Lukasiewicz on the Principle of Contradiction

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Journal of Philosophical Research, XXIV, 1999, pp. 57-112

£UKASIEWICZ ON THE PRINCIPLE OF CONTRADICTION

VENANZIO RASPA

UNIVERSITÀ DI URBINO

“Today, like in the past, we believe that the principle of

contradiction is the most reliable law of thought and being.

Certainly only a fool could deny it. The validity of this law

imposes itself on everyone with immediate evidence. It need not

be founded, nor can it be. Aristotle taught us to believe this

way. What is so surprising then, that nobody is concerned with

something so clear, unquestionable and forever resolved?” (J.

£ukasiewicz)

Introduction: the historical-philosophical context; I. The ontological, logical, and psychological formulations of

the principle of contradiction; II. The principle of contradiction is not a simple, ultimate and necessary principle;

III. The idea of a non-Aristotelian logic; IV. The ‘proof’ of the principle of contradiction; V. The principle of

contradiction and symbolic logic; Conclusion.*

INTRODUCTION

Is it possible to open in logic vistas comparable to those opened in geometry by the

introduction of non-Euclidean geometries? That is the opening question of Jan £ukasiewicz's juvenile

reflection on the principle of contradiction. On 7th March 1918 during his farewell lecture at Warsaw

University, in which he announced to have developed a three-valued logic, £ukasiewicz declared:



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“In 1910 I published a book on the principle of contradiction in Aristotle's work, in

which I strove to demonstrate that that principle is not so self-evident as it is believed to

be. Even then I strove to construct non-Aristotelian logic, but in vain”.1

The book in question is O zasadzie sprzeczno¶ ci u Arystotelesa. Studium krytyczne [On the

Principle of Contradiction in Aristotle. A Critical Study], printed in Polish in 1910. In the same

year, £ukasiewicz published, as an article, a synopsis of it in German, “Über den Satz des

Widerspruchs bei Aristoteles”. So far, only this article has been taken into consideration by Western

scholars; there exist two English translations (1971; 1979) and only recently has a French one

(1991) appeared, while the most important text has been nearly completely neglected, and has

become almost unobtainable. In 1987, Jan Woleñski edited a reprint of the major work in Polish, the

German translation of which has been published in 1993. In the following pages references will be

made mainly to the main text and, occasionally, to the article.

An analysis limited solely to the article, as it has been done up till now, risks in fact to be

misleading: it contains—for reasons of space obviously—clear-cut statements, which are not always

argued. To give an example, we read in one of the first pages:

“The psychological principle of contradiction cannot be demonstrated a priori, rather it

is at most to be induced as a law of experience”;2

we do not however find any explanation of this thesis, which instead has been given in the book (see

infra, pp. 12ff.). Moreover, the article has always been read in relation to the interpretation and tothe criticism to which the young £ukasiewicz subjects the Aristotelian texts, almost completely

neglecting the formation and the philosophical background to which he makes reference.

Nevertheless, to understand better £ukasiewicz's criticism of the principle of contradiction, I believe

we cannot leave out some considerations which are intended to contextualize his thought. These

considerations are likewise suggested by the many references present both in the main text and in the

coeval article. Moreover, this effort which is, in its double version, one of young £ukasiewicz's first

intellectual tasks, represents an interesting crossroad of different components: on one hand, he gives



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attention both to traditional as well as contemporary philosophical trends, on the other, he shows a

real enthusiasm for the recent developments of symbolic logic and the discovering of non-Euclidean

geometries.

The starting-point of £ukasiewicz's reflection is constituted by the remarkable progress

accomplished in symbolic logic starting from Boole up to Russell. Regarding the level of improvement

attained, symbolic logic, £ukasiewicz asserts, stands in the same relation to the Aristotelian logic as

the modern geometry stands to Euclid's Elements. Since the principle of contradiction occupies a

position in logic analogous to that of the parallel line postulate in Euclidean geometry, a revision of the

principle of contradiction becomes necessary, that is, a revision of the basis of Aristotelian logic in

the light of the latest results of symbolic logic.3 This is why the book, divided like the article in a pars

destruens and in a pars construens, ends with a formal-logical appendix, “The Principle of

Contradiction and Symbolic Logic”, the most important thesis of which is that the Aristotelian

principle does not correspond to the homonymous one of the new logic.4

If this is the inspiring element, the work and above all the first part consists in a consideration

of the Aristotelian text which can also be read as a critical comparison to traditional formal logic, that

is, with the logicians of the previous generation; in the text the names of Adolf Trendelenburg,

Friedrich Ueberweg and Christoph Sigwart recur, who, according to £ukasiewicz, did not bring any

substantial progress as to Aristotle's concept. The question of the different formulations of the

principle of contradiction, just to give an example (a question on which Heinrich Maier 5 had already

laid stress and whom £ukasiewicz refers to many times), does not only concern the Aristotelian text.

In fact, the problem has been set, in all its extension, even by logicians like Trendelenburg, Ueberweg

and Sigwart.6

In its fullness, the Aristotelian text lent itself to express all the different positionsseparately presented by the above-mentioned authors. Hence, not only Aristotle but also the

exponents of traditional formal logic have put to £ukasiewicz the problem of the different

formulations of the principle of contradiction, as well as the necessity of its demonstration. The latter

constitutes the pivot around which the second part (the pars construens) rotates, wherein the author

tries also to clarify the theoretical implications to which it leads.

An important author in the intellectual development of young £ukasiewicz is the Austrian

philosopher Alexius Meinong, who is quoted several times in both texts examined here, and also in



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other earlier papers. £ukasiewicz personally knew Meinong, with whom he was in correspondence

for a short time when he took part in his philosophical seminar in Graz in 1908/1909.7 £ukasiewicz

retained from Meinong several of his theories: the classification of objects with the connected theory

of impossible objects as well as those of incomplete objects and of objectives [Objektive].8 These

theories will allow him to develop the criticism of the principle of contradiction and to set out the

conditions for his ‘proof’. Referring to Meinong, £ukasiewicz asserts in fact that a proof of the

principle of contradiction is only possible on the basis of the assumption that objects are

noncontradictory. If, on the other hand, we accepted contradictory objects—just as Meinong

does—then there would be cases in which the principle is not valid.

Some other developments in Austrian philosophy, originating from Bernard Bolzano,

contribute to making £ukasiewicz's argument possible. They are first presented to him through his

teacher Kazimierz Twardowski, with whom £ukasiewicz had studied in Lwów in a period in which

Twardowski, for his express assertion, was a fervent Brentanian.9 Later on, after receiving his

doctorate in philosophy in 1902, £ukasiewicz travelled in Europe and, between 1902 and 1906,

visited several European universities, among which were Leuven and Berlin, where he attended the

lectures of Deciré Mercier and Carl Stumpf respectively10 (the latter is also cited in the article of

191011). £ukasiewicz probably came into contact with Bolzano's Wissenschaftslehre not only

through Twardowski but also through Stumpf.12 Having returned to Lwów, £ukasiewicz became

Privatdozent and started his teaching. In 1906 he also completed his work on the concept of cause,

in which the notion of concepts as abstract objects, intended as extra-spatial and extra-temporal

objects, appears. £ukasiewicz himself states that he is not able to define what these objects are, but

that he can say what they are not: abstract objects are neither psychical acts nor images existing in amind, but can be either ideal or real.13 The former (i.e., the ideal abstract objects) are mathematical

objects independent of what exists in the real world, while the latter, the real abstract objects, are

built to subsume concrete objects.14 In my opinion, the influence of Bolzanian elements can be traced

in the theory of abstract objects15.

In the frame of Austrian philosophy between the second half of the 19th and the beginning of

the 20th century, to which both Twardowski and Meinong belong, the Bolzanian concept of the in-

itself [an sich], in particular that of the ideas-in-themselves and of the objectless ideas



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[ gegenstandslose Vorstellungen], was not accepted in the terms in which it had been elaborated by

its author, but was in a sense inverted . Meinong's nonexistent objects, among which are those that

are impossible or contradictory, are the result of an elaboration that goes through a double mediation:

that of Robert Zimmermann—who at the beginning, in the first edition of the Philosophische

Propaedeutik , assumes the contradictory objectless ideas,16 which are then expunged in the second

edition17 —and the much more determining mediation of Twardowski. According to the latter, who

shares the Brentanian thesis of the intentionality of psychical phenomena, to each idea corresponds

an object, so there are no presentations without objects—which otherwise would be a real

contradictio in adjecto —there are instead presentations, the objects of which do not exist.18 From

here starts Meinong's classification of objects, including those that are nonexistent and the impossible

or contradictory objects as well.19 £ukasiewicz places himself in a moment in which this process has

been accomplished, and—as already mentioned—he makes use of several of Meinong's concepts

for the discussion of the principle of contradiction, drawing conclusions which will be exposed later

on.

In this essay I do not mean to give a complete account of the comparison between

£ukasiewicz and Aristotle—which moreover has been already widely discussed20 —as to take into

consideration the theoretical contributions (which of course concern Aristotle as well) present in the

main work but absent in the article. If we read £ukasiewicz's work only as an interpretation of the

Aristotelian texts, it is but an interpretation, though remarkable, among the many available; instead, if

we lay stress on his specific contributions with regard to the value and the significance of the principle

of contradiction, it assumes—as we shall see—a different meaning. At first I will consider some

aspects of £ukasiewicz's well-known individuation of three formulations of the principle ofcontradiction in the Aristotelian texts (I) and the criticism of the opinions which consider it as a

simple, ultimate, and necessary principle (II). I will then dwell on two aspects of the Polish

philosopher's juvenile reflection which are essential and—in my opinion—represent the final point of

his research on the principle of contradiction as well as the novelty regarding the preceding studies on

the subject. The former concerns the conception of a non-Aristotelian logic, that is, a logic operating

without the principle of contradiction, a natural consequence of the asserted independence of the

principle of the syllogism from the principle of contradiction (III). The latter concerns the attempt to



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supply a direct proof of the Aristotelian principle of contradiction, in view both of the criticism to

which £ukasiewicz subjected Aristotle's “negative demonstration [¢pode‹xai ™l egktikî j ]”,21 and

of the thesis according to which the principle of contradiction is not an ultimate principle but rather, if

it has to be accepted as true, needs a proof (IV). What clearly comes out of the analysis of these

two attempts is £ukasiewicz's intention not only to write a work on the principle of contradiction in

Aristotle but also to give life to something wider and more ambitious. In the end I will take into

consideration some topics present in the above-mentioned appendix, since they add new elements of

reflection (V).

It must be pointed out, however, that as early as in his work of 1910 £ukasiewicz expresses

some reservations about the proof of the principle of contradiction which he brought forward (see

infra, p. 36). To these reservations those relating to the attempt of constructing a non-Aristotelian

logic were added eight years later (see supra, p. 2). Later on, his judgement on his own juvenile

production had become even more critical. In a letter to I. M. Bocheñski, dated Dublin, 7th October

1947, £ukasiewicz writes:

“When I read the estimate of my activity, either in [Zbigniew] Jordan, or yours, Father,

my feeling is that I read my own necrology. And at that time different desiderata come

into my head: I would not like it would be written about my pre-logical philosophical

works. I regard my dissertation on causality as well as my book O zasadzie

sprzeczno¶ ci u Arystotelesa as weak and unsuccessful”.22

Nevertheless, although £ukasiewicz was very critical on his first book, in 1955 (less than a year before his death) he began to translate it himself into English. From here C. Lejewski, followed by V.

Wedin, infers that the book “must have stood high in the author's own estimation”.23 This is clearly in

contrast with what £ukasiewicz said above. Now, it is true that £ukasiewicz may have changed his

own opinions; however, I think that, although it is not in itself very important to know what he

thought of his book, some reflections on it could help us to explain both its fate and its value. In my

opinion, after £ukasiewicz began to be more and more involved in mathematical logic, he became

conscious that the book presents imprecisions and many analyses which are out-of-date. Actually—



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as it will be pointed out during this article—he will later on maintain ideas which are different and

even opposed to some of those asserted in the book of 1910. This explains why, as Z. A. Jordan

pointed out, “£ukasiewicz never again returned to the view on the principle of noncontradiction

expounded in his first major work”.24 However, it is probable that £ukasiewicz regarded as still valid

single parts of it and/or that he recognized its historical importance. On the one hand, the book was

very important for the logical-philosophical developments in Poland. It was very popular among

Polish philosophers, and the appendix it included, although not the first publication on mathematical

logic in Poland, was read as a handbook on this subject.25 Furthermore, its publication immediately

provoked a debate inside the Lwów-Warsaw school in the years 1912-1913, in which authors like

Tadeusz Kotarbiñski and Stanis³aw Le¶niewski took part.26 On the other hand, it aroused many

discussions among the interpreters of the Aristotelian texts, discussions that are still in progress (see

n. 20); moreover, it presents new ideas and suggestions, some of which became objects of study and

were further developed in our century. It is especially for the wealth of issues that this book

preserves for us so much of its interest.

I

In the Aristotelian texts £ukasiewicz distinguishes three formulations of the principle of

contradiction. The first, the so-called ontological formulation,27 is the classical formulation enunciated

in Met . G 3, 1005 b19-20:

“It is, that the same attribute cannot at the same time belong and not belong to the samesubject and in the same respect”.28

Moreover, in connection with the definition of contradiction given in De int . 6, 17a32-35 (but cf. also

De int . 7, 17 b26-29), Aristotle would enunciate a logical formulation29 in Met . G 6, 1011 b13-14:

“contradictory statements are not at the same time true”;



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and at last, just in the continuation of Met . G 3, 1005 b23-24, a psychological formulation:30

“it is impossible for any one to believe [Øpol amb£nein] the same thing to be and not

to be”.

In this last formulation stress is laid on Øpol amb£nein31 which just makes us think of a formulation

from the point of view of the thought and leads us to recognize a subjective moment, meaning that we

do not talk about the belonging, or not, of a predicate to a subject, but about the impossibility of the

coexistence of two opposite opinions in the same individual. It is in this sense that Sigwart makes the

Aristotelian formulation of the principle of contradiction his own,32 and it is from this kind of reading

that £ukasiewicz, criticizing the Aristotelian principle of contradiction, convinces himself of the

necessity to also criticize such a meaning of the principle.

£ukasiewicz rewrites the three formulations of the principle of contradiction individuated in

the Aristotelian texts in the following way:

(i) ontological formulation:

“ No object can possess and not possess the same property at the same time”,

where by ‘object’ he understands with Meinong, but—we can add—also with Twardowski,33

“anything that is ‘something’ and not ‘nothing’ ”, and by ‘property’ “anything that can be

predicated of an object ”;34

(ii) logical formulation:

“Two sentences, of which the one ascribes to an object exactly that property

which the other denies to it, cannot be true at the same time”,

where ‘sentence’ means “a sequence of words or other symbols, which mean that a certain

object possesses or does not possess a property”;35

(iii) psychological formulation:



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“Two beliefs, to which correspond contradictory sentences, cannot exist at the

same time in the same intellect ”,

and here £ukasiewicz means by ‘belief’ or ‘act of believing’—this is how he interprets dÒxa and

ØpÒl hyij —a “ psychical phenomenon” to which, as a logical fact, corresponds an affirmative or

negative sentence. Such a distinction between sentences as logical facts and beliefs as psychical

phenomena is also borrowed from Meinong who distinguishes beliefs as acts of judgment from the

objects of the beliefs which he calls ‘objectives’. However, £ukasiewicz wants to specify a

difference between his concept and that of the Austrian philosopher. According to Meinong,

objectives are what is in common both to beliefs and to judgments or assumptions [ Annahmen], that

is, the fact [Tatsache] that something is or is not, or the being-so [Sosein] and the not being-so. For

instance, the judgment “John is white”, the belief that John is white, or the assumption that John is

white have as object the same objective, that John is white. The judgment is a making [Tun]

provided by a moment of assertion or conviction [Überzeugungsmoment ] that has as object an

objective, while the assumption is a judgment which does not involve a conviction.36 Instead,

according to £ukasiewicz, a sentence (or judgment) is “an objective expressed in words or in

other signs”, and a belief is the psychical act to which corresponds a sentence as a logical fact. Such

a definition of a sentence, £ukasiewicz asserts, is as close as possible to Aristotle's who also makes

a clear distinction between sentence and belief, claiming that the latter, which has its seat in the soul,

has the sentence as a symbolic correlate (cf. De int . 14, 24 b1-3), that is, a meaningful speech which

can be true or false (cf. De int . 4, 17a

1-3).37

Now, the three formulations given, £ukasiewicz asserts, are not sentences of identical

meaning. However, he recognizes both that Aristotle, although he clearly distinguishes between the

ontological formulation and the psychological one, treats the logical and ontological formulations as

equivalent;38 and that “the ontological principle is the principle of contradiction kat' ™xoc»n”39 to

which the Stagirite dedicates much of his attention. An examination of £ukasiewicz's thesis requires

starting from the distinction that he operates between synonymity and equivalence. Since I will limit

my analysis to £ukasiewicz's thought, I have to specify, so that my point of view on this subject be



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clear to the reader, that I do not believe there are three distinct principles of contradiction in

Aristotle, as we could be tempted to conclude by taking £ukasiewicz's position to an extreme

degree. We must recognize, on the other hand, that the three above-mentioned statements,

paradigmatic of others that occur in the Aristotelian texts,40 have a different informative value.

£ukasiewicz takes great care in distinguishing the equivalence from the identity of meaning or

synonymity. Two propositions are synonymous, that is, they have the same meaning, if they express

the same thought by using different words, if then “O possesses p” and “O' possesses p'” express

the same object. In this sense the two propositions: “Aristotle was the founder of logic” and “the

Stagirite was the founder of logic”, are synonymous because the words “Aristotle” and “the

Stagirite” denote the same object on the basis of a convention which has now become current. No

negative sentence however can have the same meaning as a positive sentence since, £ukasiewicz

says, affirming and denying have different meanings; and moreover since each sentence, be it

affirmative or negative, is as simple as the other, neither of the two can be reconducted to the other.

Now, if “O possesses p” and “O' possesses p'” have the same meaning, then the truth of one follows

from the truth of the other and vice versa. Such sentences are called equivalent. Two sentences of

identical meaning are always equivalent; so, if two sentences are not equivalent, they are also not

synonymous: the absence of equivalence constitutes the criterion for the acknowledgement of the

diversity of sentences. On the contrary, two equivalent sentences, like “Aristotle was Plato's

disciple” and “Plato was Aristotle's teacher”, are not necessarily of identical meaning since, as in this

case, both the subjects and their predicates indicate different objects and different properties.41

The three formulations of the principle of contradiction (ontological, logical and

psychological) are not synonymous, because in the first case we take into consideration objects, inthe second sentences, and in the third beliefs: if the objects designated by the subjects and the

predicates of the propositions are different, the propositions are also different. The logical and

ontological formulations are however equivalent since the first follows from the second and vice

versa. Their equivalence is a logical consequence of the assumption of the realistic point of view

according to which “being and true sentences correspond reciprocally” (which, after acknowledging

the necessary differences, is also the point of view of Bolzano, Twardowski, Meinong and Russell).

Such a point of view is based on the definition of a true sentence:



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“ An affirmative sentence is true, if it ascribes to an object a property which it

possesses; a negative sentence is true, if it denies to an object a property which it

does not possess. Likewise, in an inverted manner: each object possesses a property

which a true sentence ascribes to it; and no object possesses a property which a

true sentence denies to it ”.42

This is attested also by Aristotle, for whom “to say of what is that it is not, or of what is not that it is,

is false, while to say of what is that it is, and of what is not that it is not, is true”.43 Nevertheless, such

equivalence is logical, not real since, according to the Stagirite, it is always reality which is the basis

for the truth of a sentence, and not the contrary. The proof of the equivalence of the logical and

ontological formulations is carried out on the basis of the definition of a true sentence. (a) To true

sentences, affirmative and negative, correspond objective facts, that is, their own relation of

inherence or non-inherence of a quality to an object (cf. De int . 9, 18a39- b1). In fact, if two

contradictory sentences were true at the same time, then the same object would have and would not

have the same property at the same time, but such a thing is forbidden by the ontological principle of

contradiction. (b) Vice versa, to objective facts correspond as many true sentences, either affirmative

or negative (cf. De int . 9, 18 b1-2; Met . Q 10, 1051 b3-4). Indeed, if the same object had and did

not have the same property at the same time, then two contradictory sentences would be true at the

same time, which is definitely forbidden by the logical principle of contradiction. 44

As to the psychological formulation of the principle, after its treatment by £ukasiewicz the

result is that his consideration assumes Aristotle as a starting point in order to reach bothcontemporary philosophy and traditional formal logic as well. £ukasiewicz takes into account the

passages of Met . G 3, 1005 b26-3245 —read in connection with De int . 14, 23a27-39—and G 6,

1011 b15-2246 which he interprets as two complementary parts of a single attempt conducted by

Aristotle to prove the validity of the principle of contradiction even for beliefs, and hence the

legitimacy of the psychological formulation. The result achieved by £ukasiewicz is that the

impossibility for a subject to have contradictory beliefs at the same time is demonstrable only

provided that we treat these as if they were sentences for which the alternative true or false is valid.



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Therefore, the psychological formulation of the principle of contradiction is nothing but a

consequence of the logical one.47 This peculiarity of the Aristotelian argumentation is interpreted by

£ukasiewicz as a falling by the Stagirite into that error which is the exact converse of “psychologism

in logic”, that is, “logicism in psychology”.48

Aristotle would then treat beliefs as if they were sentences. But a fundamental difference

between sentences and beliefs consists in the fact that the latter are “psychical phenomena” (see

supra, p. 9) which, as such, are always positive. As a consequence, it can never happen that two

beliefs are in contradiction as in affirmation and negation. Such a thing would involve that the same

belief should be present and at the same time should not be present in the same mind, but a belief

that does not exist cannot be in contradiction with another (probably this is also the reason why

Aristotle, in Met . G 3, 1005 b26-32, talks about contrary opinions and not about contradictory ones,

although the former are openly opposite). In reality, while sentences mean that something is or is not

and while they are in a relation of correspondence or of non-correspondence with their own objects

or facts, so that they can be true or false, beliefs have a different structure. As psychical phenomena

they do not assert simply that something is or is not but they rather represent an intentional relation

with something: without something that is intended, £ukasiewicz says, there is no belief. In itself this

intentional relation consists of two parts: the act of belief and the Meinongian objective (see supra, p.

9). The expression in words or in signs of the second part of the intentional relation is the sentence,

which can be true or false, but the first part, as psychical phenomenon, does not refer to any fact, so

we can say that it is neither true nor false. Beliefs then are not purely logical objects because they are

necessarily related to experience.49

These reflections make it possible to extend the subject beyond the simple reference to theAristotelian texts. £ukasiewicz marks, in fact, that the non-validity of Aristotle's argumentation does

not mean that the thesis cannot be true: there could be other argumentations capable of supporting it.

Here it becomes clear that £ukasiewicz reads Aristotle looking also at the contemporary

philosophical situation. His aim is not only to confront himself with the Aristotelian texts but, by doing

so, to confront himself with traditional logic as well. Let's ask ourselves then, £ukasiewicz continues,

which other argumentations can support the thesis that two opinions which annul each other—the

relation of mutual exclusion, as it englobes that of opposition, should render the proof easier—cannot



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exist in the same consciousness. A proof a priori, conducted on the basis of only assumptions and

definitions—that is, in the same way in which it is possible to prove that the concepts of ‘right’ and

‘equilateral’ in relation to the class of triangles annul each other—is not possible since beliefs are

psychical phenomena which necessarily have to refer to real events; if they did not, they would not

be beliefs. Only a proof a posteriori is left, a proof which starts from empirical events and, through

the observation of a certain regularity and uniformity of some phenomena, reaches the formulation of

a universal sentence. This would make the psychological principle of contradiction an empirical

law—as maintained by John Stuart Mill and Herbert Spencer. Such a law though, achieved by

induction, would not be a certain law but only probable as are all the empirical laws. On this point,

£ukasiewicz agrees with Husserl's criticisms of Mill's and Spencer's concept and of the

psychologistic interpretation of the logical principles in general.

According to Mill, the principle of contradiction together with other logical principles are

simple “generalizations from experience”. Its original foundation, he thinks, is “that Belief and

Disbelief are two different mental states, excluding one another”, as it would result from the

ascertainment that “any positive phenomenon whatever and its negative, are distinct phenomena,

pointedly contrasted, and the one always absent where the other is present”.50 This thesis is shared

by Spencer who, even though dissenting from Mill on other points (as with regard to the relation

between unthinkability and existence), asserts, taking into consideration the principle of the excluded

middle, that this is “simply a generalization of the universal experience that some mental states are

directly destructive of other states”. So the presence of a defined positive state in the consciousness

excludes the negative one that corresponds to it, and vice versa.51 Mill himself quotes Spencer for his

own support both in the Logic and in the Examination of Sir William Hamilton's Philosophy.52

On his behalf Husserl assimilates Mill's and Spencer's positions, pointing out that they intend the

impossibility that contradictory propositions can be true at the same time to be equivalent to the real

incompatibility of the corresponding acts of judgement. Since Mill asserts that what can be true or

false are acts of belief—Spencer talks about states of consciousness—Husserl states that their

principle of contradiction could be formulated in this way:



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“Two contradictorily opposed acts of belief [or states of consciousness] cannot

coexist ”.53

If this is true, the principle, then, proves to be inexact and scientifically not verified since it requires

specifications on the mental state of the subject, on the circumstances in which he thinks, etc., which

are not easy to determine.54 In fact we ask together with Husserl: in which circumstances are acts of

belief contradictory? What happens if there are two or more subjects asserting them? Is it really

impossible that some individuals do not consider two opposite beliefs true? What to say about those

beliefs that are not immediately contradictory then?

Already some years before £ukasiewicz rejected psychologism in logic stating—in the wake

of Husserl's Logische Untersuchungen but also of Meinong—that psychology cannot be a

fundament for logic, because their objects and laws are different.55 Logic does not take as object of

study the psychical processes but the relations of truth and falsity among judgements. That logical

and psychological laws have a different content means that, while the former are certain, the latter

can only be probable as they have an empirical character. The reason why logic and psychology are

associated was found by £ukasiewicz in the fact that often they use the same terminology. But in this

case, they assign different meanings to the same word. As we have already seen, ‘judgement’ means

belief for psychology, while for logic it is the objective correlate of a psychical act. In summary, logic

is an a priori science as mathematics, while psychology is based on experience.

In O zasadzie sprzeczno¶ ci u Arystotelesa £ukasiewicz continues Husserl's objection to

Mill's and Spencer's psychologistic interpretation of the principle of contradiction in order to confirm

his own criticism of the psychological formulation of the principle of contradiction, the weakness ofwhich consists in having to do not with purely logical objects, like sentences, but with objects related

to the experience, as beliefs have to be. So, £ukasiewicz concludes, a law of this type is revealed to

be inaccurate and, since specific psychological researches have not proved its validity, it remains

empirically unproved. It is however doubtful that this is possible since historically there have been

authors who have asserted with full awareness that something can be and not be at the same time.

Here £ukasiewicz quotes Hegel's passage:



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“Something moves, not because at one moment it is here and at another there, but

because at one and the same moment it is here and not here, and in this ‘here’, it at once

is and is not. The ancient dialecticians must be granted the contradictions that they

pointed out in motion; but it does not follow that therefore there is no motion, but on the

contrary, that motion is existent contradiction itself”.56

On this matter, either we agree with Aristotle that “what a man says, he does not necessarily

believe”,57 which means that Hegel wrote something that he did not believe, or we conclude that

Hegel had not fully thought out what he was writing. In both cases we have to resort—as Husserl

noticed—to subsidiary hypotheses [ Hilfshypothesen], or to specifications on the thinking subject

which complicate the principle and diminish its value.58

The conclusion of this argument is that the psychological principle of contradiction is not a

certain principle and, therefore, it cannot be assumed among the foundations of logic. It is not certain,

I repeat, because it does not have to do with sentences but with beliefs.

“The way to the foundations of logic does not go through psychology”.59

In this way the young £ukasiewicz becomes part of the European realistic stream that counted

exponents like Frege, Meinong, Husserl and Russell, strongly critical with respect to what Frege

defines as “the corrupting incursion of psychology into logic”.60

II

After £ukasiewicz has explained why he will remove the psychological formulation of the

principle of contradiction from further investigations, he has to take into consideration the other two

formulations. Another important thesis of O zasadzie sprzeczno¶ ci u Arystotelesa —which is

addressed against those who claim that the principle of contradiction is an ultimate and

indemonstrable, simple and evident principle—is the one according to which there are other

principles, simpler and more evident than it, that could hold as ultimate and indemonstrable. The



16

principle of identity (“to each object belongs that characteristic to which it belongs”), for example,

could be considered as one of these principles since it does not need, unlike the principle of

contradiction, negation and logical multiplication. On this subject, £ukasiewicz asserts, traditional

logic, which he calls “philosophical logic”, creates only confusion. In fact, (i) under the principium

identitatis it sometimes includes the principle of identity, other times the principle of contradiction;61

(ii) the latter is then confused with the principle of double negation (formulated in an inexact way like

“ A is not not - A”);62 and finally, (iii) the principle of identity, for which the ambiguous and imprecise

formula “ A is A” is generally used, is intended as the “positive counterpart [ positive Kehrseite]”63 of

the principle of contradiction or even identified with it.64 The point is that philosophical logic is lacking

in conceptual distinctions since it does not use sharply defined concepts and univocally determined

symbols;

“rather it sank into the swamp of the fluid and vague speech used in everyday life”. 65

That is why, £ukasiewicz repeats, by now a reconsideration of the principle of contradiction from the

point of view of symbolic logic is necessary.

In the case of the three mentioned principles, it is a matter of propositions not at all

synonymous, since each of them asserts a different thought. Expressed in a conditional form,66 the

principle of identity

a → a

means: “if O possesses a, then O possesses a”; the principle of double negation

a → ¬¬a

means: “if O possesses a, then O cannot not possess a”;67 the principle of contradiction

1→ ¬(a ∧ ¬a)



17

means: “if O is an object, then O cannot possess a and not possess a at the same time”.68

£ukasiewicz points out that if we consider the antecedents, we find in the principle of contradiction

the term ‘object’ which is absent in the other two principles. If we consider the consequents, we

notice—as already previously anticipated—that the principle of contradiction contains the concepts

of negation and of logical multiplication (linguistically expressed by the words ‘at the same time’ and

by the conjunction ‘and’), without which it cannot be formulated. Instead, the principle of identity

does not require such concepts and the principle of double negation can be expressed without the

logical multiplication. Not only are these last two principles different from the principle of

contradiction, they are also simpler than it; such is, in particular, the principle of identity. If this is so,

the thesis of many traditional logicians according to which the principle of identity is only the positive

formulation of the principle of contradiction would definitively fall.69 In the end it must be pointed out

that the principle of contradiction has been formulated according to what—as we shall see—

constitutes the peculiarity of the Aristotelian meaning: it refers to the properties of objects, or better it

supposes the existence of objects. (After all, in traditional logic all propositions have existential

import.) It is in regard to this peculiarity that, according to £ukasiewicz, the Aristotelian principle is

different from the one in the sense of symbolic logic (see infra, pp. 39ff.).

The question of the ultimate and indemonstrable principle still remains open. Due to the

above-mentioned differences, the principle of contradiction, £ukasiewicz asserts, cannot be deduced

from any of the other two principles; that means that it is even not equivalent to them. 70 However,

since the principle of identity came out to be simpler and more evident than that of contradiction, it

may seem that it is entitled to the qualification of ultimate. Instead, in virtue of the definition of theultimate principle—which means that a determined sentence is true “through itself” and cannot be

proved on the basis of other sentences—not even the principle of identity can be considered such, as

it can be proved on the basis of the definition of a true sentence (see supra, p. 11). From this, it

follows in fact that if an object possesses a property, then it is true that the object possesses it. At

this point, the way is open to an affirmation according to which the only principle which cannot be

proved on the basis of other principles but is true “through itself”, is the proposition which gives the

definition of a true sentence.



18

“The definition of a true sentence is true because each definition is true;71 and it is true

through itself because its truth is not based on the truth of another sentence but on its

own truth”.72

That there is no other ultimate principle is proved by the fact that all the other definitions are based

on that of a true sentence and that universal sentences cannot be ultimate principles, since they are

fundamentally hypothetical sentences (see n. 66) which need a proof conducted either on the basis of

definitions or of experience.

“Every other a priori basic law, even the principle of contradiction, must be derived

from previously demonstrated principles, if it is to count as true”.73

In fact, as the principle of contradiction is a universal sentence, and thus a hypothetical one which

asserts that “if something is an object, then it cannot possess and not possess the same property at

the same time”, and since the truth of this relation is not found in the principle itself, it therefore needs

to be proved.74

Not only is the principle of contradiction not an ultimate principle but it is not supreme in the

sense of being the necessary presupposition for each proof. In fact, many principles and theorems

are independent of it; that is to say that they would be true even if this principle were not valid

anymore. These are, according to Aristotle, the principle of the syllogism (and indeed the dictum de

omni et nullo) and, according to symbolic logic, beside the principle of the syllogism, the principle ofidentity, the principles of simplification and those of composition, the principle of distribution, the

laws of commutation, tautology and absorption, and many others.75 What has just been said is

particularly important for the results that£ukasiewicz draws from it.

III



19

Let's consider the Aristotelian statement according to which the syllogism never assumes the

principle of contradiction among its premisses except for the case in which it appears in the

conclusion.76 This is particularly put in light and discussed by £ukasiewicz, since it is from this

statement that he infers the independence of the principle of the syllogism from that of contradiction.

Actually, Aristotle himself, in the continuation of the passage from An. post . A 11, supplies us with a

proof of the thesis in question:

“Then it is proved by assuming that it is true to say the first term of the middle term and

not true to deny it. It makes no difference if you assume that the middle term is and is

not; and the same holds of the third term too. For if you are given something of which it

is true to say that it is a man, even if not being a man is also true of it, then provided only

that it is true to say that a man is an animal and not not an animal, it will be true to say

that Callias, even if not Callias, is nevertheless an animal and not not an animal. The

explanation is that the first term is said not only of the middle term but also of something

else, because it holds of several cases; so that even if the middle term both is it and is

not it, that makes no difference with regard to the conclusion”.77

What Aristotle gives as an example is a syllogism, the essential condition of which, in order to be

valid, is that the major premiss is true, that is, that the middle term is included in the extension of the

major term. At this point, it is not important, the Stagirite says, if both the middle and the minor term

“is and is not”.

Having indicated with A the major term (animal), with B the middle term (man) and with Cthe minor term (Callias), £ukasiewicz reforms the Aristotelian argumentation starting from the

following syllogism:

B is A The man is an animal

C is B Callias is a man

C is A Callias is an animal.



21

Here a problem arises of which £ukasiewicz himself is aware. While the negative term, for

ex., not - B (not-man) is not necessarily restricted to A (animal) but may extend also to not - A, the

contradiction of the minor premiss (or of the minor term) must not but can have an influence on the

conclusion. In fact, £ukasiewicz points out in regard to the syllogism (a) that, even though A has a

bigger extension than B, as to include even some not - B, it does not however include all of them, and

that, if a not - B that belongs to C would not fall in A's extension, the syllogism would not be valid.

The same holds for the syllogism (b). Consequently, both syllogisms are possible but not necessary.

£ukasiewicz does not tackle this problem any further, he nevertheless maintains that the imprecise

words of Aristotle are responsible, and finally limits himself to saying that the fact remains that it is

possible to conduct a syllogistic inference on the basis of the principle of the syllogism, leaving out of

consideration the validity or not of the principle of contradiction.81

A different reading of the Aristotelian passage in question seems to solve the problem. Some

years before £ukasiewicz, H. Maier and I. Husik had already laid stress on the same passage of the

Posterior Analytics. Here I will take into consideration only the paper of Husik and only as far as it

is useful for our matter. According to Husik, the negative term of a pair of opposite terms ( B —not-

B) does not include everything in the universe except B, but is restricted to its region; for ex., not-

man (not- B) does not include everything except man, but only all animals ( A) with the exception of

man ( B). Furthermore, “the inference of the conclusion from the premisses is based simply on the

right to repeat separately a judgment regarding an object or group of objects, which was made

before regarding the same plus others”.82 In this process the principle of contradiction is not at all

involved. The syllogism

All B is A

All C is B

∴ All C is A

would still remain, even if the principle of contradiction would no longer hold. In fact, since the

conclusion “does nothing more than repeat part of the major premiss”, “All C is A” would exclude

“All C is not - A”—as in the syllogisms (a) and (b)—only if the major premiss does so, that is, only if



22

the major premiss asserts “All B is A and not not-A”. Here it is evident that the principle of

contradiction is not taken into account. The conclusion would still hold even if the minor premiss

would assert “All C is B and is not- B”, because the major premiss does not exclude not- B from

being A and, according to Husik's concept of the negative term, not- B is limited to the region of A.

In regard to the passage of the Posterior Analytics, in which the contradiction concerns

either the minor premiss (in a) or the minor term (in b), Husik explains that

“the exclusion of not-animal in the major premiss is responsible for its exclusion in the

conclusion, even if the principle of contradiction should not hold in the minor premiss,

and in the minor term; i.e., even if it were true that Callias is man and not-man (e„ ka

m¾¥nqrwpon ¢l hqšj ), and that he is Callias and not-Callias (Kal l …an[,] e„ ka m¾

Kal l …an), still as long as man is animal and not not-animal, it would follow that Callias

is animal and not not-animal. The reason for this is, he [Aristotle] goes on to say, that

the major term is more extensive than the middle, and applies to not-man as well as to

man, and the middle term is more extensive than the minor and applies to not-Callias as

well as to Callias; and therefore even if Callias is both man and not-man (e„ tÕmšson

ka aÙtÒ™sti ka m¾aÙtÒ), this does not prevent the major term animal (and not

not-animal) from applying to it. Similarly even if the minor term is both Callias and not-

Callias, the major term still applies to it through the middle”.83

In short, in the given syllogisms the conclusion is either necessary or possible according to the

extension of the negative term. Husik's interpretation seems to be more proximate to the Aristotelian passage than does £ukasiewicz's, who does not understand the limitative condition concerning in

this case the negative term under which the syllogism becomes necessary. On the other hand, it is

questionable if such a restricted meaning of a negative term may be assumed as the very Aristotelian

meaning. Here it is not possible to give a precise and complete analysis of this question which would

require a more careful examination of the concept of negation. So we turn again to £ukasiewicz.

Later on, £ukasiewicz will completely change his opinion about the syllogism and its

independence from the principle of contradiction. He will not only claim that the dictum de omni et



23

nullo is neither a principle of the syllogistic nor an Aristotelian principle,84 but he will also deny that it

is possible to conduct any inference prescinding from the ‘metalogical’ principle of contradiction (see

infra, p. 29).85 These are but later achievements of his research. Meanwhile in 1910, to add a new

argument against the absolute unavoidability of the principle of contradiction and so to corroborate

his own thesis (not the total refusal of the principle but rather the independence of some forms of

reasoning from it), £ukasiewicz attempts to construct some inferences in a logical context in which

the principle of contradiction is insignificant, a logical context therefore called by him “non-

Aristotelian”.86

According to £ukasiewicz, even though indications that follow a non-Aristotelian logic are

already present in Aristotle's Metaphysics and in his logical works, nobody has paid attention to

them so far.87 On the contrary and nearly at the same time, a similar operation to and independent

from£ukasiewicz's was attempted by the Russian Nikolaj Aleksandrovich Vasil'év.88 This presents a

non-Aristotelian logic on the basis of the following hypotheses: an imaginary world in which the

negations, like the positive facts, are the objects of sensation, an interpretation of particular

propositions in terms of modality, and the substantial independence of the proposition and of the

syllogism structure from the principle of contradiction. It seems that an impulse for the working-out of

the “imaginary (non-Aristotelian) logic” came to Vasil'év from his encounter with some of Charles

Sanders Peirce's logical ideas; beside the reading of “The Logic of Relatives”89 which he read when

he was just seventeen, it came from an article and a short communication by Paul Carus,90 appearing

in The Monist in 1910, in which there were long quotations from Peirce's letters on his studies

concerning a non-Aristotelian logic.91 In a letter to Francis C. Russell—quoted by Carus—Peirce

asserted to have worked for a long time, before applying himself to the study of the logic of relatives,on a non-Aristotelian logic, “supposing the laws of logic to be different from what they are”. Even

though some developments were interesting, Peirce did not achieve the satisfactory results which

could have induced him to publish them.92 And in another letter, sent to The Monist as an additional

explanation to the extract of the letter to Russell, Peirce asserted that, although the continuation of his

researches in that direction would have helped him to discern features of logic that had been

overlooked, he nevertheless had decided not to pursue that line of thought.93 Unfortunately, Peirce

does not say a lot on what he meant by non-Aristotelian logic, except that it is “in the sense in which



24

we speak of non-Euclidean geometry”. Considering the example that Peirce brings forward of the

kind of “false hypotheses” analysed by him up to their consequences, he had probably tried to

modify the principle of transitivity.94 This is not the place to compare the results, or rather, Peirce's

researches with £ukasiewicz's and Vasil'év's. A common element to the three authors, though, can

be identified: the reference to non-Euclidean geometries. In particular, £ukasiewicz's and Vasil'év's

works—the latter is Nikolaj Ivanovic Lobaèevskij's fellow citizen and was also born in Kazan—

reveal a true enthusiasm for the discovery of the non-Euclidean geometries in the first half of the 19th

century. These geometries provoke a remarkable heuristic impulse and constitute the model in

relation to which they try to realise the same operation in logic.95 It is clear that the principle of

contradiction is considered as the analogon in logic to Euclid's fifth postulate; and as a geometry

without the parallel line postulate is called non-Euclidean, so a logic without the principle of

contradiction will be a “non-Aristotelian logic”.96

In the attempt to prove the possibility of building a non-Aristotelian logic, £ukasiewicz starts

from the fiction of another psychical organisation peculiar to other kinds of human beings, according

to which all negations are true. Let's imagine, £ukasiewicz says, a society of beings who live in a

world totally similar to ours, and with a psychical organisation similar to ours from which it differs,

however, for one fundamental reason: it recognizes every negative sentence as true. For example,

considering that “light of the sun”, “mortality of man” and the concepts “two”, “four”,

“multiplication” and “equality” have the same meanings both for the members of that society and for

us, they, unlike us, recognize negative sentences of the kind “the sun does not shine”, “man does not

die”, “two times two does not make four” as always true. We could ask: does it make sense to

reflect on such an absurd hypothesis? But if this is absurd in other fields, it is not so in logic.According to £ukasiewicz, a similar operation, the exclusion of certain laws valid in the ambit of the

phenomena and the enquiry on what happens when prescinding from them, leads us to understand

more clearly in which measure the laws which have been excluded influence the course of events.97

To better illustrate the fiction, that is, the way of thinking of these other human beings, and to

explain how a negation can always be true, £ukasiewicz explains that both the sun and man have

many other properties besides those, respectively, of shining or of dying. Now, these are properties

which do not necessarily include the property of dying for man, or shining for the sun, so that each



25

time we predicate a property of the sun or of man different from those mentioned (for example, that

“man lies in bed” or that “the sun rotates on its own axis”), it is also true that “the sun does not

shine” and that “man does not die”—although we know that the sun shines and that men die. In fact,

that man does die is true only when necrotic processes take place in his tissues, but not because man

lies in bed or because he presses on the sheets with the weight of his body; in this sense the negative

sentence that man does not die is true even if man dies. In other words: assuming that to the subject

A (the man) can be predicated B (dies), C (lies in bed), D, etc., to which correspond the facts b (the

beginning of the necrotic processes), c (lies on a surface), d , etc., the sentence “ A is B” is true only in

relation to the fact b, which is independent of and does not involve the other facts c, d , etc. In the

same way, if we analyse “ A is C ”, we do not find anything that will tell us of B, so it is possible to

affirm “ A is not B” just in virtue of the preceding sentence, even though b happens. It is just like

saying that, if we predicate B of A, all the other possible predicates of A can be truly denied, since

they are neither asserted nor are they included in B. Here it is implied that both the facts, including

those referring to the same subject, and the different predicates of the same subject are independent

of one another. £ukasiewicz precautionally asserts that these arguments are not intended to affect the

principle of contradiction but are meant to illustrate the fiction.

Now, since it is evident that for the beings in question each negative sentence is true, as a

consequence they are not at all worried about the negation which becomes something analogous to

zero in the addition or to the unity in the multiplication. On the contrary, probably in their language a

single expression exists to indicate all the possible negations. And certainly they do not recognize the

principle of contradiction, the assumption of which is for them as much inconceivable as its refusal is

for us. Everything that exists for them is contradictory, exactly because negations are always true. Ihave to specify that here £ukasiewicz is not talking about a world with contradictory objects for

which propositions like “ A possesses B and A does not possess B” are valid—which he will also

take into consideration (see infra, pp. 34f.)—but about a way of understanding the object by the

human beings mentioned. In their view, negations being always true, it is valid to say that for each

object which refers to b and c, or d , etc., “ A possesses B and A does not possess B”. It is not valid,

instead, for those objects, that are nonexistent or improbable, of which it is impossible to assert

anything positive. At this point, the issue is whether or not beings like those described are also



26

capable of thinking in a rational way. And for this purpose £ukasiewicz gives an example which

shows that it is possible for those beings to take note of the events of experience, to infer both in an

inductive and deductive way, and to act efficaciously on the basis of syllogistic inferences,

prescinding from the principle of contradiction. The example is structured in four phases.

(a) A doctor called by a patient afflicted with a bad sore throat diagnoses a high fever,

white-grey plaques on his tonsil's membrane, marked reddening of the adjacent membranes, swelling

of the jugular gland, in short all the symptoms of an advanced diphtheria. He also knows though that

the temperature is not high, that the patient's throat is not reddened, that the jugular gland is not

swollen, etc.; but since negations are always true, he does not pay attention to them and he takes

note only of what is and not of what is not. He verifies and asserts the above-mentioned facts

exclusively on the basis of sensible experience and without making any use of the principle of

contradiction. (b) The doctor cures the patient with a serum used in other cases, of which he is

convinced that, if taken in time, removes the disease. This therapeutical concept is the result of

previous experiences which the doctor has synthesized in the formula “all the cases with similar

symptoms up to now treated with the serum have been successful”. He also knows that the serum

does not heal since the patients have not only recovered, but lay in bed, talked to other people, and

were surrounded with attention; and just because these other things were done one does not

recover. But since this fact is evident, the doctor does not pay attention to it, instead he considers the

fact that the serum was efficacious in the previous cases. Also to bring back the set of the single

cases “ A1 is B”, “ A2 is B”, ..., “ A10 is B” to the formula “all the ten As are Bs”, he does not use the

principle of contradiction. (c) But how to explain the uniformity of all the preceding cases? The

doctor gave the explanation of the regularity of the phenomena assuming the universal sentence “all As, and not only the preceding ten considered, are Bs” as a general rule. Here we could remark that

regularity presupposes consistency, but £ukasiewicz is not contesting that reality is not contradictory,

on the contrary—as we will see in a short while—he is convinced of that. Indeed, he is showing that

it is possible to argue prescinding from the principle of contradiction. Let's return to our doctor.

Even previously he knew that a medicine does not always cure, that is, it does not cure because it is

expensive or because it was bought at a chemist's but only because it comes into contact with the

organism. Nevertheless, he did not consider the negative cases, but only worried about explaining the



27

positive cases and therefore was searching for the general rule from which to deduce the particular

sentences. For this purpose he inductively inferred the universal sentence “every A is B” from the

particular cases previously verified. The induction, in fact, consists in starting from certain sentences

(particular or singular) and reaching a universal sentence, from which the starting sentences can be

derived; and without doubt from “every A is B” follow the sentences “ A1 is B”, “ A2 is B”, ..., “ A10 is

B”, and so on. Once again, to infer inductively from this to that, the doctor did not use the principle

of contradiction. (d ) In observance of the rule previously established with his experience, the doctor

deduces that the patient cured with the serum will also recover in the present case. He also knows

that the patient will not at the same time become healthy: one is healthy in virtue of being healthy, not

because he/she was born shortly before and will soon die. Since however all the negative facts are

obvious, the doctor does not pay attention to them and builds a syllogism in virtue of which, from

“every A is B” (“every patient treated with the serum did recover”) and “C is A” (“this patient is

being treated with the serum”), he infers “C is B” (“this patient will recover”). Since he has to deal

only with positive sentences, even in this last case our doctor does not use the principle of

contradiction. Finally, he gives the serum to the patient and— £ukasiewicz ends the story—his hopes

are not disappointed.

The whole example wants to be an application to daily life of what £ukasiewicz asserted was

already present in Aristotle, i.e., the independence of the syllogism from the principle of

contradiction. It seems indeed that £ukasiewicz simplifies the Aristotelian argumentation: retaining all

the negative sentences as true is equivalent to their elimination. In fact, the doctor never takes them

into consideration. In summary, £ukasiewicz seems to reason in this way: since the principle of

syllogism is independent of the principle of contradiction and the latter implies the negation, it issufficient to build syllogisms in which negative sentences do not appear, in order to prove that it is

possible to reason and infer even without the principle of contradiction. It is on the basis of this

simple idea—held by Vasil'év as well98 —that £ukasiewicz builds his example, reaching the

conclusion that, if the mental organisation of these fictitious beings did not differ in anything else

except in the above-mentioned characteristic, then they would develop a chemistry, a physics and

even a logic like ours which however does not take into consideration the principle of contradiction.



28

It is questionable if we can consider the idea presented herein of a non-Aristotelian logic as a

first step by £ukasiewicz towards the construction of a three-valued logic. Of course, the main

purpose—as £ukasiewicz recognized himself (see supra, p. 2)—was at this time not achieved;

however, the partial resulting achievement is not trivial, even though it was asserted with a certain

emphasis, an emphasis that we find again in £ukasiewicz even in those passages where he expresses

himself on the value of a three-valued logic.99 £ukasiewicz's subject is not directed against the

principle of contradiction tout court , but rather against its presumed absoluteness and unavoidability;

in this sense he intended to show that inductions and deductions are possible even though they

prescind from the principle of contradiction, as they consist only of positive sentences.100

£ukasiewicz did not follow up on the development of a logic without the negation nor a non-

Aristotelian logic without the principle of contradiction. Later on, he will abandon definitively such

hypotheses and will turn his criticism against the metalogical principle of bivalence. Already in 1913

£ukasiewicz claims that “no proposition can be both true and false” but that there is a certain group

of propositions, i.e., the indefinite propositions, “which are neither true nor false”.101 But he did

not abandon the idea of a non-Aristotelian logic entirely. The pursuit of such an idea lead him in

1917102 to the construction of the first system of a three-valued logic which is characterised as a logic

for which the principle of bivalence is not valid.103 Moreover, he will state that the metalogical

principle of contradiction, “the principle of consistency”, must be assumed absolutely in order to

have a logic.104 Thus, he moved away from the perspective which he had outlined in 1910. His early

idea of a non-Aristotelian logic has been followed and was first realized by Stanis³aw Ja¶kowski.

Linking up to £ukasiewicz's book and following some of its suggestions, Ja¶kowski constructed in

1948 the first propositional calculus for contradictory deductive systems which is recognized as thefirst system of paraconsistent propositional calculus.105

In the preceding pages we have mentioned Vasil'év. A comparison between the two authors

would go beyond the limits and the aims of this work; nevertheless, we can point out that their

operations are not identical, because they differ not only in the results, but also in their setting out.

£ukasiewicz, by refusing all psychologistic interpolations in logic (see supra, pp. 12ff.) and claiming

that in reality there are no effective contradictions (see infra, pp. 32 and 38), sets out from the

hypothesis of another psychical organization, typical of other human beings for whom all negations



29

are true and for whom the principle of contradiction is not a part of reasoning. 106 Vasil'év, on the

other hand, by assuming Sigwart's point of view—rejected by £ukasiewicz—and claiming therefore

a subjective consistency, typical of correct reasoning, supposes another world, structured like ours,

with the only difference that negations, like positive facts, are the objects of sensation and perception:

a world in which there are contradictory objects. From this brief confrontation it emerges that

another element of distinction between the two authors consists in the different meaning assigned to

negation, which itself plays a fundamental part in the way the principle is intended.

IV

As was said previously, £ukasiewicz has affirmed that the principle of contradiction must be

proved, if it is to be considered as true. In Met . G 4 Aristotle had already enquired into that direction

supplying not an apodeictic proof, but a proof by refutation of the principle of contradiction,

intending to prove the untenability of the theses of those who deny the principle as well as the

absurdity of their consequences due to their self annulment. In particular, Aristotle's proofs are

carried out presupposing certain systems of propositions or doctrines (the existence of a substratum

to which the accidents refer, the distinction between substance and accident) as true, in relation to

which the opponent's thesis is confuted.107 Another kind of foundation consists, within the concept

which identifies thinkability with logical validity, in considering the principle of contradiction as an

element deeply rooted in thought, that is, as a fundamental law and an unavoidable condition for the

very possibility of thought and logic. Such a conception, rather than being a foundation or justification

of the principle, seems to be a way to avoid the problem. If however this path is followed by thosewho believe that logic is the science of the necessary laws of thought—among which are Kant and

William Hamilton, as well as Sigwart108 —the solution turns out to be very different when proposed

by those who conceive of logic as a science which has to do also with reality. Enunciating and

assuming the principle of contradiction as true, two paths are available: either we show ( justify) that

it is not demonstrable—but then we also must show that the principle of contradiction itself is the

very foundation searched for—or we prove it. The first way is the one suggested by Aristotle, the

second is the one run across by other strikingly different authors such as Ueberweg, Pfänder,



30

£ukasiewicz and Le¶niewski. According to these authors, in fact, the Aristotelian solution (that is, the

assumed indemonstrability of the principle of contradiction and its indirect justification) cannot be

considered satisfactory; on the contrary, even for the declared purpose of explaining the many

controversies which occurred as to the principle in question, it becomes necessary to attempt to

supply a proof .109

Friedrich Ueberweg, pupil of Trendelenburg, and Alexander Pfänder, pupil of Husserl, are

respectively placed in the second half of the 19th century and in the first half of the 20th century.

Their attempts—such are and remain—to supply a proof of the principle of contradiction110 are

important not as much for the effective results, but as for the implications that the pursuit of their aim

involves. First of all, they operate an analysis (disassembling) of the principle of contradiction in its

constitutive elements: as long as it is proved that it presupposes other notions like truth, judgement

and negation, in their turn not evident at all—the differences of opinion among different authors

confirm it—the principle can be considered neither simple nor primary. In the second place,

Ueberweg and Pfänder set as necessary the proof of the principle of contradiction. At last, they

emphasise—Pfänder in particular—the role of the ontological element both in the formulation of the

principle of contradiction and in the attempts to give it a proof: more precisely, the principle is based

on the specific notion of object, intended as existent and noncontradictory. The proof of the principle

of contradiction is demanded by both authors not because the value of the principle would be

questioned in absence of a proof, but to confirm in a definitive way its absolute validity with no

exceptions.111 The question is whether setting out from such results and further developing them, it is

possible to establish the absolute validity of the principle or whether some limitations arise where the

principle is valid only in their range. The first point has already been discussed (see supra, pp. 16ff.);now the remaining two are to be considered. Special importance is given to the last point, since it

involves a deeper probing of the notion of existence. What does exist? At which conditions? Is it also

possible to talk about what does not exist, or rather does not exist in this world, but in other possible

worlds? Is it possible to refer to other spheres of human rationality, in which the principle of

contradiction is not valid? And also, is it possible to individuate defined classes of objects which are

not subjected to it?



31

It has already been pointed out that, according to £ukasiewicz, the only principle which

cannot be proved on the basis of other principles but is true “through itself”, is the statement which

gives the definition of a true sentence; that any other a priori principle, to be counted as true, is to be

derived from principles already proved; and that therefore even the principle of contradiction needs a

proof (see supra, p. 19). However, £ukasiewicz notes, nobody seriously puts in question the

principle of contradiction which is fruitfully used both in life and in science.112 It is then a matter of

proving where its certainty comes from. For this purpose, first of all, (a) it must be shown which

proofs of the principle (already attempted or which can be hypothesised) are not valid; then, (b) the

principle has to be proved; at last, (c) it is necessary to critically reflect on the proof given, on its

validity or not, and—as it will result in short—on the reasons of its weakness.

(a) In order to result in absolute validity, a proof of the principle of contradiction has to be

conducted respecting particular conditions. This absolute validity, with no exceptions allowed,

requires a precise delimitation of the principle's sphere of application and a rigorous definition of the

conditions under which it is valid. Let's see what these conditions are in negative.

(i) The principle of contradiction cannot be proved by means of its evidence. In the first

place, the criterion of evidence is not a valid one: if ‘evident’ means something different from ‘true’,

then it means a mental state, hence the truth of a proposition cannot ever follow. In many cases, in

fact, truths which have been considered evident were not such. The use of the concept of evidence is

nothing else but a vestige of psychologism from which the step to subjectivism and to scepticism is

short. If somebody considers a proposition evident, then this is true for him; but if for another the

same proposition is not evident, then the same proposition is true for one but it is not for the other. In

the second place, if a principle is not evident for everybody, then it is not evident; in order toinvalidate the proof it is sufficient to give one case: £ukasiewicz himself or any other author like

Hegel (see supra, p. 15).

(ii) The principle of contradiction cannot be proved by founding it on a presumed necessity

based on the psychical organisation of man—in the final analysis, on his physical nature—because

man can also make false statements, and because it is not demonstrated that the principle of

contradiction is a psychical law (see supra, pp. 13ff.), or that such a necessity is real. Moreover,



32

even if it were a psychical law, the principle would not be demonstrated since nothing can assure us

that the external world really corresponds to the necessities of our intellect.113

(iii) A proof of the principle of contradiction has to be based on objective arguments, so that

from the proof given follows the truth of the object itself, not the truth of the assertion according to

which the principle is to be accepted (an implicit reference to the proof through refutation supplied by

Aristotle). Since the principle of contradiction is considered a sentence a priori such a proof can be

given, £ukasiewicz asserts, not on the basis of experience but according to definitions.114 And the

definitions (or the definition) from which the proof can originate, can be neither that of a false

sentence nor that of negation—as Sigwart tried to do—but only that of object. Let's see why the

principle of contradiction cannot be proved by means of the definition of a false sentence nor by that

of negation.

According to Sigwart, the principle of contradiction “expresses the nature and meaning of the

negation”.115 Negation corresponds to nothing real and is the refusal of a positive sentence, that is, an

act which the subject does against an attempt of assertion.116 In this way, negation is taken back to

the notion of a false sentence. Hence, we can immediately pass to the consideration of the latter.

Given the following definitions:

“ An affirmative sentence is false, if it ascribes to an object a property which it

does not possess; a negative sentence is false, if it denies to an object a property

which it possesses (a). Likewise, in an inverted manner: no object possesses a

property which a false sentence ascribes to it; and each object possesses a

property which a false sentence denies to it (b)”;117

it may seem that from the union of (a) and (b) with the definitions of a true sentence given above (see

supra, p. 11) it is possible to deduce that, if an affirmative sentence is true (or false), then the

corresponding negative sentence must be false (or true). Namely on the basis of the fact that two

contradictory sentences cannot both be true, one may think that the principle of contradiction has

been deduced.118 But this is not true, not only because the definitions of truth and falsity do not

contain the notion of logical multiplication, which is on the other hand constitutive of the principle of



33

contradiction (see supra, p. 17), but above all because the impossibility for two contradictory

sentences to both be true is based on the notion of object. In fact if we consider the given definitions

of truth and falsity, it can happen that, if we had to do with a contradictory object such as “O

possesses p and O does not possess p”, if “O possesses p” is true, the corresponding negation “O

does not possess p” would be equally true. The sentence “O possesses p” then is false only under

two conditions: that O is free from contradictions and that O does not possess p —which implies that

O either possesses or does not possess p.119 In short, £ukasiewicz asserts:

“Every proof of the principle of contradiction must take into account the fact that there

are contradictory objects (e.g., the greatest prime number). In the most general

formulation: “the same characteristic cannot belong and not belong to an object at the

same time” is in terms of the principle of contradiction most certainly false”.120

Classic examples of contradictory objects are those of ‘wooden iron’, ‘square circle’ or

‘round square’. The question which is set here is whether similar expressions represent names which

mean something, or if they are—as many believe—simple, empty, and meaningless sounds. Again

proposing an argumentation by Bolzano, Twardowski and Meinong,121 who also confront themselves

with the same problem, £ukasiewicz asserts that expressions of the type ‘round square’ have to be

distinguished from others such as ‘abracadabra’ or ‘mohatra’; the former words mean something—

of the round square we can say that it is round, that it is square and that it is a contradictory object—

while of the others it is not possible to assert anything since the word ‘abracadabra’ actually has no

meaning. If similar examples ad hoc do not suffice, then, £ukasiewicz says, we can take others fromgeometry: the construction of “a square built with the help of a line and a compass, the surface of

which is identical to that of a circle with a radius equal to 1” had engaged many minds for centuries,

until Charles Hermite and Ferdinand Lindemann in the 19th century proved that a square of that kind

is a contradictory object.122 In short, words can have meanings although they indicate something

which does not exist and which is even contradictory.123 £ukasiewicz asserts that for these objects

the principle of double negation is valid but not that of contradiction. This further confirms what has

been previously said with regard to the difference between the principle of contradiction and that of



34

double negation: that the former can be without the concept of logical multiplication while the latter is

in need of it. It is clear that the whole argumentation is valid only on the condition that we also

recognize the status of object in the contradictory objects; which is what Meinong does, and

£ukasiewicz gives him credit for being the first at expressing such an opinion.124 This anticipates the

results to which the continuation of the argumentation will lead. That's how we reach £ukasiewicz's

proof which is based on the assumption that objects are noncontradictory.

(b) If we accept contradictory objects as well, then we would have cases in which the

principle of contradiction is not valid, and this would constitute a limitation of its extension.

Consequently, the above-given definition of object as (a) “everything that is something and is not

nothing” (see supra, p. 8) is not sufficient in order to supply a proof of the principle of contradiction.

This requires an additional definition of object, intending it as (b) “everything that does not contain

contradiction”. The only possible formal proof of the principle is then the following one: if we

suppose from the beginning that an object is something that cannot at the same time have and

not have the same property, which is valid as the definition of object, it follows from this

assumption, by virtue of the principle of identity, that no object can possess and not possess the

same property at the same time.125

Here the necessity is asserted to move from the definition of (noncontradictory) object, to

justify in some way, that the principle of contradiction is right. Pfänder—as it has been said (see

supra, p. 31)—arrives at a similar result as well;126 but £ukasiewicz's text is not a repetition, not only

because it precedes Pfänder's publication by a decade, but because the two authors have different

aims. We already had a chance to say that Pfänder—as well as Ueberweg—intends to end a long

and old controversy with his proof, establishing in a definitive way the absolute validity of the principle of contradiction, with no exceptions. Briefly, Pfänder is inclined to end the question and to

do it he uses the instruments of traditional formal logic, enriched by Husserl's phenomenology.

£ukasiewicz's aim is quite different: he wants to reopen the matter on a problem which, taking into

account the means which traditional logic has and notwithstanding the many controversies and

discussions on the subject, appears to him (and it could not be otherwise) to be closed. £ukasiewicz

intends to reopen the matter in the light of the researches of the new logic, preluding even possible

developments—in the sense of a non-Aristotelian logic—which go beyond the classical symbolic



35

logic as it was elaborated by Frege, Peano and Russell, on the basis of which he tries to understand

again the meaning, the value and the limits of the principle of contradiction. £ukasiewicz himself, in

fact, does not believe much in his proof, defined by him as “too easy, economical and

superficial!”;127 however he does consider it as a starting point for further researches. In fact, two

years later will appear Le¶niewski's essay, who, however critical towards £ukasiewicz's work,

recognizes the large debt he owes to it and the many stimuli that he has received from it.128 Unlike

other authors, £ukasiewicz has an open attitude toward further possible courses of research. That is

attested by the chapters which follow the one just examined: he does not try to defend his own

proof, but he screens it through his own criticism.

(c) In order to demonstrate the principle of contradiction effectively, £ukasiewicz points out,

it is necessary to supply not only a formal proof but also a real one which shows the

correspondence between the two given definitions of object. At this point, it is a matter of seeing

whether what is an object in the first sense (“everything which is something and is not nothing”, that

is, things, people, phenomena, events, relations, thoughts, feelings, theories, etc.) is such in the

second sense as well and then is noncontradictory. For this purpose, £ukasiewicz says, it is not

necessary to analyse all the single objects, but it is sufficient to consider some large groups and,

among these, those which have greatest importance for research on the principle of contradiction.

The first problem, therefore, concerns the classification of objects. Here £ukasiewicz makes a

distinction—taken from Meinong, of whom he quotes not a specific work, but the lectures of the

winter semester 1908/1909129 —between two large groups of objects: the complete, about which it

is possible to formulate propositions which are either true or false, and the incomplete, that is to say

objects not sufficiently defined in all their aspects, about which we can formulate propositions but forwhich it is not possible to tell if they are true or false. To give an example: if I talk about a triangle, it

is determined in relation to its essential qualities (for ex., in relation to the fact that it has three sides),

but not in relation to its accidental qualities (for ex., in relation to the equilaterality or non-

equilaterality). In this case, the proposition “the triangle has three sides“ is submitted to the principle

of the excluded middle, while the proposition “the triangle is equilateral” is not. In the same way, if I

talk about the Caryatids in Athens, I can say that they are lady shaped, marble columns, etc.; for

each of these, as for other sentences on the Caryatids, it is always possible to decide if they are true



36

or false. If instead I talk about ‘the column in itself’, and I say that “the column is made of bronze”,

this sentence is not necessarily true or false because the subject is not enough defined and because in

reality there are columns of bronze and others which are not. The principle of the excluded middle

requires that, in order to be valid, the objects are clearly defined.130 In other words, complete are

those real objects provided with a spatial-temporal existence while incomplete are those abstract

objects which do not exist in reality and are products of the human mind.131

The latter can be divided, in their turn, into two other classes, (i) “constructive objects

[ przedmioty konstrukcyjne]”, that is to say the objects of the concepts a priori, which belong

mainly to mathematics and logic and are independent of experience, and (ii) “reconstructive objects

[ przedmioty rekonstrukcyjne]”, that is to say the objects of the empirical concepts which refer to

experience and count as instruments to understand real objects (man, plant, crystal, etc.).132

(i) Constructive objects are “free creations of the human mind”. Even though they depend

upon the respect we have of the principle of contradiction and are constructed in a noncontradictory

way, some of them have appeared nonetheless to be contradictory: the squaring of the circle, the

trisection of any given angle, the highest prime number. Therefore, they have been excluded from

science, but this does not prevent the fact that we can have others which today are considered to be

noncontradictory.133 As an example £ukasiewicz mentions Russell's antinomy which touches on the

logical foundations of mathematics.134 The fact that constructive objects, the contradictoriness of

which is not obvious at the present, can exist implies the acceptation of a mediate contradiction, that

is, the acceptation of a contradiction which, while it may not be evident at the moment, may turn out

as such with time and after an accurate study.

(ii) Also with regard to reconstructive objects it is a question of verifying whether they are soeven in the sense of the second definition of object, that is, if they are free from contradictions, or

not. On the basis of the assumption of the realist point of view, according to which “being and true

sentences correspond reciprocally” (see supra, p. 11), it is clear that a contradiction inside a

reconstructive object corresponds to a real contradiction; therefore, £ukasiewicz concludes, rather

than constructions of mind, it is better to take into consideration real objects. With regard to these,

he believes that they do not contain any contradiction.



37

“ In fact there is known to us no single case of a contradiction existing in reality.

Indeed it is generally impossible to suppose that we might meet a contradiction in

perception; the negation which inheres in contradictions is not at all perceptible.

Actually existing contradictions could only be inferred ”.135

However, if we take into consideration the continuous change, in which regard the existence of

contradictions has always been hypothesised, though it is improbable that such hypothesis will find a

verification, it is not conclusively assured that real objects cannot contain contradictions.136

At this point it is clear that on the basis of the results of (c.i) and (c.ii) a real proof of the

principle of contradiction which assures the perfect correspondence of what is real and possible with

what is noncontradictory cannot be given. In the same way, as we cannot assert with certainty that all

constructive objects are noncontradictory, so we are not assured that all real objects are

noncontradictory as well. On the other hand, by the joined conclusions of (b) and (c) the result is

that, even though the principle of contradiction needs a proof, it nevertheless remains difficult to give

one, and the formal proof supplied by £ukasiewicz has been considered weak by the author himself.

The attitude of the young Polish philosopher is distinguished from that of the opponent of the

principle of contradiction hypothesised by Aristotle in Met . G 4, since £ukasiewicz does not intend

to deny the validity of the principle in all the cases. On the one hand, he intends to warn about the

uncritical assumption of the principle of contradiction as first principle and, on the other hand, to

question its absolute validity for each and every case. He believes in fact that the reason why the

principle of contradiction for centuries has been considered self-evident, supreme, absolute, and

indemonstrable does not consist so much in its logical value as in its practical-ethical value: for inconsequence of the intellectual and moral imperfection of man it constitutes the only weapon that man

has against error and falsehood.137

V

In the appendix “The Principle of Contradiction and Symbolic Logic”, £ukasiewicz tries to

give, together with some outlines of mathematical logic, an essay on the principle in light of the new



38

science, aware that it could not be exhaustive as symbolic logic is in continuous evolution and is

subject to unforeseeable developments. The theses expressed here partly corroborate and partly

integrate the arguments made in the main text. At first £ukasiewicz enounces seven principles (the

principle of identity, that of the syllogism, the principles of simplification, the principles of

composition, and the principle of distribution) and eighteen theorems (the laws of commutation, of

tautology, of absorption, and others), with their relative proofs; both the principles and the theorems

are considered independent of the principle of contradiction. Consequently, he enounces four other

principles (0 → a, a → 1, the principle of contradiction and that of excluded middle), as well as the

consequential theorems (the laws of double negation, of contraposition, De Morgan's laws, and

many others).138 The important result for the purpose of this work is not the expositive part, 139 but

the distinction, pointed out by £ukasiewicz, between the Aristotelian principle of contradiction and

that of symbolic logic.

In fact, the formal proof of the principle of contradiction given above is conducted on an

ontological basis and—as we shall see in a short while—concerns the Aristotelian ontological

formulation of the principle, which is equivalent with the logical one. Following Couturat's L'Algébre

de la Logique,140 £ukasiewicz puts among the axioms of symbolic logic the principle of

contradiction. Thus, in order to not invalidate his research, he is led to make a difference between the

Aristotelian principle of contradiction and that of symbolic logic, and then to try to derive the former

from the axioms and theorems of symbolic logic.

Leaving out then the expositive part, we turn our attention directly to this distinction, taking

into consideration the principle according to symbolic logic. Its formulation is given by £ukasiewicz in

this form:

(1) (a ∧ ¬a)→ 0

with the meaning: “if O possesses both a and ¬¬a at the same time, then O is not an object”. Joined

to the principle

(2) 0→ a



39

which means: “if O is not an object, then it possesses any quality”, in other words: from the empty

set (or from a contradictory object) follows anything, the proposition (1) can be expressed in form of

an equivalence:

(3) (a ∧ ¬a)↔ 0.141

This principle, as £ukasiewicz asserts, is different from the one expressed in the Aristotelian

formulation, which is not a principle of symbolic logic, but a simple theorem deduced from other

principles and theorems. £ukasiewicz translates the Aristotelian proposition “the same attribute

cannot at the same time belong and not belong to the same subject in the same respect” as “no

object can possess and not possess the same property at the same time”, which is equivalent to the

hypothetical proposition “if O is an object, then O cannot possess a and not possess a at the same

time” (see supra, pp. 8, 17 and n. 66), which can be expressed symbolically as:

(4) 1→ ¬(a ∧ ¬a).

Under which conditions is this proposition demonstrable? Since (4) contains the negation of a

multiplication, if we want to consider well-known principles and proved laws, there are only two

ways we can resort to in order to demonstrate it: either (i) the law of contraposition [(a→ b)↔ (¬

b→ ¬a)], or (ii) De Morgan's formulas [¬(a ∨ b)↔ (¬a ∧ ¬b) and ¬(a ∧ b)↔ (¬a ∨ ¬b)].

(i) From the application of the law of contraposition and the law of double negation we have:

(1→ ¬(a ∧ ¬a))↔ ((a ∧ ¬a)→ 0),

from which it follows, on the basis of the definition of equivalence [(a ↔ b) → (a → b) ∧ (b →

a)],142 that:

((a ∧ ¬a)→ 0)→ (1→ ¬(a ∧ ¬a));



40

(4) is therefore deducible from the principle of contradiction (1) with the help of the law of

contraposition and the equations ¬1 = 0; ¬0 = 1.

(ii) From the application of De Morgan's second law and the law of double negation we

have:

(1→ ¬(a ∧ ¬a))↔ (1→ (a ∨ ¬a)),

from which it follows, on the basis of the same definition of equivalence:

(1→ (a ∨ ¬a))→ (1→ ¬(a ∧ ¬a));

(4) is therefore deducible from the principle of the excluded middle [1 → (a ∨ ¬a)], with the help of

De Morgan's formula. In both ways the Aristotelian principle can be deduced from some of the

principles and theorems of symbolic logic.143

£ukasiewicz's argumentation in support of his own thesis goes on, asking what are the

principles of symbolic logic previously announced by him in the same appendix that constitute the

foundation of the principle of contradiction. Through the analysis of the principles and the theorems

on which the law of contraposition and De Morgan's second law can be based,144 £ukasiewicz

concludes that the Aristotelian principle of contradiction has as its foundation, both in (i) and in (ii),

all the eleven previously enounced principles of symbolic logic (see supra, p. 39),145 which, taken

one by one, express each a thought simpler than the one of the principle which together they justify.146

It may be objected, as £ukasiewicz remarks, that the whole discussion concerns only words;

in fact, if we consider that, in virtue of the law of contraposition, it is not only valid that (4) follows

from (1), but its opposite holds as well, then the two propositions would be equivalent:

(1→ ¬(a ∧ ¬a))↔ ((a ∧ ¬a)→ 0).



41

Moreover, considering the fact that (2) and (1) together also constitute the principle of contradiction

(3) according to symbolic logic, we could assert that (1) and (4) are not only equivalent, but also

synonymous. At this point £ukasiewicz takes on the task of proving, on the contrary, that (4) and (1)

are not synonymous.

The distinction between (4) and (1) which £ukasiewicz translates respectively as (4) “what is

an object cannot possess and not possess the same property at the same time” and (1) “what

possesses and does not possess the same property at the same time cannot be an object” is based

on the definitions of synonymous and equivalent propositions given above at pp. 10-11. Since both

propositions have different subjects and different predicates which mean different things, they are not

synonymous.147

£ukasiewicz does not stop here. Supposing that the above-given definition of synonymity is

too limited and may allow cases of synonymous propositions with different subjects, or with

predicates of not identical meaning, he attempts to translate (1) and (4) into two propositions of the

form “no A is B” and “no B is A”, both of which indicate, on the basis of the law of commutation,

that classes A and B have no elements in common. When applied to our two propositions it turns out

that: (1) “what is at the same time a and ¬a, is not 1 (that is, it is 0)” and (4) “what is 1, is not at the

same time a and ¬a (that is, it is ¬(a ∧ ¬a))”. If these two propositions are synonymous, then the

Aristotelian formulation of the principle of contradiction (4) is not a simple deduced theorem but a

principle. In this case too, £ukasiewicz tries to prove—against appearances—that (1) and (4) are

not synonymous.

To prove synonymity among two propositions it is necessary, in the first place, to compare

them with a third one, but, in our specific case, the third proposition to which the first two could bereferable, in any form is to be intended (“ A and B annul each other reciprocally”, “ A and B have no

element in common”, or “there does not exist an A which is at the same time B”), contains however

the additional concept of logical multiplication, thus it cannot be synonymous with either of the two

propositions. Furthermore, there is no other way to prove synonymity apart from the comparison of

two sentences to a third one.

In the second place, synonymity takes place among propositions the difference of which

consists in signs, not in what they indicate. In other words, the equivalence between two propositions



42

is an unconditional fact. It has been noted though, that in our case in virtue of the law of

contraposition the equivalence is conditional to the acceptance of all eleven principles of symbolic

logic mentioned by £ukasiewicz. Now the refusal of even a single one of those principles—e. g.,

(2)—may result in cases in which the two propositions are no longer equivalent. This occurs just with

regard to contradictory objects. If A is “a number belonging to the series of natural numbers” and B

is “the last number of the series of natural numbers”, since B is a contradictory object, it belongs and

does not belong to the series of natural numbers; therefore, if it is true that no number which is the

last of a series belongs to the series of natural numbers (“no A is B”), it is also true that B in virtue of

the law of simplification, because it is the last number of the series, belongs to the series of natural

numbers. “No B is A” will then be true, and it will also be equivalent to “no A is B”, only if we

assume (2), that is, “from an empty set follows anything”. But then the equivalence in question is an

equivalence conditioned. And such is the equivalence between (4) and (1) as well. As noted, it

depends on the validity of the law of contraposition which presupposes all the principles of symbolic

logic, including (1). Consequently, (4) and (1) cannot be synonymous. By doing so, £ukasiewicz

claims to have proved, in another manner, that the Aristotelian principle of contradiction does not

constitute an ultimate and indemonstrable law since it is a proposition deduced from other

propositions and therefore much more complex. Moreover, it is not a necessary law because—even

in the case of a very close connection among (1), (2), the principle of the excluded middle and (4)—

many other laws (see supra, pp. 19 and 39) would be true even if it were not valid. However, these

laws would still be sufficient for yielding inferences that are both deductive and inductive, and hence

suitable for building a science.148

CONCLUSION

Some points that emerged from the discussion of £ukasiewicz's texts must now be recalled.

The first remark regards the interpretation which in general is given of them: it turns out that his

subject is more complex than commonly believed, and that his three formulations of the principle of

contradiction, however deeply discussed, constitute only a part of the work and certainly not the

most ambitious. Furthermore, in interpreting Aristotle, £ukasiewicz never loses touch with the



43

contemporary philosophical situation (his journeys through Europe between 1902 and 1906 are an

evident symptom of such an interest); in particular, through the criticism of the principle of

contradiction in its psychological formulation, he decidedly places himself (at that time) in the stream

of European logical realism.

Moreover, he settles an issue, traceable in the traditional formal logic of the 19th century with

offshoots up until the beginning of the 20th century, achieving the following results: the principle of

contradiction is not a simple principle, as it presupposes some logical notions not present in

simpler laws; it is not an ultimate principle since it is not true “through itself”, a characteristic to

which only the definition of a true sentence is entitled; it is not a necessary principle because other

laws are independent of it. All this does not mean though that the principle of contradiction is not

valid; on the contrary, it maintains its validity but, if it is to be founded, it is necessary to resort to the

notion of object and give it an ontological foundation. Besides, what is given as a basis cannot be

arbitrarily assumed, but needs at least a justification to motivate its assumption and to say why

exactly it constitutes the foundation and not something else.

With regard to this, £ukasiewicz does not supply a real proof, but—as one of the last

exponents of a process dating back to Bolzano and inherited by Meinong—proves that the principle

of contradiction is valid only on the basis of a defined meaning of object, intended as what is

something and is not contradictory; and that, if we assumed contradictory objects, we would have

some exceptions to the principle. Hence it becomes possible to hypothesize worlds in which there

are contradictory objects, or minds which reason apart from the principle of contradiction, as we

previously noted. In both cases the principle is not valid, in the sense that the negation does not mean

exclusion or refusal of the affirmation. On this basis, we can proceed to elaborate some logicalsystems that are alternative to the one of classical logic and, when the operation is successful, to see

whether even in our world there are objects corresponding to those constructed, or argumentations

for which are valid the inferences according to the system built.

However, we could be tempted to say that £ukasiewicz promises much more than he

maintains; above all if, besides the quite emphatic way with which he announces some of his ideas,

we consider that two of his main goals, the proof of the principle of contradiction and the

construction of a non-Aristotelian logic, are not actually achieved. With regard to this it has to be



44

kept in mind that £ukasiewicz's operation does not want to abolish the principle of contradiction, but

debate it as a principle, or rather reopen a discussion, begin a new course of research. And

£ukasiewicz does so in two ways: he proposes a first treatment of the principle in light of symbolic

logic and as a consequence declares the end to the attempts which came from (and still come from)

the traditional formal-logical background; and he pursues, contemporaneously with Vasil'év and

Peirce ( supra, pp. 24f.), the idea of constructing a non-Aristotelian logic. In this way, £ukasiewicz

presents a “pioneer“ work compared to the logical developments of the 20th century as to the

principle of contradiction.149

At last, another result implied in £ukasiewicz's argument (of which probably £ukasiewicz

himself was not fully aware) is that in reality it is not possible to prove absolutely the principle of

contradiction starting from other principles which do not presuppose it. This is possible instead only

in the realm of a defined logical system, but this needs a general definition of a logical system which

itself contemplates the possibility of the assumption of other principles besides the one in question; in

any case, this proof would always be relative to the adopted logical system . In particular, the

interdefinability of ∧ and ∨ (via ¬) and the properties of ¬ assumed by £ukasiewicz are too strong

to discriminate between the principle of the excluded middle and the principle of contradiction.

£ukasiewicz does not take into account the possibility of assuming weaker notions of ¬ or of

considering principles as rules and not as laws. We know today that it is possible to formulate

propositional systems (of intuitionistic or co-intuitionistic logic) in which alternatively one and only

one of the principles is demonstrable. But this is possible only by weakening ¬. Another way of

proceeding consists in an approach of a semantical type which separates justification from proof. If

we want to give a strong foundation to the principle, we attempt to prove it, but in the absence ofmore certain and surer principles we are forced to resort to an ontological foundation based on the

assumption that objects are not contradictory. This points out the impossibility of giving a proof

which is not circular. A semantical justification instead accepts the circularity: it does not start from

more secure principles but, assuming a defined universe of objects in which obviously all laws

mutually entail one another, connects the nature of the logical objects to the validity of the principles:

this is precisely shown by the deductions (5.i) and (5.ii) conducted by £ukasiewicz.



45

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ENDNOTES

* I wish to thank Silvio Bozzi for critical remarks and suggestions.

1 £ukasiewicz (1918/1970: 86). The works will be quoted in accordance to the original date of publication

or, if they are in translation or in another edition, also with the date of the latter. The references to the English



58

translation of the sources are given in square brackets. In the absence of a standard English translation I have

provided the translation of the texts myself. Collective works appear under the name of the editor.

2 £ukasiewicz (1910b: 21 [1971: 492; see also 1979: 53]).

3 Cf. £ukasiewicz (1910a/1987: 7-8 [1993: 6-7]; 1910b: 15 [1971: 486; 1979: 50]).

4 Cf. £ukasiewicz (1910a/1987: 153-196 [1993: 187-245]); on this, see infra, pp. 39ff.

5 Cf. Maier (1896-1900: I, 41-45).

6 Cf. Trendelenburg (18401/18622: I, 23, 31-32; II, 153), Ueberweg (18571/18825: § 77, pp. 234-237), and

Sigwart (1873-18781/19114: I, § 23, pp. 191ff. [1895: 139ff.]).

7 Cf. £ukasiewicz's letter to Meinong of the 23.XII.1908 (cit. in Simons 1992: 219-220), and £ukasiewicz

(1913/1970: 16, n. 1, 49). Cf. also Sobociñski (1956: 4), Woleñski and Simons (1987: 68), Simons (1992: 202), and

Jadacki (1994: 228). Upon his return to Lwów, £ukasiewicz held a lecture on Meinong's philosophy, of which a

summary has been published (cf. £ukasiewicz 1909); on this, cf. Simons (1992: 203).

8 £ukasiewicz explicitly quotes Über die Stellung der Gegenstandstheorie im System der Wissenschaften

(cf. Meinong 1906-1907) and Meinong's lectures of the winter semester 1908/1909. I could not say whether he had

learned about impossible objects and the objectives only by the lectures and the work above mentioned, or even

by the reading of “Über Gegenstände höherer Ordnung und deren Verhältnis zur inneren Wahrnehmung” (cf.

Meinong 1899 [1978: 137-208]), in which we find Meinong's first classification of the nonexistent objects including

those which are contradictory or impossible (cf. Meinong 1899: 382 [1978: 141]), or of Über Annahmen (cf.

Meinong 1902), chapter VII of which is wholly dedicated to the analysis of the objective. On the page of Über die

Stellung der Gegenstandstheorie im System der Wissenschaften (p. 16) referred to by £ukasiewicz (1910a/1987:

110 n. [1993: 135, n. 1]), Meinong quotes in a footnote “Über Gegenstandstheorie” (cf. Meinong 1904b [1960]), but

even such indication is not sufficient to attest £ukasiewicz's direct knowledge of Meinong's text. What is certain

is that the young £ukasiewicz knew and, to a great extent, shared several theories expressed in the mentioned

Meinong's texts.

9 Twardowski (1991: 11, 14) declares to have written Zur Lehre vom Inhalt und Gegenstand der

Vorstellungen (cf. Twardowski 1894 [1977]) in the sense of Brentano and of Bolzano, whose theories he

constantly confronts himself with.

10 Cf. £ukasiewicz (1956: 44), Sobociñski (1956: 3-4), Skolimowski (1967: 56), and Jadacki (1993: 430).



59

11 Cf. £ukasiewicz (1910b: 17 [1971: 488; 1979: 51]).

12 This is what Winter (1975: 30) supposes. To confirm £ukasiewicz's knowledge of Bolzano, cf. also

£ukasiewicz (1913/1970: 52ff.). Here in footnote 20 £ukasiewicz states that for the reference to Bolzano he is

indebted to Twardowski.

13 Cf. £ukasiewicz (1906/1961: 9ff.), to which refer Borkowski and S³upecki (1958: 12-13), Skolimowski

(1967: 57-59), Woleñski (1989: 54-55), Trzêsicki (1993: 255-256), and Coniglione (1994: 75ff.).

14 These theories are continued by £ukasiewicz in chapters XVIII and XIX of O zasadzie sprzeczno¶ ci u

Arystotelesa . Later (see infra , pp. 36ff.), the distinction only mentioned here will be better clarified. About four

years passed between the dissertation on the concept of cause and the essay on the principle of contradiction in

Aristotle. It is during these years that £ukasiewicz's stay in Graz took place. £ukasiewicz probably improved his

theory on the matter after his meeting with Meinong.

15 Also Bolzano (1837: I, § 19, pp. 76ff. [1972: 20ff.]; §§ 48-49, pp. 215ff. [61ff.]) had negatively defined the

logical objects (ideas-in-themselves [Vorstellungen an sich] and propositions-in-themselves [Sätze an sich]) with

regard to the reality, the thought, and the language; cf. Raspa (1995/96: 120ff.).

16 Cf. Zimmermann (18531/1975: § 16 and note, p. 42).

17 Cf. Zimmermann (18602: § 23, pp. 21-22), where no mention is made of the objectless ideas. Moreover, i t

seems that Zimmermann—though his thought in this regard is not very clear—accepts the non-factual objectless

ideas and refuses the contradictory ones. This, without being said explicitly, is a logical consequence of the

thesis according to which any content or idea, in order to subsist, must conform itself to the principles of

contradiction, identity, and excluded middle (cf. § 31, pp. 26-27).

18

Cf. Twardowski (1894: § 5, pp. 20-29 [1977: 18-26]).19 For the illustration of these issues, cf. Raspa (1995/96).

20 Cf. e. g. Zwergel (1972), Seddon (1981), Cassin and Narcy (1989: 10-17; 1991), and Schiaparelli (1994).

21 A discussion in this regard is present in Schiaparelli (1994: 57-76). Also Zwergel (1972: 110 and passim)

takes into account £ukasiewicz's criticisms.

22 The passage of the letter is cited in Jadacki (1993: 440). £ukasiewicz probably refers to Jordan (1945),

and Bocheñski (1947: 237-238).

23 Cf. Lejewski (1967: 104), and Wedin (in £ukasiewicz 1971: 485, n. 1).



60

24 Jordan (1963: 13). Actually, except for some rare exceptions in writings of the same period (cf. for ex.

£ukasiewicz 1912/1970: 12, n. 20), £ukasiewicz makes no mention of it. This even holds for his works on the

history of logic and for the book on Aristotle's syllogistic.

25 Cf. Sobociñski (1956: 11), Skolimowski (1967: 39), and Woleñski (1989: 82-83). The first Polish book on

mathematical logic was Stanis ³aw Piatkiewicz's Algiebra w logice [ Algebra in logic] (Lwów 1888); cf. Sobociñski

(1956: 6). Cf. also Le¶niewski (1927-1930/1992: I, 181): “In the year 1911 (still in my student years) I came across a

book by Jan £ukasiewicz about the principle of contradiction in Aristotle. This book, which in its time had a

considerable influence upon the intellectual development of a number of Polish ‘philosophers’ and the

‘philosophising’ scholars of my generation, became a revelation for me in many respects and for the first time in

my life I learned of the existence of the ‘symbolic logic’ of Bertrand Russell as well as his ‘antinomy’ regarding the

‘class of classes, which are not elements of themselves’”.

26 Cf. esp. Kotarbiñski (1913/1968), and Le¶niewski (1912/1992; 1913a/1992; 1913b/1992). Regarding the

dispute refer to Woleñski (1989: 120; 1990b).

27 Cf. £ukasiewicz (1910a/1987: 9-10 [1993: 9]; 1910b: 16 [1971: 487; 1979: 51]). As an ontological

formulation of the principle of contradiction £ukasiewicz interprets even the one given in Met . B 2, 996 b30. Cf.

also Maier (1896-1900: I, 41-42), and Zwergel (1972: 88-90).

28 This and the following translations of passages from the Metaphysics are by W. D. Ross.

29 Cf. £ukasiewicz (1910a/1987: 10-11 [1993: 10]; 1910b: 16 [1971: 487; 1979: 51]).

30 Cf. £ukasiewicz (1910a/1987: 11-12 [1993: 11]; 1910b: 16 [1971: 487; 1979: 51]). Cf. also Zwergel (1972:

90ff.).

31

The word Øpol amb£nein is translated in many ways, for. ex. as ‘to believe’ (Ross; Barnes 1969;

Kirwan: 7) or ‘credere’ (Reale: II, 145), ‘to suppose’ (Tredennick: 163) or ‘supporre’ (Russo: 94), ‘ritenere’ (Viano:

273), ‘annehmen’ (Bonitz/Seidl: 137; Golke: 116), ‘penser’ (Colle: 12), ‘concevoir’ (Tricot: 195), ‘soutenir’ (Cassin

and Narcy 1989: 41, 125). According to £ukasiewicz (1910a/1987: 12 [1993: 11-12]), “here Øpol amb£nein does not

mean ‘to assume’, that is, ‘to suppose’, but (compared with l šgein, ‘to talk’, ‘to reveal an opinion’) expresses the

psychical act which usually—if not always—accompanies the expression of an opinion. This act precisely is a

conviction (przekonanie), belief (wierzenie)”. £ukasiewicz finds confirmation to his thes is in A. Schwegler (II, 54),

who translates Øpol amb£nein with ‘glauben’, and in H. Maier (1896-1900: I, 46, 104), according to whom in



61

Aristotle the word Øpol amb£nein sets out, in the same way as the noun ØpÒl hyij , the psychical state of the

conviction, that is, an act of believing combined with a subjective decision.

32 Cf. Sigwart (1873-18781/19114: I, § 23, pp. 192-193 [1895: 139-140]), who reads the Aristotelian text in

the view of the psychologistic logic, which was very widespread in the 19th century.

33 Cf. Twardowski (1894: § 7, pp. 37-38 [1977: 35]): “in short, everything which is not nothing, but which

in some sense is ‘something’, is an object”. Cf. also Bolzano (1837: I, § 60, p. 259 [1972: 76]; § 99, p. 459 [145-146]).

34 £ukasiewicz (1910a/1987: 10, 149 [1993: 9-10, 182]; cf. also 1910b: 16 [1971: 488; 1979: 51]). In his article

£ukasiewicz writes “property p belongs to object O” instead of “object O possesses property p”. In the present

work the second formulation is preferable in regard of £ukasiewicz's theory of synonymity which will be referred

to in short (see infra, pp. 10-11). However, we will keep in mind that—in accordance with Meinong— £ukasiewicz

uses ‘object’ in a very wide sense which includes also the concept of the object (see supra , pp. 4f.).

35 £ukasiewicz (1910a/1987: 11, 12, 149 [1993: 11, 12, 182]; cf. also 1910b: 16-17 [1971: 488; 1979: 51]). Cf.

also Bolzano (1837: I, § 81, p. 393 [1972: 126]): “I mean by form a certain concatenation of words or signs in

general, which can represent a certain kind of idea, proposition or argument”.

36 Cf. Meinong (1910: 42ff., 340). Cf. also £ukasiewicz (1910a/1987: 13-14 [1993: 13-14]; 1910b: 17 [1971:

488; 1979: 51]).

37 Cf. £ukasiewicz (1910a/1987: 12-14, 27-28 [1993: 12-15, 32]; 1910b: 17-18 [1971: 488-489; 1979: 51]).

Probably during £ukasiewicz's stay in Graz, Meinong and £ukasiewicz discussed these subjects. Cf. the letter of

the 12.IV.1910, enclosed to which £ukasiewicz sends Meinong a quotation from Russell's Mathematical Logic as

based on the Theory of Types regarding the notion of assertion (cf. Simons 1992: 222-223).

38

Cf. £ukasiewicz (1910a/1987: 16-18 [1993: 18-20]; 1910b: 18 [1971: 489; 1979: 51-52]).39 £ukasiewicz (1910a/1987: 13 [1993: 13]).

40 Cf. Aristotle, De int . 12, 21 b17-18; An. pr . A 46, 51 b20-22; B 2, 53 b15-16; An. post . A 11, 77a10-11; Soph.

el . 25, 180a26-27.

41 Cf. £ukasiewicz (1910a/1987: 15-16 [1993: 16-17]).

42 £ukasiewicz (1910a/1987: 18 [1993: 20]).

43 Aristotle, Met . G 7, 1011 b26-27.

44 Cf. £ukasiewicz (1910a/1987: 16-18 [1993: 18-20]).



62

45 “If it is impossible that contrary attributes should belong at the same time to the same subject (the

usual qualifications must be presupposed in this proposition too), and if an opinion which contradicts another is

contrary to it, obviously it is impossible for the same man at the same time to believe the same thing to be and not

to be; for if a man were mistaken on this point he would have contrary opinions at the same time”.

46 “Now since it is impossible that contradictories should be at the same time true of the same thing,

obviously contraries also cannot belong at the same time to the same thing. For of the contraries, one is privation

no less than it is a contrary—and a privation of the essential nature; and privation is the denial of a predicate to a

determinate genus. If, then, it is impossible to affirm and deny truly at the same time, it is also impossible that

contraries should belong to a subject at the same time, unless both belong to it in particular relations, or one in a

particular relation and one without qualification”.

47 Cf. £ukasiewicz (1910a/1987: 19-24 [1993: 22-28]; 1910b: 18-19 [1971: 489-491; 1979: 52]). For an analysis

of the Aristotelian argumentations, cf. Barnes (1969), Zwergel (1972: 91ff.), Stevenson (1975), Schiaparelli (1994:

49-55), and Raspa (1996: 46ff.).

48 Cf. £ukasiewicz (1910b: 20 [1971: 491; 1979: 53]). £ukasiewicz's thesis is discussed by Corradini (1985:

237ff.).

49 Cf. £ukasiewicz (1910a/1987: 25, 29-30 [1993: 29, 34-35]).

50 Mill (18431/18728/1973: II, vii, § 5, pp. 277-278).

51 Cf. Spencer (1865: 533; 1966: 191-192).

52 Cf. Mill (18431/18728/1973: II, vii, § 5, pp. 278-279; 18651/18724/1979: 381 n).

53 Husserl (1900-19011/19223: I, 81 [1970: I, 113]).

54

Cf. Husserl (1900-19011/19223: I, 81-82 [1970: I, 113-114]).55 A lecture on “Husserl's thesis on the relationship between logic and psychology” held by£ukasiewicz

at the Polish Philosophical Society testifies to this (for a short report of the lecture, cf. £ukasiewicz 1904). More at

length he speaks about this subject in “Logika a psychologia [Logic and Psychology]” (cf. £ukasiewicz 1907). On

this, cf. also Borkowski and S³upecki (1958: 46-47), Kuderowicz (1988: 142-143), Sobociñski (1956: 8-9), and

Woleñski (1989: 194).

56 Hegel (1812-1813/1978: 287 [1969: 440]).

57 Aristotle, Met . G 3, 1005 b25-26.



63

58 Cf. £ukasiewicz (1910a/1987: 30-34 [1993: 37-41]; 1910b: 21 and nn. 1-2 [1971: 492-493 and nn. 6-7; 1979:

53-54 and nn. 4-5]).

59 £ukasiewicz (1910a/1987: 36 [1993: 43]).

60 Frege (1893: XVI [1964: 12]).

61 On the subject, £ukasiewicz (1910a/1987: 41 n. [1993: 49-50, n. 2]; 1910b: 22, n. 1 [1971: 493, n. 8; 1979:

54, n. 6]) mentions Trendelenburg (18401/18622: I, 31), Sigwart (1873-18781/19114: I, § 23, pp. 194-195 [1895: 141]),

and Ueberweg (18571/18825: § 76, pp. 232-233).

62 Here £ukasiewicz refers to Höfler (1890: § 57, p. 135): “The principle of contradiction is usually

expressed in this way: A is not not- A”

63 Sigwart (1873-18781/19114: I, § 23, p. 194 [1895: 141]). H. Dendy translates “positive rendering”.

64 Cf. Trendelenburg (18401/18622: I, 23; II, 153), and Lotze (18741/19123: 76).


66 £ukasiewicz (1910a/1987: 43 [1993: 52-53]) shares Russell's thesis —well established in mathematical

logic—according to which the universal proposition “every A is B” is in reality a hypothetical proposition, which

asserts “if something is A, then it is also B”. In “On Denoting”, Russell (1905/1956: 43) asserts to have taken such

thesis from Bradley (18831/19222: 82).

67 Here, in virtue of the whole argument £ukasiewicz (1910a/1987: 43 [1993: 52]) is using the wake

principle of double negation: [a → ¬¬a]; while in the appendix (p. 174 [217]) he writes the strong formulation of

the principle: [¬¬a ↔ a].

68 Cf. £ukasiewicz (1910a/1987: 43, 155, 174, 185 [1993: 52, 189, 217, 231]). The symbolism adopted by

£ukasiewicz (1910a/1987: 154 [1993: 188]) is in this presentation modernised according to the current use.

Moreover, ‘a’ means “O possesses a”, the symbol ‘1’ (or logical unit) expresses the sentence “O is something, is

an object”, while ‘0’ (or logical zero) means the sentence “O is nothing, is not an object”.

69 Cf. £ukasiewicz (1910a/1987: 44-46 [1993: 53-56]; 1910b: 22, 27 [1971: 493, 498; 1979: 54, 56]).

70 Cf. £ukasiewicz (1910a/1987: 60-62 [1993: 74-76]).

71 According to £ukasiewicz (1910a/1987: 48-49 [1993: 59-60]), the definition is a singular sentence, which

states the fact that someone determines an object in a certain way; this fact is produced with and it is contained in



64

the definition itself. This is why it is always true. We must distinguish, however, the principle from the definition

on which it is founded; the principle is always a universal sentence.

72 £ukasiewicz (1910a/1987: 49 [1993: 60]).


74 Cf. £ukasiewicz (1910a/1987: 47-51 [1993: 57-62]).

75 Cf. £ukasiewicz (1910a/1987: 94-95, 191-192 [1993: 116-117, 239]; 1910b: 32-33 [1971: 504; 1979: 59-60]).

On this, see infra, pp. 39ff.

76 Cf. Aristotle, An. post . A 11, 77a10-12: “No demonstration assumes that it is not possible to assert and

deny at the same time—unless the conclusion too is to be proved in this form”.

77 Aristotle, An. post . A 11, 77a12-21. For a discussion of the whole passage, cf. Mignucci (1975: 221-237),

and Barnes (1994: 144-147). In his translation of the article, Barnes notes (cf. £ukasiewicz 1979: 59, Translators

note) that £ukasiewicz's translation of the lines 77a15-19 is different from that of the Oxonian edition—edited

moreover by Barnes himself; now, Barnes's observation is referred to £ukasiewicz (1910b: 32), while the version of

£ukasiewicz (1910a/1987: 92 [1993: 113]) is much more similar to the Oxonian reading than to the English

translation given in the text.

78 Cf. £ukasiewicz (1910a/1987: 92-93 [1993: 113-114]). In the article, £ukasiewicz (1910b: 32 [1971: 504;

1979: 59]) presents the two syllogisms uniting them:

B is A (and not also not- A)

C , which is not-C , is B and not- B

C is A (and not also not- A).

Furthermore, the negation in the minor premiss concerns only the predicate.

79 Cf. Couturat (1905: 8). In this book Couturat uses the same symbol for both the inclusion relation

between classes and the notion of consequences between propositions.

80 Cf. T2 in £ukasiewicz (1910a/1987: 155 [1993: 190]); but cf. also £ukasiewicz (1912/1970: 7; 1920/1970:

88; T33 in 1921/1970: 108), wherein he states that the formulation given above is the most common but not the

only one to express such a logical relation.



65

81 Cf. £ukasiewicz (1910a/1987: 93-95 [1993: 115-116]; 1910b: 32 [1971: 504; 1979: 59]). Cf. also Bocheñski

(19561/19703: 72 [1961: 61-62]), who shares £ukasiewicz's interpretation, and Zwergel (1972: 21-28) and Seddon

(1981: 203-206), who contest it.

82 Husik (1906: 216).

83 Husik (1906: 219-220).

84 Cf. £ukasiewicz (19511/19572: 46-47, 73-74). Here he states (p. 47): “It is not true that the dictum de

omni et nullo was given by Aristotle as the axiom on which all syllogistic inference is based”.

85 In 1910 £ukasiewicz (1910a/1987: 8, 9 [1993: 7, 8]) is using the word ‘metalogical’, but he does not take

it in the meaning it will have later and still has today in logic.

86 To this, chapter XVI (1910a/1987: 95-100 [1993: 118-124]) of the book is dedicated, thus ending the

critical part.

87 Cf. £ukasiewicz (1910a/1987: 8 [1993: 7]).

88 Although it was for long time unnoticed, this was already pointed out by a pupil of £ukasiewicz,

Antoni Korcik (1955); cf. Kline (1965: 316), and Dahm and Ignatow (1996, eds.: 281). Of Vasil'év, cf. esp. Vasil'év

(1910/1989; 1912/1989; 1912-1913/1989).

89 Cf. Peirce (1897/1933).

90 Cf. Carus (1910a; 1910b).

91 This has been pointed out by Bazhanov (1992: 48, 50).

92 Cf. Carus (1910a: 45).

93 Cf. Carus (1910b: 158).

94

Cf. Bazhanov (1992: 49).95 Cf. Mangione and Bozzi (1993: 17).

96 Cf. £ukasiewicz (1910a/1987: 8 [1993: 6-7]): “then it will become clear, [...] whether the principle of

contradiction can be transformed, or whether—without taking the principle at all into account—a system of non-

Aristotelian logic can be developed, so as through the transformation of the parallel line postulate arose a system

of non-Euclidean geometry”; Vasil'év (1912/1989: 54): “A non-Euclidean geometry is a geometry without Euclid's

fifth postulate, the so-called parallel line postulate. A non-Aristotelian logic is a logic without the law of



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contradiction. It is not useless to add that it was the non-Euclidean geometry that provided us with an example for

the construction of a non-Aristotelian logic”.

97 Cf. £ukasiewicz (1910a/1987: 96 [1993: 118]). In a similar way Peirce (1878/1986: 266-267), too, says:

“Suppose, then, that a diamond could be crystallised in the midst of a cushion of soft cotton, and should remain

there until it was finally burned up. Would it be false to say that that diamond was soft? This seems a foolish

question, and would be so, in fact, except in the realm of logic. There such questions are often of the greatest

utility as serving to bring logical principles into sharper relief than real discussions ever could. In studying logic

we must not put them aside with hasty answers, but must consider them with attentive care, in order to make out

the principles involved”.

98 Cf. Vasil'év (1912-1913/1989: 115ff. [1993: 346ff.]).

99 Cf. £ukasiewicz (1930/1970: 175-176; 1961a/1970: 126).

100 In the article, £ukasiewicz (1910b: 33 [1971: 504; see also 1979: 60]) does not talk about a non-

Aristotelian logic but restricts himself to hint at the possibility of building inferences of the type above-

mentioned and claims: “Moreover, it would be not at all difficult to show in words, as well, that the basic

principles of deduction as well as induction do not on the whole presuppose the principle of contradiction.

Indeed there are innumerable deductions and inductions which proceed only by affirmative propositions;

consequently, the principle of contradiction finds no application to these because it always meets an affirmative

proposition and its contradictory negative. On my view, we must give up the false, though widely spread view

that the principle of contradiction is the highest principle of all demonstrations! That holds only for indirect

proofs; for the direct ones, it is not true”.

101

Cf. £ukasiewicz (1913/1970: 37-38); but see also n. 130 and Simons (1992: 198-199).102 As he says in £ukasiewicz (1918/1970: 86).

103 Cf. £ukasiewicz (1920/1970: 87; 1930/1970: 164 ss.; 1961a/1970: 126).

104 Cf. £ukasiewicz (1929/1963: 67-68; 1937/1970: 243, 248); cf. also Sobociñski (1956: 11ff.), and Jordan

(1963: 13). Bocheñski (1951: 39) regards the logical principle of contradiction (in £ukasiewicz's sense) as

metalogical.

105 Cf. Ja¶kowski (1948/1969); cf. also D±mbska (1978/1990: 26), Arruda (1980: 9f.; 1989: 103f.), and Priest

and Routley (1989: 44ff.).



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106 The same hypothesis, that the nature of reason could be different from how it was generally believed,

has been taken into consideration also by Peirce (cf. Carus 1910b: 158).

107 Already Maier (1896-1900: I, 56) caught a glimpse of a presupposition in the proof, a presupposition

which however he accepts and recognizes in the defined meanings of the words.

108 Cf. Kant (1781-17872: A 52 = B 76 [1929: 93]; 1800: Ak . IX, 13 [1974: 15]), Hamilton (1837-18381/18662:

III, 4, 26), and Sigwart (1873-18781/19114: § 1, pp. 1-11 [1895: 1-10]).

109 Cf. Ueberweg (18571/18825: § 77, p. 237), and Pfänder (19211/19633: 201, 204).

110 Cf. Ueberweg (18571/18825: § 77, pp. 234-235, 238-239), and Pfänder (19211/19633: 205-207).

111 I have developed such arguments in Raspa (1996: 139ff.).

112 Cf. £ukasiewicz (1910a/1987: 103 [1993: 126]).

113 Cf. £ukasiewicz (1910a/1987: 103-105 [1993: 126-128]; 1910b: 33 [1971: 505; 1979: 60]).

114 Cf. £ukasiewicz (1910a/1987: 105-106 [1993: 129]).

115 Sigwart (1873-18781/19114: I, § 23, p. 191 [1895: 139]).

116 Cf. Sigwart (1873-18781/19114: I, § 20, pp. 158-160 [1895: 119-120]).

117 £ukasiewicz (1910a/1987: 106 [1993: 130]).

118 Such an operation has been attempted by Ueberweg (18571/18825: § 77, p. 235).

119 Cf. £ukasiewicz (1910a/1987: 106-109 [1993: 130-134]; 1910b: 34-35).


121 Cf. Bolzano (1837: I, § 70, pp. 317-318 [1972: 93]), Twardowski (1894: § 5, pp. 23ff. [1977: 21ff.]), and

Meinong (1906-1907: § 3, pp. 15-16).

122

Cf. £ukasiewicz (1910a/1987: 60-61 [1993: 74-75]; cf. also 1910b: 27 [1971: 498; 1979: 56]).123 Cf. £ukasiewicz (1910a/1987: 65-66 [1993: 80-81].

124 Here £ukasiewicz (1910a/1987: 110 n. [1993: 135, n. 1]; 1910b: 35, n. 1 [1971: 506, n. 14; 1979: 61, n. 13])

refers explicitly to the controversy between Meinong and Russell; with regard to this cf. Russell's (1905/1956)

criticism and Meinong‘s (1906-1907: 14-16) answer. To Russell who accused him of having infringed upon the

principle of contradiction, admitting objects which neither exist nor subsist, Meinong replied that the principle of

contradiction applies only to the real and possible, and that it would be in fact surprising if it would hold also for



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the impossible. I have supplied both a brief exposition of the controversy between Meinong and Russell, and an

indication of the main writings on the subject in Raspa (1995/96: 181ff.).

125 Cf. £ukasiewicz (1910a/1987: 110-111 [1993: 135]).

126 Cf. Pfänder (19211/19633: 207): “The general theoretical-objective [ gegenstandstheoretische] or

ontological-formal fact that an object cannot be at the same time P and not P is the ultimate foundation of the

truth of the principle of contradiction. Thus this truth is firmly tied to the reaction of the object in general and

completely independent of the nature of all thinking beings, even of that of man”.

127 £ukasiewicz (1910a/1987: 111 [1993: 136]).

128 Cf. Le¶niewski (1912/1992: 20f.); see also n. 25. For an analysis and a comparison of the respective

positions of the two philosophers, cf. Betti (199*).

129 Cf. £ukasiewicz (1910a/1987: 112 n. [1993: 138, n. 1]).

130 This passage is important because we find here a mention of the principle of the excluded middle;

otherwise there would arise the impression that £ukasiewicz disregarded it in this period completely. In fact,

parallel to the research on the principle of contradiction, he was also developing some theories on the principle of

the excluded middle, as testifies a report held at Lwów, the 26 February 1910, at the Polish Philosophical Society.

In the English translation of the short summary which remains of his talk, £ukasiewicz (1910c/1987) reaches, with

regard to the principle of the excluded middle, results similar to those already considered in connection with the

principle of contradiction: the excluded middle is not a fundamental principle, it is not self-evident, and it cannot

be proved logically, but it is necessary to practical ends and it has to be considered specifically in relation to real

objects. Moreover, he repeats what has just been said: that the principle is not valid for general objects (like man

and triangle), which are determined only in relation to essential properties, but not to those which are accidental;

and he adds: “With regard to real objects, the principle of excluded middle seems to be closely connected with the

postulate of universal determination of phenomena, not only present and past but also future ones. Were

someone to deny that all future phenomena are today already predetermined in all respects, he would probably

not be able to accept the principle in question” (p. 69). On this, cf. Borkowski and S³upecki (1958: 14-15),

Woleñski and Simons (1987: 67-68), Woleñski (1990b), and Simons (1992: 197-198).

131 Cf. £ukasiewicz (1910a/1987: 112-114 [1993: 137-139]).



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132 In accordance with Meinong, £ukasiewicz (1910a/1987: 114 [1993: 140]) uses the word ‘objects

[ przedmioty]’ in the main text, while in the article (cf. £ukasiewicz 1910b: 35-36) he talks about Begriffsbildungen

[‘abstractions’ (1971: 507) or ‘concepts’ (1979: 61)]; but see also supra , pp. 4f. and n. 34.

133 Cf. £ukasiewicz (1910a/1987: 114-115 [1993: 139-141]; 1910b: 35-36 [1971: 507; 1979: 61]).

134 Cf. £ukasiewicz (1910a/1987: 119-122 [1993: 145-149]; 1910b: 36 [1971: 507; 1979: 61]). It is known that

Russell discovered in the naive set theory, which is based on the comprehension principle, the following

contradiction: given the set R of all sets that are not members of themselves, we have that X j ∈ R iff X j ∉ X j, and

that so, in particular, R ∈ R iff R ∉ R —which is contradictory.

135 £ukasiewicz (1910b: 36 [1971: 507-508; see also 1979: 61-62]; cf. also 1910a/1987: 129-131 [1993: 158-

159]).

136 Cf. £ukasiewicz (1910a/1987: 125-128 [1993: 153-156]; 1910b: 36 [1971: 508; 1979: 62]).

137 Cf. £ukasiewicz (1910a/1987: 131-142, 173 [1993: 161-173, 215]; 1910b: 36-37 [1971: 508; 1979: 62]). As

we saw, £ukasiewicz makes a similar affirmation even with regard to the principle of the excluded middle (see n.

130).

138 Cf. £ukasiewicz (1910a/1987: 155-180 [1993: 189-225]). On the symbolism here used, see n. 68.

139 This is based nearly exclusively on Couturat's Algébre de la Logique, which £ukasiewicz (1910b: 33,

n. 1 [1971: 504, n. 12; 1979: 60, n. 11]) regards as “the best introduction to symbolic logic”. Further, as it has been

already pointed out, £ukasiewicz seems to not make any distinction between propositional calculus and Boolean

algebra (cf. Sobociñski 1956: 11), because “at that time £ukasiewicz probably did not know the propositional

calculus or, at least, did not recognize its importance” (Sobociñski 1956: 13).

140

Cf. Couturat (1905: 24).141 Cf. £ukasiewicz (1910a/1987: 170-171 [1993: 211-212]).

142 Cf. T1b in £ukasiewicz (1910a/1987: 160 [1993: 197]). But probably £ukasiewicz is thinking here of two

other propositions, that is, [(a ↔ b) → (a → b)] and [(a ↔ b) → (b → a)], or he applies the rule of conjunction

elimination.

143 Cf. £ukasiewicz (1910a/1987: 185-186 [1993: 231-233]).

144 Cf. the proofs in£ukasiewicz (1910a/1987: 176, 178 [1993: 220, 222]), which I do not quote here.

145 Cf. £ukasiewicz (1910a/1987: 155-158, 170-171 [1993: 189-194, 211-212]).



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146 Cf. £ukasiewicz (1910a/1987: 187 [1993: 233-234]).

147 Cf. £ukasiewicz (1910a/1987: 187-188 [1993: 234-235]).

148 Cf. £ukasiewicz (1910a/1987: 188-192 [1993: 235-239]).

149 According to Priest and Routley (1989: 25ff.), although£ukasiewicz avoids such conclusions they are

found to be implicit in his subject—we have seen that he did not reject the principle of contradiction but only

degraded it to a theorem—the work of 1910 “opens the way for paraconsistent enterprise”.

Lukasiewicz on the Principle of Contradiction

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