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.

Re: FOM | TerminologyIrving Anellis

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.

This entry was posted in Foundations of Mathematics, Higher Order Propositions, Irving Anellis, Logic, Mathematics and tagged , , , , . Bookmark the permalink.

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