[FOM] Intensionality

[Michael Zeleny]

Truth-value-preserving intersubstitutivity (salva veritate) of concurrent
terms, i.e. terms that denote the same object, is neither a necessary nor
a sufficient criterion of intensionality. Thus on the Frege-Churchian
view, oblique contexts such as those arising in the contexts of beliefs,
inquiries, strivings, and other instances of what Russell characterized as
propositional attitudes, induce a shift in denotation of terms occcurring
therein, from their ordinary denotations to their ordinary senses. Under
the circumstances, terms that are synonymous, and a fortiori concurrent in
ordinary contexts, retain their concurrence in oblique contexts. However
even such terms may be distinguishable in nested oblique contexts such as
those in Benson Mates' example:
	Nobody doubts that whoever believes that the seventh consulate
	of Marius lasted less than a fortnight believes that the seventh
	consulate of Marius lasted less than a period of fourteen days.
The English terms `fortnight' and `fourteen days' accurately translate as
the same expression into all languages lacking a single word for denoting
a period of two weeks. This fact demonstrates that they are synonymous as
to their ordinary senses. But Mates' intuition that his example is false 
suggests that their oblique or second-order senses are distinct.

In connection with his formulation of the Logic of Sense and Denotation,
Church informally characterized intensionality as a relative phenomenon
correlated with the strength of identity criteria. Thus the set-theoretic
study of functions as individuated by the course of values is extensional
with respect to their study as objects that define their course of values
through a certain process of computation. Ever finer individuation of the
same domain, with a corresponding increase in the grade of intensionality,
is achievable through further distinctions in the objects and procedures
of computation.

cordially,                         -- Mikhail Zeleny at math.ucla.edu
7576 Willow Glen Rd, Hollywood, CA 90046 323-876-8234 323-363-1860
All of old. Nothing else ever. Ever tried. Ever failed. No matter.
Try again. Fail again. Fail better.              -- Samuel Beckett

On Fri, 14 Mar 2003, Nikolaj Oldager wrote:

> > -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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