Identity and univocity, by Olavo de Carvalho, is a draft for a philosophy seminar on the topic of identity and univocity. It begins with definitions of metaphysics, necessity, critical metaphysics, and dogmatic metaphysics. The author presents a series of axioms, including the idea that self-evident propositions are those whose contradictories cannot be formulated in a logically univocal way. The text then introduces the supreme metaphysical principle called the Principle of Integrity, which asserts that every subject of a proposition that can be the object of an action or the subject of an action by another subject is one and the same. The text explores various forms of suppression and reduction in relation to this principle. It also discusses self-evident propositions and argues that the principle of identity is self-evident because its contradictory lacks univocal meaning. Olavo claims that self-evident propositions are necessarily true and that evidence cannot be hypothetical. The self-evidence and truth of the Principle of Integrity are emphasized. The text provides an example of another self-evident proposition: “I am here.” The self-evidence and necessary truth of St. Anselm’s proof are also discussed. The author argues against the existence of purely formal logical self-evidences and distinguishes between logic and metaphysics. The text concludes with the statement that logic is based on self-evident principles but does not encompass the entire domain of truth. The draft indicates that it is part of a larger work in progress called “The Eye of the Sun” and will be used for an oral presentation in the Philosophy Seminar.
The appendix includes a discussion between a participant named Villiers de L’Isle-Adam and Olavo de Carvalho in the Sapientia Forum. Villiers raises the topic of the principle of non-contradiction formulated by Aristotle and presents a study by Jan Lukasiewicz that examines different formulations of this principle. Lukasiewicz’s analysis focuses on the ontological and logical formulations and explores their validity. Villiers discusses Lukasiewicz’s arguments, including the challenge to the principle of non-contradiction posed by contradictory objects and the distinction between real objects, constructive abstractions, and reconstructive abstractions. Villiers acknowledges the practical importance of the principle in everyday life but questions its complete validity in relation to certain objects and situations. Olavo de Carvalho responds by stating that Lukasiewicz’s challenge is not metaphysically valid and fails to distinguish between logical and ontological aspects. He argues that Lukasiewicz’s arguments actually presuppose the validity of the principle of identity. Olavo criticizes the separation of logic and ontology and warns against drawing ontological conclusions from purely constructive formalisms. He suggests referring to his forthcoming book, “The Eye of the Sun,” for further clarification on the topic.
(The book “The Eye of the Sun” was never published, and might never be published now that Olavo has died.)
Identity and univocity
Handout for the Philosophy Seminar
Draft for a class at the Philosophy Seminar
June 15, 1998
This draft is part of the work in progress, The Eye of the Sun, where it makes up, in the bulk of the 700 pages written so far, the first section of the chapter “From dogmatic metaphysics to critical metaphysics - and vice versa”. It will soon be used as a basis for oral presentation in the Philosophy Seminar and is therefore published here for notification of students. – O. de C.
1. Definitions
Metaphysics is the science of supreme necessities that encompass and subordinate all others.
Necessity (from nec cedo = do not yield) is to be, not to be able not to be. Necessity is the impossibility of the contrary.
Critical metaphysics is the part of this science that addresses the problems and difficulties that arise for the investigator in the search for supreme necessities.
Dogmatic metaphysics is the discrimination and affirmation of supreme necessities, as well as the unfolding of their immediate consequences for various sectors of human knowledge.
Metaphysics is responsible for the study of possibility as such and impossibility as such, as well as the various gradations and modes of possibility, which viewed quantitatively will be called probabilities.
2. Axioms
A self-evident proposition is one whose contradictory cannot be formulated in a logically univocal proposition.
Pure metaphysical propositions, that is, those that express supreme necessities, must all be self-evident.
All proof is based on self-evident principles.
A principle is self-evident or it is not. You cannot simply “take as” self-evident a principle that is not. In other words: there cannot be a hypothetically self-evident principle (although there can naturally be hypothetically true principles).
The psychological conditions that allow one to grasp the evidence of a principle can vary from person to person, so the feeling of certainty has nothing to do with self-evidence.
3. First statement of the supreme metaphysical principle, or Principle of Integrity.
- Every subject of a proposition, to the extent that it may also be the subject of an action or the object of an action carried out by another subject also capable of being an object of action, is one.
Subjects that are merely logical-formal, or ideal, are not objects of action, not even the “action” of being thought; because what is thought is only their concept, or the term that designates it, and not the object as such.
An impossible subject is one whose definition implies its non-existence, not just logically but self-evidently; that is, a subject is impossible when the affirmation of its existence cannot be logically univocal.
Therefore, every subject is complete, and anything that real or hypothetically opposes its completeness requires, real or hypothetically, its suppression.
Suppression has two forms: 1st negation, 2nd, reduction.
Negation can be definitive or conditional. Definitive negation is that which deprives the subject, real or hypothetically, of the possibility of being the subject of action or passion. Conditional negation is that which, real or hypothetically, deprives the being of being the subject of some actions or passions (determined or undetermined).
Reduction has two forms: 1st reduction to its elements, or analytical reduction; 2nd, reduction to another subject, or synthetic reduction.
An absolutely necessary subject is one whose very definition excludes, in a self-evident manner, its analytical or synthetic reduction. Put another way: it is that whose analytical or synthetic reduction cannot be stated in a logically univocal proposition.
4. Of self-evident propositions
The principle of identity A = A is self-evident, not because it seems so to us or because we have a feeling of certainty that it is self-evident, but because its contradictory, A ¹ A, has a double meaning: if A ¹ A, the subject of the proposition is not equal to its predicate, but, being the proposition reversible — the predicate becoming the subject, and the subject predicate —, we then have two different subjects, which are both subjects of the same proposition: A1 ¹ A2. Therefore, the sentence A ¹ A is not univocal and cannot be univocal, hence it is patent that A = A is self-evident.
The silly objection that this demonstration in turn presupposes the principle of identity falls before the verification that the objection also presupposes it. The purpose, however, is not here to “demonstrate” the principle of identity but to demonstrate the impossibility of its univocal negation. If in the old logic it was said that a self-evident proposition neither requires nor admits proofs, this was what was fundamentally meant, without saying it outright, perhaps for not having perceived it clearly: There is nothing to object to the principle of identity, except propositions of double meaning, that is, without sense.
Therefore, if there is no logical demonstration of a self-evident principle, there is, indeed, of the impossibility of its contradictory. This applies to all logical and metaphysical principles.
5. That the Principle of Integrity is self-evident
Action is a change of state in time and/or space.
I provisionally adopt the definition of time as the form of successions and of space as the form of simultaneity, to which I will return later.
State is a stage of change.
There are only three types of change: the change of state or the two reductions.
The change of state presupposes the permanence of the subject.
The analytical reduction presupposes that the parts belong to the same subject.
The real synthetic reduction presupposes that the one in which the subject was absorbed was not it.
The hypothetical synthetic reduction either presupposes the possibility of real synthetic reduction or is impossible.
Therefore, every subject that is the object of action (that is, subject of passion) is one and the same, not many or another.
The action consists in changing another or changing oneself, or even in changing the other by also changing oneself.
The three hypotheses presuppose the unity and sameness of the subject, as already demonstrated in items 1 to 9. If the subject that changes the other does not change its state, it remains the same. If it changes its state, it is the same in another state. Therefore, the subject of any action is one and the same.
These propositions are not only logically certain but self-evident: their contradictories are not univocal. Let’s see: A1 changes to the state A2. If the subject in state A2 is not the same A from the previous state, then it was not A1 the subject of change; if, inversely, state A2 does not refer to the same subject A, then A2 is not the predicate of the proposition referring to the change from A1. It is impossible to decide if the negation of A’s continuity from A1 to A2 says that there was no change or that the subject was another. The negation is therefore ambiguous, or equivocal. It makes no sense. Therefore, the unity of the subject of change (subject of action or passion) is self-evident.
6. That there is no hypothetical self-evidence
For evidence to be hypothetical, its contradictory would also have to be admitted as hypothetical.
But the contradictory of an evidence is ambiguous, so its formulation would contain not only the negation of the evidence but also its affirmation.
Therefore, the evidence cannot be hypothetical. Either a proposition is evident, or it is not. The criterion of the impossibility of a univocal contradictory will resolve all doubts that arise.
7. That the self-evident is necessarily true
Not being hypothetically true, the self-evident can only be categorically true.
It makes no sense to formulate a sentence like “x is hypothetically categorically true”, which would fall into the objections of item 2 of § 6.
Therefore, there is no alternative but to accept the truth of the evidence.
The mind, however, can refuse to do so. Why can man refuse evidence? Because he can refuse to understand. Because the exercise of intelligence in man is free and not necessary, since, if it were necessary, man would necessarily understand everything, which is seen, by experience, not to happen, but the very definition of man, later, will clarify to us in its deepest metaphysical sense.
The refusal of evidence may have moral and psychological significance, but intellectually it means nothing and falls outside the sphere of interest of metaphysics.
8. Another example of a self-evident proposition
“I am here”: This proposition is self-evident whenever it is uttered by a subject about himself, it is not tautological and is univocal.
Its contradictory, “I am not here” means “It is not me who is here”, or “This place is not here”? Being impossible to decide, the proposition is ambiguous, and therefore “I am here” is self-evident.
9. That the proof of St. Anselm is self-evident and necessarily true
An absolutely necessary being necessarily exists, says St. Anselm’s proof.
Kant’s objection is that the being so defined is defined by us, therefore its existence is hypothetical, based on the assumption - made by us - that the being defined therein is absolutely necessary.
The contradictory is “An absolutely necessary being does not necessarily exist” or “An absolutely necessary being necessarily does not exist?” Being impossible to decide, it is an equivocal proposition and does not make sense.
Therefore, the proof of St. Anselm is self-evident.
There being no hypothetical self-evidence (7:1-5), the proof of St. Anselm is necessarily true.
10. That there are no purely formal logical self-evidences, that is, that are not also ontological
Purely formal truth is that which necessarily verifies itself in the field of logical relations, but not necessarily in the field of experience. It is, therefore, a hypothetical proposition.
There being no hypothetical self-evidences, no self-evident proposition is purely formal.
11. The domain of Logic
Every logical proposition is ultimately based on self-evident principles. So why doesn’t the domain of logic coincide entirely with the true? It is because the set of logically necessary consequences, being able to start from any premise and not from self-evident premises, is not self-evident, only logically consistent.
It identifies, therefore, with the extension of what is necessarily possible, not necessarily true. That is, it is impossible for a logical consequence deduced from self-evident principles to be impossible, but not all that is possible is necessary.
Logic, therefore, distinguishes itself from metaphysics to the extent that the latter positively affirms the necessary, while the former only affirms the necessary possibility.
The necessary possibility is based on the necessary as such and is not an independent domain, since the “hypothetical necessary” only exists as an impossible hypothesis. Now, logic without a metaphysical foundation could only be based on the hypothetical necessary, and therefore it only exists as an impossible hypothesis. The fragmentation of modern logics is precisely due to the impossibility of reducing impossible hypotheses to the unity of the necessary.
[To be continued]
Appendix: a discussion in the Sapientia Forum
I reproduce below a message sent to the forum of this homepage by the participant who adopted the pseudonym Villiers de L’Isle-Adam and my response to him. This message was what motivated the publication of the above text on this homepage and the decision to address the subject in the next Philosophy Seminar class. – O. de C.
Message from Villiers
Dear friends,
In this topic, I intend to discuss some issues related to the famous ‘principle of non-contradiction’ formulated by Aristotle. To this end, I plan to present to you an article on the aforementioned topic by the notable Polish logician, mathematician, and philosopher Jan Lukasiewicz (1878-1956), one of the leading figures, alongside Kazimierz Twardowski (1866-1938) and Stanislaw Lesniewski (1886-1939), of the renowned school of logic that emerged in the universities of Lvov and Warsaw. Lukasiewicz’s study, “O Zasadzie Sprecznosci u Arystotelesa: Studium Krytyczne,” was originally published in 1910 but can be found in issue XXIV of the Review of Metaphysics, translated by Michael V. Wedin under the title “On the Principle of Contradiction in Aristotle: A Critical Study.”
In Book IV of Metaphysics, Aristotle presents the principle of non-contradiction in three different ways, which Lukasiewicz will refer to as ‘ontological,’ ‘logical,’ and ‘psychological’ formulations. However, the analytical effort of the Polish logician will mainly focus on the ontological and logical formulations. For Aristotle, they are equivalent, keeping in mind that a proposition, to be true, must conform to objective reality. The ontological and logical formulations would therefore be true because the world is metaphysically as it is. It should also be noted that, from Aristotle’s perspective, the principle of non-contradiction is a final, indemonstrable law. Demanding a demonstration, an ultimate foundation of the ‘principle,’ would be a regress that could only be infinite, an exigency that, by the nature of the question itself, could not be satisfied. And if there is something that can be known without proof, what could be more appropriate to this kind of knowledge than the law of non-contradiction, a principle that we cannot doubt when we think?
However, in order to highlight the necessity of the principle of non-contradiction, the Stagirite proposes a series of arguments that, refuting the possibility of contradiction in the order of Discourse, seek to justify the principle. Lukasiewicz refers to these arguments as “elentic and apagogic demonstrations,” although it should be emphasized that Aristotle never thought of this set of deductions in terms of ‘positive’ demonstrations of the principle. It seems evident to me that the objective of Aristotle’s strategy is to prove that by admitting contradiction, discourse is destroyed, rational communication is disrupted, since symbols cease to act as symbols and can no longer reflect Reality in Discourse. Furthermore, Aristotle seeks to demonstrate, especially in apagogic demonstrations, the absurd consequences to which we are led when we deny the principle of non-contradiction.
It is neither reasonable nor desirable to reproduce here all the steps of Lukasiewicz’s meticulous analysis. However, I would like to examine the most relevant considerations that the Polish logician derived from his argumentative journey.
Firstly, Lukasiewicz notes that the principle of non-contradiction cannot be demonstrated based on its evidence; indeed, ‘evidence’ itself is not a reliable criterion of truth. On the other hand, attempting to derive the Principle from our psychic structure would be inconsequential since psychological laws can only be verified through experimental methods, which do not even allow us to formulate the Law of non-contradiction as a valid principle in a preliminary approach. A third possibility would be to deduce the Principle from the definition of ‘negation’ or ‘falsity.’ If “A is not B” simply expresses the falsity of “A is B,” it would naturally follow that this definition entails the Principle. However, Lukasiewicz tells us that this does not occur in reality: even if we accept the aforementioned definition of falsity as correct, nothing prevents the propositions “A is B” and “A is not B” from both being true; it only follows, as a consequence, that the proposition “A is B” is simultaneously false and true. The Law of non-contradiction involves the notion of conjunction and does not solely derive from the definition of falsity (or negation). The Polish logician draws our attention to another definition of ‘truth’ and ‘falsity’ which, in a certain way, seems to be more fruitful than the traditional one: the proposition “A is B” is true if it corresponds to something objective, false otherwise. Similarly, “A is not B” is a true proposition if it represents an objective link, false if that is not the case. Taking into account these criteria, there is no a priori obstacle for both the propositions “A is B” and “A is not B” to be true, provided they represent objective situations.
Lukasiewicz also observes that any defense of the principle of non-contradiction must necessarily take into account the fact that there are ‘contradictory objects,’ such as Meinong’s Square Circle. For such objects, it is clear that the Principle is not valid. Obviously, the Polish logician does not assume that Aristotle could have worked based on such considerations, which are part of a body of studies that began to develop only in the mid-19th century, following the flourishing of symbolic logic. However, this does not prevent us from highlighting the intrinsic relevance of Lukasiewicz’s observation: the existence of ‘contradictory objects’ has been confirmed by recent developments in logic, particularly by the Theory of inconsistent formal systems. Today we can attest to the existence of logical-mathematical theories where contradictory objects appear, thus derogating the principle of non-contradiction. Considering such perspectives, the Principle does not appear as absolute and unassailable as it may seem at first glance. Moreover, Lukasiewicz states that even for Aristotle, the principle of non-contradiction could not be a supreme law, at least in the sense that it constitutes the necessary presupposition of all other logical axioms. Citing a famous passage from Aristotle’s Posterior Analytics (An. Post. A, 11, 77a 10-22), the Polish logician asserts that the following syllogism would be valid according to the Stagirite’s postulates:
B is A (and is also not non-A)
C, which is not-C, is B and not-B
C is A (and is not non-A)
Therefore, the above syllogism is valid, although the law of non-contradiction is violated. My limited knowledge of syllogistics does not allow me to verify whether Lukasiewicz’s proposed syllogism is valid or not within the framework of Aristotelian logic. However, if the Polish logician is correct, it becomes imperative to accept the existence of valid laws of reasoning that are independent of the principle of non-contradiction.
The central question we now arrive at can be presented as follows: are there ‘objects’ in relation to which we are certain of the validity of the principle of non-contradiction? In his analysis, Lukasiewicz distinguishes three types of objects: 1) real objects; 2) “constructive abstractions,” free creations of the intellect, such as objects in classical mathematics; 3) “reconstructive abstractions,” which are concepts developed to represent real things.
Regarding constructive abstractions, paradoxes such as the one Bertrand Russell (1872-1970) discovered in 1901, when considering the Set of all sets that are not members of themselves, indicate that in most cases, we will never be certain that they will not violate the principle of non-contradiction. Concerning reconstructive abstractions, which reflect objective reality well, and real objects, they seem to be protected from contradiction. In fact, there seems to be certainty that there are no directly perceptible contradictions in Reality, as the negations correlated to perception judgments are not themselves perceptible, at least in our everyday experience. At the current stage of our knowledge, we tend to accept as correct the observation that any ‘real’ contradiction can only be ‘mediate,’ resulting from inferences. On the other hand, we cannot forget the fact that since the early days of philosophy, the thesis that ‘movement’ and ‘change’ necessarily involve contradictions has been recurrent (in this regard, we can mention the aporia of Zeno of Elea). Although these logical difficulties have always been avoided through theoretical schemes, as they result from inferences, there does not seem to be any definitive proof that there are no contradictions in the objective ‘world.’ Therefore, there is also no positive and unequivocal proof that the principle of non-contradiction is fully valid concerning real objects and reconstructive abstractions. However, to the extent that we can see the Principle as ‘useful,’ we should consider it only as a supposition or hypothesis that guides and shapes scientific inquiry, regulating certain theorizations of the Real.
According to Lukasiewicz, the principle of non-contradiction lacks any a priori logical dignity; nevertheless, it has an ethical and ‘practical’ value of utmost importance. As the Polish logician emphasizes, if we did not accept the validity of the Principle for ‘practical’ activities, we would be subject to all sorts of problems. Therefore, for ordinary life (communicative, social activities, etc.), as Aristotle had already pointed out, the principle of non-contradiction constitutes a fundamental presupposition. However, it is necessary to underline that the practical-ethical indispensability of the Principle is entirely different from its logical-theoretical validity. Lukasiewicz’s conclusion in this regard is somewhat disturbing: the need to recognize the ‘validity’ of the law of non-contradiction is merely a symptom of the ethical and intellectual imperfection of Man. The Polish logician argues that Aristotle perceived the practical-ethical importance of the principle of non-contradiction, even if this realization was not clearly formulated in his work. In an era when Greece’s political decline was already evident, the Stagirite became the founder and main promoter of a systematic and rigorous philosophical-scientific work. It is highly likely that the Greek philosopher, speculates Lukasiewicz, regarded all this intellectual effort as a powerful instrument for the future greatness of his nation. The denial of the Principle would thus clear the path for all sorts of falsehoods and uncertainties, undermining the then fragile structures of scientific inquiry. For this reason, the Polish logician notes that Aristotle vehemently opposed opponents of the Principle, using an uncommonly vehement language in his work. In a singular analogy, Lukasiewicz tells us that the Greek philosopher fought for the principle of non-contradiction as if he were dueling for personal goods.
In conclusion, in this already too lengthy message, I must say that, as a mere beginner in the study of Aristotle, I do not possess the necessary qualifications to assert the relevance of Jan Lukasiewicz’s positions regarding Aristotelian logic. While I cannot guarantee the truthfulness of his criticisms, I would like to praise, first and foremost, the unusual conceptual subtlety of the analytical engineering developed by the Polish logician, as well as the creativity and audacity of his propositions. I would like to have the opportunity to discuss these ideas with knowledgeable scholars of Aristotle and, above all, I would like to know how Professor Olavo de Carvalho, being a profound connoisseur of Aristotelian philosophy, would evaluate Lukasiewicz’s thoughts.
Sincerely,
Villiers de L’Isle-Adam
Reply from Olavo de Carvalho
Dear friend,
You and the other participants are raising this forum to the level of the most important Brazilian cultural debate in recent years, perhaps the only important one if by this word we mean something that touches on essential problems rather than what is touched by the graces of illiterate media.
Regarding your observations, I do not currently have the famous study by Lukasiewicz at hand, nor can I provide the extensive response they deserve. What I can say for now is:
The principle of identity is of a metaphysical order, and its valid challenge must be metaphysically valid. Lukasiewicz’s challenge is neither nor intends to be. It merely intends to demonstrate that in constructivist logic, we can deal with contradictory objects (something that Aristotle not only does not contest but firmly affirms), and obviously, all objects of this logic exist only as hypothetical definitions and have no metaphysical significance. The possibility of constructing contradictory reasonings is the very basis of Aristotle’s dialectic, but Aristotle would never fall into the trap of confusing the ratio arguendi with the ratio essendi. When Lukasiewicz asserts that “contradictory propositions A is B” and “A is not B” can logically coexist, he not only fails to distinguish between coexistence “in re” and “in verbis” (a distinction beyond the reach of pure constructivism) but also presupposes as constant and identical to themselves the definitions of A and B because if he applied to them the same principle of coexistence of the contradictory that he just affirmed, he would not have two definitions but four, and so on indefinitely, which shows that his alleged contestation of the principle of identity takes for granted the validity of that same principle, only showing that its negation is thinkable, but thinkable precisely as self-contradiction that multiplies itself indefinitely.
All of Lukasiewicz’s arguments aimed at challenging the principle of identity presuppose the identity of the propositions and concepts that express it. This is a typical case of a general rule that I have adopted as a criterion for the critical examination of philosophical theories: when the fact of a theory being enunciated contradicts the content of that theory, the theory can be discarded as a simple case of mental confusion. When Lukasiewicz states that the propositions “A is B” and “A is not B” can coexist logically, he not only fails to distinguish between coexistence “in re” and “in verbis” (a distinction beyond the reach of pure constructivism) but also presupposes as constant and identical to themselves the definitions of A and B because if he applied to them the same principle of coexistence of the contradictory that he just affirmed, he would not have two definitions but four, and so on indefinitely, which shows that his alleged contestation of the principle of identity takes for granted the validity of that same principle, only showing that its negation is thinkable, but thinkable precisely as self-contradiction that multiplies itself indefinitely.
All this confusion arises from the bad habit of severing the connections between logic and ontology, obtaining a logic of pure constructivist invention from which ontologically valid conclusions are then derived, surreptitiously introducing terms like “existence” into the discourse. All of this is unbelievably foolish, combined with formidable malice.
To say, for example, that the notion of identity involves the notion of conjunction is valid in pure constructivist logic, but not in metaphysics. In the identity of a being with itself, there is no conjunction whatsoever. Conjunction comes into play only in the construction of the logical proposition that translates this identity into the verbal microcosm. To retroactively attribute to the identity of the being the formal qualities of the proposition that designates it is the same as combing, instead of your own hair, its image in the mirror.
It is true that Lukasiewicz admits the distinction between logical and ontological validity, but to the extent that he also admits a non-ontological logic that can at the same time serve as a criterion of truth in the sciences, this admission becomes ineffective, so he can continue to draw ontological conclusions from pure constructive formalisms with impunity. In short, it’s a devilish confusion.
The further clarifications I can provide on this matter are in the text on “Identity and Univocity” – an excerpt from my forthcoming book “The Eye of the Sun” – which I intended to release later, but this discussion suggests that it is opportune to unload it on my homepage right now.
A big hug from
Olavo de Carvalho
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