## Constants, Inconstants, and Higher Order Propositions

A question arising on the Foundations Of Math List gives me an opportunity to introduce the subject of higher order propositions, which I think afford a better way to handle the situations of confusion, doubt, obscurity, uncertainty, and vagueness often approached by way of variations in the values assigned to propositions.

IA:
My own preference for t-definite, t-indefinite or f-indefinite, and f-definite, as opposed to tautology, contingent, and contradiction lies in allowing application of those terms for truth as well as for validity, for semantic and syntactic uses.

If we start with a universe of discourse $X$ and think of propositions as being (or denoting) functions of the form $f : X \to \mathbb{B},$ where $\mathbb{B} = \{ 0, 1 \},$ then what we are charged with is choosing suitable names for higher order propositions of the form $m : (X \to \mathbb{B}) \to \mathbb{B}.$

The term tautology or 1-definite is true of exactly one $f : X \to \mathbb{B},$ namely the constant function $1 : X \to \mathbb{B}.$

The term contradiction or 0-definite is true of exactly one $f : X \to \mathbb{B},$ namely the constant function $0 : X \to \mathbb{B}.$

The term contingent or indefinite is true of all the functions $f : X \to \mathbb{B}$ which are neither of the above.

Here is a place where I took the trouble to think up names for higher order propositions over a 1-dimensional universe.

I called the contingent propositions either informed or non-uniform.

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