A content semantics for quantified relevant logics. II (Q910392)

This is the sequel to Part I [ibid. 47, 111-127 (1988; Zbl 0666.03017)]. As promised there, additional postulates are given here to specify the content semantics (an algebraic style semantics) of almost any first- order extension of the system BBQ which one might care to consider, e.g., TWQ, RWQ, TQ, RQ and KQ (classical first-order logic). (See the author [Logique Anal., Nouv. Sér. 27, 355-377 (1984; Zbl 0559.03011)] for a larger list of suitable extensions.) EWQ and EQ are the only significant systems notably absent. Generalised consistency and completeness proofs are given to cover all of the systems included. The author then goes on to produce ``reduced'' content semantics. (See \textit{R. Routley}, \textit{R. K. Meyer}, \textit{V. Plumwood} and the author [Relevant logics and their rivals (1982; Zbl 0579.03011)] for the original notion of reduced modelling and its motivation.) The point here is to give models certain plausible properties which they lack otherwise, to wit, primeness and saturation properties for the truth filter: If \(A\vee B\) is an element of the truth filter, then so is either A or B ; and if (\(\exists x)A\) is an element of the truth filter, then so is the result of substituting some object of the domain for x in A. To this end the original system BBQ and its extensions are modified by adding the ``meta-rules'': MR1. If \(A\Rightarrow B\), then \(C\vee A\Rightarrow C\vee B,\) MR2. If \(A\Rightarrow B\), then (\(\exists x)A\Rightarrow (\exists x)B,\) yielding the system \(BB^ dQ\) and its extensions. The model structures for these systems are just like the originals except for the addition of these two postulates: (p6c) The union of c and d is in T only if either c is or d is. (p11b) The generalised union of \(\{F^ ns^ b/k(i_ 1,...,i_ n):\) b is in \(D\}\) is in T only if \(F^ ns^ b/k(i_ 1,...,i_ n)\) is in T for some b in D. Soundness and completeness are then proved as required.