The P-W problem
Date
1978
Authors
Martin, Errol Peter
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Abstract
This 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.
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