Comments on some completeness theorems of Urquhart and Méndez \& Salto (Q5935786)

This paper diagnoses and corrects an error in claims of \textit{A. Urquhart} [in: D. Gabbay et al. (eds.), Handbook of philosophical logic. Vol. III. Dordrecht: Reidel. Synth. Libr. 166, 71-116 (1986; Zbl 0603.03001)] and \textit{J. M. Méndez} and \textit{F. Salto} [Notre Dame J. Formal Logic 36, 407-413 (1995; Zbl 0838.03019), J. Philos. Log. 27, 75-84 (1998; Zbl 0904.03011)] to have proved completeness for the positive many-valued logic \({\mathbf C}\) and some negation extensions of it, respectively. Valid in the semantics but not provable in these logics is the principle (i) \(((\psi\to \varphi)\wedge (\psi\to\theta)) \to(\psi\to (\varphi\wedge \theta))\). If this is added to the negation extensions of Méndez and Salto then their completeness proofs will go through. To repair Urquhart's proof, add (i) and also (ii) \(((\varphi^k\to \psi)\wedge (\theta^k\to \psi))\to ((\varphi\vee \theta)^k \to \psi)\) to \({\mathbf C}\). There is an unfortunate misprint in the matrix argument for the unprovability of distribution. Page 53, line 5 from the bottom should read: \(2\wedge(3\vee 2)= 2\wedge 5=2\), rather than \(2\wedge(3\vee 2)= 2\wedge 2=1\), and the last line on the page should read: \((2\wedge (3\vee 2))\to ((2 \wedge 3)\vee (2\wedge 2))= 2\to 1=5\neq 6,\) rather than \((2\wedge (3\vee 2)) \to ((2\wedge 3)\vee (2\wedge 2))= 1\to 1=1\neq 6\). This does not vitiate the result of the paper, however.