term paper on 'intensional logic

7/30/2019 Term Paper on 'Intensional Logic'

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The Intension of Intensional Logic

In Theaetetus, when Socrates asks Theaetetus to define knowledge, instead of providing a definition

Theaetetus tries to explain knowledge in terms of certain paradigm examplesknowledge is geometry,

astronomy, and shoemakingand Socrates is quick to point out the qualitative difference between

definition and examples. Though intension and extension were coined at a much later stage in the

human history, the difference that Socrates points out in Theaetetus is qualitatively similar, if not

identical, to the difference between the two. The objective of this essay is to identify the difference

between intension and extension and share an understanding (acquired by reading pertinent literature,

which will be duly acknowledged at the end) of what Intensional Logic is all about, the problem the

system of Intensional Logic tries to address, and the methodology the system adopts to realize its goals.

Finally, the paper will conclude with a non-technical critique of some of the concepts involved in

Intensional Logic.

To understand what intension and extension mean, lets consider a set S whose members are whole

numbers less than 10. In set-theoretic language, the set S can be represented as {x: xW & x


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objective world, independent of the human mind. It may be noted that the extension of a term may in turn

have an intension and extension: 1, which is an extension of whole numbers, has the intension, the

smallest mathematical object used to count and measure, and has innumerable extensions, such as a pen,

a living being, or a planet.

Like singular terms, definite descriptions also have intensions and extensions. A definite description is a

collection of words that categorically identifies an individual class or element, living or non-living: thus,

The author ofHamlet means (intensionally) the person who wrote the play, Hamlet and refers

(extensionally) to Shakespeare; rational animal means a living being endowed with the faculty of

rationality and refers to a human being. Natural languages tend to have multiple terms to designate the

same object. In English, for instance, featherless biped who is not a plucked chicken and human being

are sometimes employed to refer to the same sentient creature. The two terms have different intensions

(featherless biped who is not a plucked chicken = a two-footed animal with no feathers; human being = a

living being who can rationalize), but they have the same extension. Then, on the basis of co-

extensionality, would one be justified to say that the two expressions can be used interchangeably,

without altering the meaning of a sentence? If yes, then is interchangeability or substitutivity the criterion

for determining identity? A simple mathematical equation provides a classic example: we know that 1+1= 2. But, what kind of equality are we looking at here? Is it the same as saying 2 = 2? Of course, to know

that 1+1 equals 2, one must have knowledge of the mathematical concepts 1, +, =, and 2; whereas

to know 2 = 2, one only needs to know 2 and =. Perhaps, the two mathematical equations represent

two different claims: they may designate the same value (2); however, they are intensionally different

the former (1+1 = 2) is a predication relation like Platos teacher is Socrates, whereas the latter (2 = 2) is

an identity relation like Socrates is Socrates.

So far, we have discussed intension and extension with respect to singular terms and definite descriptions.

Do sentences, too, have intension and extension? Here, we are only concerned with declarative sentences

that describe a state of affairs. In On Sense and Reference, Frege says that the sense of a declarative

sentence is the proposition4 it expresses, whereas the reference of the sentence is its truth-value.

According to Freges Principle of Compositionality5, the meaning of a complex expression is a function

of the meanings of its constituent parts, and this explains why he thinks the sense of a sentence is given

by its proposition. He uses the following example to illustrate his point:

The sentencesThe evening star is a body illuminated by the Sun and The morning star is a

body illuminated by the Sunhave different propositional content, which is due to the intensional

distinction between evening star and morning star.

Then, he considers the possibility that sentences may not have any reference at all. It is certainly true that

there are sentences whose constituent parts may not have an extension at all. Consider Achilles is about

to reach India, for example. The sentence definitely has a sense; however, one of its constituent parts(Achilles) may or may not have a reference. Upon hearing such a claim, one would either affirm the

statement if Achilles refers to a definite object (a ship or an individual named so) or deny it if Achilles

refers to the Greek mythological character. The truth-value of the sentence seems to depend on whether

the constituent part, Achilles, has a reference or not. Frege concludes, We are therefore driven into

accepting the truth value of a sentence as constituting its reference.6 In addition, Frege distinguishes

between two types of sense and referencecustomary and indirect7. He classifies the sense and reference

of declarative sentences as customary. In indirect speech, however, the reference of a sentence coincides


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with the corresponding sentences customary sense. The distinction between the two types is crucial to an

understanding of the intensionality of sentences, and I shall return to the topic shortly.

Now, since the truth-value of a sentence is a function of the extensions of its constituent parts, it follows

that the truth-value of the sentence should remain unchanged if a constituent part is substituted by an

extensionally equivalent expression8 (that is, a co-extensional expression). A sentence whose truth-value

does not change after substitution of its parts with extensionally equivalent expressions is called an

extensional sentence. In the article, Intentionality and Intensionality, James W. Cornman formulates the

following two individually necessary and jointly sufficient conditions for sentential extensionality:

1. The truth-value of a sentence formed after substitution of an expression in the original sentence

by an extensionally equivalent expression will unconditionally remain the same as that of the

original sentence.

2. The truth-value of a compound or complex sentence is a function of the truth-values of the

constituent simple sentences, such that the truth-value will remain unchanged after substitution

of a coordinate or subordinate clause by a co-extensional clause.9

The two rules can be symbolized as:

1. ((x y) -> (S(x) S1

(y))): If x and y are co-extensional (materially equivalent) expressions,

then the simple sentence (S) containing x and the new sentence (S1) formed after substituting x

with y are also co-extensional. Sonia Gandhi is the head of INC and The wife of Rajiv Gandhi

is the head of INC are co-extensional sentences.

2. ((p q) -> (C(p) C1(q))): If p and q are co-extensional sentences, then the original complex or

compound sentence (C) that contains p as a subordinate or co-ordinate clause and the new

complex or compound sentence (C1) formed after substituting p with q are also co-extensional. If

Sonia Gandhi becomes the Prime Minister of India, then the Indian economy will grow and Ifthe wife of Rajiv Gandhi becomes the Prime Minister of India, then the Indian economy will

grow.

Once we have these conditions for extensionality, the condition for intensionality automatically follows

any sentence that fails to meet any of the two criteria is non-extensional or intensional. For example,

consider the complex sentence, Emily believes that the morning star is the Venus. It is possible that

Emily may not be aware that evening star refers to the same object as morning star. In that case, while

the proposition about Emilys belief about the morning star being the Venus may be true, the sentence,

Emily believes that the evening star is the Venus, may turn out to be false. Clearly, each of the two

complex sentences is intensional because the truth-value of each sentence changes even though its

subordinate clause is replaced by a co-extensional sentence. Note that each of the two complex sentences

cited here is a reported/indirect speech. We earlier noted that Frege draws a distinction between the sense

and reference of a direct speech and the sense and reference of an indirect speech. The sense of the

evening star is the Venus, is its proposition and its reference is its truth-value. Similarly, the sense of

Emily believes that the evening star is the Venus, is the proposition about Emilys belief; however, the

reference of this indirect speech is the customary sense of the subordinate clausethe evening star is the

Venusand not its truth-value. In other words, the reference of the sentence is the object of Emilys


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belief. Further note that the intensional sentence reports on Emilys belief, which is a mental/conscious

state. Typically, sentences that use a that-clause to express a proposition towards which a mental attitude

(believing, desiring, fearing, knowing, or perceiving, for example) is held are calledpropositional

attitudes10.

Propositional attitudes express a relation between a person (Emily) and a proposition (the evening star is

the Venus). These propositions report on peoples mental states that are necessarily about (directed upon)

something: Tanvir loves watching movies; a capitalist wants to maximize profitsin each of these

sentences, the verb (loves/wants) expresses a mental state (an act of consciousness), which is directed

towards an object. Such sentences expressing a mental state are also called intentional sentences and the

objects the mental state is directed upon are called intentional objects. The difference between

propositional attitudes and intentional sentences seems to be that propositional attitudes are necessarily

complex sentences containing a that-clause, whereas not all intentional sentences are complex sentences

for example, Tanvir likes Physics is a simple sentence. The term intentional in this context is used a

strictly technical sense, and is neither to be confused with the homophonous intensional, nor to be

thought of as a derivative of intention, which minimally means a purpose or motive that guides an

action. Intentionality, on the other hand, is an essential property of different mental states in that everymental state is directed upon/towards an object. Considered this way, intentionality can be understood as

directedness11(or about-ness12) of mental states or events towards an object, real (Tanvir hopes that

his friend will recover from his injuries soon) or imaginary (Tanvir believes that unicorns exist). In his

book, Psychology from an Empirical Standpoint, Franz Brentano says, Every mental phenomenon is

characterized bythe intentional (or mental) inexistence of an object. This intentional inexistence is

characteristic exclusively of mental phenomena.. We can, therefore, define mental phenomena by

saying that they are those phenomena which contain an object intentionally within themselves.13

Brentanos use of the expression, intentional inexistence, has been diversely interpreted, a discussion of

which is beyond the scope of this essay, and I shall restrict myself to briefly mentioning a few lines of

investigations that I personally find intriguing.


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Notes:

1. On Sense and Reference, Gottlob Frege

2. Though logicians often make a distinction between Fregean sense and reference and intensionand extension, in this essay I have assumed minimal correspondence between the two sets of

terms and have used them interchangeably throughout.

3. An Approach to Intensional Analysis, Pavel Tichy

4. Frege uses the term, thought, and not proposition: By thought I understand not the subjective

performance of thinking but its objective content, which is capable of being the common property

of several thinkers (On Sense and Reference).

5. http://www.sfu.ca/~jeffpell/papers/FregesPrincipleEarlyVersion.pdf



8. Any two expressions (name/sign/definite description) are extensionally equivalent if they refer to

the same object. Thus, morning star and evening star are extensionally equivalent expressions.

9. Intentionality and Intensionality, James W. Cornman, The Philosophical Quarterly, Vol. 12,

pp. 44-52

10.Propositional Attitudes, http://people.pwf.cam.ac.uk/kmj21/PropositionalAttitudes.Enc.pdf

11.http://plato.stanford.edu/entries/intentionality/

12.http://en.wikipedia.org/wiki/Intentionality

13. http://plato.stanford.edu/entries/intentionality/

References:

Freges On Sense and Reference

Pavel Tichys An Approach to Intensional Analysis

James W. Cornmans Intentionality and Intensionality

http://www.sfu.ca/~jeffpell/papers/FregesPrincipleEarlyVersion.pdf

http://people.pwf.cam.ac.uk/kmj21/PropositionalAttitudes.Enc.pdf
http://www.sfu.ca/~jeffpell/papers/FregesPrincipleEarlyVersion.pdfhttp://people.pwf.cam.ac.uk/kmj21/PropositionalAttitudes.Enc.pdfhttp://plato.stanford.edu/entries/intentionality/http://en.wikipedia.org/wiki/Intentionalityhttp://www.sfu.ca/~jeffpell/papers/FregesPrincipleEarlyVersion.pdfhttp://people.pwf.cam.ac.uk/kmj21/PropositionalAttitudes.Enc.pdfhttp://www.sfu.ca/~jeffpell/papers/FregesPrincipleEarlyVersion.pdfhttp://people.pwf.cam.ac.uk/kmj21/PropositionalAttitudes.Enc.pdfhttp://plato.stanford.edu/entries/intentionality/http://en.wikipedia.org/wiki/Intentionalityhttp://www.sfu.ca/~jeffpell/papers/FregesPrincipleEarlyVersion.pdfhttp://people.pwf.cam.ac.uk/kmj21/PropositionalAttitudes.Enc.pdf


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http://plato.stanford.edu/entries/intentionality/

http://en.wikipedia.org/wiki/Intentionality
http://plato.stanford.edu/entries/intentionality/http://en.wikipedia.org/wiki/Intentionalityhttp://plato.stanford.edu/entries/intentionality/http://en.wikipedia.org/wiki/Intentionality

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