Strict Implication in A Sequence of Extensions of S4

Abstract

Zeitcrhr. I . malh. Lugtk und Grundhf6w d . Math. Rd. 27, s. 301--212 ( 1 9 8 1 ) STRICT IMPLICATION IN A SEQUENCE OF EXTENSIONS O F S4 by DOLPH ULRICH in West Lafayette, Indiana (U.S.A.) l) Introduction Work on the problem of isolating the strict, implicational fragment of various modal logics began with [7], from which MEREDITH'S normal-form proof of completeness for the strict fragment of S 5 was extracted and included in [9]. The development of CENTZEN techniques next permitted HACKING [5] to provide axiomatizations of the strict fragments of T, 5 2 , S 3 and 54 (the latter also in [l], fn. 12), along with a simpler proof of MEREDITH'S S5-result. I n this paper the largely syntactic methods of these earlier authors are set aside in favor of a semantic approach along the lines of KRIPKE, et. al. The reader is presumed t o be familiar with some of the standard works on relational semantics and HENKINstyle completeness proofs for-full modal logic, e.g., [6], [S] and/or [ll].8 1. Theories, frames, and related preliminaries Let S be the set of wffs built in the usual way from sentential letters p , y, r , s, t , u , p l , p,, p , , . . . and the binary connective C. Wffs that contain at least one C will be called strict and for each subset W of S, %(W) is to be t h e set of all strict wffs in W.The weakest system we shall here consider, C4, has as axioms the substitution instances of ( A l ) C p p , (-42) CCpqcrCpq, (A3) CCpCqrCCpyCpr, and as sole rule of inference detachment ("from N and Cq3 infer 8"). Standard terms concerning axiomatic systems (theorem, deductive consequence, and the like) will be employed, with normal supporting notation, in the ordinary way and so without individual explanation.A system L' whose only rule is detachment is an extension of a similar system L provided its set of theorems is closed under substitution and includes the theoremsof the latter. For extensions of C4, we have available the well-known L e m m a 1 (Deduction theorem for extensions of C4). Let L be any extension of C4 and let W be any set of strict wffs. Then for all wffs 01 and tf? in S, if W, N kI,p thenThe straightforward proof may be found, for example, in [l], and so is not repeated.Defining an L-theory to be any set of wffs containing the theorems of L and closed under detachment, hence, under deductive consequences in L, we assure ourselves also of w t 1, cap. l) Some of the results presented here were announced, albeit with more complicated constructions and unnecessarily intricate proofs, in [14] and [15]. I owe general thanks to J. MICHAEL DUNN, with whom a number of these resultg have been discussed over the years. I am indebted also to the members of my Spring 1973 seminar, Mssrs. HADDEN, LYMAN, PARKER, PAVLOVIC and, especially, RALPH MOON. The latter's skill and enthusiasm, through our continuing discussions, have greatly improved the presentation.Lemma 2. Let W be a n L-theory, L any extension of C4. Then for all (x, L is to be the set of all L-theories. S is clearly the strongest member of L ; let TL be the...

Shortest Axiomatizations of Implicational S4 and S5

“…Moreover, no single axiom for C4 has been discovered; indeed, this is stated as an open problem in [2, page 83]. Ulrich [16] has shown that C4 is also the strict implicational fragment of each modal logic between S4 and S4.3; hence, our bases are new and shortest bases for the strict implicational fragments of these extensions of S4 as well.…”

A Descending Chain of Incomplete Extensions of Implicational S 5

“…With "P" short for " p -+ r"> C 5 has as axioms tjhe siibstitut'ion instances of ( A l l P-+ P . there is an alternate proof in [12]. whose numbering is followed above-that C 5 is the st,rict implicational fragment, of S5: if -+'s are removed by replacing wffs of the form DI -+ , 8 with the corresponding expressions of the form o(i.…”

“…Familiarity with [12] would be helpful in 5 1 below for matters having to do with frames, models and the like, and with Ell] in the remaining sections for terminology and notation concerning matrices. But most of what is used is more or less standard and the rest easily guessed?…”

“…But most of what is used is more or less standard and the rest easily guessed? so consultation of [ l l ] and [12] should not often be necessary.') 5 1.…”