in West Lafayette, Indiana (U.S.A.)l) = Cu~+lumn-l. Then for i < j it is known (MCKAY [2]; cf. ANDERSON [l], N I S~U R A [3]) that Cajai is not provable in IC+, the positive fragment of the intuitionistic sentential calculus, IC. It follows, of oourse, by a method of argument dubbed the MCKINSEY method in [4], that IC+ has no finite characteristic matrix. Indeed (of.[4]), that argument establishes the nonexistence of finite chara&&stic matrices for all subsystems of IC+ whose theses include Cpp: any m-valued model of such a subsystem would "identify" some ai and uj with i < j 6 mm' + 1 and so, validating Cpp7 would validate Cupi as well.Even this result can be generalized, however, for a role similar to that played here by the wff Cpp can in fact be played by any thesis of ICf.Where 9 is a wff b d t up from the single letter p and the connectives C, R and A of IC+, and u and @ are any wf3s of IC+, we shall say that a wff y (a, , 9) is an u, @-affiliate of y just in case q(m, @) can be obtained from y by replacing zero or more occurrences of p with occmnces of a and the rest (if any) with occurrences of j3.
Lemma. For each tuff'p hilt wp from the single letter p and the mnectives of IC+, and for all tuffs a and /I of IC+, there exists an a,/?-affiliate q(a,B) of Q such that For proof we induce on the length of y , letting q(a, 8) be j3 in the base case, where Q is p. If p is Cxy we let ~( a , @) be the result of putting a for p throughout x, use the induction hypothesis to obtain y(a, /I) such that F~c+Cy(u, j3) Cap, and let y(a, B) be C&x, p) y(a, j3). It is clear from [3] that l-lctCpx, whence l-rc+Ca~(a, b), and we have tIc+Cy(u, /?) C@ by hypothesis. Substitution in the thesis CCr$CqCrsCCpqCrsa) provides CCax(a, B) CCy(a, 8) Ca@CCx(a, B)y(., 8) Cab, whereupon two detachments deliver CCx(a7 p ) y(or, j3) Cap, that is, Cy(a, 8) Cap. If is Kxy [respectively, AX^] we let Q(K B) be Kx(a7 8) y(a, @) [respectively, Ax(", @) y(a, !)I, where t~+c&, 8) Cup and t r&,u(u,B) Cap, and use the thesisCCprCCqrCKpqr [respectively, CCprCCqrCApqr].Of course the trivial subsystems of IC+-those which have, whatever their rules, no theorem-re characterized by finite matrices with no designated values. But:Theorem. No nommpty subsystem of IC+ laas a finite characteristic wzqtfix.Proof. Let H be any nonempty subsystem of IC+ and suppose that it has an m-valued characteristic matrix, %. Then for some i < j mm' + 1, ui and aj must receive identical values for each assignment of values of % to the letters occurring in them. Replacement of zero or more occurrences of either by the other in any W-tautology, then, must yield an '91-tautology, and so such replacement in a theorem of H can yield only a theorem of H. Let a1 = p, = q, = Au,-,a, and h c + C y b , B) cap. l) The theorem eetablished below wss annonncad in [5], The terminology is thst of [el.e, Writing 1, 2 and 3, respectively, for the familiar theses CCrpCCpqCrq, CCqCrsCrCqs and ccdq&~rpcrS of IC+, and Dzy for the most general rewlt of detaching the wff y, or a substitution insta...
