Form equivalence class Howard-Rubin Number: 14
Statement: Structural Completeness Theorem for PropositionalLogic: Assume \(At\) is a set of propositional variables and \(S\)is the set of propositional formulas generated by \(At\). Assume\(X\subseteq S\) and \(a\in S\) satisfy the following: For any function\(e: At\rightarrow S\) if \(Cn(\{ b[e]: b\in X\})\subseteq Cn(\emptyset)\)then \(a[e]\in Cn(\emptyset)\). Then \(a\in Cn(X)\). (\(Cn(Y)\) is thedeductive closure of \(Y\ \cup\) all substitutions of axioms for theclassical predicate calculus and for \(a\in S\), \(a[e]\) is \(a\) with eachpropositional variable \(p\) replaced by \(e(p)\).) Pogorzelski/Prucnal [1974].
Howard-Rubin number: 14 AX
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