historically been made to get an axiom system that somehow describes definite kinds of objects. But now the main way this has been done is by progressively adding axioms so as to get closer to having a system that is complete—with only a rather vague notion of just what underlying objects one is really expecting to describe.
In studying basic processes of proof multiway systems seem to do well as minimal idealizations. But if one wants to study axiom systems that potentially describe definite objects it seems to be somewhat more convenient to use what I call operator systems. And indeed the version of logic used on page 775—as well as many of the axiom systems on pages 773 and 774—are already set up essentially as operator systems.
The basic idea of an operator system is to work with expressions such as (p ∘ q) ∘ ((q ∘ r) ∘ p) built up using some operator ∘, and then to consider for example what equivalences may exist between such expressions. If one has an operator whose values are given by some finite table then it is always straightforward to determine whether expressions are equivalent. For all one need do, as in the pictures at the top of the next page, is to evaluate the expressions for all possible values of each variable, and then to see whether the patterns of results one gets are the same.
And in this way one can readily tell, for example, that the first operator shown is idempotent, so that p ∘ p == p, while both the first two operators are associative, so that (p ∘ q) ∘ r == p ∘ (q ∘ r), and all but the third operator are commutative, so that p ∘ q == q ∘ p. And in principle one can use this method to establish any equivalence that exists between any expressions with an operator of any specific form.
But the crucial idea that underlies the traditional approach to mathematical proof is that one should also be able to deduce such results just by manipulating expressions in purely symbolic form, using the rules of an axiom system, without ever having to do anything like filling in explicit values of variables.
And one advantage of this approach is that at least in principle it allows one to handle operators—like those found in many areas of mathematics—that are not based on finite tables. But even for operators given by finite tables it is often difficult to find axiom systems that can successfully reproduce all the results for a particular operator.