R. Carnap’s view on Method of Extension and Intension from the approach of logics

The article studies the basic contents of the above method from the approach of logic through clarifying the logical relationship between the content and appearance in linguistic entities, thereby showing its meaning. with linguistic philosophy and logic.

TRƯỜNG ĐẠI HỌC SÀI GỊN SAIGON UNIVERSITY TẠP CHÍ KHOA HỌC SCIENTIFIC JOURNAL ĐẠI HỌC SÀI GÒN OF SAIGON UNIVERSITY Số 68 (02/2020) No 68 (02/2020) Email: tcdhsg@sgu.edu.vn ; Website: http://sj.sgu.edu.vn/ QUAN NIỆM CỦA R CARNAP VỀ PHƯƠNG PHÁP NỘI HÀM VÀ NGOẠI DIÊN TỪ CÁCH TIẾP CẬN LOGIC HỌC R Carnap’s view on Method of Extension and Intension from the approach of logics ThS.NCS Phạm Minh Hoàng Trường Đại học Khoa học Xã hội Nhân văn – ĐHQG Hà Nội TÓM TẮT Carnap (1891 – 1970) nhà triết học, nhà logic học vĩ đại triết học phương Tây kỷ XX Phương pháp nội hàm ngoại diên ông xây dựng coi phương pháp phân tích ngữ nghĩa độc đáo Mục đích viết vào nghiên cứu nội dung phương pháp từ cách tiếp cận logic học thông qua việc làm rõ mối tương quan logic nội hàm ngoại diên thực thể ngơn ngữ, từ cho thấy ý nghĩa triết học ngơn ngữ logic học Từ khóa: ngoại diên logic học, nội hàm, triết học ngôn ngữ ABSTRACT Carnap (1891 – 1970) is a great philosopher, logician of Western philosophy in the twentieth century The method of extension and intension that he developed is considered to be a very unique method of semantic analysis The purpose of this paper is to study the basic content of the method based on the approach of logics through clarifying the logical correlation between intension and extension in linguistic entities, thereby showing its implications for philosophy of language and logics Keywords: extension logics, intension, philosophy of language An overview of the intension and extension of the concept in logics The concept plays role as a form of thinking which is the object of study of logic, including formal logic and dialectical logic In contemporary logics textbooks in Vietnam, the concept of “concept” that the authors give is basically unified In Formal Logic monograph, author Nguyen Gia Tho defines the concept as follows: “Concept is a form of thinking in which objects or classes of objects are generalized and differentiated from other objects by their basic properties” (Nguyen, 2016, p 26) In the Textbook of Dialectical Logics work, when talking about the concept, the authors argue that “concept is the form of thinking that reflects the common, nature, and different properties of the object in their unity” (Nguyen & Nguyen, 2015, p 172) Thus, the authors agree that the concept is the product of human thinking which contains contents reflecting the reality through Email: phamminhhoang.ussh@gmail.com 36 PHẠM MINH HỒNG TẠP CHÍ KHOA HỌC ĐẠI HỌC SÀI GÒN basic properties If there is a discrepancy in the approach to the concept, it is the difference between formal logic and dialectical logic in studying the status of concept If formal logic research the concept in an immovable status, i.e., the concept is considered as an available concept and reasoned on the basis of the available content provided by the concept, then dialectical logic studies the concept in a motive, transformation and continuous status, following the motive logic of concepts in thinking However, as analyzed above, both formal logic and dialectical logic share the common approach that concept “simulates” the object in thinking through signs which are expressed through the intension and extension of the concept In Formal Logic, the author defines the intension and extension as follows: “The intension of the concept is a collection of basic signs of objects reflected in the concept [ ] The extension of the concept are the objects, the set of objects that have signs reflected in the intension of given concept.” (Nguyen, 2016, p 30-31) Thus, intension and extension can be considered as two aspects that constitute the whole body which is named concept and has a close relationship with each other Formal logic refers to the conceptual structure through the inversely proportional relationship between the intension and the extension That is, a concept has rich intension whereas its extension is poor, i.e there are little objects to satisfy the signs which has set in the intension Those concepts that are considered “the most abundant” are unique concepts because their extension contain only one element that satisfies the criteria mentioned by the intension The inverse proportional relationship between intension and extension is also the most basic feature of the concept in formal logic However, in dialectical logic, the relationship between the intension and the extension is a directly proportional relationship, which means that the richer the intension is, the more objects the extension contains This derivatives from the dialectical nature of abstract thinking with continuous movement process At that time, the intension of the concepts will be increasingly enriched and penetrated deeper into the essence of the object, while the extension will not narrow but will be further expanded, when the number the amount of new nature signs on the subject is increasingly perceived by people to be discovered more It can be said that the intension and the extension of the concept are considered the qualitative aspect and the quantitative aspect of a concept The concept wishes to “show up” in awareness and in practical activities must be through the intension and the extension In essence, the human thinking process is done through connecting and linking concepts to each other Therefore, that for not grasping fully, consistently about the intension and the extension of the concept may lead to causing “thinking errors” because thinking does not accurately reflect the exist as it is present When building his method of extension and intension, Carnap provided a new and different approach to the intension and extension Accordingly, both of them are now understood in a relatively different 37 SCIENTIFIC JOURNAL OF SAIGON UNIVERSITY No 68 (02/2020) sense and not only limited in the scope of a concept The root cause here is that Carnap's intensional and extensional approach comes from considering them as the basic elements of linguistic entities, but not the two parts of a private form which is the concept Understanding the intension and the extension as well as their role in linguistic entities and in the semantic system will help us acquire the basic content of his thoughts on the method of extension and intension from the approach of logics Basic content of R Carnap's method of extension and intension from the approach of logics 2.1 The difference between equivalent and L-equivalent Carnap introduced the concept of equivalent and L-equivalent which played role as semantic concepts that Carnap used in its system These are the two key concepts that Carnap uses to introduce the method of extension and intension Carnap uses the symbol “≡” to talk about the equivalent relationship instead of the symbol “=” often used Regarding the above symbol and the concepts of equivalent and L-equivalent, Carnap originally applied to predicates in a linguistic expression and was generalized by him in the formula: featherless is Fx, bipeds is Bx, rational animal is RAx Carnap asserted that sentence (x)[Hx ≡ (F•B)x] is a sentence of getting the true truth-value in the Si but is not logical true (L-true) Therefore, of course, the predicates “H” and “F•B” are equivalent, not logical equivalents (Lequivalent) However, sentence (x)[Hx ≡ RAx] according to him is logical true, that is, to receive L-true value and of course H  L RAx According to Carnap, the property of “human” and the property of “featherless bipeds” need to have a reference to reality, but in particular, direct observations to conclude that these two attributes are equivalent Therefore, the two sentences constructed from these two properties can only receive true values in the fact, or Carnap is also called F-true because the inference therefrom cannot be based only on the internal logic rules of the semantic system However, “human” and “rational animal” are L-equivalent because, according to Carnap, it is possible to determine the L-true value of two sentences made up of these two predicates without reference to objects outside the language Each predicate refers to one or more attributes or certain relationships between objects Therefore, each predicate will normally be attributed to a class of objects with the same attribute or relationship Therefore, from applying equivalence and logical equivalence to the predicate, Carnap built the application of these two truth-functional operators into constructing equivalence classes and logical equivalence classes: “a The equivalence class of Ai =Df the Ai ≡ Aj because (x1)(x2) (xn)[Aix1x2 xn ≡ Ajx1x2 xn] To illustrate the equivalence between the predicates, Carnap gave an example of equivalent and logical equivalents expressed through different predicates: “All human beings are featherless bipeds and vice versa” (Carnap, 1948, p 15) Human is designated by Hx, 38 PHẠM MINH HỒNG TẠP CHÍ KHOA HỌC ĐẠI HỌC SÀI GỊN class of those expressions (in Si) which are equivalent to Ai b The L-equivalence class of Ai = Df the class of those expressions (in Si) which are L-equivalent to Ai” (Carnap, 1948, p 16) The concept of classes and properties are two concepts that have a close relationship with each other and involve the definition of the intension and the extension of Carnap, so further analysis is needed 2.2 The intension and the extension play role as class and property of predicate According to Carnap, distinguishing classes and properties is crucial when studying predicates in sentences He said that a sentence “Scott is human” is denoted by Hs, can be expressed in two completely different ways An expression highlighted attribute of human existence of Scott, also an expression, as Carnap has shown, determines the class that Scott belonged to, i.e specifying named Scott element that belong to which set Going back to the example of the equivalence and logical equivalence (Lequivalent) discussed above, it can be seen that there is a difference between class and property If Carnap considered that what is attributed to a class is often referred to as “belonging to” or “being an element of” then the property leaves another way of speech when it emphasizes elements such as “having” or “owning” a certain attribute In addition, according to our view, we would like to explain one more step of Carnap's understanding hereunder In case of speech about the class, any element mentioned will establish its position in a group, a combination of other elements, i.e the definition of the selected element is only clarified by identifying other elements having a common relationship with the considered member In the case of speech about property, however, the subject of perception does not consider the object in relation to other groups which have common connections, but merely pays attention to the “qualitative” aspects of the object, that is, the attribute that the object holds when it faces to the perceived subject Thus, it can be said that the distinction between classes and properties here of Carnap is the separation of two different aspects of the object to recognize, and the result of this cognitive process is expressed through two different ways of speech At the same time, Carnap also said that, when talking about classes, we often grasp the principle of identity, through this principle, Carnap wanted to highlight the overlap between the two classes if they have similar elements, and this in some cases has to have a comparison and beyond the internal structure of the language in order to draw conclusions However, according to Carnap, this principle does not show up clearly when talking about attributes Because if we want to examine the perfect equivalence or uniformity between the two properties, we can simply rely on logical semantics that can “straighten out” their similarity From this point of view, Carnap came to important conclusions about classes and properties: “4-7 The class Human is the same as the class Featherless Biped […] 4-9 The property Human is the same as the property Rational Animal” (Carnap 39 SCIENTIFIC JOURNAL OF SAIGON UNIVERSITY No 68 (02/2020) 1948, p 18) From this starting point, Carnap also takes a clearer view of class and properties: “4-10 Classes are identical if and only if the predicators for them are equivalent 4-11 The properties are identical when and only if the predicates for them are L-equivalent” (Carnap, 1948, p 18) Accordingly, the two concepts of intension and extension were also introduced by him from these points This view, however, was only within the scope of predicates but not beyond this limit The concept of equivalent is used to talk about two predicates with the same extension while the concept of logical equivalent is used to talk about two predicates with the same intension The structure of the predicate consists of intension and extension, corresponding to the two objects that are classes and attributes: “4-14 The extension of a predicator (of degree one) is the corresponding class “4-15 The intension of a predicator (of degree one) is the corresponding property.” (Carnap, 1948, p 19) According to Carnap, the predicate (λx)(Hx) can now use two ways of speech to interpret, in which the intension of H is the human property, while its extension is the human class 2.3 Intension - extension relations and its role in linguistic entities Before Carnap mentioned about the intension and the extension of his works, Gottlob Frege had discussed two similar concepts in his famous work Sense and Reference The term of reference that Frege used corresponds to the term of extension, and the term of sense corresponds to the term of intension of Carnap According to the contemporary semantics, meaning consists of two elements that are sense and designation The denotation contains the intension and the extension, so as mentioned above, the method of speech by class or by property that Carnap understood, is according to the intension or the extension, essentially, are just two ways to express the same meaning From this point, Gottlob Frege used this thesis to solve the very famous paradox of Venus The morning stars are Venus and the evening stars are Venus, but if you say “The morning star is the evening star” then this is an absolutely false statement, because the morning star is Venus as seen in the morning, and the evening star is Venus as seen in the evening Likewise, we can say “the morning star is the morning star”, “the evening star is the evening star”, but we also cannot say “the morning star is the evening star” and vice versa because these two concepts are different on the intension but belong to the same class, i.e Venus As mentioned above, this paradox comes from two different ways of speech intension and extension, but in essence, they all have the same meaning which is about Venus However, compared to the interpretation of Frege's sense and reference, the explanation of Carnap's intension and extension is more complicated He divided the linguistic entity into certain categories and each had its own the intension and the extension In addition to the intension and the extension of a predicate as analyzed above, Carnap also referred to the intension and the extension of a sentence, individual 40 PHẠM MINH HỒNG TẠP CHÍ KHOA HỌC ĐẠI HỌC SÀI GÒN expression and extensional and intensional contexts He argued that the affirmation of the extension as a class of a predicate could easily be recognized, because obviously the domain of the objects that have common attributes would play role as a class containing elements with the same properties However, according to Carnap, unlike the predicate, the truth-value of a sentence is the extension of that sentence He himself admitted that this initially sounds ridiculous but it is in fact necessary to acknowledge that possibility The truthvalue of a sentence that is identified by Carnap is a great similarity with the extension of a predicate Because when studying a predicate, it can be divided into many steps, but the predicate of zero will be equivalent to one sentence If we consider Ai and Aj to be any n-order predicates (n ≥ 1) then they have the same intension if and only if: thought that there were no unity among thinkers when they used of the term “proposition” Within the scope of a sentence Carnap gave an example of a comparison between two sentences ~(Pa•Qb) and ~Pa˅~Qb These are two different sentences but get the same value as L-equivalent because they both express a proposition And therefore, they have same the intension, which is the same proposition expressed Unlike the intension and extension of a sentence, the intension and extension of an individual expression is more easily grasped The term of individual sometimes said by Carnap is often used to refer to single thing, in other words, this is the object that language expressions reflect Features of objects such as names, properties, relationships will form the content reflected in the language Therefore, it is quite understandable that Carnap considers that the extension of an individual expression is the individual in which the expression “contains”: “9-3 The extension of an individual expression is the individual to which it refers (hence the descriptum, if it is a description” (Carnap, 1948, p 40) When talking about the intension of the individual expression, Carnap returned to his conception of the intension of the designator: “5-2 Definition Two designators have the same intension (in Si) =Df they are L-equivalent (in Si).” (Carnap, 1948, p 23) Thus, the individual expression is considered a kind of designators, so its intension must also be related to L-equivalent, like the term of people and the phrase of higher animals capable of thinking which express (x1)(x2) (xn)[Aix1x2 xn ≡ Ajx1x2 xn] According to Carnap, repeating this string with n ≥ will certainly result in two predicates of level as well as two sentences S1 and S2 equivalent and having the same extension, i.e having the same truth-value For the intension of a sentence, Carnap called it a proposition He argued that linguistic entities, including sentences, which, in particular, Carnap stressed that sentences are the object of study of logic only encapsulated in the scope of declarative sentences, are often not referred to entities outside the language However, the proposition contains references to such entities Carnap also 41 SCIENTIFIC JOURNAL OF SAIGON UNIVERSITY No 68 (02/2020) to introduce to intensional context and extensional context These two concepts are built on the basis of equivalence and logical equivalence Accordingly, the interchangeability is the substitution (or replacement) of a linguistic expression in a sentence with another expression but the sentence still receives the primitive truthvalue compared to the original sentence That is, the extension of the sentence has not been changed and therefore, the original sentence and the interchanged sentence have an equivalent relationship Contrary to the interchangeability, Linterchangeability conserves the intension of the sentence, i.e the proposition and the sentence express Thus, the original sentence and the interchanged sentence are in a logical equivalence (L-equivalent) Based on this foundation, Carnap used the equivalence and the logical equivalence to distinguish three different types of context, namely extensional contexts, intensional contexts and sentences about beliefs Accordingly, in a context called an extension context, equivalent linguistic expressions can be interchangeable, whereas in an intensional context, logical equivalent expressions can be interchanged for each other For sentences about beliefs, this is a quite complicated case compared to two above contexts Carnap gave an example of the two sentences about beliefs he studied: “(i) ‘John believes that D’ (ii) ‘John believes that D´’.” (Carnap, 1948, p 53) The difference between the sentences about beliefs and the two contexts analyzed is that these sentences contain psychological terms The intension and the extension of ordinary sentence according to the analysis of Carnap will correspond to the individual relationship of L-equivalent because they have the identical intension and not need references to non-linguistic entities From the point, Carnap proposed that when talking about individuals, we have to use individual concepts and they are the intension of the individual expression, then he took the example to confirm his thesis, in combination with using the iota operator to talk about an individual: “9-4 The intension of an individual expression is the individual concept expressed by it Examples: 9-5 The intension of ‘s’ is the individual concept Walter Scott 9-6 The connotation of ‘(ɩx)(Axw)’ is the individual concept The Author Of Waverley.” (Carnap, 1948, p 41) In order to understand more about the concept of individual as the intension of the individual expression about which Carnap mentioned, we consider that there should be a comparison with the unique concept from the perspective of formal logic: “The unique concept that the its extension has only one” (Nguyen, 2016, p.34) The unique concept reflect unique objects and attributes and it has the same intension with the concept of the individual at this point However, if the criteria for distinguishing a unique concept with the general concept is in the number of elements in the extension of the two concepts, then here, Carnap did not mention about the extension of the concept of individual but only considered it as being the intension of the individual expression After analyzing meticulously the intension and the extension of entities language, Carnap used two concepts of interchangeability and L-interchangeability 42 PHẠM MINH HỒNG TẠP CHÍ KHOA HỌC ĐẠI HỌC SÀI GÒN concepts and truth-value of the sentence, but in the sentences about beliefs, the establishment of the intension and the extension for sentences based on the two objects mentioned above contain ambiguity because these sentences are considered to contain two meaning classes A class meaning related to the content of propositions D (and D’), the other meaning class refers to John's belief in the content of propositions D (and D') In this case, not only the internal semantic rules mentioned in the semantic system can be solved, but also the reference to the object beyond the language, such as observation and experimentation are not possible to conclude whether John's belief in D and D’ is “true” or “false” In John's “belief network”, there may appear to be beliefs that are true to reality and may also appear to believe that John thinks it is true, but it is unlikely that the process of reference to non-linguistics entities outside language, the belief still retains the true truth-value Thus, we see clearly in the beliefs derived from John, there will be sentences that receive true truth-values and those that receive true logical-value (L-true - true in logical aspect), but it may appear to be false (false) in practice because they have not been to refer to reality for testing Based on Carnap's conventions of interchangeable, one sentence is true and true in scope of logic, or one sentence is true and one sentence is not interchangeable, and therefore, they are neither in the intensional context nor the external context Faced with such psychological statements, Carnap developed the concept of intensionally isomorphic (the term that Carnap named) to determine the intension and value of sentences about beliefs Accordingly, two sentences are considered intensionally isomorphic if they have the same intenstional structure The structural isomorphism is based on the logical equivalence between the most basic components of the two sentences that Carnap called “synonymous” To illustrate the isomorphism of the intensional structure, Carnap gave his classic example If “+” is the equal “sum”, “2” and “5” are equivalent to “II” and “V” then the expression “2 + 5” will be intensionally isomorphic with the expression “sum(II, V)” If “>” and “Gr” are similar, then we will have two isomorphic expressions about the intensional structure: “5> 3” is intensionally isomorphic with “Gr(V, III)” So, which case will not be considered as intensionally isomorphic? Carnap argued that if two expressions “Gr [sum (II, V), III]” and “2 + 5> 3” are intensionally isomorphic, then the two expressions “Gr [sum (II, V), III]” and “7> 3” is not isomorphic because of “7” and “sum (II, V)” are not isomorphic Thus, based on the analysis of intensionally isomorphic, Carnap asserted that: “There is a sentence Si in a semantical system S’ such that (a) Si is intensionally isomorphic to ‘D’ and (b) John is disposed to an affirmative response to Si” (Carnap, 1948, p 62) When we prove that the two sentences are isomorphic, they are logically equivalent and thus can use semantic rules in semantic system S to operate the reasoning process Conclusion It can be said that Carnap's method of extension and intension is a comprehensive and thorough analysis method of each level of language entities, from predicates to individual expressions and to sentences Through the process of defining the inner 43 SCIENTIFIC JOURNAL OF SAIGON UNIVERSITY No 68 (02/2020) and external functions of these language objects, Carnap wanted to devise a sharp and appropriate method to adopt the thinking process based on semantic rules so that people can achieve logical truth The method of extension and intension of R Carnap has also provided us with other approaches to intuition and externality compared to the traditional approaches of current Vietnamese logic on internal and extension of concept This method has elicited a number of problems and applied new ideas of the meaning of Carnap in logic and other sciences REFERENCES Carnap, R (1948) Meaning and necessity: A Study in Semantics and Modal Logic Chicago: The University of Chicago Press Frege, G (1948) “Sense and Reference”, The Philosophical Review, Vol 57, No 3, pp 209 – 230 Nguyen, G Tho (2016) Formal Logic Hanoi: Thế Giới Publisher Nguyen, Th Van & Nguyen, A Tuan (2015) Textbook of Dialectical Logics, Hanoi: Vietnam National University Press Hanoi Ngày nhận bài: 18/7/2019 Biên tập xong: 15/02/2020 44 Duyệt đăng: 20/02/2020 ... acquire the basic content of his thoughts on the method of extension and intension from the approach of logics Basic content of R Carnap's method of extension and intension from the approach of logics. .. with the general concept is in the number of elements in the extension of the two concepts, then here, Carnap did not mention about the extension of the concept of individual but only considered... intension of the concept is a collection of basic signs of objects reflected in the concept [ ] The extension of the concept are the objects, the set of objects that have signs reflected in the intension