On a Property of Matrices for Subsystems of IC⁺

Abstract

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...

Answer to a Question Suggested by Schumm

“…His result does not hold for sentential calculi gcnemlly. [9] providing a subsystem of intuitionism l) SCHUNM [4] describes a modal logic as a "class of formulas containing all classical tautologies and closed under the rules of substitution and material detachment." But it is the finite model property which is here under discussion, and since ANDERSON [l] has shown the latter to be a property of sentential calculitheorems together with rulesrather than merely of their theorem sets, it has seemed judicious to be explicit on this point.…”