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.

ĐẠI HỌC SÀI GÒN OF SAIGON UNIVERSITY

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 và Nhân văn – ĐHQG Hà Nội

TÓM TẮT

Carnap (1891 – 1970) là một nhà triết học, nhà logic học vĩ đại của triết học phương Tây thế kỷ XX Phương pháp nội hàm và ngoại diên ông xây dựng được coi là một phương pháp phân tích ngữ nghĩa hết sức độc đáo Mục đích của bài viết này đi vào nghiên cứu nội dung cơ bản của phương pháp trên từ cách tiếp cận của logic học thông qua việc làm rõ mối tương quan logic giữa nội hàm và ngoại diên trong các thực thể ngôn ngữ, từ đó cho thấy ý nghĩa của nó đối với triết học ngôn ngữ và 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

1 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

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

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

2 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:

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)

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

(L-equivalent)

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:

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

(L-equivalent) 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

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,

“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

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

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

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

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

truth-value 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:

(x1)(x2) (xn)[Aix1x2 xn ≡ Ajx1x2 xn]

According to Carnap, repeating this

string with n ≥ 1 will certainly result in two

predicates of level 0 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

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

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

relationship of L-equivalent because they

have the identical intension and do 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

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

truth-value 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,

L-interchangeability 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

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

3 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

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 Duyệt đăng: 20/02/2020