Sentence examples for first-order valuation from inspiring English sources
A first-order valuation in a model is a mapping \ v\) that assigns a member of \ \bD_{O}\) to each variable.
A first-order valuation in FOIL model \ \cM\) is a mapping that assigns to each object variable a member of \ \bD_{O}\), as before, and to each intension variable a member of \ \bD_{i}\).
Truth, at a state \ \Gamma\) of a model \ \cM = \langle \bG, \bR, \bD_{O}, \bI\rangle\), with respect to a first order valuation \ v\), is characterized as follows, where \(P(x_{1}, \ldots,x_{n})\) is an atomic formula: Call a formula valid if it is true at every state of every first order model with respect to every first-order valuation, as defined above.
A first order valuation \(w\) is an \(x\ -variant of valuation \(v\) if \(v\) and \(w\) agree on all variables ex\ -variantbly fof \(x\).
Note that first order valuations are not state-dependent in the way that interpretations are.
For example we define classical propositional consequence by means of truth valuations, first-order consequence relation by means of structures, the intuitionistic consequence relation by means of Kripke models, etc.
Hence a part of the specification of an alethic dialogue in the case of propositional logic will be a valuation function, and in the case of first-order logic a τ-structure for an appropriate vocabulary τ.
A back-of-the-envelope calculation suggests that excluding the Jewish managers reduced the aggregate market valuation of firms listed in Berlin by 1.8% of German gross national product, a first-order economic loss.
Tax is not a first-order consideration.
These are of course first-order intensions.
