Ultralogics and probability models (Q5957224)

Assume that \(L\) is a nonempty set that represents a language and let \(\mathcal P\) denote the set-theoretic power set operator. A mapping \(\mathcal C : \mathcal P (L)\to \mathcal P (L)\) is a general consequence operator (or closure operator) if for each \(X,Y\in\mathcal P(L)\):NEWLINENEWLINENEWLINE(i) \(X\subset \mathcal C(X)=\mathcal C(\mathcal C(X))\subset L\), and NEWLINENEWLINENEWLINE(ii) \(X\subset Y\), then \(\mathcal C(X)\subset\mathcal C(Y)\). NEWLINENEWLINENEWLINEA consequence operator is called finite whenever \(\mathcal C(X)=\bigcup\{\mathcal C(A)\mid A\in F(X)\}\) where \(F\) is the finite power set operator. In this paper, the author discusses, by using nonstandard analysis, how nonstandard consequence operators, ultralogics, can generate the informational content determined by probability models.