Martin, Errol Peter2017-12-012017-12-011978b1216838http://hdl.handle.net/1885/136657This thesis investigates the properties of two systems of pure implication, with axioms a~d rules corresponding to the principles of transitivity of implication in the one system, and these axioms and rules plus the law of identity in the other. The main part of the text is devoted to a semantic analysis of these two systems. A class of models, called S-models, is defined and studied. The original construction of these S-models is due to R.K. Meyer. They derive t heir strength and interest from the fact that they employ a three-valued interpretation of the metalogical operations. By suitably determining the notion of validity using these three values, both of the systems mentioned above can be accommodated at once within the same class of models. The results obtained include soundness and completeness, decidability, arid, most importantly, the solution to a problem posed by Anderson and Belnap in their treatise, Entailment (Princeton U.P. 1975). Briefly, it is shown that, in the system o f pure transitivity axioms (and corresponding rules) , no instance of the axiom of identity is provable. When the axiom of identity is added to the system , one obtains as a corresponding result that no two distinct formulas co-entail each other.1venLogic, Symbolic and mathematical197810.25911/5d70ee7d4baf32017-11-22