GENERAL SOLUTION OF LOGICAL PARADOXES

 

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.