Extending Metacompleteness to Systems with Classical Formulae

In honour of Bob Meyer, the paper extends the use of his concept of metacompleteness to include various classical systems, as much as we are able. To do this for the classical sentential calculus, we add extra axioms so as to treat the variables like constants. Further, we use a one-sorted and a two-sorted approach to add classical sentential constants to the logic DJ of my book, Universal Logic. It is appropriate to use rejection to represent classicality in the one-sorted case. We then extend these methods to the quantified logics, but we use a finite domain of individual constants to do this. In [5], Meyer introduced the notion of coherence for logics that “can be plausibly interpreted in their own metalogic” (p. 658). Meyer set up this idea by introducing  v for a modal logic L such that v( A) = T iff ` A in L, with classical-style valuations for the connectives ‘⊃’ and ‘∼’. He defined a formula A to be  iff v(A) = T , for all metavaluations v of L, and the logic L to be  iff each theorem of L is metavalid. He went on to show that a wide range of modal logics are coherent, and that, in particular, the property ‘if ` A ∨ B then ` A or ` B’ holds for these logics. He also showed that the relevant logics NR and E are coherent, with NR satisfying the above property and with E satisfying a similar disjunctive property, ‘if ` (A→ B)∨ · · ·∨(An → Bn)∨C, for C ‘→’-free, then ` Ai → Bi, for some i, or ` C, where C is then a tautology.’ Ross T. Brady, “Extending Metacompleteness to Systems with Classical Formulae.”, Australasian Journal of Logic (8) 2010, 9–30 http://www.philosophy.unimelb.edu.au/ajl/2010 10 Meyer in [6] used a preferred metavaluation for a logic L (defined below), which essentially has the effect of expressing the theorems of L in an inductive semantic-style form. Meyer then introduced the notion of  for L to mean that L is sound and complete with respect to this metavaluation. He went on to prove meta-completeness for a wide range of positive relevant logics, including their quantified logics, thus showing the priming property, α if ` A∨ B then ` A or ` B, for the theorems of such logics. For the quantified logics, the additional property, β if ` ∃xA then ` At/x , for some term t, was shown to hold. Meyer set up the (preferred)  v (called the canonical quasivaluation v ′ in [6]) on the formulae of a quantified relevant logic L as follows : (i) v(p) = F, for all sentential variables p. (ii) v(A & B) = T iff v(A) = T and v(B) = T . (iii) v(A∨ B) = T iff v(A) = T or v(B) = T . (iv) v(A→ B) = T iff ` LA→ B and, if v(A) = T then v(B) = T . (v) v(∀xA) = T iff v(At/x) = T , for all terms t, i.e. all individual variables and constants. (vi) v(∃xA) = T iff v(At/x) = T , for some term t. Meyer used a simple induction on formulae to prove completeness, i.e. if v(A) = T then ` LA, and he proved soundness, i.e. if ` LA then v(A) = T , using the usual induction on proof steps, thus establishing metacompleteness for the quantified relevant logic L. Before going on, we present some of the main relevant logics and their quantified forms, including the ones explicitly referred to in this paper. Primitives: ∼, &, ∨, →, ∀, ∃ (connectives and quantifiers); p, q, r, . . . (sentential variables); f, g, h, . . . (predicate variables); x, y, z, . . . (individual variables).