The Intention of Intensional Logics
In Theaetetus, when Socrates asks Theaetetus to define knowledge, instead of providing a definition
Theaetetus tries to explain knowledge in terms of certain paradigm examples—knowledge is geometry,
astronomy, and shoemaking—and 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 similar, if not identical, to the
difference between the two. The objective of this essay is to capture the intention of Intensional Logics—
the essay will begin with a discussion of the difference between intension and extension; then, it will
relate the common understanding as regards intension and extension of sentences—what determines
extensionality and non-extensionality or intensionality of sentences, for example; next, the essay will
point out the distinction between intentionality and intensionality; finally, the essay will end with a brief
overview of the problem that Intentional Logics try to address.
To understand what intension and extension mean, let’s consider a set S whose members are whole
numbers less than 10. In set-theoretic language, the set S can be represented as {x: x∈W & x<10} (let’s
call this R1), where W stands for whole number, or as {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} (let’s call this R2). Here,
R1 captures the intension and R2 captures the extension of the set S. Every linguistic term has two
distinct types of meaning, intension and extension. This is similar to what Gottlob Frege calls the ‘sense’
and ‘reference’1 of a term, where ‘sense’ corresponds to ‘intension’ and ‘reference’ corresponds to
‘extension’2. The intension of a term captures the term’s definition (what the term means), whereas the
extension captures what the term is applied to. Consider the question, ‘What is a planet?’ Two valid
responses to the question are:
A. A planet is an enormously large celestial body in space moving around a star.
B. The Earth is a planet.
‘A’ provides the conditions that must be satisfied for an object to be considered a planet; ‘B’ gives us an
example of a planet. The Earth is a planet because it satisfies the conditions individually necessary and
jointly sufficient for an object to qualify as a planet: the Earth is an extremely large object; it is in space;
it is a celestial (pertaining to the Sky) object; and it orbits the Sun. Intension may be understood as what a
term ‘means’ and extension as what the term ‘refers to’. Note that ‘B’ ascribes the property of ‘being-aplanet’ to ‘Earth’, presumably because Earth satisfies the required conditions (or, possesses all the
properties necessary for an object to be called a planet). In the terminology of the French linguist,
Ferdinand de Sassure, ‘planet’ is a ‘signifier’ (visual or auditory sign) that ‘signifies’ (means) the set of
conditions (intension) necessary to identify a ‘referent’ (Earth). Thus, an intension of a term/sign can be
thought of as a certain identification procedure3 that is executed to determine/identify the individual
objects (extensions) the term/sign can be extended (or applied) to.
The intension of a term can be likened to the Platonic Form—just as a Form may not necessarily be
instantiated, so an intension may not have an extension at all. For example, the intension (sense) of
‘unicorn’ is ‘a white horse with a horn on its forehead’. However, the term does not designate (refer to) an
object, unless we want to regard human-made pictures/paintings depicting ‘unicorns’ as extensions—thus
‘unicorn’ has a/an null/empty extension. This indicates that ‘extension’ is an existentially loaded term—
that is, we say that a certain term has an extension if and only if the object it refers to exists as part of the
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 of Hamlet’ 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 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 coextensionality, 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 ‘Plato’s 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 Frege’s 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 sentences—‘The evening star is a body illuminated by the Sun’ and ‘The morning star is a
body illuminated by the Sun’—have 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 reference—customary and indirect7. He classifies the sense and reference
of declarative sentences as customary. In indirect speech, however, the reference of a sentence coincides
with the corresponding sentence’s 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 ‘If
the 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 Emily’s 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 direct speech and the sense and reference of 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 Emily’s belief; however, the
reference of this indirect speech is the customary sense of the subordinate clause—the evening star is the
Venus—and not its truth-value. In other words, the reference of the sentence is the object of Emily’s
belief. Further note that the intensional sentence reports on Emily’s 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 called propositional
attitudes10.
Propositional attitudes express a relation between a person (Emily) and a proposition (the evening star is
the Venus). These propositions report on people’s mental states that are necessarily about (directed upon)
something: Tanvir ‘loves’ watching movies; a capitalist ‘wants’ to maximize profits—in 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 in 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 to the extent
that every mental 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 “Intentionality and Intensionality”, John W. Cornman argues that a certain relation obtains between
intentionality and intensionality of sentences. A demonstration of the relation is beyond the scope of this
essay. Here, it suffices to note the conceptual distinction between intentionality and intensionality—
whereas ‘intentionality’ pertains to mental states, ‘intensionality’ is a logical property of sentences. So far
we have noted what it takes for two terms, expressions, or sentences to be co-extensional. Next, we must
address the question—when are two terms, expressions, or sentences co-intensional? Logicians have tried
to provide an account of co-intensionality of sentences, terms, and expressions by employing the Possible
Worlds theory, according to which two terms/expressions are co-intensional if and only if (iff) they
designate the same object (or the same set of objects) in each and every possible world, including the
actual world. As regards sentences, any two sentences are co-intensional iff they share the same truthvalue in every possible world. For example, let U be a set of unicorns and G be a set of golden deer. In the
actual world, both the sets have a null extension, {}, and hence are co-extensional; however, U and G are
not co-intensional because conceivably in some other possible world both unicorn and golden deer may
refer to actual creatures, in which case their extensions will be different—that is, the members that
constitute U will differ from the members of G. Similarly, in the actual world, both the English language
sentences—‘Unicorns are found in Paris’ and ‘Golden Deer are seen in Cuba’—have the same extension
(truth-value), False. However, in some other possible world, it is not inconceivable that these propositions
pick out True as their truth-value, and hence these cannot be considered co-intensional.
Definitionally, a proposition is necessarily true iff the proposition is true in every possible world,
including the actual world. That is, the falsity of a necessary truth is inconceivable, as it leads to
contradiction (A & ~A). Since any two necessarily true propositions always designate the same object
(True), they must also be co-intensional. In terms of extensionality, we can say that two propositions are
co-intensional (one can always be substituted for another without altering the truth-value) if they are
necessarily, and not merely possibly, co-extensional. This intensionality of sentences is captured by
sentential as well as first-order logic minimally in terms of the relation of logical equivalence that holds
between certain propositions—(p & q)  ~(~p V ~q), (p -> q)  (~p V q)  (~q -> ~p), for instance.
However, neither sentential nor first-order logic is adequately equipped to deal with the various aspects of
natural language sentences. It is known ‘Equilateral triangle’ and ‘equiangular triangle’ refer to the same
geometrical shape/object, and it is impossible to conceive of a possible world where the two terms may
have different extensions; however, it is possible that a person with minimal understanding of geometry,
may deny the sentence ‘it is known that equilateral triangle and equiangular triangle refer to one and the
same thing.’ Even if one assumes that the definition of triangle, equilateral-ness, and equiangular-ness
will never change, so that equilateral triangle and equiangular triangle will necessarily denote the same
geometrical shape, there is no denying the fact that the two expressions differ in intension. As another
example, consider the very notion of the necessary truth of self-identity: A -> A. Does it mean
‘necessarily, A is A’? Or, does it mean ‘A is necessarily A’? To begin with, are these two sentences
intensionally different—that is, do they express different propositions? Classical logic, which is only
concerned with the extension of propositions, does not have the tools to deal with these problems. Formal
logical systems that try to take into consideration such intensional contexts are generally referred to as
Intensional Logics. And, as the plurality of the name suggests, there is no single system of Intensional
Logic that provides a comprehensive solution to the problem of intensionality.
Notes:
1. “On Sense and Reference”, Gottlob Frege
2. Though logicians often make a distinction between Fregean ‘sense and reference’ and ‘intension
and 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
6. “On Sense and Reference”, Gottlob Frege
7. “On Sense and Reference”, Gottlob Frege
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
References:
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Frege’s “On Sense and Reference”
Pavel Tichy’s “An Approach to Intensional Analysis”
James W. Cornman’s “Intentionality and Intensionality”
http://www.sfu.ca/~jeffpell/papers/FregesPrincipleEarlyVersion.pdf
http://people.pwf.cam.ac.uk/kmj21/PropositionalAttitudes.Enc.pdf
http://plato.stanford.edu/entries/intentionality/
http://en.wikipedia.org/wiki/Intentionality
http://plato.stanford.edu/entries/logic-intensional/