By Innocentius Marie Bochenski
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The following scheme, in which the relative extension of the terms is indicated by lines, illustrates the point: Division 1st figure Major Middle Minor 8 D. AXIOMATIZATION The assertoric syllogism is probably the most important discovery in all the history of formal logic, for it is not only the first formal theory with variables, but it is also the first axiomatic system ever constructed. Aristotle's theory of the axiomatic system belongs to methodology and cannot be treated here; we shall limit ourselves to the remark that according to him there must be some undemonstrated axioms (dEdpata) l9 while other theorems are deduced; that in each axiomatic system the number of steps must be finitez0; and that, as far as logical axioms are concerned, they must be intuitively evident.
Pr. A 15, 35 a l l f . , 3 1 f . ; A 25, 42 b 9-10; An. Post A 19, 82 b G f . ; 22, 8 4 a 35; 23, 8 4 b 14. Atcimvpa (relation) is opposed t o neltaotg 42 b 20, but in most texts the terms are synonimow, especially in An. Post. ASSERTORIC SYLLOGISTIC 43 product of which forms the antecedent, are called “premisses” (neoz&asLs or Giaozrjpaza 6 ) ; the substitution for “r” is called “conclusion” (ovpndeaupa ; often simply : ovMoy~opds). (4) What is substituted for “A” and “B” in the premisses and in the conclusion - indeed the letters themselves - are called “terms” (6’eot).
41. * 10. 42. * 10. 43. * 10. 44. * 10. 45. * 10. 46. 1 prim. of double neg. (0) sentences N(XeP) 3 N(PeS)27 N(#aP( 3 N(PiS)2* N(8iP) 3 N(Pi8)29 O ( f J e P )3 O ( P e J 930 ()(#aP)3 ()(Pi#)31 ()(XiP) 3 ()(Pi#). 41-43 meet a serious difficulty if the structure 10. 21 ff. is presupposed. z4 An. Pr. A 13, 32 a 37f. - 85 ib. 38f. - 26 ib. 40. - 27 An. Pr. A 3, 25 a 298. - An. Pr.