[FOM] Intensionality

Nikolaj Oldager sno at imm.dtu.dk
Fri Mar 14 08:43:05 EST 2003

> -o- Is there anything like a "universally agreed" *formal definition* of
> what it means for a LOGIC to be called INTENSIONAL (as opposed to
> extensional)??

In the literature there are (at least) two concepts of intensionality.

1) A language is INTENSIONAL if we distinguish between the extension and
the intension of an expression (different nomenclature is used: extension
is also known as reference, and intension is also known as sense,
comprehension, or meaning). For example, the extension of 'vertebrate' is
the mammals, birds, fishes, amphibians and reptiles, whereas the intension
is the denotation of ``animal with backbone or spinal column''. This
informal, albeit intuitive concept of intensionality is found in older
literature like Port-Royal logic.

2) A language is EXTENSIONAL if co-extensional expressions may be
substituted for each other, and it is INTENSIONAL if co-extensional
expressions may not be substituted for each other. This more precise
concept of intensionality may be credited to Frege due to his famous 1892
paper. (Several closely related formulations of this concept exist, see
e.g. Carnap's Meaning and Necessity or a dictionary of philosophy like the
Cambridge). This concept of intensionality has the advantage that it may be
formulated in the logic without employing dubious notions (like intension)
in the metalanguage. However, it presupposes a well-defined concept of
CO-EXTENSIONALITY, and it is disputable whether this is definable in
general. For instance, let us say that formulae are co-extensional if they
are equivalent in the logic, then propositional logic is extensional since
co-extensionality implies substitutability, i.e.

(A <-> B) -> (C <-> C[B/A])

is a tautology for all propositional formulae A,B,C, where C[B/A] denotes
the result of substituting a possible occurrence of A with B in C.
Moreover, propositional modal logic is intensional since

(A <-> B) -> (\Box A <-> \Box A[B/A]),

that is,

(A <-> B) -> (\Box A <-> \Box B)

is NOT valid in modal logic, at least not in a normal modal logic like S5.
However, this definition of co-extensionality does not work for first-order
predicate logic, because formulae may contain free variables, i.e. let A
and B be wfs, then

(A(x) <-> B(x)) -> ((x)A(x) <-> (x)B(x))

is NOT valid. Instead, co-extensional formulae must be equivalent for all
individuals, then we get that predicate logic is extensional since

((x1)(x2)...(A <-> B)) -> (C <-> C[B/A])

is valid for all wfs A,B,C.

The relation between these two concepts of intensionality was noted by
Church (Carnap's Introduction to Semantics, 1943). No matter whether we
adopt the first or the second concept of intensionality, it does not give a
precise and general definition of intensionality (at least not as long as
co-extensionality is considered undefined). This may seem unsatisfying,
nevertheless, we ARE able to precisely formulate whether well known logics
are intensional as shown above. Finally, it should be noted that presenting
a general definition of intensionality for any logical system remind me
about defining what a logic is, and to that P. Aczel (What is a Logical
System, 1994) has said: ``But there is no generally accepted account of
what a logic is. Perhaps this is as it should be. We need imprecision in
our vocabulary to mirror the flexible imprecision of thinking.''


Nikolaj Oldager

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