W I S D O M 3(16), 2020

Home , Law of identity, Law of thought

S C I E N T I F I C J O U R N A L Published by the decision of the Scientific Council of Khachatur Abovian Armenian State Pedagogical University

Department of Philosophy and Logic named after Academician Georg Brutian

W I S D O M

3(16), 2020

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YEREVAN – 2020

Robert DJIDJIAN, Hasmik HOVHANNISYAN

DOI: 10.24234/wisdom.v16i3.395 Robert DJIDJIAN,

Hasmik HOVHANNISYAN

AXIOMS OF PHILOSOPHY IN ARISTOTLE‟S METAPHYSICS

Abstract

The goal of this article is to study Aristotle‟s concept of philosophical principles. Metaphysics re- quired from a philosopher to reveal the axioms of his teaching. It declared the law of contradiction as the most certain of all principles and axioms. This article proves that the Aristotelian definition of truth makes it necessary to accept the ontological formulations of all the three main laws of thought as axioms of first philosophy. This article points out the absence of any reference in Metaphysics on Categorias and vice versa. This circumstance questions if could Aristotle be the author of both works? Authors of this article underline that the modern trend of meta-philosophical studies requires investigating the possibility of building the system of axiomatic philosophy. EPISTEMOLOGY, PHILOSOPHY OF SCIENCE, METAPHILOSOPHY, ARGUMENTATION, COMMUNICATION Keywords: axiom, the axiom of philosophy, the choice of an axiom, the set of axioms, the priority of ontology.

Introduction ments and its rigorous proofs did not yet come to light. Developing the theory of syllogisms and Axioms – the most valuable acquisition in revealing that his theory was based on the gen- the treasury of scientific knowledge – have been eral syllogisms of the first figure, Aristotle called highly appreciated from the time of Aristotle. them perfect syllogisms and never qualified them They were rightly regarded as the unshakable as axioms. These two general syllogisms of the foundation of the colossal temple of demonstra- first figure in medieval logic were denoted as tive scientific knowledge. The clearest concept of Barbara and Celarent and were called “dictum the function of axioms was presented in Posteri- de omni et de nullo”. And only much later, in the or analytics (chapter 10 of the book I). Moreo- frame of traditional formal logic the principle ver, Jan Lukasiewich proved that Aristotelian “dictum” started to be called “the axiom of syllo- syllogistics was the first axiomatic theory in the gism” though in the traditional formal logic this history of sciences (Lukasiewicz, 1951). axiom was not used in the theory of syllogisms. Yet the axiomatic method of building scien- Axioms of fundamental natural sciences tific theories by introducing a set of axioms and could be called the highest principles of the natu- deducing all other statements of the theory from ral world. Albert Einstein devoted the last dec- axioms and definitions with the help of rules of ades of his life to the study of the general theory valid inferences was not yet put into scientific of the field. If Einstein had succeeded in his great practice by the days of Aristotle. Euclid‟s Ele- endeavour, then the physical science could claim

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that it has revealed the ultimate essence of exist- Principia Mathematica. Its famous laws of me- ence, the first principles and the highest causes chanics can serve as illustrating analogues for the of the physical world. Now substituting in this “set of axioms of a theory”. Each modern edu- formulation the concept “physical world” by Ar- cated man is able to learn and remember these istotelian “being as being” we‟ll get the task of there laws. On the contrary, the task “creating a the first philosophy – to reveal the first principles complete axiomatic theory” means proving the and the highest causes of being as being. Thus entire body of the statements of Newtonian me- we come to an understanding that in essence the chanics, and this task is out of reach of any mod- Aristotelian first philosophy could be viewed as ern educated man if he is not helped by New- the most basic level of physical science, a true ton‟s famous work. meta-physics, the most valuable knowledge of In short, the task of “setting out of axioms of natural science, and in the latter sense – the high- a theory” means suggesting a number of state- est and ultimate knowledge of all the existence – ments that presumably could serve a basis for the the true meta-physics. theory. While “creating a complete axiomatiza- Commentaries on Metaphysics are traced tion” means building the theory as a whole, pre- back by historians to Alexander Aphrodisias, and senting a proof of its every statement based on its their number is constantly growing in our days axioms and definitions. and times to come (Fraser, 2002; Wedin, 2002; “The most certain axiom” of the first phi- 2009; Sharples, 2010; Yu, 2003; Bell, 2004; Gill, losophy is clearly stated in the book IV of Meta- 2005; Gill, 2006; Anagnostopoulos, 2009; Per- physics: “the same attribute cannot at the same amatzis, 2011; Kotwick, 2016; van Inwagen, & time belong and not belong to the same subject Sullivan, 2018). Our study is an element in this and in the same respect” (Metaphysics, 1005b huge domain of research in first philosophy ded- 20). In the following sentence, Aristotle under- icated to a quite narrow branch of axioms of Me- lines that this “is the most certain of all princi- taphysics that, as we hope, could become a sig- ples” and then suggests another formulation for nificant trend in modern philosophy. the first axiom of his philosophical system: “it Preparing this article for publication, we is impossible that contrary attributes should be- have revealed that some current time philoso- long at the same time to the same subject” (natu- phers even those who are involved in neo-Aristo- rally, this both formulations of the first axiom of telian research do not understand the significant the first philosophy are accompanied by traditi- difference between the concepts “axioms of a onal precautions against possible “dialectical ob- theory” and “axiomatic theory”. One of them jections” by pointing out that they are considered even stated proudly: “I didn‟t distinguish clearly “in the same respect”, “at the same time”, and enough between setting out axioms and creating other “usual qualifications”). a complete axiomatization”. We hope that the following simple example The Law of Contradiction as the will be helpful to understand clearly the huge Axiom of the First Philosophy difference between “set of axioms of a theory” and “creating a complete axiomatic theory”. Let It is quite evident that the law of contradic- us consider Newton‟s Philosophiae Naturalis tion is formulated in Metaphysics as a principle

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that it has revealed the ultimate essence of exist- Principia Mathematica. Its famous laws of me- of ontology since both formulations of the axiom the same subject”) unite for further analysis of ence, the first principles and the highest causes chanics can serve as illustrating analogues for the speak about things and subjects, about “existing the axioms of philosophy in the following formu- of the physical world. Now substituting in this “set of axioms of a theory”. Each modern edu- things qua existing” aimed to reveal the most lation: “It is impossible that the same attribute formulation the concept “physical world” by Ar- cated man is able to learn and remember these certain principles of “all things”. belong and not belong to the same subject at the istotelian “being as being” we‟ll get the task of there laws. On the contrary, the task “creating a On the other hand, the law of contradiction same time and in the same respect” (1). This the first philosophy – to reveal the first principles complete axiomatic theory” means proving the is much wider recognized in the language of sci- formulation we‟ll call ontological formulation of and the highest causes of being as being. Thus entire body of the statements of Newtonian me- ence as the main law of human thought rather the law of contradiction (the first axiom of Aris- we come to an understanding that in essence the chanics, and this task is out of reach of any mod- than the first axiom of the first philosophy deal- totle‟s first philosophy). Respectively, we‟ll call Aristotelian first philosophy could be viewed as ern educated man if he is not helped by New- ing with the most certain principles of “all logical-cognitional the law of contradiction of the most basic level of physical science, a true ton‟s famous work. things”. Unavoidably, this twofold nature of the traditional logic: “It is forbidden to assert and meta-physics, the most valuable knowledge of In short, the task of “setting out of axioms of law of contradiction raises the question of “prior- simultaneously to negate the same property of natural science, and in the latter sense – the high- a theory” means suggesting a number of state- ity”: which one of these two natures of the law of the same subject” (2). The most laconic wording est and ultimate knowledge of all the existence – ments that presumably could serve a basis for the contradiction is the primary? In its turn, a con- of the logical-cognitional formulation is given in the true meta-physics. theory. While “creating a complete axiomatiza- vincing answer to this principle question could the first book of Posterior Analytics: “it is im- Commentaries on Metaphysics are traced tion” means building the theory as a whole, pre- have a decisive impact in evaluating Aristotle‟s possible to affirm and deny simultaneously the back by historians to Alexander Aphrodisias, and senting a proof of its every statement based on its refutation of Plato‟s idealism. Indeed, if the onto- same predicate of the same subject” (Posterior their number is constantly growing in our days axioms and definitions. logical principle of “all things” were deducible Analytics, 77a10). and times to come (Fraser, 2002; Wedin, 2002; “The most certain axiom” of the first phi- from the law of thought it would significantly The impression is that in Metaphysics, there 2009; Sharples, 2010; Yu, 2003; Bell, 2004; Gill, losophy is clearly stated in the book IV of Meta- strengthen Plato‟s idealism, and on the contrary, was not made a strict distinction between onto- 2005; Gill, 2006; Anagnostopoulos, 2009; Per- physics: “the same attribute cannot at the same if one demonstrates that the logical formulation logical and logical formulations of the law of amatzis, 2011; Kotwick, 2016; van Inwagen, & time belong and not belong to the same subject the law of contradiction is deducible from its on- contradiction even when this law became the Sullivan, 2018). Our study is an element in this and in the same respect” (Metaphysics, 1005b tological formulation, it would essentially sup- subject of analysis. As it was emphasized above, huge domain of research in first philosophy ded- 20). In the following sentence, Aristotle under- port Aristotle‟s postulate that existence is inher- Metaphysics considers the law of contradiction icated to a quite narrow branch of axioms of Me- lines that this “is the most certain of all princi- ent in particular things (compare Makin, 2003). as an ontological principle. Chapter 4 of the book taphysics that, as we hope, could become a sig- ples” and then suggests another formulation for The text of Metaphysics contains an attempt IV is fully devoted to the substantiation of the nificant trend in modern philosophy. the first axiom of his philosophical system: “it of proving the law of contradiction grounded on law of contradiction. Summing up the discussion Preparing this article for publication, we is impossible that contrary attributes should be- the concept of belief: “it is impossible for anyone of this issue, Metaphysics concluded: “it has have revealed that some current time philoso- long at the same time to the same subject” (natu- to believe the same thing to be and not to be”. been shown that contradictories cannot be predi- phers even those who are involved in neo-Aristo- rally, this both formulations of the first axiom of And just this belief is thought to be at the basis of cated at the same time” (Metaphysics, 1007b18). telian research do not understand the significant the first philosophy are accompanied by traditi- all proofs: “all who are carrying out a demonstra- Some passages later this same chapter 4 suggests difference between the concepts “axioms of a onal precautions against possible “dialectical ob- tion reduce it to this [belief] as an ultimate belief; a clear formulation of the law of contradiction: theory” and “axiomatic theory”. One of them jections” by pointing out that they are considered for this is naturally the starting-point even for all “it will not be possible to assert and deny the even stated proudly: “I didn‟t distinguish clearly “in the same respect”, “at the same time”, and the other axioms” (Metaphysics, Book IV, chap- same thing truly at the same time” (Metaphysics, enough between setting out axioms and creating other “usual qualifications”). ter 3). 1008a35). But both the two latter formulations of a complete axiomatization”. Let us the above-cited Aristotelian two on- the law of contradiction are evidently logical We hope that the following simple example The Law of Contradiction as the tological formulations of the law of contradiction formulations. will be helpful to understand clearly the huge Axiom of the First Philosophy (“the same attribute cannot at the same time be- Now let us investigate the above mentioned difference between “set of axioms of a theory” long and not belong to the same subject and in twofold nature of the law of contradiction. and “creating a complete axiomatic theory”. Let It is quite evident that the law of contradic- the same respect” and “it is impossible that con- Namely, is it possible to deduce the logical-cog- us consider Newton‟s Philosophiae Naturalis tion is formulated in Metaphysics as a principle trary attributes should belong at the same time to nitional formulation of the law of contradiction

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(2) from its ontological formulation (1)? (Posterior Analytics, 76b22). We would like to mention that we are not We would like to point out that, unlike Met- going to prove the law of contradiction. We are aphysics, that presents axioms as the most cer- just investigating the claim of “priority”: should tain principles, Posterio Analytics speaks in a be the ontological formulation of the law of con- quite modern language characterizing axioms as tradiction considered “prior” to the logical-cogni- the primary premises of demonstrations. Never- tional formulation if one succeeds to demonstrate theless, due to the ontological formulation of the that the logical-cognitional formulation of the law of contradiction in Metaphysics, readers law of contradiction (2) is deducible from its on- should realize that contradiction is not only for- tological formulation (1)? bidden in human cognition but also is impossible We fully agree with Metaphysics point of in the material world. Moreover, Aristotle‟s con- view that axioms as the most basic statements of cept of truth requires that thoughts should be ad- science (in our time – of a scientific theory) equate to reality, attributes combined with the could not be proved, but should be accepted. things, in reality, should be presented combined Considering a principal case of contradiction, in thought, and the separated in reality should be Metaphysics emphasizes its position: “But we presented in thought as separated: “he who have now posited that it is impossible for any- thinks the separated to be separated and the com- thing at the same time to be and not to be, and by bined to be combined has the truth, while he this means have shown that this is the most in- whose thought is in a state contrary to that of the disputable of all principles” (Metaphysics, 1006a objects is in error” (Metaphysics, 1051b5). As it 5). Those who demand that even this statement often happens in abstract sciences, Aristotle shall be demonstrated show “want of education”. makes things ultimately clear using an example: People in want of education could learn ma- “It is not because we think truly that you are ny important things concerning the most basic, pale, that you are pale, but because you are pale primary statements and axioms of sciences from we who say this have the truth” (Metaphysics, the book I of Posterio Analytics. Aristotle ex- 1051b9). plained there: “The fact of their existence as re- Aristotle‟s definition of truth using “connec- gards the primary truths must be assumed; but it tion/separation” may be thought as a bit clumsy: has to be proved of the remainder, the attributes” “the true judgment affirms where the subject and (Posterior Analytics, 76a33). Then Aristotle con- predicate really are combined, and denies where tinues his explanation: “every demonstrative sci- they are separated” (Metaphysics, 1027b20). Me- ence has three elements: (1) that which it posits, dieval logicians suggested an aphoristic defini- the subject genus whose essential attributes it tion: “Truth is the adequacy of thought and reali- examines; (2) the so-called axioms, which are ty”. But we would like to mention that this lacon- primary premises of its demonstration; (3) the ic definition requires an additional definition of attributes, the meaning of which it assumes” the concept of adequacy. And this additional def- (Posterior Analytics, 76b10). In short, the essen- inition brings us back to connectedness: “A tial elements of the demonstration are three: the thought is adequate to reality if it “connects” subject, the attributes, and the basic premises what is connected in reality”.

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(2) from its ontological formulation (1)? (Posterior Analytics, 76b22). The Law of Contradiction and Aristotelian Then, “A truth should state that it is impos- We would like to mention that we are not We would like to point out that, unlike Met- Definition of Truth sible for any attribute to belong and not belong to going to prove the law of contradiction. We are aphysics, that presents axioms as the most cer- the same subject at the same time and in the just investigating the claim of “priority”: should tain principles, Posterio Analytics speaks in a Now we are going to demonstrate that the same respect” (4). be the ontological formulation of the law of con- quite modern language characterizing axioms as logical-cognitive law of contradiction (2) could It is quite evident that the conclusion (4) is tradiction considered “prior” to the logical-cogni- the primary premises of demonstrations. Never- be deduced from the ontological formulation (1) equivalent to the logical-cognitive law of contra- tional formulation if one succeeds to demonstrate theless, due to the ontological formulation of the using the Aristotelian definition of truth. This diction (2) “It is forbidden to assert and simulta- that the logical-cognitional formulation of the law of contradiction in Metaphysics, readers will be done by building a syllogism, the first neously to negate the same property of the same law of contradiction (2) is deducible from its on- should realize that contradiction is not only for- premise of which is a slight reformulation of the subject”. tological formulation (1)? bidden in human cognition but also is impossible ontological formulation of the law of contradic- Thus, the syllogism we have built deduces We fully agree with Metaphysics point of in the material world. Moreover, Aristotle‟s con- tion (1): “All the things exist in the way that it is the logical-cognitive formulation of the law of view that axioms as the most basic statements of cept of truth requires that thoughts should be ad- impossible for any attribute to belong and not contradiction “It is forbidden to assert and simul- science (in our time – of a scientific theory) equate to reality, attributes combined with the belong to the same subject at the same time and taneously to negate the same property of the could not be proved, but should be accepted. things, in reality, should be presented combined in the same respect” (1a). same subject” (2) from the ontological formula- Considering a principal case of contradiction, in thought, and the separated in reality should be Metaphysics suggests the definition of the tion “It is impossible that the same attribute be- Metaphysics emphasizes its position: “But we presented in thought as separated: “he who concept truth in the last chapter of the book YI: long and not belong to the same subject at the have now posited that it is impossible for any- thinks the separated to be separated and the com- “the true judgment affirms where the subject and same time and in the same respect” (1). thing at the same time to be and not to be, and by bined to be combined has the truth, while he predicate really are combined, and denies where This success in proving the leading role of this means have shown that this is the most in- whose thought is in a state contrary to that of the they are separated” (Metaphysics, 1027b20). We the ontological formulation of the law of contra- disputable of all principles” (Metaphysics, 1006a objects is in error” (Metaphysics, 1051b5). As it meet a similar definition of the truth in the last diction encourages us to examine the priority 5). Those who demand that even this statement often happens in abstract sciences, Aristotle chapter of the book IX: “he who thinks the sepa- problem in regard of the other two main laws of shall be demonstrated show “want of education”. makes things ultimately clear using an example: rated to be separated and the combined to be human thinking – the law of excluded middle and People in want of education could learn ma- “It is not because we think truly that you are combined has the truth, while he whose thought the law of identity. The law of excluded middle ny important things concerning the most basic, pale, that you are pale, but because you are pale is in a state contrary to that of the objects is in is formulated in Metaphysics in the first sentence primary statements and axioms of sciences from we who say this have the truth” (Metaphysics, error” (Metaphysics, 1051b4). Later on, medie- of Chapter 4 of Book VII: “there cannot be an the book I of Posterio Analytics. Aristotle ex- 1051b9). val logicians improved the wording of Aristoteli- intermediate between contradictories, but of one plained there: “The fact of their existence as re- Aristotle‟s definition of truth using “connec- an definition by using the concept adequacy in- subject we must either affirm or deny any one gards the primary truths must be assumed; but it tion/separation” may be thought as a bit clumsy: stead of combined/separated: “Veritas est ade- predicate.” has to be proved of the remainder, the attributes” “the true judgment affirms where the subject and quatio of ratio et rei” (3). As the second premise The study of the law of excluded middle (Posterior Analytics, 76a33). Then Aristotle con- predicate really are combined, and denies where of our syllogism, we take the latter definition of could be essentially helped by the approach of tinues his explanation: “every demonstrative sci- they are separated” (Metaphysics, 1027b20). Me- truth in its following modification: “A truth modern propositional logic. In the frame of ence has three elements: (1) that which it posits, dieval logicians suggested an aphoristic defini- should be adequate to things [under study]” (3a). propositional logic, the law of contradiction is the subject genus whose essential attributes it tion: “Truth is the adequacy of thought and reali- Using the premises (1a) and (3a) we com- presented by the following statement: “the con- examines; (2) the so-called axioms, which are ty”. But we would like to mention that this lacon- pose modus Barbara of Aristotelian syllogistics: junction of propositions p and non-p is false” (5). primary premises of its demonstration; (3) the ic definition requires an additional definition of “All the things exist in the way that it is impossi- According to de Morgan‟s rule and the law of attributes, the meaning of which it assumes” the concept of adequacy. And this additional def- ble for any attribute to belong and not belong to double negation, the statement (5) is equivalent (Posterior Analytics, 76b10). In short, the essen- inition brings us back to connectedness: “A the same subject at the same time and in the to the following disjunction: “p or non-p” (6) tial elements of the demonstration are three: the thought is adequate to reality if it “connects” same respect” (1a). which is just the law of excluded middle in the subject, the attributes, and the basic premises what is connected in reality”. “A truth should be adequate to things [under language of the propositional logic. study]” (3a). Metaphysics deliberates in detail regarding

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the problem of several meanings of the words in demand of the law of identity to have a certain chapter 4 of the book IV. This consideration of qualitative identity. But if the law of identity fails the issue in Metaphysics is summed up in a short in this “high speed changing world” there woul- conclusion: “it is impossible to think of anything dn‟t be the place for the law of excluded middle if we do not think of one thing”. Taking into ac- also, since these “fluid” objects couldn‟t be con- count the context of the above analysis of this sidered as having or not having a given property. issue in Metaphysics we come to the following This failure of the law of excluded middle would formulation of the law of identity: “During a con- bring with it the abolishment of the law of con- crete discourse, words should be used in one and tradiction since as we have shown above the law the same meaning”. of the excluded middle can be deduced from the At first sight, the impression could be that law of contradiction. the law of identity is completely separate from We would like to mention also that all the the other two laws of thought – the law of con- above-revealed relations of “priority” of the on- tradiction and the law of excluded middle. At tological in regard to the cognitive is in full ac- least, the law of identity, unlike the other two cord with Aristotle‟s general view of human cog- laws of thought, could be expressed in proposi- nition. He emphasized in Posterior Analytics that tional logic only as a tautology “p is equivalent cognition begins with the sensorial perception of to p”. Actually, the Aristotelian law of identity external objects. In Metaphysics the priority of deals with words and concepts that in the lan- the things with material substrate also is stated guage of mathematical logic are expressed with clearly: “that the substrata which cause the sensa- the help of predicates, the latter being the instru- tion should not exist even apart from sensation is ment of the logic of predicates but not of propo- impossible. For sensation, it is surely not the sen- sitional logic. sation of itself, but there is something beyond the Modern meta-logic considers propositional sensation, which must be prior to the sensa- logic as the basis of entire mathematical logic. It tion…” (Metaphysics, 1010b 33). couldn‟t be another way since the simple subject- For the sake of proper understanding of Ar- predicate judgments of natural language in math- istotle‟s philosophical heritage, we have to delib- ematical logic are presented as formulas of the erate on some peculiar aspects of the cognitive logic of predicates necessarily using logical con- function of Metaphysics and Aristotelian Cate- junctions of propositional logic. For this reason, gories. According to Aristotle‟s intention and the modern meta-logic considering the issue of prior- content of Meta-physics its goal was to reveal the ity of laws of thought has to award priority to the most general predicates and principles of all ma- law of contradiction. Yet this order of priority terial world, or in Aristotle‟s words, of “being as could be radically changed if we apply the philo- being”. And what could be said about Categories sophical-cognitional approach. in this regard? Categories, included into the Or- Indeed, let us consider a material world ganon, – the scope of Aristotelian logical works where things were changing with such a speed – have no chance to be considered as a meta- that no object could be considered as having a physical work about the most general character- qualitative identity. Obviously, these objects be- istics of “being as being”. Yet, by its content, reaved of qualitative identity couldn‟t satisfy the Categories deal only with the most general pred-

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the problem of several meanings of the words in demand of the law of identity to have a certain icates applicable to the realm of separate sub- Categorias did not mention Metaphysics. This chapter 4 of the book IV. This consideration of qualitative identity. But if the law of identity fails jects, the things of the material world. In actuali- kind of mutual negligence in regard to basic the issue in Metaphysics is summed up in a short in this “high speed changing world” there woul- ty, Categories and Metaphysics cover the same components of the language of sciences is com- conclusion: “it is impossible to think of anything dn‟t be the place for the law of excluded middle field of study, just with different intention (completely unexplainable. Especially if we take into if we do not think of one thing”. Taking into ac- also, since these “fluid” objects couldn‟t be con- pare Stough, 1972; Matthews, 2009). account that both famous works were considered count the context of the above analysis of this sidered as having or not having a given property. The modern view of the statue of Catego- obligatory in understanding the essence of cate- issue in Metaphysics we come to the following This failure of the law of excluded middle would ries could be expressed by Christian Shields‟ gories of science and philosophy. In light of the formulation of the law of identity: “During a con- bring with it the abolishment of the law of con- accurate sentence: “Aristotle does not say explic- above said, this kind of mutual negligence of crete discourse, words should be used in one and tradiction since as we have shown above the law itly, but his examples make reasonably clear that Metaphysics and Categories could have hap- the same meaning”. of the excluded middle can be deduced from the he means to categorize the basic kinds of beings pened providing Aristotle hadn‟t been the author At first sight, the impression could be that law of contradiction. there may be” (Shields, 2016). Nevertheless, the of one of them. the law of identity is completely separate from We would like to mention also that all the short statement “Categories were meant to cate- the other two laws of thought – the law of con- above-revealed relations of “priority” of the on- gorize the basic kinds of beings there may be” The Problem of Categories tradiction and the law of excluded middle. At tological in regard to the cognitive is in full ac- (7) comprises an essentially problematic point. If least, the law of identity, unlike the other two cord with Aristotle‟s general view of human cog- the phrase “to categorize the basic kinds of be- Another problem arises in regard of Meta- laws of thought, could be expressed in proposi- nition. He emphasized in Posterior Analytics that ings there may be” were understood as a task “to physics “lexicon” presented in the book V. The tional logic only as a tautology “p is equivalent cognition begins with the sensorial perception of reveal the most general predicates of being as main thesis of the book V is expressed here by to p”. Actually, the Aristotelian law of identity external objects. In Metaphysics the priority of being” then the statement (7) would interpret the assertion “The name has a meaning and has deals with words and concepts that in the lan- the things with material substrate also is stated Categories as an Introduction to Metaphysics. one meaning” (Metaphysics, 1006b11). But how guage of mathematical logic are expressed with clearly: “that the substrata which cause the sensa- Since 10 categories of Categories, in essence, are we should deal with a word that has several the help of predicates, the latter being the instru- tion should not exist even apart from sensation is the most general predicates of the conceptual meanings? The answer to this question is quite ment of the logic of predicates but not of propo- impossible. For sensation, it is surely not the sen- human language, these categories and their con- definite: “It makes no difference even if one sitional logic. sation of itself, but there is something beyond the tents could be very helpful for Metaphysics. Yet, were to say a word has several meanings if only Modern meta-logic considers propositional sensation, which must be prior to the sensa- there is a principle point. Categories speak of they are limited in number; for to each definition logic as the basis of entire mathematical logic. It tion…” (Metaphysics, 1010b 33). language as language never descending to the there might be assigned a different word” (Meta- couldn‟t be another way since the simple subject- For the sake of proper understanding of Ar- level of the material world and never considering physics, 1006a34). Thus, the problem of word predicate judgments of natural language in math- istotle‟s philosophical heritage, we have to delib- issues of being as being. This specific feature of ambiguity is solved by introducing for each sepa- ematical logic are presented as formulas of the erate on some peculiar aspects of the cognitive Categories means that it should be evaluated as rate meaning of the word an appropriate defini- logic of predicates necessarily using logical con- function of Metaphysics and Aristotelian Cate- an independent branch of science – meta-theory tion and, if necessary, assigning a peculiar name junctions of propositional logic. For this reason, gories. According to Aristotle‟s intention and the of natural language. for each definition. modern meta-logic considering the issue of prior- content of Meta-physics its goal was to reveal the The puzzling thing is that Metaphysics does But how should we deal with the words ex- ity of laws of thought has to award priority to the most general predicates and principles of all ma- not evaluate and appreciate ideas and concep- pressing categories of Metaphysics? And which law of contradiction. Yet this order of priority terial world, or in Aristotle‟s words, of “being as tions of Categories in a proper extent. For in- words of the “lexicon” of the book V are catego- could be radically changed if we apply the philo- being”. And what could be said about Categories stance, in Book V, Metaphysics presents about ries of Metaphysics? None of these two questions sophical-cognitional approach. in this regard? Categories, included into the Or- three dozen terms used in discussions of its is- is answered. The term “category” is not used at Indeed, let us consider a material world ganon, – the scope of Aristotelian logical works sues. These set of terms involves many catego- all in Metaphysics. Definitely, the book V didn‟t where things were changing with such a speed – have no chance to be considered as a meta- ries of Categorias like substance, quantity, quali- realize the principle role of definitions and cate- that no object could be considered as having a physical work about the most general character- ty, relative, disposition, deprivation etc. Yet Me- gories in building an axiomatic theory. qualitative identity. Obviously, these objects be- istics of “being as being”. Yet, by its content, taphysics didn‟t mention any interpretation of Factually, the term “category” was not used reaved of qualitative identity couldn‟t satisfy the Categories deal only with the most general pred- these categories in Categorias, and vice versa, in any of Aristotle‟s theoretical works. The term

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“category” was not used even in the text of Cat- This means that it is improbable that there would egorias, forming only the title of this work. We be revealed an additional axiom for the first phi- know only of two cases of using the term “cate- losophy in the foreseeable future. gory” in Aristotle‟s theoretical works. One of On the other hand, the above-suggested in- them happens in Prior Analytics (Book I, Chap- terpretation of the first philosophy as of true me- ter 37). This case is involved in a single sentence ta-physics opens a new way for the study of the asserting that the expression “one thing is truly issue of axioms of Metaphysics. In modern un- predicated of another” must be assumed in as derstanding, meta-physics could be interpreted as many ways as categories are divided (Prior Ana- the philosophy of natural sciences that is, in lytics, 45a6). gross, identical to the modern concept of philo- The second case is presented in Topics sophical ontology. So, the modern philosophical (chapter 3 of the book I). This one-page chapter ontology if presented as an axiomatic theory, just repeats the famous chapter 4 of Categorias may open new approaches to the study of Meta- that introduced Aristotelian 10 categories, with- physics. out any connection with previous or subsequent Of course, the task of building philosophy chapters of Topics. axiomatically should interest modern researchers too, independent of the historical value of Meta- The Problem of Axiomatic physics. Especially, taking into account the huge Philosophy Today wave of meta-theoretical and meta-philosophical investigations during the last decades (Lazero- Returning to the main issue of this article, it witz, 1970; Marsoobian, 2007; Hovhannisyan, should be stated that Metaphysics explicitly stat- 2007; Hovhannisyan, 2008; Overgaard, Gilbert, ed as the axiom of the first philosophy the law of & Burwood, 2013; Hovhannisyan, 2014; Hov- contradiction in its ontological formulation. As it hannisyan, 2015). was shown above, Aristotle‟s definition of truth But philosophers persistently circumvent presumes that the other two “main laws of hu- the problem of the axiomatic representation of man thought” – the law of the excluded middle philosophical knowledge. Alfred Whitehead and and the law of identity – should be considered as his former student, Nobel laureate Bertrand Rus- the axioms of the first philosophy also (of course, sell wrote the three-volume “Principia Mathe- in the ontological formulation of these laws). matica” (1911-1913) were presented the formal- Texts of Metaphysics do not suggest, even ized system of axiomatic mathematics. But did in an implicit form, any other axiom for the first not write even a few lines on the axiomatization philosophy. Moreover, even if statement A were of philosophical theories. Rudolf Carnap did a suggested as an additional axiom of the first phi- serious job in the direction of the axiomatic rep- losophy, this same statement A, according to the resentation of relativistic mechanics, but he did Aristotelian definition of truth, should also be not show any sort of interest to the problem of considered as an additional “main law of tho- the axiomatization of philosophy. Over the past ught”. But in the long history of philosophy and few decades, when everything that is written is logic, there was no convincing attempt of extend- somehow published, no article has been written ing the set of the three famous laws of thought. on the problem of the axiomatization of philo-

WISDOM 3(16), 2020 14 14 Axioms of Philosophy in Aristotle’s Metaphysics

“category” was not used even in the text of Cat- This means that it is improbable that there would sophical teachings. The exception is the interest- allow proving the following important statements egorias, forming only the title of this work. We be revealed an additional axiom for the first phi- ing book by Eberhard Rogge “Axiomatik alles of philosophical ontology rigorously: know only of two cases of using the term “cate- losophy in the foreseeable future. moglichen Philosophierens” (1950). Note that E. Theorem 1. Form and matter possess exist- gory” in Aristotle‟s theoretical works. One of On the other hand, the above-suggested in- Rogge‟s research did not have the task of axio- ence only insofar as they are two main sides of them happens in Prior Analytics (Book I, Chap- terpretation of the first philosophy as of true me- matizing philosophical knowledge but only re- objects of reality. ter 37). This case is involved in a single sentence ta-physics opens a new way for the study of the vealed and explicitly formulated the principles Theorem 2. The source of all motion and asserting that the expression “one thing is truly issue of axioms of Metaphysics. In modern un- (in this sense, axioms) of the main philosophical change in the world is the internal quality (attrib- predicated of another” must be assumed in as derstanding, meta-physics could be interpreted as directions. ute) of the matter presented by physical charges. many ways as categories are divided (Prior Ana- the philosophy of natural sciences that is, in We published two joint articles on axiomat- Theorem 3. Matter is the cause of itself lytics, 45a6). gross, identical to the modern concept of philo- ic philosophy in the journal of Moscow State Pe- (Materia est causa sui). The second case is presented in Topics sophical ontology. So, the modern philosophical dagogical University (Hovhannisyan & Djidjian, Theorem 4. Each phenomenon has its cause. (chapter 3 of the book I). This one-page chapter ontology if presented as an axiomatic theory, 2018; Djidjian & Hovhannisyan, 2019) that, pos- Theorem 5. Nothing arises from “nothing” just repeats the famous chapter 4 of Categorias may open new approaches to the study of Meta- sibly, would serve a starting point for new inves- (Ex nihilo nihil fit). that introduced Aristotelian 10 categories, with- physics. tigations in the axiomatic presentation of philo- Theorem 6. The question of the origin of the out any connection with previous or subsequent Of course, the task of building philosophy sophical theories. Now we would like to present material world as a whole, in principle, cannot chapters of Topics. axiomatically should interest modern researchers our system of axioms of philosophical ontology have a proof in the framework of natural science. too, independent of the historical value of Meta- (Hovhannisyan & Djidjian, 2018) that could be Theorem 7. The origin of the material world The Problem of Axiomatic physics. Especially, taking into account the huge considered as the basis of first philosophy and can only be postulated. Philosophy Today wave of meta-theoretical and meta-philosophical meta-ontology. Theorem 8. Evolution as development is ac- investigations during the last decades (Lazero- Axiom 1. The existence (essence) is inher- cidental and can occur in nature only in favoura- Returning to the main issue of this article, it witz, 1970; Marsoobian, 2007; Hovhannisyan, ent, first of all, to individual sensually perceived ble conditions. should be stated that Metaphysics explicitly stat- 2007; Hovhannisyan, 2008; Overgaard, Gilbert, objects of the external world. ed as the axiom of the first philosophy the law of & Burwood, 2013; Hovhannisyan, 2014; Hov- Axiom 2. All objects of the world are in Conclusions contradiction in its ontological formulation. As it hannisyan, 2015). constant change and motion. was shown above, Aristotle‟s definition of truth But philosophers persistently circumvent Axiom 3. The source of all motion and Aristotle highly appreciated axioms as the presumes that the other two “main laws of hu- the problem of the axiomatic representation of change in the world is the interaction of material unshakable foundation of the demonstrative sci- man thought” – the law of the excluded middle philosophical knowledge. Alfred Whitehead and objects with each other. entific knowledge. Yet the axiomatic method of and the law of identity – should be considered as his former student, Nobel laureate Bertrand Rus- Axiom 4. Objects of the external world have building scientific theories by introducing a set the axioms of the first philosophy also (of course, sell wrote the three-volume “Principia Mathe- a 3D volume and are in a certain mutual disposi- of axioms and deducing all other statements of in the ontological formulation of these laws). matica” (1911-1913) were presented the formal- tion. the theory from axioms and definitions with the Texts of Metaphysics do not suggest, even ized system of axiomatic mathematics. But did Axiom 5. The phenomena of the external help of rules of valid inferences was not yet put in an implicit form, any other axiom for the first not write even a few lines on the axiomatization world have a duration and succession. into scientific practice by the days of Aristotle. philosophy. Moreover, even if statement A were of philosophical theories. Rudolf Carnap did a Axiom 6. The world as a whole cannot be Euclid‟s Elements, with its impressive demon- suggested as an additional axiom of the first phi- serious job in the direction of the axiomatic rep- conceived by human limited empirical experi- strations, came to light decades later. Developing losophy, this same statement A, according to the resentation of relativistic mechanics, but he did ence. his theory of syllogistics, Aristotle called the Aristotelian definition of truth, should also be not show any sort of interest to the problem of Axiom 7. Atheistic faith: the material world basic principles of his theory perfect syllogisms considered as an additional “main law of tho- the axiomatization of philosophy. Over the past is eternal in the sense that the chain of interac- and never qualified them as axioms. ught”. But in the long history of philosophy and few decades, when everything that is written is tions and transformations of material objects is Metaphysics declared the law of contra- logic, there was no convincing attempt of extend- somehow published, no article has been written infinite in time and had no beginning. dicttion as “the most certain of all principles” of ing the set of the three famous laws of thought. on the problem of the axiomatization of philo- These 7 axioms supported by 17 definitions the first philosophy, of being qua being. “The

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most certain axiom” of the first philosophy had Aristotle. Metaphysics. The internet classics ar- clear ontological formulation: “the same attribute chive: http://classics.mit.edu/Aristotle- cannot at the same time belong and not belong to /metaphysics.html. the same subject and in the same respect”. In ac- Aristotle. Posterior Analytics. The internet clas- tuality, the practice to consider the law of contra- sics archive: http://classics.mit.edu/A- diction as the main principle of human thought ristotle/pos terior.html. got its beginning much later in the frame of tradi- Aristotle. Prior Analytics. The internet classics tional formal logic. archive: http://classics.mit.edu/Aristotl- In Metaphysics, there was not made a strict e/prior.ht ml. distinction between ontological and logical for- Bell, I. (2004). Metaphysics as an Aristotelian mulations of the law of contradiction even when Science. Sankt Augustin: Academia this law became the subject of analysis in its Verlag. book IV. Nevertheless, in general, epistemologi- Djidjian R. & Hovhannisyan, H. (2019). Opyt cal approach Metaphysics clearly proved the pri- aksiomaticheskogo postroeniya teorii ority of ontology: “It is not because we think tru- poznaniya (Building an Axiomatic Sys- ly that you are pale, that you are pale, but beca- tem of Epistemology, in Russian). Bul- use you are pale we who say this have the truth”. letin of Moscow City Pedagogical Uni- As we have shown in this paper, the Aristotelian versity, series Philosophical sciences, 1 definition of truth allows deducing each one of (29), 31-40. doi: 10.25688/2078-9238.- the three “main laws of thought” of traditional 2019.29.1.04. logic from its ontological formulation. This de- Fraser, K. (2002). Demonstrative Science and the monstrative solution of the problem of priority Science of Being qua Being. Oxford awards to ontological formulations of the tradi- Studies in Ancient Philosophy, 22, 43- tional laws of thought the high statue of axioms 82. of first philosophy. Gill, M. (2006). First Philosophy in Aristotle. In Guided by Aristotle‟s call for building axi- M. L. Gill & P. Pellegrin, A Compan- omatic philosophy, we have suggested the axio- ion to Ancient Philosophy (pp. 347- matic bases of the two main branches of philoso- 373). Malden: Blackwell. phy – of the philosophical ontology and of epis- Gill, M.-L. (2005). Aristotle‟s Metaphysics Re- temology. The philosophical ontology if present- considered. Journal of the History of ed as an axiomatic theory, may open new ap- Philosophy, 43, 223-251. proaches to the study of Metaphysics. Hovhannisyan, H., & Djidjian, R. (2018). Aksi- omatizaciya gumanitarnikh nauk – im- References perativ epokhi tsifrovykh tekhnologii (Axiomatization of Humanitarian Sci- Anagnostopoulos, G. (Ed.) (2009). A Companion ences – Imperative of the Epoch of Di- to Aristotle. Chichester: Wiley-Black gital Technologies, in Russian). Bulle- well. tin of Moscow City Pedagogical Uni- Aristotle. Categories. http://classics.mit.edu/A- versity, series Philosophical Sciences, ristotle/categories.html. 3(28), 47-57. https://www.elibrary.ru/-

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most certain axiom” of the first philosophy had Aristotle. Metaphysics. The internet classics ar- item.asp?id=35617771. ion to Aristotle (pp. 144-161). Chiches- clear ontological formulation: “the same attribute chive: http://classics.mit.edu/Aristotle- Hovhannisyan, H. (2007). Argumentation and ter: Wiley-Blackwell. cannot at the same time belong and not belong to /metaphysics.html. Meta-Argumentation. In G. A. Brutian Overgaard, S., Gilbert, P., & Burwood, S. the same subject and in the same respect”. In ac- Aristotle. Posterior Analytics. The internet clas- (Ed.), Philosophy and Meta-philoso- (2013). An Introduction to Metaphilos- tuality, the practice to consider the law of contra- sics archive: http://classics.mit.edu/A- phy, Book 1 (pp. 64-67). ophy. Cambridge University Press. diction as the main principle of human thought ristotle/pos terior.html. Hovhannisyan, H. (2008, April). Argumentation Peramatzis, M. (2011). 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