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

TRƯỜNG ĐẠI HỌC SÀI GÒN
SAIGON UNIVERSITY
TẠP CHÍ KHOA HỌC
SCIENTIFIC JOURNAL
ĐẠI HỌC SÀI GÒN
OF SAIGON UNIVERSITY
Số 68 (02/2020)
No. 68 (02/2020)
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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

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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
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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:

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,
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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
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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
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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 ≥ 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
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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 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
interchangeability and L-interchangeability
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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
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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