Matrix functions

In this chapter we develop the elements of an algebraic theory of matrices, the results of which will be needed later.

A truth-function is any function (singulary, binary, k-ary) defined over the truth-values of a matrix and taking truth-values for its values. A truth-function is a matrix function if it can be obtained from the basic functions and the projective functions λx1…xn.xi by a finite number of operations of composition. Thus if f is a k-ary matrix function and g1,…,gk are n-ary matrix functions, λx1…xn.f(g1(x1,…,xn), …,gk, (x1,…,xn)) is also a matrix function. In general the matrix functions will not exhaust the truth-functions: if they do the matrix is said to be functionally complete. Let p1,…,pn be distinct propositional variables that include all those in the formula A, and let f be the n-ary truth-function such that f(x1,…,xn) is the value taken by A when each pi is assigned the value xi. It is easy to see that the matrix functions are precisely the functions associated with formulae in this way, and their logical significance derives from this. For example, where two matrices differ in their basic functions but have the same matrix functions, the corresponding calculi will merely be versions of each other, based on a different choice of connectives.

A submatrix M′ of M is determined by any subset of the values of M that is closed under the basic functions (or, equivalently, under the matrix functions).