Arithmetic of logic
Eric Temple Bell · Transactions of the American Mathematical Society · 1927
This is probably the first attempt to construct an arithmetic for an algebra of the non-numerical genus.fIn his classic treatise, An Investigation of the Laws of Thought,J Boole developed the thesis that "Logic (is) • • • a system of processes carried on by the aid of symbols having a definite interpretation, and subject to laws founded on that interpretation alone.But at the same time they exhibit those laws as identical in form with the laws of the general symbols of Algebra, with this simple addition, viz., that the symbols of Logic are further subject to a special law, to which the symbols of quantity as such, are not subject."The special law is what Boole calls the law of duality, x(l-x)=0, or the excluded middle; here the indicated multiplication is logical, 1-a: is the supplement of x.Boole showed therefore that abstractly logic is contained in common algebra.Taking rational arithmetic Si ( = the theory of numbers in reference to the positive rational integers 0, 1, 2, 3, • • • , only) and certain parts of the theory of algebraic numbers, particularly the rudiments of Dedekind's theory of ideals and those of Kronecker's modular systems as our guides, we shall see to what extent Boole's algebra of logic may be arithmetized in a precise sense to be defined presently.Although it will be unnecessary to refer explicitly anywhere to the theory of algebraic numbers, it may be mentioned that this theory, which includes rational arithmetic, is a surer guide than the latter in problems of arithmetization.In rational arithmetic the essential abstract structure of the concepts to be extended beyond 0, 1, 2, • • • is often quite ingeniously concealed.This is true, for example, of the G.C.D., L.C.M., and residuation.The theory of ideals, on the other hand, often indicates immediately what transformations by formal equivalence must first be applied to operations or relations of rational arithmetic in order that they shall be significant for sets of elements for which order relations are either irrelevant or meaningless.