ̷Lukasiewicz view is correct, then Aristotle cannot be regarded 

as the founder of the science of logic. Indeed Aristotle would 

merit the 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 man set down axiomatizations 

of bodies of information without explicitly developing 

the underlying logic [in Church (p. 317) sense].” (Corcoran, 1974a, 

p. 280) 

If logic is actually the underlying part of axiomatic sciences, then it is a formal 

epistemology. The direct consequence is that ̷Lukasiewicz is wrong, all the way 

down. Also, it would show that the proponents of formal ontology are wrong in 

saying that axiomatized propositional logic is the ultimately primitive system 

of logic. 

After this discussion, it seems to be objectively clear that the Corcoran- 

Smiley reconstruction works much better than ̷Lukasiewicz’s. Being myself more 

inclined to adopt the view of logic as formal epistemology, it is a satisfying fact 

to have a pillar of philosophy such as Aristotle on my side. This may certainly 

suggest that this view is philosophically more sound. However, even if it is 

accepted that Aristotle stands better on the side of formal epistemologists, he 

may well have been wrong on this point. Deciding which one of these conception 

is preferable is extremely difficult, as it is always the case when a comparison 

between different research programmes is required. However, considering the 

long-standing presence of Aristotle’s philosophy of logic, versus the relatively 

very short-standing presence of the Fregean philosophy of logic, we are confident 

that adopting the view of logic as formal epistemology is better grounded.


References 

Austin, J.L. (1952). Critical Notice on Aristotle’s Syllogistic, from the Standpoint 

of Modern Formal Logic, by J. ̷Lukasiewicz. Mind, 61(243): 395–404. 

Banks, P. (1948). On the Philosophical Interpretation of Logic: An Aristotelian 

Dialogue. In: Menne (1962), pp. 1–14. D. Reidel Publishing Company. 

Bocheński, I.M. (1948). On the Categorical Syllogism. In: Menne (1962), pp. 

15–39. D. Reidel Publishing Company. 

——— (1961). A History of Formal Logic. Notre Dame: University of Notre 

Dame Press. Tr. I. Thomas. 

——— (1963). Ancient Formal Logic. Amsterdam: North-Holland Publishing 

Company. 

Boehner, P. (1952). Review of Aristotle’s Syllogistic from the Standpoint of 

Modern Formal Logic, by Jan ̷Lukasiewicz. The Journal of Symbolic Logic, 

17(3): 209–10. 

Copi, I.M. (1961). Introduction to Logic. New York: The MacMillan Company. 

Corcoran, J. (1972). Completeness of an Ancient Logic. The Journal of 

Symbolic Logic, 37(4): 696–702. 

——— (1974a). Aristotelian Syllogisms: Valid Arguments or True Universalized 

Conditionals Mind, 83(330): 278–81. 

——— (1974b). Aristotle’s Natural Deduction System. In: J. Corcoran, 

(Ed.) Ancient Logic and Its Modern Interpretations, pp. 85–131. Dordrecht- 

Holland: D. Reidel Publishing Company. 

——— (1994). The Founding of Logic: Modern Interpretations of Aristotle’s 

Logic. Ancient Philosophy, 14: 9–24. 

Corcoran, J. et al. (1982). The Contemporary Relevance of Ancient Logical 

Theory. The Philosophical Quaterly, 32(126): 76–86. 

Fillion, N. (2006). The Frege-Hilbert Controversy on the Foundations of Geometry. 

Master’s thesis, Université Laval, Quebec City. 

Frege, G. (1895). Letter from Frege to Hilbert dated 1 October 1895. In: 

Frege (1980), pp. 32–4. The University of Chicago Press. 

——— (1980). Philosophical and Mathematical Correspondence. Chicago: The 

University of Chicago Press. 214 p. 

——— (1994). Review of E.G. Husserl, Philosophy of Arithmetic I. In: 

B. McGuinness, (Ed.) Gottlob Frege: Collected Papers on Mathematics, 

Logic, and Philosophy. Oxford: Basil Blackwell.


Gasser, J. (1991). Aristotle’s Logic for the Modern Reader. History and 

Philosophy of Logic, 12: 235–40. 

Geach, P.T. (1972). Logic Matters. Los Angeles: University of California 

Press. 

Goldblatt, R. (1979). Topoi. The Categorial Analysis of Logic. New York: 

North-Holland Publishing Company. 486 p. 

van Heijenoort, J. (1967). Logic as Calculus and Logic as Language. Synthese, 

17: 324–30. 

Hintikka, J. (1988). On the Development of the Model-Theoretic Viewpoint 

in Logical Theory. Synthese, 77: 1–36. 

——— (1997). Hilbert Vindicated Synthese, 110: 15–36. 

Kleene, S.C. (1967). Mathematical Logic. New York: John Wiley & Sons Inc. 

Kotarbiński, T. (1958). Jan ̷Lukasiewicz’s Works on the History of Logic. 

Studia Logica, VIII: 57–62. 

Lear, J. (1980). Aristotle and Logical Theory. Cambridge: Cambridge University 

Press, 123 pp. 

̷Lukasiewicz, J. (1929). Elementy Logiki Matematycznej. Warsaw: PWN – 

Polish Scientific Publishers, 1st edn. 

——— (1957). Aristotle’s Syllogistic from the Standpoint of Modern Formal 

Logic. Oxford: Clarendon Press, 2nd edn. 

——— (1963). Elements of Mathematical Logic. Oxford: Pergamon Press, 1st 

edn. 1st Polish edition in 1929. 

Menne, A., (Ed.) (1962). Logico-Philocophical Studies. Dordrecht-Holland: D. 

Reidel Publishing Company, 136 pp. 

Novak, J.A. (1980). Some Recent Work on the Assertoric Syllogistic. Notre 

Dame Journal of Formal Logic, XXI(2): 229–42. 

Patzig, G. (1968). Aristotle’s Theory of the Syllogism: A Logico-Philological 

study of book A of the Prior Analytics. Dordrecht-Holland: D. Reidel Publishing 

Company. 

Peckhaus, V. (2004). Calculus Ratiocinator vs. Characteristica Universalis 

The Two Traditions in Logic, Revisited. History and Philosophy of Logic, 

25(1): 3–14. 

Pelletier, F.J. (1999). A Brief History of Natural Deduction. History and 

Philosophy of Logic, 20: 1–31.


Popper, K.R. (1972). The Objective Knowledge: An Evolutionary Approach. 

Oxford: Clarendon Press. 

Quine, W.V.O. (1951). Mathematical Logic. Cambridge, Mass.: Harvard 

University Press, Revised edn. 346 p. 

Raymond, D. (2006). Qua. Ph.D. thesis, The University of Western Ontario, 

London. 

Russell, B. (1919). Introduction to Mathematical Philosophy. New York: 

Dover Publications. 1991. 208 p. 

Shapiro, S. (2005). Categories, Structures, and the Frege-Hilbert Controversy: 

the Status of Meta-Mathematics. Philosophia Mathematica, 13(1): 61–77. 

Smiley, T.J. (1973). What is a Syllogism Journal of Philosophical Logic, 2: 

136–54. 

Smith, R. (1982). What is aristotelian ecthesis 

Logic, 3: 113–27. 

History and Philosophy of 

Smith, R., (Ed.) (1989). Prior Analytics. Cambridge: Hackett Publishing 

Company. 

Thom, P. (1979). Aristotle’s Syllogistic. Notre Dame Journal of Formal Logic, 

XX(4): 751–9. 

Tournier, F. (1987). Les Enjeux du Troisième Monde. Les Cahiers 

d’épistémologie, UQAM, no 8702, Montréal.

Two Accounts of Aristotle's Logic - Nicolas Fillion

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