Logic Of Contradiction: On Łukasiewicz’s critique of the Aristotelean formulations of the principle of contradiction

II — On the Surpassing of the Aristotelean Logic through the Refinement of Symbolic Logic

§3. Aristotelean attempts at demonstration and the so-called philosophical logic

At this point, where the psychological formulation has been eliminated, and a logical proof has been provided that the logical and ontological formulations are equivalent, we can proceed to Aristotle’s attempts at demonstration, and to the formulation of this principle from the standpoint of modern symbolic logic. In this section, I shall expose the attempts at demonstration made by Aristotle, and expose the limitations of Aristotle’s traditional logic. And in the next section, I shall establish how the formulation of this principle is carried out through modern symbolic logic — as opposed to traditional logic —, and introduce fundamental concepts to proceed to the refutation of Aristotle’s formulations. This refutation, in turn, will constitute the last section of this chapter, which will close the negative part of my work.

As it was already established, Aristotle considered the principle of contradiction to be indemonstrable. However, he strived to demonstrate its validity, nonetheless. He listed elenctic and apagogic proofs in order to prove the principle of contradiction. Elenctic proofs are of a dialectic nature, which proceed by establishing contradictions and inconsistencies within an opponent’s belief. Apagogic proofs, in turn, proceed by establishing the impossibility or the absurdity of the contrary, i.e. the validity of a conclusion is established by demonstrating that all other alternatives are impossible or absurd. The first elenctic proof consists of:

“If one can truly say of something that it is man, it is necessary that it be a two-legged living creature; for it was that which the word “man” signified. If, however, this is necessary, so it is impossible that the same thing not be a two-legged creature. For necessity means just the impossibility of not being. Accordingly, it is not possible to assert at the same time that the same thing is man and is not man (respectively, two-legged living creature).” (Met. IV 4, 1006b 28-34, translated by V. Wedin)

The second elenctic proof consists of:

“Suppose a word to be given which signifies something and in particular something singular. Then it is not possible that being a man means the same as not being a man, so far as the word “man” signifies something singular. Consequently, one and the same thing can be and not be only homonymously, as when that which we call man others want to call not man. But the point does not turn on whether one and the same thing can be named man simultaneously, but whether it can be so.” (Met. IV 4, 1006b 11-22, translated by V. Wedin)

As for the apagogic proofs, there are three most important ones. The first apagogic proof consists of: “if all contradictory propositions were true at the same time in respect to the same thing, then clearly everything will be one. For a trireme, a wall, and a man would then be the same” (Met. IV 4, 1007b 18-21, translated by V. Wedin); the second one consists of: “it follows that every one speaks the true and the false and must admit that lie speaks the false. (Met. IV 4, 1008a 28-30, translated by V. Wedin); and the third one consists of:

“One can clearly see that no one believes such a thing, neither anybody else nor one who practices such rhetoric. For why does such a one still go to Megara instead of quietly sitting at home with the thought that he is going? Or why does he not one fine morning immediately throw himself into the well or abyss, when it is directly come upon; rather he takes care, as if he is of the opinion that falling in is not equally good and not good.” (Met. IV 4, 1008b 12-19, translated by V. Wedin)

Furthermore, having established the most important Aristotelean attempts at demonstration, it is fundamental for our understanding to distinguish between symbolic logic and the so-called “philosophical logic”, which Łukasiewicz considered to be “nothing more than bold phraseology” (1971, p. 493). What Łukasiewicz calls “philosophical logic” not only encompasses Aristotelean discourse, but also extends to any extrapolation of natural language that aims to achieve the same clarity as symbolic logic. This philosophical logic has no commitment to the clarity of the questions it seeks to answer, precisely because it lacks conceptual refinement. To clearly distinguish between concepts, it is fundamental to prioritise the delimitation of concepts and their determination through the use of specific symbols. Thus, it is possible to avoid ambiguity in discourse and not succumb to error. In Łukasiewicz’s words, it is precisely because of this that the philosophical logic “sank into the swamp of the fluid and vague speech used in everyday life” (1971, p. 494).

The error Aristotle made — when attempting to determine the position of the principle of contradiction among the axioms of logic — stems from the ambiguity of natural language, which was the sole tool at his disposal in his time. This becomes quite evident when considering the concepts of negation, double negation, logical multiplication, logical addition, and so forth. These concepts, when viewed separately from the framework of symbolic logic, can easily be confused, as is the case in Aristotle’s work. That the principle of identity should not be confused with the principle of contradiction, nor the principle of contradiction with the principle of double negation, will be demonstrated in the following sections. To achieve this, given that it has been established that traditional logic lacks conceptual refinement and, therefore, cannot achieve surgical clarity, it is fundamental to demonstrate the efficiency and precision of symbolic logic.

§4. Symbolic logic, algebraic logic and the surpassing of traditional logic

Symbolic logic, also known as mathematical logic, “is the set of logical theories elaborated in the course of the last Nineteenth Century with the aid of an artificial notation and a rigorously deductive method” (Bochenski, 1959). In symbolic logic, the algebra of logic is the specific method of representation and manipulation of logical propositions as algebraic objects. It was founded by Boole, developed by Schröder and Couturat, and one of its main characteristics is that it allows the simplification of complex logical structures. This simplification was impossible in traditional logic, for it lacked an artificial notation that could precisely replace natural language, as already established. One of the fundamental elements in algebraic logic which made this simplification possible is the use of “0” and “1”, special terms introduced in logical calculus. The “0” “is the ‘null’ or ‘void’ class which contains no element (Nothing or Naught)”, the “1”, in turn, “is the totality of the elements which are contained within it. It is called, after Boole, the ‘universe of discourse’ or simply the ‘whole’” (Couturat, 2004). In other words, the “0” denotes the “false” or the “absurd”, and the “1” denotes the “true”. Another element which made the simplification of logical structures possible is that “the algebra of logic admits of three operations, logical multiplication, logical addition, and negation. The two former are binary operations, that is to say, combinations of two terms having as a consequent a third term which may or may not be different from each of them” (Couturat, 2004). Due to the introduction of “0” and “1” in logical calculus, it became possible to define many principles of logic through algebraic logic. The definition of negation, for example, is possible through an operation “which transforms a single term into another term called its negative” (Couturat, 2004, emphasis in original). Let us use the letter a to denote a proposition, and let us call the negation of a “not-a”, which is written as a’. Thus, we are left with the following axiom: whatever the term a may be, there is also a term a’. Such that:

a x a’ = 0

a + a’ = 1

It should be noted that, besides the “0” and “1”, the formulas also contain the basic operations of algebraic logic, i.e. logical multiplication, addition and negation. The proposition a stands for “1”, for it is true and denotes the whole, the universe of possible discourse. Its negation a’, in turn, denotes the false and the absurd, that which cannot be the case. Thus, the first formula, which is a logical multiplication, results “0”, for “0 x 1 = 0”; the second formula, in turn, is a logical addition and results “1”, for “0 + 1 = 1”. As for the former, one could think of the classical propositional logic operation of conjunction, which is true if, and only if, both propositions are true. As for the latter, one could think of the classical propositional logic operation of disjunction, which is true if, and only if, at least one of the propositions is true. Hence the efficiency of algebraic logic in employing its basic operations in order to simplify logical calculus, and still clearly communicate the operations of logic. So much so that it is possible to establish two main principles of logic after what has been established: the principle of contradiction and the principle of the excluded middle. These two principles derive, by definition, from the two preceding formulas, i.e. logical multiplication and addition:

a x a’ = 0

a + a’ = 1

As for the first formula, it is not possible for a and a’ to be true at the same time. Thus, the simultaneous affirmation of a and a’ is false. As for the second formula, for every proposition, either the proposition or its negation is true. Thus, the alternative affirmation of a and a’ is true, for one of the propositions is true. This conforms with the fact that “0” and “1” are the negatives of each other. Thus:

0 × 1 = 0

0 + 1 = 1

It is primarily in this way that algebraic logic, as a specific branch of symbolic logic, is able to avoid ambiguity and clarify the nature of logical principles. Obviously, there is an extensive list of principles demonstrated through algebraic logic, especially in Couturat’s seminal work (2004). However, it is sufficient to retain some of the points that have been outlined in this section, among which the concept of logical multiplication and the definition of negation — both demonstrated through the formulas employed — are the main ones. Thus, by establishing these operations, the understanding of the concepts that Łukasiewicz employs when dealing with the refutation of Aristotle is facilitated. Moreover, one can also identify and confirm — through the application of these sophisticated concepts of modern symbolic logic — the surpassing of traditional logic itself.

§5. Refutation of the Aristotelean attempts at demonstration

Now, let us return to the proofs provided by Aristotle. As for the first elenctic proof, it should be noted that it does not prove the principle of contradiction, but at most the principle of double negation, i.e. if something is A, then that thing cannot be Not-A. This is due to what has been established so far: given its refinement, symbolic logic shows that the principle of double negation is more basic and prior in relation to the principle of contradiction, because the principle of contradiction depends on logical multiplication, whereas the principle of double negation does not. This is to say that the principle of double negation depends only on the operation of negation, whereas the principle of contradiction depends on the operation of negation and logical multiplication. Furthermore, there are contradictory objects for which the principle of double negation is valid, but not the principle of contradiction. An example of this is “the greatest prime number”. Hence, Łukasiewicz establishes that “an inference concerning the principle of contradiction cannot be made from the principle of double negation” (1971, p. 498). As for the second elenctic proof, which concerns the concept of “substance” or “essence”, one must note that there is a formal problem with the following formulation: “if in its essence an object could be and not be A simultaneously, then it would not be unitary” (Łukasiewicz, 1971, p. 499). The premise in this formulation can only be demonstrated ad impossible, i.e. it is an apagogic proof, which leads us to the next moment of this refutation: all apagogic proofs are insufficient to prove the principle of contradiction, for they contain formal problems.

In each apagogic proof provided by Aristotle there is a petitio principii.5placeholder All apagogic proofs depend on the principle of contraposition, which presupposes the principle of contradiction. Given the delimitation of my work, it is not possible for me to delve deeply into what the principle of contraposition is and its relationship with the principle of contradiction. What is relevant here is to establish that it is necessary, considering a sequence of more fundamental principles of logic, for the principle of contradiction to be prior and more basic than the principle of contraposition. Hence, one must conclude two fundamental points from this. The first one is that the apagogic mode of inference consists of the principle of contraposition. This is to say that the logic behind the apagogic proof itself derives from the principle of contraposition. And the second one is that, in this specific case, Aristotle uses an apagogic proof to try to prove the principle of contradiction. Therefore, apagogic proofs are not sufficient to prove the principle of contradiction because the very logic behind these proofs presupposes the principle of contradiction. In an analogous manner, it is the same as organising a grand race in which a certain car named Bertoldito is supposed to race alone, in order to prove its incredible speed and power. However, despite much discussion about how fast and strong the Bertoldito car can be, there is no concrete evidence that it even exists. Hence, organising a grand race is, at best, senseless, for there is no proof that the car even exists.

Furthermore, all Aristotelean apagogic proofs face the problem of the ignoratio elenchi.6placeholder For Łukasiewicz, “Aristotle proves not that the mere denial of the principle of contradiction would lead to absurd consequences, rather he attempts to establish the impossibility of the assumption that everything is contradictory” (1971, p. 499, emphasis in original). This is crystal clear in many passages of his Metaphysics, as with the following: “if all contradictory propositions were true at the same time in respect to the same thing” (Met. IV 4, 1007b 18-21, translated by V. Wedin). This is an evident problem, for Aristotle presupposes that if someone admits the possibility of contradictions in the world, that person must admit that everything is contradictory, which is clearly not the case. Neither does it mean that if a person sees that there are no proofs of the unlimited generality of the principle of contradiction, that person should admit full contradictions in all everyday situations. In conclusion, based on all that has been established so far, Aristotle failed to prove the principle of contradiction. At this point, we are back to square one and facing an elephant in the room.

III — On the Horizon of Possibilities Regarding the Principle of Contradiction and Contemporary Logic

§6. Łukasiewicz’s pragmatic solution to the problem of the principle of contradiction

The final and absolutely controversial conclusion of Łukasiewicz is that the principle of contradiction has no logical value whatsoever. Naturally, this conclusion raises many questions. One of the most important might be: are there real contradictions in the world? Łukasiewicz’s answer is that “there is known to us no single case of a contradiction existing in reality” (1971, p. 507). However, it is necessary to proceed with caution on this question. Łukasiewicz makes a distinction between constructive abstractions — which relate to numbers, logical and ontological concepts, etc. — and reconstructive abstractions — which relate to reality and concepts which correspond to it. As for the constructive abstractions, he states that “the possibility is by no means excluded that constructions which count today as free of contradiction nevertheless contain a deeply hidden contradiction which we have not yet been able to discover” (1971, p. 507). As for the reconstructive abstractions, in turn, he states that: “one will never be able to assert with full definiteness that actual objects contain no contradictions. Man did not create the world and he is not in a position to penetrate its secrets; indeed, he is not even lord and master of his own conceptual creations” (1971, p. 507). Both kinds of abstractions, for Łukasiewicz, are not immune to the possibility of being contradictory, which leads to some kind of scepticism of his. Therefore, even if his answer is negative, i.e. that there is no single case of a real contradiction known to us, it points in only one direction: that of human limitation.

Another fundamental question that is raised by Łukasiewicz’s conclusion is: should one then abolish the principle of contradiction? And again, the answer is negative. One must distinguish between the logical and the practical value of this principle. The Polish logician argues that “the principle of contradiction is the sole weapon against error and falsehood” (1971, p. 508, emphasis in original). He uses the example of someone who, having been falsely accused of murder, must prove his innocence somehow. This example brings one back to the impossibility of fully rejecting the principle of contradiction, for there would be no means to prove someone’s innocence if it was the case that one could affirm and negate things simultaneously. However, for Łukasiewicz, if one keeps in mind that the principle of contradiction has no logical value, and yet is constrained to hold it as a dogma, then this is nothing other than a sign of the “intellectual and ethical incompleteness of man” (1971, p. 508). Therefore, the fact that the principle of contradiction holds enormous practical value, and no logical value at all, reinforces the fragility of human reason. Łukasiewicz’s solution is given in an analogous manner to the solution Reichenbach provides to the problem of induction,7placeholder in the context of philosophy of science: faced with the lack of justification of a principle or law, one must still assess its practical significance in everyday life.

In the present conjuncture, with the advancement of non-classical logics, I tend to believe that pragmatism truly is the inevitable tone in this context of destruction of the principle of contradiction as a logical principle. In paraconsistent logic, for example, one might point out that this pragmatism is inherited when discussing the foundations of physics. For Da Costa, “the mathematical structures depend on the logic in which they are immersed; if one changes the logic, one changes the properties of the structures” (2009). And thus, naturally, the foundations of physics also change as one changes the underlying logic of physics. It is in this way that, for a portion of paraconsistent logicians, there may not necessarily be real contradictions in the world, and yet it is worth tolerating certain inconsistencies within specific formal systems. This pragmatic heritage enables a more careful and inclusive approach in various areas of science, aligning more or less with Feyerabend’s pluralistic stance in philosophy of science.8placeholder Therefore, this pragmatism becomes fundamental. Thus, I do not take as absolute the theses of those who arrogantly claim to have grasped the ultimate essence of the things in the world, whereas the history of science has tirelessly shown us just the opposite. Lastly, regarding the non-classical logics, of which I have appropriated paraconsistent logic to discuss the value of pragmatism, it is worth briefly establishing their relevance and applications in contemporary contexts.

§7. The role of the principle of contradiction within contemporary logic

Łukasiewicz was the main responsible for a great turn of events in logic, which took place due to the rejection of certain “foundations” of traditional logic. Nowadays, one thinks of logics, in the plural, and not in a single unified and unbeatable logic, largely because of the innovative work of the Polish logician. In 1920, he published “On Three-Valued Logic”, in which he addressed Aristotle’s paradox of the sea battle. The paradox deals with future contingents, and consists of the fact that today, the sentence “there will be a sea battle tomorrow” is neither true nor false. Hence, a third value “possible” was introduced by Łukasiewicz, for the values “true” and “false” are not sufficient in this case. This was the first time in history that the principle of bivalence was rejected within a formal system. Naturally, much of the formulation of Łukasiewicz’s innovative three-valued modal logic is owed to his work on the principle of contradiction, which defied the notion of irrevocability attributed to it. Indeed, contemporary logic as a whole owes much to Łukasiewicz’s work on the principle of contradiction. Furthermore, shortly after the formulation of the first three-valued modal logic, many logicians, such as Post, Kleene, and Reichenbach, also formulated logics known as many-valued logics. Within the group of many-valued logics, there is a considerably recent one called fuzzy logic, in which the truth value of variables might be a real number between 0 and 1 (Novák, Perfilieva and Močkoř, 1999). This is to say that the concept of partial truth is comprehended within a fuzzy system. As for its applications, one might be amazed by the number of successful innovations implemented in concrete examples from the real world. Some of them include the improvement of the precision of metro system controls, fuel efficiency, handwriting recognition in Sony pocket computers, early recognition of earthquakes through the Institute of Seismology Bureau of Meteorology of Japan, neural networks for machine learning, etc. (Bansod, Kulkarni and Patil, 2005).

Although fuzzy logic does not necessarily have as its foundation the rejection of the principle of contradiction, it is important to establish that the general emphasis of contemporary logic is in the very spirit of Łukasiewicz’s work, namely, the revocation of principles once taken as dogmas of human thought. Now, to address the role of the principle of contradiction in contemporary logic directly, one can think of paraconsistent logic as an example. In paraconsistent logic, inconsistency is tolerated within certain formal systems. In this way, one might not conclude, from two contradictory propositions, the trivialisation of the system. In 1974, for example, the Brazilian logician Newton da Costa formulated the system C1, in which contradictions are tolerated and yet the system is not trivialised. Nowadays, there are some advocates of this recent and revolutionary logic, among whom are Priest, Omori, and D’Ottaviano. As for the applications of this logic, one can emphasise its use in artificial intelligence, engineering, control systems, quantum physics, and so forth. However, as already mentioned, I cannot afford to delve deeply into each logic established throughout this article, given the nature of the writing I set out to produce. Nevertheless, it is sufficient to note the numerous applications of non-classical logics. Moreover, while I have provided only “useful” examples thus far, I do not believe that the value of these logics should be measured solely by their “utility”. Obviously, the advancement of non-classical logics also resonates in mathematics, epistemology, philosophy of science, and many other areas of human knowledge. In conclusion, the work of Łukasiewicz occupies a special place in the history of logic because it is profoundly revolutionary. Łukasiewicz is undoubtedly one of the most important logicians in history, with extremely fruitful contributions. Thus, it is essential to understand the nuances of his logic, as little can be grasped about contemporary logic without its foundations, which were so well established by him and, I hope, analysed with extreme thoroughness in this article of mine.