Remark on Gronwall's inequality (Q1198394)

The present note is devoted to abstract analogues of the Gronwall inequality in classical propositional calculus. Let \(a_ 1,a_ 2,\dots,b_ 1,b_ 2,\dots,\) and \(x_ 1,x_ 2,\dots\) be infinite sets of statement variables and \(\sim\), \(\land\), \(\lor\), \(\supset\), and \(\equiv\) connectives in classical propositional calculus. The main results of the note are embodied in Theorems 1 and 2 below. Theorem 1. If \(a_ n\lor\bigvee^ n_{j=1} b_ j\land x_ j\supset x_{n+1}\) holds with logical value true for every \(n\in\mathbb{N}\), then: (i) \(a_ 1\lor b_ 1\land x_ 1\supset x_ 2\); and (ii) for \(n\geq 2\), we have \(a_ n\lor\left[\bigvee^{n-1}_{j=1} b_{j+1}\land a_ j\lor\left(\bigwedge^ n_{j=1} b_ j\right)\land x_ 1\right]\supset x_{n+1}\). Theorem 2. If \(a_ n\lor\bigwedge^ n_{j=1}(b_ j\lor x_ j)\supset x_{n+1}\) has logical value true for every \(n\in\mathbb{N}\), then \(a_ n\lor(b_ 1\lor x_ 1)\supset x_{n+1}\), \(n\in\mathbb{N}\). The terminology used is from the book ``Elementary formal logic'' by \textit{G. N. Georgacarakos} and \textit{R. Smith} (McGraw-Hill, 1979). Two other similar results are stated in the note.