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Intermediate propositional logics.

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posted on 19.11.2015, 08:55 by C. G. McKay
The main object of the thesis is to investigate a variety of questions relating to the set of intermediate prepositional logics. Let H denote the set of words which are intuitionist theses and let K denote the set of words which are classical theses. Then a set of words X is an intermediate (prepositional) logic iff 1) HcXcK and 2) X is closed wrt modus ponens and substitution. Of special interest among intermediate logics, are those which are characterised by a finite pseudocomplemented lattice. We prove the important result that every such finite logic is finitely axiomatisable. This result is one of the many consequences of the fundamental representation theorem for pseudocomplemented lattices (PLs) whereby every PL is subdirectly reducible to a set of so-called strongly compact PLs. In addition we provide a neat syntactic characterisation of finite logics, and show that H is the limit of a certain sequence of explicitly axiomatised finite logics. In addition we consider more restricted types of intermediate logics, in particular intermediate ICN logics. By generalising a result of DIEGO, to show that every ICN algebra with a finite number of generators, is finite, we manage to prove that every finitely axiomatised intermediate ICN logic is decidable with primitive recursive bound. This generalises and completes earlier work of BULL. The same methods are then applied to obtain a proof of the decidability of all those intermediate logics, obtained by adding a finite set of disjunction-free words, as additional axioms to H. Many older results in the literature are then seen to be special cases of this general result. We introduce the new concept of strong undefinability of a prepositional connective, and examine its relation to McKINSEY'S related notion. It is shown that the connectives of implication, disjunction and negation, are all strongly undefinable in H, whereas conjunction is weakly definably. Lastly we investigate the scope of the so-called Separation theorem in the field of intermediate logics. It is shown that certain intermediate logics treated in the literature do not possess any axiomatisation for which the Separation theorem can be proved.


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University of Leicester

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