8/11/2019 Lukasiewicz on the Principle of Contradiction http://slidepdf.com/reader/full/lukasiewicz-on-the-principle-of-contradiction 1/70 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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8/11/2019 Lukasiewicz on the Principle of Contradiction
“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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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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“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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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.
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“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
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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.
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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
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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
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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
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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
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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
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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.
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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
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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,
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£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?
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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,
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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
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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
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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
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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.
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“ 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
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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
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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
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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.
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- (1961): A History of Formal Logic, trans. and ed. by I. Thomas, Notre Dame (Indiana):
University of Notre Dame Press, 1961; repr. New York: Chelsea P. C., 1970.
Bolzano, Bernard (1837): Wissenschaftslehre. Versuch einer ausführlichen und größtentheilsneuen Darstellung der Logik mit steter Rücksicht auf deren bisherige Bearbeiter , 4 Bde.,
Sulzbach: J. E. v. Seidelschen Buchhandlung, 1837 [partial Engl. trans.: Bolzano (1972)].
- (1972): Theory of Science. Attempt at a Detailed and in the Main Novel Exposition of
Logic with Constant Attention to Earlier Authors, ed. and trans. by R. George, Berkeley-
Los Angeles: University of California Press, 1972.
Borkowski, Ludwik and Jerzy S³upecki (1958): “The Logical Works of J. £ukasiewicz”, Studia
Logica 8 (1958), 7-56.
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- (1970): Logical Investigations, 2 vols., trans. by. J. N. Findlay, London: Routledge &
Kegan Paul, 1970; repr. 1976.
Jadacki, Jacek Juliusz (1993): Critical notice of: Peter Simons, Philosophy and Logic in Central Europe from Bolzano to Tarski, Dordrecht-Boston-London: Kluwer, 1992, Axiomathes 4
(1993), No. 3, 427-440.
- (1994): “Warsaw: The Rise and Decline of Modern Scientific Philosophy in the Capital City of
Poland”, Axiomathes 5 (1994), No. 2-3, 225-241.
Ja¶kowski, Stanis³aw (1948/1969): “Rachunek zdañ dla systemów dedukcyjnych sprzecznych”,
Studia Societatis Scientiarum Torunensis, Sectio A, vol. I, No. 5, Toruñ 1948, pp. 57-77;
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Kotarbiñski, Tadeusz (1913/1968): “Zagadniene istnienia przysz³o¶ci”, Przegl ± d Filozoficzny 16(1913), 74-92 [Engl. trans.: “The Problem of the Existence of the Future”, The Polish
Review 13 (1968), No. 3, 7-22].
Kuderowicz, Zbigniew (1988): Das philosophische Ideengut Polens, Bonn: Bouvier, 1988.
Lejewski, Czes³aw (1967): “Jan £ukasiewicz”, in The Encyclopaedia of Philosophy, ed. by P.
Edwards, New York-London: Collier-MacMillan, 1967, vol. 5, pp. 104-107.
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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.
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