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Journal of Logic and Computation 1998 8(1):119-130; doi:10.1093/logcom/8.1.119
© 1998 by Oxford University Press
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Original Articles

Adding the Everywhere Operator to Propositional Logic

DAVID GRIES and FRED B. SCHNEIDER

Computer Science, Cornell University Ithaca, NY 14853, USA E-mail: gries{at}cs.cornell.edu

Sound and complete modal propositional logic C is presented, in which {square}P has the interpretation ‘P is true in all states’. This interpretation is already known as the Camapian extension of S5. The new axiomatization for C provides two insights. First, introducing an inference rule textual substitution allows integration of the propositional and modal parts of the logic in a way that gives a more practical system for writing formal proofs. Second, the two following approaches to axiomatizing a logic are shown to be not equivalent: (i) give axiom schemes that denote an infinite number of axioms and (ii) write a finite number of axioms in terms of propositional variables and introduce a substitution inference rule.

Keywords: Propositional logic; calculational logic; everywhere operator


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