The
considerations below sketch an extremely concise exposition of the solution.
Reflexivity
is a kind of indexicality. Indexical expressions depend on the context because
they are affected by some lack of information: and the context is just the
integrative source of information which converts the indexical (open)
expression into an absolute (closed)
one. On this ground a basic distinction can be focused between linguistic
conversions (Socrates was a great man; he ...) and ostensive conversions
(“yesterday”, “down there”). Ostensive conversions can be neglected for the
present purpose.
The
aforesaid lack of information is symbolizable by a free variable whose
conversion is carried out by its substitution with a constant. Of course that
conversion cannot be effective if the free variable to substitute continues to
appear in the substitutor. Here I do not dwell on the improvement that the
introduction of a reflexive variable can bring
to a formalization of the entire theme.
The
current criterion for the classification of open sentences (Skolem) is the
number of different free variables occurring
in the sentence under examination. We
are interested in monovariable sentences. Skolem criterion does not account for
the basic difference between sentences where the free variable occurs either in
the subject or in the predicate, and sentences where the same free variable
occurs both in the subject and the predicate (reflexive sentences). Logical
paradoxes arise with self-conversions. Whatever sentence resulting from a
self-comversion cannot be closed (the
conversion cannot be effective), as the free variable to substitute continues
occurring in the substitutor. Paradoxical dilemmas are defective and no
defective dilemma can admit a well-founded answer; the usual argument (if it is
were so, then it ought to be the contrary) is nothing but the intrinsically
vain attempt to fill through an arbitrary hypothesis the lack of information
created by our same definitions.
This
general solution of logical paradoxes entails very wide consequences.
