TY - JOUR
AU - Kuhn, H. W.
AB - SOLVABIUTY AND CONSISTENCY FOR LINEAR EQUATIONS AND INEQUALITIES* H. W. KUHN, Byrn Mawr College 1. Introduction. The term consistent is in common use in two contexts that appear to be quite different at first sight. It is often applied to systems of linear equations as a synonym for solvable. Thus, Dickson says in [3]: "We shall call two or more equations consistent if there exist values of the unknowns which satisfy all of the equations." Again, BOcher writes in [1]: "The equations may have no solution, in which case they are said to be inconsistent." Elsewhere, it is applied by logicians to deductive systems as a synonym for non-contradicuwy. Thus, Tarski defines the term in [8]: "A deductive theory is called consistent or non-contradictory if no two asserted statements of this theory contradict each other." Our preliminary purpose is to reconcile these two usages, agreeing informally that "solvable" means "satisfiable" and that "consistent" means "non-contra­ dictory." The reconciliation is brought about by setting forth explicitly the definition of consistency that has been employed implicitly in ordinary treat­ ments of linear equations. This definition is based on a non-effective character­ ization of the logical consequences of the system, and is 
TI - Solvability and Consistency for Linear Equations and Inequalities
JF - The American Mathematical Monthly
DO - 10.1080/00029890.1956.11988793
DA - 1956-04-01
UR - https://www.deepdyve.com/lp/taylor-francis/solvability-and-consistency-for-linear-equations-and-inequalities-wuyRH0mtfC
SP - 217
EP - 232
VL - 63
IS - 4
DP - DeepDyve
ER -