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Lukasiewicz on the Principle of Contradiction
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"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. Lukasiewicz)
Introduction: the historical-philosophical context - 1. The ontological, logical, and psychological formulations of the principle of contradiction - 2. The principle of contradiction is not a simple, ultimate and necessary principle - 3. The idea of a non-Aristotelian logic - 4. The 'proof' of the principle of contradiction - 5. 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 Lukasiewicz'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, Lukasiewicz declared:
"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".
The book in question is O zasadzie sprzecznosci u Arystotelesa. Studium krytyczne [On the Principle of Contradiction in Aristotle. A Critical Study], printed in Polish in 1910. In the same year, Lukasiewicz 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 Wolenski 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";
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 to the criticism to which the young Lukasiewicz subjects the Aristotelian texts, almost completely neglecting the formation and the philosophical background to which he makes reference. Nevertheless, to understand better Lukasiewicz'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 Lukasiewicz's first intellectual tasks, represents an interesting crossroad of different components: on one hand, he gives 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 Lukasiewicz'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 - Lukasiewicz asserts - stands in the same relation to the Aristotelian logic as the modern non-Euclidean 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. 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.
If this is the inspiring element, than 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 Lukasiewicz, 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 had already laid stress and whom Lukasiewicz 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. In its fullness, the Aristotelian text lent itself to express all the different positions separately presented by the above-mentioned authors. Hence, not only Aristotle but also the exponents of traditional formal logic have put to Lukasiewicz 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 Lukasiewicz is the Austrian philosopher Alexius Meinong, who is quoted several times in both texts examined here, and also in other earlier papers. Lukasiewicz 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. Lukasiewicz 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]. 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, Lukasiewicz asserts in fact that a proof of the principle of contradiction is only possible on the basis of the assumption that objects are non-contradictory. 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 Lukasiewicz's argument possible. They are first presented to him through his teacher Kazimierz Twardowski, with whom Lukasiewicz had studied in Lwow in a period in which Twardowski, for his express assertion, was a fervent Brentanian. Later on, after receiving his doctorate in philosophy in 1902, Lukasiewicz 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 respectively (the latter is also cited in the article of 1910). Lukasiewicz probably came into contact with Bolzano's Wissenschaftslehre not only through Twardowski but also through Stumpf. Having returned to Lwow, Lukasiewicz 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. Lukasiewicz 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 a mind, but can be either ideal or real. 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. In my opinion, the influence of Bolzanian elements can be traced in the theory of abstract objects.
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 [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 non-existent 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, which are then expunged in the second edition -, 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. From here starts Meinong's classification of objects, including those that are non-existent and the impossible or contradictory objects as well. Lukasiewicz 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 Lukasiewicz and Aristotle - which moreover has been already widely discussed - 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 Lukasiewicz'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 Lukasiewicz's well-known individuation of three formulations of the principle of contradiction in the Aristotelian texts (1) and the criticism of the opinions which consider it as a simple, ultimate, and necessary principle (2). 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 (3). The latter concerns the attempt to supply a direct proof of the Aristotelian principle of contradiction, in view both of the criticism to which Lukasiewicz subjected Aristotle's "negative demonstration ", 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 (4). What clearly comes out of the analysis of these two attempts is Lukasiewicz'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 (5).
It must be pointed out, however, that as early as in his work of 1910 Lukasiewicz 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. Bochenski, dated Dublin, 7th October 1947, Lukasiewicz 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 sprzecznosci u Arystotelesa as weak and unsuccessful".
Nevertheless, although Lukasiewicz 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". This is clearly in contrast with what Lukasiewicz said above. Now, it is true that Lukasiewicz 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 Lukasiewicz 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 - 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, "Lukasiewicz never again returned to the view on the principle of non-contradiction expounded in his first major work". However, it is probable that Lukasiewicz 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. Furthermore, its publication immediately provoked a debate inside the Lwow-Warsaw school in the years 1912-1913, in which authors like Tadeusz Kotarbinski and Stanislaw Lesniewski took part. 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.
1. In the Aristotelian texts Lukasiewicz distinguishes three formulations of the principle of contradiction. The first, the so-called ontological formulation, is the classical formulation enunciated in Met. G 3, 1005b19-20:
"It is, that the same attribute cannot at the same time belong and not belong to the same subject and in the same respect".
Moreover, in connection with the definition of contradiction given in De int. 6, 17a32-35 (but cf. also De int. 7, 17b26-29), Aristotle would enunciate a logical formulation in Met. G 6, 1011b13-14:
"contradictory statements are not at the same time true";
and at last, just in the continuation of Met. G 3, 1005b23-24, a psychological formulation:
"it is impossible for any one to believe  the same thing to be and not to be".
In this last formulation stress is laid on  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, and it is from this kind of reading that Lukasiewicz, criticizing the Aristotelian principle of contradiction, convinces himself of the necessity to also criticize such a meaning of the principle.
Lukasiewicz 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, "anything that is 'something' and not 'nothing'", and by 'property' "anything that can be predicated of an object";
(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";
(iii) psychological formulation:
"Two beliefs, to which correspond contradictory sentences, cannot exist at the same time in the same intellect",
and here Lukasiewicz means by 'belief' or 'act of believing' - this is how he interprets and - 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, Lukasiewicz 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. Instead, according to Lukasiewicz, 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 - Lukasiewicz 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, 24b1-3), that is, a meaningful speech which can be true or false (cf. De int. 4, 17a1-3).
Now, the three formulations given - Lukasiewicz 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; and that "the ontological principle is the principle of contradiction " to which the Stagirite dedicates much of his attention. An examination of Lukasiewicz's thesis requires starting from the distinction that he operates between synonymity and equivalence. Since I will limit my analysis to Lukasiewicz's thought, I have to specify, so that my point of view on this subject be 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 Lukasiewicz'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, have a different informative value.
Lukasiewicz 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 - Lukasiewicz 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.
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, in the 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:
"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".
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". 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, 18b1-2; Met. Q 10, 1051b3-4). 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.
As to the psychological formulation of the principle, after its treatment by Lukasiewicz the result is that his consideration assumes Aristotle as a starting point in order to reach both contemporary philosophy and traditional formal logic as well. Lukasiewicz takes into account the passages of Met. G 3, 1005b26-32 - read in connection with De int. 14, 23a27-39 - and G 6, 1011b15-22 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 Lukasiewicz 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. Therefore, the psychological formulation of the principle of contradiction is nothing but a consequence of the logical one. This peculiarity of the Aristotelian argumentation is interpreted by Lukasiewicz as a falling by the Stagirite into that error which is the exact converse of "psychologism in logic", that is, "logicism in psychology".
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, 1005b26-32, 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 - Lukasiewicz 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.
These reflections make it possible to extend the subject beyond the simple reference to the Aristotelian texts. Lukasiewicz 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 Lukasiewicz 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 - Lukasiewicz 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 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, Lukasiewicz 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". 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. Mill himself quotes Spencer for his own support both in the Logic and in the Examination of Sir William Hamilton's Philosophy. 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:
"Two contradictorily opposed acts of belief [or states of consciousness] cannot coexist".
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. 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 Lukasiewicz 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. 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 Lukasiewicz 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 sprzecznosci u Arystotelesa Lukasiewicz 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 of which 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, Lukasiewicz 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 Lukasiewicz quotes Hegel's passage:
"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".
On this matter, either we agree with Aristotle that "what a man says, he does not necessarily believe", 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.
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".
In this way the young Lukasiewicz 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".
2. After Lukasiewicz 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 sprzecznosci 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 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 - Lukasiewicz 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; (ii) the latter is then confused with the principle of double negation (formulated in an inexact way like "A is not not-A"); 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]" of the principle of contradiction or even identified with it. 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".
That is why - Lukasiewicz 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, the principle of identity
means: "if O possesses a, then O possesses a"; the principle of double negation
means: "if O possesses a, then O cannot not possess a"; the principle of contradiction
1 (a a)
means: "if O is an object, then O cannot possess a and not possess a at the same time". Lukasiewicz 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. 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 Lukasiewicz, 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 - Lukasiewicz asserts - cannot be deduced from any of the other two principles; that means that it is even not equivalent to them. 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 the ultimate 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.
"The definition of a true sentence is true because each definition is true; and it is true through itself because its truth is not based on the truth of another sentence but on its own truth".
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".
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.
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 of identity, the principles of simplification and those of composition, the principle of distribution, the laws of commutation, tautology and absorption, and many others. What has just been said is particularly important for the results that Lukasiewicz draws from it.
3. 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. This is particularly put in light and discussed by Lukasiewicz, 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".
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 C the minor term (Callias), Lukasiewicz 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.
Since the syllogism presupposes the principle of contradiction only if the conclusion has to assert that C is A and not at the same time not-A, it is necessary then that the major term affirms that B is A and not at the same time not-A. For the rest the syllogism is however possible either if C is B and is not B at the same time, or if it is C and is not C. Consequently there are two possible syllogisms:
(a) B is A (and is not at the same time not-A)
C is B and is not B
C is A (and is not at the same time not-A);
(b) B is A (and is not at the same time not-A)
C, which is not C, is B
C is A (and is not at the same time not-A).
(a) is correct - Lukasiewicz explains - since C is B. The fact that at the same time C is not B neither compromises the conclusion "C is A", nor compromises the addition "C is not at the same time not-A" since the extension of A is such as to include both B and a certain number of objects which are not B. (b) is correct as well since here also C is B. The fact that at the same time C is not C neither compromises the conclusion "C is A", nor the addition "C is not at the same time not-A" since the extension of B is greater than that of C. Despite the contradiction of the minor premiss, the truth of the conclusion "C is A" follows directly from the truth of the premisses "B is A" and "C is B". That shows the independence of the principle of the syllogism from the principle of contradiction. With the principle of the syllogism Lukasiewicz means - following Couturat - the law of transitivity: (a b)(bc)(ac), the meaning of which - he says - is founded on the dictum de omni et nullo. The word "is" simply means the inclusion relation, which is transitive. Therefore, if B is included in A and C is included in B, then C is included in A.
Here a problem arises of which Lukasiewicz 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, Lukasiewicz 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. Lukasiewicz 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.
A different reading of the Aristotelian passage in question seems to solve the problem. Some years before Lukasiewicz, 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 (Bnot-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". 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 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 (), and that he is Callias and not-Callias (), 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 (), 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".
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 Lukasiewicz'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 Lukasiewicz.
Later on, Lukasiewicz 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 nullo is neither a principle of the syllogistic nor an Aristotelian principle, but he will also deny that it is possible to conduct any inference prescinding from the 'metalogical' principle of contradiction (see infra, p. 29). 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), Lukasiewicz 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".
According to Lukasiewicz, 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. On the contrary and nearly at the same time, a similar operation to and independent from Lukasiewicz's was attempted by the Russian Nikolaj Alexandrovich Vasil'év. 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" which he read when he was just seventeen, it came from an article and a short communication by Paul Carus, appearing in The Monist in 1910, in which there were long quotations from Peirce's letters on his studies concerning a non-Aristotelian logic. 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. 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. Unfortunately, Peirce does not say a lot on what he meant by non-Aristotelian logic, except that it is "in the sense in which 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. This is not the place to compare the results, or rather, Peirce's researches with Lukasiewicz's and Vasil'év's. A common element to the three authors, though, can be identified: the reference to non-Euclidean geometries. In particular, Lukasiewicz's and Vasil'év's works - the latter is Nikolaj Ivanovic Lobachevskij'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. 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".
In the attempt to prove the possibility of building a non-Aristotelian logic, Lukasiewicz 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 - Lukasiewicz 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 Lukasiewicz, 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.
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, Lukasiewicz 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 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. Lukasiewicz 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. I have to specify that here Lukasiewicz 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 non-existent or improbable, of which it is impossible to assert anything positive. At this point, the issues is whether or not beings like those described are also capable of thinking in a rational way. And for this purpose Lukasiewicz 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 Lukasiewicz 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 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 - Lukasiewicz ends the story - his hopes are not disappointed.
The whole example wants to be an application to daily life of what Lukasiewicz asserted was already present in Aristotle, i.e., the independence of the syllogism from the principle of contradiction. It seems indeed that Lukasiewicz 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, Lukasiewicz 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 is sufficient 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 well - that Lukasiewicz 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.
It is questionable if we can consider the idea presented herein of a non-Aristotelian logic as a first step by Lukasiewicz towards the construction of a three-valued logic. Of course, the main purpose - as Lukasiewicz 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 Lukasiewicz even in those passages where he expresses himself on the value of a three-valued logic. Lukasiewicz'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. Lukasiewicz 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 Lukasiewicz 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". But he did not abandon the idea of a non-Aristotelian logic entirely. The pursuit of such an idea lead him in 1917 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. Moreover, he will state that the metalogical principle of contradiction, "the principle of consistency", must be assumed absolutely in order to have a logic. 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 Stanislaw Jaskowski. Linking up to Lukasiewicz's book and following some of its suggestions, Jaskowski constructed in 1948 the first propositional calculus for contradictory deductive systems which is recognized as the first system of paraconsistent propositional calculus.
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. Lukasiewicz, 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 are true and for whom the principle of contradiction is not a part of reasoning. Vasil'év, on the other hand, by assuming Sigwart's point of view - rejected by Lukasiewicz - 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.
4. As was said previously, Lukasiewicz 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. 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 those who believe that logic is the science of the necessary laws of thought - among which are Kant and William Hamilton, as well as Sigwart -, 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, Lukasiewicz and Lesniewski. 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.
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 contradiction 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 non-contradictory. 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. 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?
It has already been pointed out that, according to Lukasiewicz, 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 - Lukasiewicz notes - nobody seriously puts in question the principle of contradiction which is fruitfully used both in life and in science. 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 to invalidate the proof it is sufficient to give one case: Lukasiewicz 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 demonstrable that the principle of contradiction is a psychical law (see supra, pp. 13ff.), or that such a necessity is real. Moreover, even if it were a psychical law, the principle would not be demonstrable since nothing can assure us that the external world really corresponds to the necessities of our intellect.
(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 - Lukasiewicz asserts - not on the basis of experience but according to definitions. 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". 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. 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)";
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. 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 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. In short, Lukasiewicz 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".
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, who also confront themselves with the same problem, Lukasiewicz 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 - Lukasiewicz says - we can take others from geometry: 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. In short, words can have meanings although they indicate something which does not exist and which is even contradictory. Lukasiewicz 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 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 Lukasiewicz gives him credit for being the first at expressing such an opinion. This anticipates the results to which the continuation of the argumentation will lead. That's how we reach Lukasiewicz's proof which is based on the assumption that objects are non-contradictory.
(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.
Here the necessity is asserted to move from the definition of (non-contradictory) 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; but Lukasiewicz'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. Lukasiewicz'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. Lukasiewicz 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 logic as it was elaborated by Peano, Frege and Russell, on the basis of which he tries to understand again the meaning, the value and the limits of the principle of contradiction. Lukasiewicz himself, in fact, does not believe much in his proof, defined by him as "too easy, economical and superficial!"; however he does consider it as a starting point for further researches. In fact, two years later will appear Lesniewski's essay, who, however critical towards Lukasiewicz's work, recognizes the large debt he owes to it and the many stimuli that he has received from it. Unlike other authors, Lukasiewicz 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 - Lukasiewicz 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 non-contradictory. For this purpose - Lukasiewicz 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 Lukasiewicz makes a distinction - taken from Meinong, of whom he quotes not a specific work, but the lectures of the winter semester 1908/1909 - 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 for which 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 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. 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.
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.).
(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 non-contradictory 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 non-contradictory. As an example Lukasiewicz mentions Russell's antinomy which touches on the logical foundations of mathematics. 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 so even 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 - Lukasiewicz 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.
"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".
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.
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 non-contradictory cannot be given. In the same way, as we cannot assert with certainty that all constructive objects are non-contradictory, so we are not assured that all real objects are non-contradictory 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 Lukasiewicz 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 Lukasiewicz 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 in consequence of the intellectual and moral imperfection of man it constitutes the only weapon that man has against error and falsehood.
5. In the appendix "The Principle of Contradiction and Symbolic Logic", Lukasiewicz tries to give, together with some outlines of mathematical logic, an essay on the principle in light of the new 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 Lukasiewicz 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). The important result for the purpose of this work is not the expositive part, but the distinction, pointed out by Lukasiewicz, 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 Algébre de la Logique, Lukasiewicz 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 Lukasiewicz 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
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.
This principle - as Lukasiewicz 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. Lukasiewicz 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) (ba)], 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)], that:
((a a) 0) (1 (a a));
(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.
Lukasiewicz'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 according to the two reconstructions above, Lukasiewicz 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), which, taken one by one, express each a thought simpler than the one of the principle which together they justify.
It may be objected - as Lukasiewicz 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);
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 Lukasiewicz takes on the task of proving, on the contrary, that (4) and (1) are not synonymous.
The distinction between (4) and (1) which Lukasiewicz 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.
Lukasiewicz does not stop here. Assuming 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, Lukasiewicz 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 be referable, 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 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 Lukasiewicz. 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, Lukasiewicz 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.
Conclusion. Some points that emerged from the discussion of Lukasiewicz'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, Lukasiewicz never loses touch with the 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, Lukasiewicz 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 logical systems 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 Lukasiewicz 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 kept in mind that Lukasiewicz'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 Lukasiewicz 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, Lukasiewicz presents a "pioneer" work compared to the logical developments of the 20th century as to the principle of contradiction.
At last, another result implied in Lukasiewicz's argument (of which probably Lukasiewicz 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 Lukasiewicz are too strong to discriminate between the principle of the excluded middle and the principle of contradiction. Lukasiewicz 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 of more 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 Lukasiewicz.
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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.
 This passage is important because we find here the only mention of the principle of the excluded middle; otherwise there would arise the impression that Lukasiewicz 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 Lwow, the 26 February 1910, at the Polish Philosophical Society. In the English translation of the short summary which remains of his talk, Lukasiewicz (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 Slupecki (1958: 14-15), Wolenski and Simons (1987: 67-68), Wolenski (1990b), and Simons (1992: 197-198).
 According to Priest and Routley (1989: 25ff.), although Lukasiewicz 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".