Papers on Lukasiewicz


Lukasiewicz, Aristotle, and Contradiction

Owen LeBlanc



In 1910 the Polish philosopher Jan Lukasiewicz (1878-1956) published his first book, 0 Zasadzie Sprzecznosci u Arystotelesa: Studium Krytyczne, which means On the Principle of Contradiction in Aristotle: A Critical Study. I would like to discuss this book and present some of its contents. I hope that I can interest you in learning more about it.


1. Tributes

For many years my own knowledge of the book was based on three brief accounts of it. In 1975 I read an article by Boleslaw Sobocinski , written at the time of Lukasiewicz's death. Speaking of this book, he says:

We find a detailed analysis of all the passages in Aristotle which relate to this topic [i.e., to the principle of contradiction]. All possible interpretations of these texts are presented and discussed thoroughly. The problems raised by the principle of contradiction are presented in so clear and precise a way that even a mind untrained in philosophy can understand them at once. Moreover, Lukasiewicz for the first time used the methods of symbolic logic in order to obtain a strict analysis of the problems under investigation.... The influence of this book was extremely great on Polish philosophers of that generation, regardless of their philosophical schools. This book set Lukasiewicz at the forefront of the Polish philosophers of his time. Looking back on it over a span of forty-six years, we can say that its historical and analytic parts are still the best that have appeared on this subject. Even now, if this book were translated into other languages, it would gain fresh renown (Sobocinski 1956, pp. 10-11).

He goes on to say that on its logical side the book is dated and sometimes in error; for our knowledge and understanding of logic have grown, partly of course in consequence of Lukasiewicz's own later discoveries. Because Sobocinski's article is admittedly a eulogy of Lukasiewicz, we must read it with some caution to filter out some of its excessive enthusiasm.

Later in 1975 I was reading the philosophical autobiography of Stanislaw Lesniewski, where I found the following:

In 1911, during my years as a student, I came across Jan Lukasiewicz's book On the Principle of Contradiction in Aristotle. In its time this book significantly influenced the intellectual development of a succession of Polish 'philosophers' and 'philosophising' scholars of my generation, and it was in many respects a revelation to me personally. From it I learned for the first time of the existence of 'symbolic logic', of Bertrand Russell, and of his 'antinomy' concerning 'the class of the classes which are not elements of themselves’ (Lesniewski 1927, p. 169).

Finally in the Encyclopedia of Philosophy I found an article on Lukasiewicz by Czeslaw Lejewski , which states:

On the Principle of Contradiction in Aristotle was one of the most influential books in the early period of the twentieth-century logical and philosophical revival in Poland. It must have stood high in the author's own estimation, for in 1955 he began translating it into English (Lejewski 1967, p. 104).

These tributes aroused my interest, but for many years I was unable to obtain a copy of the book, and there matters rested.

Lukasiewicz did publish in 1910, the year in which the book appeared, a German summary of it, which is twenty-three pages long, about one tenth of the length of the book (Lukasiewicz 1910a). This summary has been reprinted (in German), and has appeared in two English translations, and once in French. These summaries have been the primary means of access to the original for most European readers. Clearly a short summary must omit most of the content of the longer work.


2. History

I now have some understanding of why this book was so influential. Like many of Lukasiewicz's other works, it covers a wide variety of topics. Some readers might have felt the book contains too much peripheral matter, but most found something in it that encouraged them to read it. Shortly after its publication, Lukasiewicz became very well known. In 1915 he was appointed Professor of Philosophy in the Faculty of Mathematics at the newly reopened University of Warsaw. Two years later he was elected Rector of the University. For a year in 1919 he joined the Cabinet as Minister of Education. Later his textbook, The Elements of Mathematical Logic, was popular in schools and universities. These circumstances surely encouraged and maintained interest in his works.

Moreover, Lukasiewicz was a good stylist. He was always clear and persuasive. He was a gifted orator, and his writings are said to read well aloud. His works thus appealed to many readers who knew little of the subjects he discussed. There were articles in Polish which mention symbolic logic as early as 1888, but this book, which appeared in the same year as Whitehead and Russell's Principia Mathematica, became the first popular introduction to the theory in Poland. One copy, which was part of a library for Polish soldiers in England during the Second World War, contains many underlinings and marginal notes by at least two of its readers.

At the end of that war, Lukasiewicz and his wife escaped from Warsaw only two weeks before the Uprising, which destroyed everything they left behind. They intended to go to Switzerland, but ended up in Belgium, where Lukasiewicz taught children in a Polish refugee camp. He brought a few of his papers with him, but no books. Early in 1946 the Irish government invited him to Dublin, to lecture on logic at University College. On his way to Dublin, he spent two days in London, where he happened to meet one of his former students, Czeslaw Lejewski, who later completed his second doctorate at the University of London under Karl Popper. Lukasiewicz was asked to come to London to be Lejewski's external examiner, and there Lejewski presented him with a copy of The Principle of Contradiction in Aristotle which he had obtained there. It was this copy that Lukasiewicz used when he began his English translation of the book less than a year before his death.

Lukasiewicz suffered from severe kidney problems, and he was unable to complete the translation before he died in Dublin on February 13, 1956. 1 believe his wife placed the Polish copy of the book in his coffin. Lejewski was asked to take charge of Lukasiewicz's remaining papers. It was he who saw the second edition of Aristotle's Syllogistic through to publication, correcting the proofs of the new chapters and enlarging the index. He brought Lukasiewicz's incomplete translation of the older book back to England, and in 1994 he asked me to revise and complete the project.


1.3. Lukasiewicz's English version

Lukasiewicz was a methodical worker. He produced at least two draughts of everything he wrote, even letters. The first draught of his English version is lost, and was probably not much longer than the second draught, of which we have what appears to be a carbon copy. It begins on April 6, 1955-almost every page is dated at the top-and it ends in the middle of a sentence at the bottom of a page on September 27 of the same year. It is possible that further pages have been lost. He wrote one page of translation every day, except apparently when travel or illness prevented him. There are 100 pages, which cover pages 1 to 118 of the original book, which was 212 pages long.

The first page may serve to illustrate the difficulties of revision. I think you can 'taste' Lukasiewicz's oratorical style even in translation:

There are two moments in the history of philosophy when contr6versy about the principle of contradiction excited the minds: with one of them is connected the name of Aristotle, with the other the name of Hegel. Aristotle has formulated the principle of contradiction as the highest law of thinking and being. In a stubborn polemic, in which indignation and contempt is vibrating, he persecuted all those who would not accept this law.. .. [I skip over their names.] He has won this fight, and so great was the force of his arguments, or so right the cause defended by him, that through whole centuries nobody dared to deny that highest principle. Only, Hegel revived the opinions buried by Aristotle, i [the Polish word for 'and'] let us believe that reality is reasonable and contradictory at the same time. He restored the esteem to the Greek Sophists, and included the statements of Heraclitus into his system of logic. This doctrine caused again an ardent discussion: they strived to bury Hegel with the words of Aristotle. These contr6versies have now died away, and the problem of the principle of contradiction is not actual. The better, as it can be considered sine ira.

Instead of reading out my own version of this passage, I shall point out some of the difficulties in this one:

1. The word 'moments' does not express 'extended periods of time'.

2. 'Excited the minds of philosophers'.

3. It is Aristotle, not the polemic, who is stubborn.

4. 'Vibrating' does not work. Lukasiewicz intends to suggest that Aristotle is unusually upset, that he has lost his self-control.

5. 'So great was the force': the word 'evidently' has been omitted.

6. 'So right the cause': it would be better to say 'so just the cause' to make it clear that moral values are involved.

7. 'Only Hegel': the original actually says 'but Hegel', but the translation sounds as if it means 'Hegel alone'.

8. 'Opinions buried by Aristotle': the phrase 'long before' is omitted.

9. 'The problem of the principle of contradiction is not actual': the word 'actual' in English does not mean 'a subject of current interest'.

10. The expression 'sine ira' is not a Latin phrase familiar to most English readers.

Besides these specific points, Lukasiewicz has the usual difficulties with definite and indefinite articles and with word order in English, though I must admit that some other Poles are worse.

Lukasiewicz's translation is fairly intelligible and does present a valuable guide to the original text. Moreover it expresses his wish to change the format of the text slightly. For example, in the original book Greek, German, Latin, and other foreign quotations appear in the text, followed by a Polish translation. But in the manuscript Lukasiewicz moves most of the foreign text into footnotes; only the English translations appear in the text.

Incidentally, Lukasiewicz's knowledge of English dates back many years. He had at least a reading knowledge of it by 1905, when in collaboration with Kazimierz Twardowski he produced a Polish translation of Hume's Essay concerning Human Understanding, which (being a translation) does not usually appear in the lists of Lukasiewicz's publications. I don't believe he tried to speak English before he went to Dublin in 1946, and his first article actually written in English was published two years later, in 1948.

Before we continue I should like to point out that the epigraph at the top of the first page is a twist on a phrase by Aristotle. He wrote

M G 6 1011 a 12 They look for a proof of things which have no proof.

and Lukasiewicz says

We look for a proof of things, which do have a proof.


4. Summary of the book

The book consists of an introduction, twenty-one sections each about eight pages long, and an appendix containing nine further sections.

In the introduction, Lukasiewicz remarks that the study of the philosophical foundations of arithmetic and geometry led to a better understanding of those fields and to new developments in them. So too reexamining the foundations of logic should prove fruitful. He knew that such work had begun-he mentions Russell, Couturat, Frege, Peano, Hilbert, and others-but he saw in Aristotle the historical origin of logic in a way in which Western logicians did not.

Western (by which I mean mostly English-speaking) logicians (British, American, and later Australian logicians) often hold Aristotle in contempt, but in Poland, at least before the Second World War, Aristotle was more highly respected, and most Polish logicians felt that they were building on and developing his tradition rather than replacing it. If the Western view is largely due to Bertrand Russell, the Polish view is certainly indebted to Lukasiewicz. Besides Lukasiewicz himself, Stanislaw Lesniewski, Tadeusz Kotarbinski , Jan Slupecki , Czeslaw Lejewski, and others lectured on and wrote about Aristotle's logic. Lukasiewicz also promoted a new interest in and understanding of the Stoics and of medieval logicians. A respect for the logical tradition is evident throughout the Principle of Contradiction in Aristotle.

The first fifteen sections of the book are concerned more directly with Aristotle; we shall look at these more closely later. The next five sections are more concerned with contemporary ideas. For example, section 18, 'The principle of contradiction and mental constructs', discusses Meinong's complete and incomplete objects, abstraction, Dedekind's idea that mathematical theories are 'free creations of the human spirit', the foundations of set theory, Russell's and other antinomies. Here Lukasiewicz suggests that, while the principle of contradiction does obtain in the 'real world', it may not apply to 'free creations of the human spirit', that there may possibly be 'contradictory a priori objects'. This view he later rejected.

At the end of the main part of the book, Lukasiewicz sees the principle of contradiction as our only weapon against error and falsehood. He wants it to be true, but feels that we can never establish it with certainty. Still, he says, we ought to accept it as probable, at least as far as the 'real world' is concerned. Aristotle feared the consequences of rejecting it, and so accepted it as an axiom, as unassailable dogma.

The Appendix, an introduction to symbolic logic, was evidently written before Lukasiewicz came to be familiar with the 'theory of deduction', later known as the 'propositional calculus', as a distinct branch of logic. For this reason he based his presentation on the better known turn-of-the-century logicians, and particularly on a textbook by Couturat. Nevertheless it contains some ideas which Lukasiewicz originated, and which had some influence on later Polish philosophers. For example, in the following section he classifies valid reasoning into four divisions. He says

The relation of consequence is an asymmetric relation; that is, when it occurs in the direction from a to b, it can but need not occur in the direction from b to a. This feature is the basis for dividing reasoning into deduction and reduction. Deductive reasonings run in the direction of the relation of consequence, and reductive reasonings run contrary to the direction of this relation. So I reason deductively when from a given judgment a I infer a judgment b, which follows from a. I reason reductively when for a given judgment b I find a judgment a from which b follows.

The two kinds of deductive reasoning are inference and testing. Inference occurs when the point of departure of reasoning is a judgment, which is certain, and I prove some consequences from it. [I omit the examples Lukasiewicz gives for each kind of reasoning.] Testing occurs when the point of departure of reasoning is an uncertain judgment, which I try to make probable by finding true consequences. [He means also that if he finds false consequences, the starting point fails the test.] The two kinds of reductive reasoning are proof and explanation. Proof occurs when the point of departure of reasoning is an uncertain judgment for which I seek true grounds. [In other words, if he can infer it by valid inferences from premises, which are certain, he has proved it.] Explanation occurs when the point of departure of reasoning is one or more certain judgments, and I seek some reason from which these judgments follow.

Note that here Lukasiewicz regards the traditional process of induction as just one kind of ‘explanation’.

I hope you can see that the Principle of Contradiction in Aristotle covers a wide range of subjects, most of which continued to interest Lukasiewicz for the rest of his life. And now I would like to look more closely at his interpretation of Aristotle.


5. Formulations of the principle of contradiction

Aristotle has several formulations of the principle of contradiction, which Lukasiewicz classifies into three groups.

The ontological principle of contradiction states that no object can have and not have the same property. Here are two of the texts which Lukasiewicz says express this form of the principle:

1. M G 3 1005 b 19-20 For the same [property] cannot together belong and not belong to the same [object] and in the same respect.

2. M B 2 996 b 29-30 And it is impossible together to be and not to be.

The logical principle of contradiction states that two contradictory sentences cannot both be true. A text expressing this formulation is:

3. M G 6 1011 b 13-14 So that this is the most certain of all opinions: contradictory sentences are not together true.

The psychological principle of contradiction states that two beliefs corresponding to contradictory sentences cannot exist together in the same mind. A text expressing this formulation is:

4. M G 3 1005 b 23-26 For it is impossible for someone to believe that the same is and is not, as some people think Heraclitus said, but it isn't necessary that someone believes what he [someone] says.

Lukasiewicz translates upolambanein as 'believes', but he takes great pains to justify this, referring not only to the commentators Schwegler and Maier but also to a series of passages taken mostly from the de Interpretatione. He might have used the word 'judgment' instead of 'belief' if he were not already using this term to mean 'sentence' in harmony with the usage of late nineteenth century philosophers. He says that a belief (doxa or upolepsis) is a psychic act or decision accepting something as true. I would distinguish this belief from pistis, which seems to be a state of mind resulting from making a judgment.

Lukasiewicz asks whether it is possible that the three formulations, which Aristotle does not explicitly distinguish, express three distinct principles or just one principle in three different ways. To answer this he asks, when do two sentences express the same thought in different ways? The sentences 'Aristotle was the creator of logic' and 'the Stagyrite was the creator of logic' appear to express the same thought in different words, because they both refer to the same object, the same man. Two sentences are equivalent when the first follows from the second and the second follows from the first. (In Aristotle's Syllogistic Lukasiewicz presents a more mature version of the same notion as deductive equivalence.) Two sentences are synonymous when they both attribute or deny the same property to the same object.

From this it follows that synonymous sentences are always equivalent to each other, but some equivalent sentences are not synonymous. For example, 'Aristotle was a pupil of Plato' and 'Plato was a teacher of Aristotle' are clearly equivalent, but since the first sentence is about Aristotle, while the second is about Plato, they are not synonymous. (In his later works, Lukasiewicz proposes a more complex notion of synonymity because of his rejection of the Aristotelian notion that every sentence asserts or denies that some object has some property. Even here he states that he feels his definition of synonymity is not generally adequate, but that it is sufficient for his present purpose.)

If we accept these ideas, however qualified, we can see that the three formulations of the principle of contradiction are not synonymous, because the ontological formulation refers to objects and properties, the logical formulation to sentences, and the psychological to beliefs. This argument has a considerable influence on other Polish philosophers; for example, a few years later Lesniewski argues against Whitehead and Russell that the sentences 'p' and 'it is true that "p" cannot be synonymous, since, for example, the sentence 'Warsaw is a city on the river Vistula' refers to Warsaw, while the sentence 'it is true that "Warsaw is a city on the river Vistula"' does not refer to Warsaw, though it does refer to another sentence which refers to Warsaw.

Even if two sentences are not synonymous, they may be equivalent. Lukasiewicz argues that although Aristotle does not state this explicitly, the ontological and logical principles of contradiction are equivalent on Aristotelian grounds. For example,

5. E 9 18 a 39-18 b 1 If it is true to say that something is white or not white, then it must be white or not white.

The context, says Lukasiewicz, makes it clear that this example is typical; that is, that any property at all can replace 'white'. Therefore

a) If we had an exception to the logical principle of contradiction, we would have two true contradictory sentences.

b) One of these sentences would affirm that some object has some property, and the other would deny that the same object has the same property.

c) Following Aristotle's example, we must infer from the first true sentence that the object does have the property, and from the second that it does not have the property. This violates the ontological principle of contradiction.

In other words, if the logical principle fails, so does the ontological principle. Therefore if the ontological principle holds, so does the logical principle. The converse is also true if we accept certain other statements by Aristotle. For example,

6. E 9 18 b 1-2 And if [something] is white or not white, it would be true to affirm or deny this.

7. M Q 10 1051 b 3-4 So that he speaks truly who regards what is disjoint as being disjoint, and what is conjoint as being conjoint.

From examples like these Lukasiewicz infers that for Aristotle, if an object has a property, a sentence which affirms that the object has the property is true, and that if an object does not have a property, a sentence which denies that the object has the property is true. Therefore

a) If there were an exception to the ontological principle of contradiction, some object would both have and not have some property.

b) According to Aristotle, a sentence affirming that the object has the property, and a sentence denying that the object has the property, are both true.

c) These true sentences are contradictory. This violates the logical principle of contradiction.

In other words, if the ontological principle fails, the logical principle fails as well. Therefore if the logical principle holds, so does the ontological principle. If Aristotle would accept these arguments based on his own views, he would accept that each of the principles, ontological and logical, follows from the other, so that, while they are not synonymous, they are deductively equivalent. Lukasiewicz feels this is in complete harmony with Aristotle's well-known definition of truth: 1011b

8. M G 7 1011 b 27 Truth is to say of what is that it is, and of what is not that it is not.

Moreover Aristotle makes statements which suggest that he would say that, while the two principles are logically equivalent, they are in some sense not really equivalent. For example,

9. M Q 10 1051 b 6-9 For you are not white because we hold truly that you are white, but because you are white we who say so tell the truth.

Hence if a sentence is true which ascribes some property to some object, it follows logically that that object has that property, but the circumstance that the object has the property is not only a logical reason for the sentence being true, but also the real cause why it is true. Lukasiewicz says that, if Aristotle had been clearly aware of this distinction, he would have accepted it, as passage 9 shows.


6. The psychological principle of contradiction

Aristotle assumes tacitly that the ontological and logical principles of contradiction are equivalent, and emphasises that they are ultimate principles and not provable. But he does try to prove the psychological principle explicitly from the other two. The proof has two parts, of which the first is found in this passage:

10. M G 3 1005 b 26-32 If it is not possible for contrary properties to belong together to the same [object], and contrary beliefs correspond to contradictory sentences, then clearly it is impossible for someone to believe together that something is and is not, for he would then have contrary beliefs together, being mistaken in that respect.

At first glance this passage appears to confuse contradiction (antifasis) with contrariety (enantiousis). A solution was found by Alexander of Aphrodisias and accepted by Maier; it is based on a long passage in the de Interpretatione, chapter 14, p. 23 a 27-24 b 3. Lukasiewicz explains the passage as follows: According to Aristotle, the relation of contrariety exists between properties at extreme ends of a series. Sentences are not properties, so we cannot strictly speaking say that they are contraries. But sentences expressed in words correspond to beliefs in the psyché and Aristotle treats beliefs as properties of the mind in which they exist. Being properties, beliefs can be contraries. Aristotle then tries to show that beliefs which correspond to an affirmation and a negation about the same object are contraries; for example, the belief that Callias is just and the belief that Callias is not just. Lukasiewicz concludes that the following represents Aristotle's view:

A doxa or upolepsis is a psychic act which exists in thought (en te dianoia) or in the soul (en te psyché). In language (en te foné) of this as a symbol (symbolon) there is a sentence, which is either an affirmation (katafasis) or a negation (apofasis).

Lukasiewicz believes that the following statement in the de Anima supports his interpretation:

11. Y G 3 428 a 20-21 For it is not possible for someone who has an opinion not to believe in his opinion.

I must say that to me this suggests that 'belief' is the wrong way to translate ‘doxa’, since it is a different kind of belief from 'pistis', unless Lukasiewicz means that 'pisteuein' and 'doxazein' and 'upolambanein' are the same, which I don't think he does. At any rate, it is the above interpretation which he feels justifies him in translating 'enantia d’esti doxa doxe e tes antifaseos’ in passage 10 as 'contrary beliefs correspond to contradictory sentences'.

Bearing all this in mind, Lukasiewicz rephrases the first part of Aristotle's proof of the psychological principle of contradiction as follows:

No object can have contrary properties at the same time. If someone could believe that something is and at the same time believe that it is not, he would have two contrary beliefs, so that his mind would have two contradictory properties. Therefore no one can believe that something is, and at the same time believe that it is not. In other words, two beliefs corresponding to contradictory sentences cannot exist together in the same mind.

To prove the psychological principle of contradiction, Aristotle has only to prove that no object can have two contrary properties at the same time. To do this he makes use of the logical principle of contradiction, reasoning as follows (This is my paraphrase of Lukasiewicz's paraphrase of Aristotle):

Whenever two properties are contrary, there must be some property which one contrary involves having, while the other contrary involves lacking it. Thus, for example, white and black are contrary properties. There must be some property, such as reflecting light, that being white involves having, while being black involves not having. So if some object is white and the same object is black, it follows that it is true that the object has the property of reflecting light, and it is true that it does not have the property of reflecting light. This violates the logical principle of contradiction. Therefore no object can be together black and white. Similar considerations obtain for any other contrary properties.

To support this claim, Lukasiewicz cites passage 12:

12. M I 4 1055 b 18 Every contrariety contains lack of one of the contrary properties.

though he admits that this is not as clear as it might be. But he feels that this same principle is invoked in passage 13:

13 M G 1011 B 15-21 Since it is impossible for contradictory sentences to be together true of the same object, it is evident also that contrary properties cannot together belong to the same object. For one of two contraries is no less a defect, a defect of a thing. [That is, it is the lack of something as well as being a contrary.] But defect is negation of a certain kind. Therefore, if it is impossible together to affirm and to deny truly, is also impossible for contrary properties to exist together.

The two parts of the proof establish the psychological principle of contradiction from the logical principle.


7. The proof of the psychological principle of contradiction

Is Aristotle right to say that two beliefs which correspond to contradictory sentences must contain a contrariety, a hidden contradiction? In the de Interpretatione Aristotle defines contraries as follows:

14. E 14 23 b 22-23 Contrary properties are those most different in the same respect.

So Aristotle needs some principle which establishes an order among beliefs, and hence determines two extreme ends of a series, which are contrary to each other. In chapter 14 of the de Interpretatione, just before passage 14, he argues that there is a scale of truer and falser beliefs, with the truest belief being one which ascribes an essential property to an object, while the falsest belief denies to an object an essential property which it actually has. Lukasiewicz argues:

It is impossible to agree with this. We cannot accept that there are differences of degree in truth or falsity. If a sentence ascribes to an object a property which it has, the sentence is true regardless of whether the property is essential or accidental, or belongs to the object permanently or temporarily.. .. If we want to accept the existence of sentences which are more true or less true, we must change the definition of truth.

He has a more serious objection to Aristotle's argument about contrary beliefs and contrary sentences. He says,

In chapter 14 of the de Interpretatione there appears, probably for the first time in the history of philosophy, a now too common confusion of logical with psychological questions. . .. In order to solve... a logical problem, Aristotle tacitly assumes the false supposition that the relations between sentences are the same as the relations between beliefs.

He goes on to say that, being unable to give an accurate psychological analysis of the relations between beliefs, Aristotle actually treats them as sentences. He ascribes to beliefs relations which exist only between sentences, such as the relation of consequence, and properties which belong only to sentences, that is, truth and falsity. After several pages of argument, Lukasiewicz concludes:

The continuous confusion of beliefs and sentences is why the psychology of knowledge consists chiefly of logical analyses based on a priori suppositions instead of on experience. The argument in chapter 14 of the de Interpretatione belongs to such pseudo psychological analyses.

And therefore Aristotle's proof of the psychological principle of contradiction is a failure.

The fact that an argument is bad does not make its conclusion false. But Lukasiewicz argues that beliefs are mental phenomena, and that such objects of experience have relations which cannot be investigated adequately by reasoning from a prioristic premises. We could, of course, assume axioms and infer conclusions from them. We could demonstrate that such an axiom is false by deriving false conclusions from it, but even if we found only true conclusions, the axioms would never become certain, only probable at best, since we would be employing the kind of reasoning he calls 'testing'. In the light of these considerations, the psychological principle of contradiction is at best an empirical law. After examining some arguments from Husserl, Lukasiewicz comes to doubt that the psychological principle of contradiction is in fact true. Looking at the history of philosophy, he finds three possible counter examples:

a) Heraclitus, although no actual instance of contradictory beliefs appears to have survived in Heraclitus's own words.

b) Hegel, who says clearly and unambiguously, 'Something is in movement ... because in one and the same 'now' it is here and not here, and in that here it simultaneously is and is not.

c) Lukasiewicz, because he accepts the Athanasian creed, which contains apparently contradictory statements.

He admits that none of these examples disproves the psychological principle of contradiction to anyone else, because after all, counter examples must exist in one person's mind, and so cannot be exhibited to someone else. You may claim to have contradictory beliefs, but, as Aristotle says in passage 4, it is not necessarily the case that someone believes what he says. But whether you accept Lukasiewicz's arguments or not, he feels that the psychological principle of contradiction is uncertain and doubtful, and hence not suited to be a fundamental principle of logic, so that we can dismiss it from further consideration. He concludes, 'The way to the fundamental principles of logic does not lead through psychology'. And this conclusion was readily accepted by all the Polish philosophers whom I have studied.


8. Conclusion

In the remainder of his book, Lukasiewicz explains why he feels that Aristotle becomes upset by the possibility that someone might reject the logical or ontological principle of contradiction, why he insists that they are indemonstrable, how he nevertheless tries to prove them in five different ways, and that each proof either proves something else, something which is not the principle of contradiction, or else commits some logical error which makes it invalid. After examining the relations between the principle of contradiction and other laws, Lukasiewicz observes that it is not in fact very useful as a logical tool, and consequently he calls into question its status at the most fundamental of all principles.

We do not have the time to examine these arguments in detail. But I hope that what I have said has convinced you of three things:

First, that Lukasiewicz's disagreements with Aristotle are based on a careful examination of a large number of texts in several of Aristotle's works. Consequently the summary of this book (which is more accessible to those who do not read Polish) does not give a complete or adequate account of its contents.

Second, that it is reasonable to accept the statements by Sobocinski, Lesniewski, Lejewski, and myself. this book appealed to a wide variety of readers, and so had a considerable influence on the development of Polish thought in the early part of this century.

Finally, that this work is still of interest today, so that if I am able to complete a translation of the book, you might be willing to read it.



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