classically complete modal relevant logics

Math. Log. Quart. 39 (1993) 165 - 177

Mathematical Logic Quarterly

@ Johann Amhrosius Bart11 1993

Classically Complete Modal Relevant Logics

Edwin D. Mares

Department of Philosophy, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5l)

Abstract A variety of modal logics based on the relevant logic R are presented. Models are given for each of these logics and completeness is shown. It is also shown that each of these logics admits Ackermann's rule y and as a corollary of this it.. is proved that each logic is a conservative extension of its counterpart based on classical logic, hence we call them "classically complete". MSC: 03B45, 03B46. Keywords: Modal logic, relevance logic.

0. In MARES and MEYER [ll], we provided the iiiodal relevant logic R4 with a semantics and proved completeness. R4 is obtained by adding S4isli principles governing the modal operators to the relevant logic R. Coinbiiiing R wit11 S4 is of special interest because of the intrinsic philosophical interest of this project and because of its long history, originating with ORLOV'S little known [16] (see also [4]). This project was reintroduced, in effect, in ACKERMANN [l] aiid lins been carried on by ANDERSON and BELNAP and their students. Although combining R 1vit.h S4- ish principles is perhaps the best kiiown project. to amalgamate relevance logic w i t h modal logic, i t is by no means the only such project presented in the literature. For example, BACON [3] suggests combining R with principles from various nornial iiiodal logics, PARK and BYRD [17] add S5-like principles to R, and FUHRMANN [8] adds modal principles to relevant logics weaker than R (as well as to R itself).

In this paper I also extend R by adding various iiiodal principles t.o it.. In particular, I amalgainate the modal logics D, T, the Brourversche System (Br). S4, and S5 with R. The resulting systems are called RD, RT, RBr, R4, aiid R5. The novelty of the present approach is that all of the systeiiis produced contain the following asiom:

o(A V B ) - ( O A V OB).

')My greatest debt in writing this paper is to ROBERT K. MEYER, who taught inr most of wllat I know about modal relevant logic. I am also grateful to Iim FINE, JOHN SLANEY, ANGRE FUHRMANN, ERROL MARTIN, JACQUES RICHE, and JILL LEBLANC for discussionsrelating to the topic of this paper. hloreover, I must acknowledge my debt to tlle Social Sciences and Humanities Research Council of Canada who enabled me to keep my body and soul together while writing this paper by giving ine postdoctoral fellowships 15G-S9-0738 and 457-90-OOSI.

166 Edwiii D. Mares

This axiom was suggested by BELNAP as an addition to E and, later, to MEYER’S system NR so that they would contain all the theorems of S 4 on a direct traiislation (see [U]). As I shall show below, each of the systems formulated in the present paper also contains its classical counterpart on a direct translation. Thus, for example, R4 contains S4, R5 contains S5, and so on. For this reason, I call these logics “classically complete”.

1. In this section we set out the syntax of our systems. Our language is a standard modal sentential language & that contains propositional constants p1, p 2 , p 3 , . . . connectives A, -, 4, and the modal operator 0. It is governed by the usual formation rules. By Wff(C) or merely Wff we denote the set of formulas o f t . In our metalanguage, we use lower case italic letters from the latter half of the alphabet to range over propositional constants and capital italic letters from the early part of the alphabet to range over formulas in general. We use three defined connectives, naiiiely,

A + B =df O ( A - + B ) , A V B = d f - ( - A A - B ) , O A =df-O^.A.

The logic RD has the following axiom schemes and rules: A x i o m s .

AO.

A l .

A2.

A3.

A4.

A5.

A6. A7.

AS.

A9.

A10.

A l l .

A12.

A13.

A14.

A + A ,

( A - B ) - ( ( B -+ C) - ( A - C)), A --+ ( ( A - B ) -+ B ) ,

( A - ( A - B ) ) - ( A + B ) ,

( A A B ) - A, ( A A B ) - B ,

( ( A -+ B ) A ( A - C)) -+ ( A .-+ ( B A C ) ) ,

( A A ( B V C)) -+ ( ( A A B ) V ( A A C)), --A - A ,

( A +-B) + ( B --A), O(A -+ B ) -+ ( O A - o B ) ,

(UA A OB) - O ( A A B) , O(A V B ) -+ (OA V OB),

OA + OA,

If A is an axiom, OA is an axiom.

R u l e s .

AI. From A and B , infer A A B , -E. From A + B and A , infer B .

Classically Complete Modal Relevant. Logics 167

RT results from replacing A13 with A15. O A - - , A ,

RBr is obtained from adding to RT the scheme A16. A + OOA,

R4 results from adding A17. O A - O O A

to RT and, finally, R5 is obtained from adding A16 to R4. In any of these systems we can derive the following rule:

01. From A, infer O A .

2. In this section, we present the semantics for the systems set out in section 1 above. For a logic L, we call an Gmodel structure an Lms. An RDlns is a quintuple K = (K, P, R , S, *), where I< is a non-empty set, P is a non-empty subset of K , R is a ternary relation on K, S is a binary relation on K, and * is a unary operator taking members of K to members of I<, satisfying definitions and postulates to come. We use the following abbreviations for relational products: Let M he an m-ary and IV an n-ary relation; then when there is an x such that Ma1 . . .am- lx and Nbl . . . b i - l x b i + l . . . bn, we may write Nbl . . . bi-l(Ma1 . . . a,-l)bi+l. . . b n , and where M(Mb1. . . bm-l)az . . .an we may write in particular M’bl . . . b m - 1 0 3 . . . an. Using this notation, we set out the following definitions:

R2abcd =df 3x(Rabt & Rxcd),

Eabc =df R(Sa)bc,

S’ab =df 3x(Sax 8 Sxb). We also define

a 5 b =df 3x(Px& Rxab), Tab =df SabOSa’b’. Moreover, we sometimes write ‘P(a) ’ instead of ‘a E P’.

The postulates on a RKms are the following: P1. Raaa, P2. R’abcd j R’acbd, P3. S(Rab)c 3 R(Sa)(Sb)c, P4. 5 is a partial order, P5. a _< b & Rbcd 3 Racd, P6. a 2 b & Sbc + Sac, P7. Rabc + Rac*b*, P8. a** = a , P9. Sab + 3x(z 5 b&Ta+),

P10. 3x(P(x ) &Txa) P(a ) , P11. 3yTay.

P12. Saa. An RTms is an RDms that also satisfies the postulate

An R4ms is an RTms that also satisfies P13. Sab& Sbc =+ Sac.

168 Edwin D. Mares

An RBrms is an RTms that satisfies

P14. Tab =+ Tba. An R5ms is an R4ms that is also an RBrms.

A valuafion u of L in an Lms K is a function from ordered pairs of propositional variables and worlds into {TI F} (the set of truth values). Every valuation v obeys the following heredifan'ness postulate:

(HP) Each valuation uniquely determines an associated inferprefafion, which is a function from ordered pairs of formulas and worlds into {TI F}, according to the following truth conditions:

If v (p ,a ) = T and a 5 b , then v (p ,b ) = T.

T A . T-. TO.

T-.

I ( A A B, a) = T iff I (A , a) = T and I ( B , a ) = T, I(-A,a) = T iff I(A,a') = F, I (OA, a ) = T iff Vz(Saz a I(A, z) = T), I ( A - B, a) = T iff VzVy(Ra+y a. I(A, z) = T I (B , y) = T).

For any of our logics L, a formula A is true at a world u on an inierprefdion I iff I(A,a) = T and it is otherwise false at a; A is verified on I iff V z ( r E P 3 I (A , Z) = T); A i s L-valid iff A is verified on all interpretations in all Lms.

We can also derive the following truth conditions:

T V . TO.

I ( A V B, a) = T iff I(A, a) = T or I (B , a) = TI I(OA, a) = T iff 3z(Sa*z* & I(A, z) = T).

In our soundness proof, we will use the lemmas below. We direct the reader to

L e m m a 2.1 (Hereditariness). If I(A, a) = T and b 5 a, then I(A, b ) = T. L e m m a 2.2 (SemEnt). A - B is verified on I iff, for all a E I<, i f I ( A , a ) = T,

fhen I ( B , a ) = T. Borrowing some terminology from FINE [7], let us call T O and TO the generic

f ru fh condifions for necessify and possibility, respectively. Using Lemma 2.1 above, we can show that these generic truth conditions can be replaced with the following orthodoz Kripke-style conditions, viz.,

ROUTLEY and MEYER [18] and MARES and MEYER [ll] for their proof.

To'. TO'.

I(A, a) = T iff Vz(Taz * I ( A , z) = T), I(A, a) = T 8 3z(Taz & I(A, z) = T).

For a proof that this is so, see MARES and MEYER [ll].

3. Where LE {RD, RT, RBrIR4,R5}, we now show that all theorems of L are valid in the class of Lms. To do so, we translate some lemmas from ROUTLEY and MEYER [la] into an idiom appropriate to the present context and prove them.

L e m m a 3.1. Lef LE {RD,RTlRBr,R4,R5}. IfA is L-valid fhen so i s OA. P r o o f . Suppose A is L-valid. Then, in all Lms and on all interpretatiolls I, for

all a E P, I ( A , a ) = T. By TO', for all such I in all Lms, if Tab, then I (A ,b ) = T. But, if Tab, then b E P. So, for all a E PI I(A,a) = T. 1.e. O A is valid.


The preceding lemma proves that 01 is valid on all Lms. It also shows that A14 is valid.

L e m m a 3.2. If K = (K, PI R, S, *) is an RDms and I is an interpretation on li, then A1-A13 are verified on I .

P r o o f . Clearly, K' = ( K , PI R, *) is an unreduced Rms in the sense of ROUTLEY et al. [20] and I is an R-interpretation, so Al-A9 are verified on I . Moreover, the proof that A10 and A l l are valid in all RDms are the same as those given for Nkns in MARES and MEYER [ll] . So we go on now to the verification of A12. Suppose I (O(A V B ) , a ) = T. Assume that I (OA, a ) = F. We show that I (OB, a ) = T. By TO', for all z E I< such that Taz, I ( A , z) = F. But, by TO', I ( A V B , E ) = T. So, I ( B , z ) = T for all z such that Taz. By TO', I (OB,a ) = T. So, by SemEnt, A12 is verified. The verification of A13 is easy and left to the reader.

The two preceding lemmas together show that A1-A14 are valid in all Lms. More- over, it is a trivial matter to show that +E and A1 are also valid in all Lms. Thus all theorems of RD are valid on the class of RDms.

The following lemmas are proven in the usual fashion. L e m m a 3.3. Every instance of A15 is valid on the class of RTms. L e m m a 3.4. Every instance of A16 is valid on the class of RBnns . L e m m a 3.5. All instances of A17 are valid over the class of R4ms. C o r o l l a r y 3.6. All theorems of RBr are valid over the class of RBnns . C o r o l l a r y 3.7. All theorems of R4 are valid over the class of R4ms. C o r o l l a r y 3.8. All theorems of R5 a n valid over the class of R5ms.

4. In this section we show completeness. Our proof relies on a version of LINDEN- BAUM'S lemma due originally to BELNAP and, independently, to GABBAY. Before we can state the lemma, we need to set out a few definitions. For a logic L and formulas A and B , A I-L B iff FL A 3 B . Where A and I' are sets of formulas, 4 I-L I' iff for some A l l . . . , A , E A and some B 1 , . . . , B, E I', A1 A . . . A A,, I-L B1 V - . - V B,. (Where I' has only one member B , we may write A I-L B instead of A I-L I' or A I-L { B } . ) A set of sentences A is said to be an L-theory iff for any formula B of L, if A I-L B, then B E A. Note that every L-theory is closed under A1 and -E. A theory A is prime iff for all formulas A and B , if AV B E A, then A E A or B E A. A theory is L n g u l a r if it is a theory and contains all theorems of L. For sets of formulas A and I', (A,I') is said to be a consistent L pair iff A i L I'; (A,I') is an inconsistent L pair otherwise. We can now state the priming lemma.

L e m m a 4.1 (Priming). For L E {RD,RTIRBr,R4,R5}, let ( A , r ) be a consistent L pair. Then A can be eztended to a prime theory A' such that (A',I') as a consistent L pair.

To state our next lemma, we need the following definition. Where a, b and c are sets of sentences,

a o b = c iff c = { A I3B(B -+ A E a & B Eb)}

The operator o is "fusion" in the sense of FINE [GI. Now we can state our squeezing lemma.

170 Edwin D. Mares

L e m m a 4.2 (Squeezing). For L E {RD,RT,RBr,R4,R5}, if a and b are L-theories, c' is a pr ime L-theory, and a o b C c', then there i s a p r i m e L- theory a' such that a E a' and a' o b

The proof of Lemma 4.2 is the same as the proof of the squeezing lemma for R

If L E {RD, RT,R4,RBrlFt5} and a and b are L-theories, then a o b = b 0 a (i.e. fusion is commutative). Given the preceding lemma, this implies that if a and b are L-theories, c' is a prime L-theory, and a o b E c', then there is a prime L-theory a' such that a C a' and a prime L-theory b' such that b E b' and a' o b' E C .

To formulate our completeness proof, we need the two definitions below. Let a be an arbitrary prime theory, then

c'.

(see ROUTLEY and MEYER [19]).

O-'a =df { A I OA E a}, O-'U =df { A I O A E a } .

We call 0"a the depossibilization of a and O-'a the denecessitation of a.

We define a canonical model following ROUTLEY and MEYER [18]. The canonical model is a quintuple KL = ( K L , pL, RL, SL, *L) such that the following conditions hold:

DKL. D4,. DRL. DSL. D*L.

D I L .

K L is the set of prime L-theories, 4, is the set of prime regular L-theories, RL = { ( a , b , c ) I a, b , c E KL,ao b C c } . SL = {(a, b) I a, b E KL, 0-'a C - b } , a*= = { A 1-A 4 a},

LL= {(a,b) I a,b E K L , ~ C b } . As before, we define

TLab iff SLab and SLa*Lb*L.

It is easy to show that, for a, b E KL , T L a b iff 0"a C b 2 0"a.

We now go on to demonstrate that, for each of our logics L, KL is an Lms. Using the techniques of ROUTLEY and MEYER [18], [19], and ROUTLEY et. al. [20],

it can be shown easily that for any L E {RD, RT, RBr, R4, R5} the canonical model KL satisfies P1, P2, P4, P5, P6, P7, and P8. We now show that it satisfies P3, P9, and P10, hence we show that it is an RDms.

L e m m a 4.3. For any of our logics L an.d a, b E I ~ L , if a E 4, and TLclb, then b E h.

P r o o f . Suppose &(a) and T L a b . By Dh, a is regular. By definition of TI,, 0 - l ~ C b. By 01, therefore, b is also regular. So, by D f i , 4 , ( b ) , concluding the lemma.

L e m m a 4.4. For L E {RD, RT, RBr, R4, R5) and a, b, c E I<L, if s L ( R ~ u b ) c , then RL ( SL a) ( SL b)c .

P r o o f . Same as for NR or R4 - see ROUTLEY and MEYER [18] or MARES and MEYER [ll].

Classically Complete Modal Relevant. Logics 1'71

L e m m a 4 . 5 . F o r L E {RD,RT,RBr,R4,R5} a n d a , b € I i L , z f S L a b , there IS a c E I i L such thaf c S L b and TLac. That is, the canonical model sattsfies P11.

P r o o f . Let a and b be members of I<L such that S L a b . Thus, by DS;, 0- 'a E b. We now show that b contains a prime L-theory c such that 0 - ' a C c 5 O-'a. We first note that, since all the logics we are dealing with contain the theorem 0.4 - 0.4, 0- 'a C_ O-'a. Now we show that ( O - l a , (Wff - b ) u (Wff - O- 'a ) ) is a consistent L pair. For suppose that it is not. Then there are some A ' , . . . ,A, E O-'a, some B1,. . ., Bn E (Wff - b ) , and some C1,. . . , C o E (Wff - 0 " a ) such that

(1) A ~ A . . . A A , I - L B1 V . * . V B n V C l V . . . V C o . By (l), 01, A10, and -E, we obtain

(2) o ( A l r \ . * * A A m ) k ~ o(B1 V . . . V B n V C l V * . . V C o ) . By (2), A12, A l , and +E, we derive

(3) o(Al A . * . A Am) I-L o(B1 V . * . V Bn) V O(C1 V . . . V Co). Thus, since a is a prime L-theory and, by assumption, O(A1 A e - 0 A Am) E a, either n(B1 V - . V Bn) E a or O(C1 V . - .v Co) E a. Suppose first that O(C1 V . . . V Co) E a. Then OCI V . - . V OC, E a. So, for some i such that 1 5 i 5: 0, C, E a , since a is prime. But our assumption says that this is not so. Thus O(B1 V . . . V Bn) E u . Then, since O-'a b, B1 V ... V B, E b . So, since b is prime, for soiiie i such that 1 5 i 5: n, Bj E b , contra assumption. Therefore, (O- 'a, (Wff - b ) U (Wff - O - ' u ) ) is a consistent L pair. By the priming lemma, 0 - ' a can be extended to a prime L-theory c such that (c, (Wff - b) u (Wff - O- 'a ) ) is a consistent L pair. In other words, there is a c E h ' ~ such that c b (i.e. c <L b ) , S ~ a c , and SLa*c*, and this is what we wanted to show.

C o r o l l a r y 4.6. For L E {RDIRT,RBr,R4,R5} nnd a ,€ I ~ L . there I S a b E I i L such that T L a b .

P r o o f . Let a E I i L . By A13, OA I-L 0.4, so 0 - ' a O-'a. By Lemma 4.5 above, there is a prime theory b such that 0- 'a c b E O-'a. Thus, by DTL, T L u b .

L e m m a 4.7. For L E {RT, RBr, R4, R5}, SL zs refierive. P r o o f . Obvious. L e m m a 4.8. For L E {RBr, R5} the canonical model satisfies T L a b =+ T L b a . P r o o f . For some a , b E K L , suppose T L a b . Then 0-'a C b C O-'u. We show

that O-'b C a 2 O - l b . First, suppose that OA E 6 . By assumption, O O A E a. But, contraposing the Brouwersche axiom, OOA + A. By the fact that u is an RBr- theory, A E a. Thus, O - l b C a. Now suppose that A E a. Since a is an RBr-theory, OOA E a. SO O A E b, by assumption. Hence A E O - ' b . Thus, a C O-'b, concluding the lemma.

L e m m a 4.9. For L E {R4, R5} the canonical model saf isf ies S i u b =+ S L o b . P r o o f . Suppose that S L a b and S L b c . We show tshat SLac. Suppose that

A E O-'a, i.e., OA E a. Then, by the R4 axiom and -E, OOA E a. By our assumption that S L a b , OA E a. By our assumption that S L b c , A E c . In other words, 0- 'a E c. By DSL, SLac, concluding the lemma.

172 Edwiii D. Mares

C o r o l l a r y 4.10. Let L be RD. Then the canonical model is an RDnis. P r o o f . Follows from Lemmas 4.3-4.6. C o r o l l a r y 4.11. Let L be RT. Then the canonical model is an RTms. P r o o f . Follows from Corollary 4.10 and Lemma 4.7. C o r o 11 a r y 4.12 Let L be FLBr. Then the canonical model is an RBrms. P r o o f . Follows from Corollary 4.11 and Lemma 4.8. C o r o 11 a r y 4.13. Let L be R4. Then the canonical model is an R4ms. P r o o f . Follows from Corollaries 4.10-4.12. C o r o l l a r y 4.14. Let L be R5. Then the canonical model is an R5ms. P r o o f . Follows from Corollaries 4.12-4.13. We now set a canonical valuation u such that u(p, a ) = T iff p E a. Again following

the proof set out in ROUTLEY and MEYER [18], it can easily be showii that for the associated interpretation I, I ( A , a) = T iff A E a, for any formula A. By the priming lemma, for any non-theorem B of L, there is a prime regular L-theory that does not contain B. So, by Dh, B is not verified in the canonical model. Since the caiioiiical model is an Lms, for each non-theorem of L there is an Lms and L-interpretation that does not verify it. In other words, only theoreins of L are valid in the class of Lms. So we have shown completeness.

5. In this section, we make further use of some of the techniques we used to prove completeness to show that A C K E R M A N N ~ rule 7 is adinissible each of our logics. We say that a rule of the form

From A1 , . . . , An infer B is admissible in a logic iff, if all of A1 , . . . , An are theorems of the logic, then so is B. ACKERMANN'S rule 7 is the following:

From A and -A V B infer B.

In what follows, we generalize the proof of MARES and MEYER [lo] that shows that y is admissible in R4 to treat all the logics of the present paper.

We begin by creating a structure ML = ( M L , T L ) , where L is one of our logics, M L is the set of prime regular L-theories, and, as in our completeness theorem, TL is a binary relation on M L such that TLab iff 0-'a C b O- 'a . Furthermore, let M i = ( M ~ , T L ) be such that, if a E M L , then a* E Mi, where * is defined as for the canonical model in Section 4 above, and let TL be as for ML. It is clear that for each a* E M i , a* is a prime L-theory.

Before we proceed, we need to prove the followiiig lemma: L e m m a 5.1. Let a be a prime L-fheory and OA E a. Then there is a prime

L-theory b such that A E b and TLab. P r o o f . Suppose a is a prime Gtheory and A E O-'a. For the sake of a reductio,

let us assume that there is no prime theory b such that A E b and TLab. By the priming lemma, this can happen only if

(1) O-'O U { A } I-L B1 V - - V B, ,

Classically Complete Modal Relevant Logics 173

for some B 1 , . . . , B, 4 O - l a , for m > 0 . In other words, there are C1,. . . , C'n E O-'a, for an n E w , such that

(2) C ~ A . - * A C ~ I - L B l V * * * V B m .

Moreover, (2) implies

(3) O(C1 A . . . A Cn A A ) I-L O(B1 V * * . V Bm). But, it is a theorem that O(C1 A .. .Cn) A O A + O(C1 A * . . A Cn A A ) and it is a theorem that O(B1 V . . . V Bm) + (OB1 v . . . v OB,), so (3) implies that

(4) o ( C 1 A . - . A C ~ ) A O A I - L O B , V . . . V O B , . Now, CI,. .., C n E O - l a , so OC1,. . . , OCn E a. So, since a is closed under conjuiic- tion,OC1A...AOC,Ea,andbyAll, O ( C I A . - . A C , ) E a . Moreover,A€O-'a, whence OA E a. Thus, O ( c l A - - .AC,)AOA E a, whence, by (3), 0BlV.e .VOB,, E a. Since a is prime, OBi E a for some i such that 1 2 i 5 m. But this means that for some i , 1 5 i 5 m, Bi E O"a, and this contradicts our assumption, concluding the lemma.

L e m m a 5.2. TLab i f fTLa*b*. P r o o f . =+. Suppose TLab. Suppose also that A E b'. We show that 0.4 E a*.

By the definition of *, -A 4 b. By TLab, 0 - A 4 a. By the fact that a** = a and the definition of *, -0 - A E a*, whence OA E a* , as required. Now suppose t.hat OA E a* and we show that A E b'. By the definition of *, - 0 A 4 a. Since u is an L-theory, 0 -A 4 a. By TLab, -A 4 b , so, by the definition of *, ---A E b', wheiice A E 1' as required. Moreover, since a** = a, the right to left direction of the proof is similar.

Extending MEYER'S method of metaualuat io~~s , we employ a function L froin members of M L and formulae into the set {T, F} such that

Trp. T r A . Tr-. Tr-. TrO.

~ ( p , a) = T iff p E a.

L(A A B , a) = T iff L(A, a) = T and L(B, a) = T, L(-A,Q) = T iff (i) L ( A , o ) = F and (ii) -A E a,

L(A - B , a) = T iff (i) L(A, a) = F or v ( B , a) = T and (ii) A - B E a.

L(OA, a) = T iff V z ( T L a z L(A, t) = T). Moreover, for each a E M L , we set n ( a ) = { A I L(A, a) = T}.

L e m m a 5.3 (Mediating Lemma). For a E AIL, u* P r o o f . The proof is the same as for R4. See MARES and MEYER [lo]. C o r o l l a r y 5.4. (i) A E Tr(a) or -A E Tr(a), and (ii) it is nof the case that both

P r o o f . (i) Suppose A 4 Tr(a). By Lemma 5.3, A 4 a*. Thus, by the definition

L e m m a 5.5. A V B E Tr(a) iff A E Tr(a) or B E Tr(a). P r o o f . Suppose A V B E Tr(a), i.e. - (-=1A -B) E Tr(a). By the precediiig

corollary, - A A -B 4 %(a). By T r A , -A 4 Tr(a) or -B 4 Tf(u). so, by Clorollary 5.4, A E Tr(a) or B E Tr(a). Now suppose that A E Tr(a). By the same moves as before, - ( -A A -B) E Tr(a), concluding the proof of the lemma.

Tr(u) 5 a.

A E Tr(a) and -A E Tr(a).

of *, -A E a, whence, by Tr-, -A E Tr(u). (ii) follows directly from Tr-.

174 Edwin D. Mares

L e m m a 5 . 6 . O A E T r ( a ) i f f 3 z ( T ~ a z & A A E ( z ) ) . P r o o f . =+. Suppose O A E Tr(a). By definition of 0, -0 - A E Tr(a). By

Corollary 5.4, 0 - A 4 Tr(a). By TrO, there is some b such that T ~ a b and -A 4 Tr(b). By Corollary 5.4, A E Tr(b) as required. e. Suppose there is some 6 such that A E T r ( b ) and TLab. Thus -A 4 T r ( b ) . By TrO, O - A 4 Tr(a). By Corollary 5.4, -0 -A E Tr(a); SO O A E Tr(a).

The following lemmas show that Tr(a) is regular. L e m m a 5.7. I f A is an instance of azioms AO-A9, then A E Tr(a). P r o o f . As usual. See MEYER [13] and DUNN [5]. L e m m a 5.8. If A is an instance o f A l l or A13, then A E Tr(a). P r o o f . Trivial. L e m m a 5.9. I f A is an instance ofAl0 or A12, then A E Tr(a). P r o o f . Ad A10. (i) Suppose O(A --L B) E Tr(a). Now suppose that O A E Tr(a).

We show that OB E Tr(a). By TrO, for all b such that T'ab, A + B E Tr(b). Also by TrO, for all such b , A E Tr(b ) . By Tr+, either A 6 T r ( b ) or B E T r ( b ) , so B E T r ( b ) . By 20, oB E Tr(a) as required. (ii) Since a is regular, o ( A - B) - ( o A - OB) belongs to a. So, by (i), (ii), and n+, O ( A + B ) - (CIA -. O B ) E Tr(a).

Ad A12. (i) Suppose that O ( A V B) E n ( a ) . By TrO, for all b such that T ~ a b , A V B E T r ( b ) . By Corollary 5.5, either A E Tr(b) or B E T r ( b ) . Suppose for one such b, A E Tr(b) ; then OA E Tr(a) and, so OAVOB E Tr(a), as required. Suppose, 011 the other hand that for all such b, A 4 T r ( b ) . Thus, B E T r ( b ) for all b such that T ~ a b . By TrO, OB E Tr(a). By Corollary 5.5, OA V OB E Tr(a). (ii) Since a is regular, O ( A V B) + ( O A V OB) E a. Therefore, by Tr-, O(A V B ) - ( O A V O B ) E Tr(a).

The following lemmas are easy to prove. We leave their proof to the reader. L e m m a 5.10. IfL E {RTlRBr,R4,R5} and a E M L , then O A - A E Tr(a). L e m m a 5.11. IfL E {R4,R5} and a E M L , then OA - OOA E Tr(a). L e m m a 5.12. IfL E {RBr,R5} and a E M L , then A - OOA E Tr(a). The following lemma shows that A14 holds in Tr(a). L e m m a 5.13. I f A is a an'om ofL and a E ML, then 0.4 E Tr(a). P r o o f . Suppose A is an axiom of L other than A14. Then, by the preceding

lemmas, A E 2 ( b ) for all b such that T ~ a b . Thus, by 2 0 , O A E Tr(a). Similarly, O O A E Tr(a), OOOA E n ( a ) , and so on. Thus every instance of A14 is in Tr(a), concluding the proof of the lemma.

L e m m a 5.14. (i) I f A 4 B E Tr(a) and A E Tr(a), then B E Tr(a). (ii) J" A E Tr(a) and B E Tr(a), then A A B E Tr(a).

P r o o f . (i) follows directly from Tr-c and (ii) follows directly froin T r A . We say that a theory is consistent iff, for any formula .4, it does not. contain both

A and - A ; it is normal if it is prime, regular, and consistent. By Corollary 5.4, for each a E ML, n ( a ) is consistent. Furthermore, by Leininas 5.7-5.14, Tr(a) is regular and, by Corollary 5.5, Tr(a) is prime. Hence, Tr(a) is normal. Now suppose that FL-A V B and I-L A. Thus, for each prime regular L-theory a, '4 E Tr(o) and -A V B E Tr(a). Since Tr(a) is consistent, -A 4 Tr(a). Thus, since Tr(a) is prime,

Classically Complete Modal Relevant. Logics 17.5

B E " r ( a ) . By Lemma 5.3, T r ( a ) Thus, by the extension lemma, I-L B. I n other words, 7 is admissible in L.

a. So B E u , for every prime regular L-theory n.

6. Given the proof of the preceding section, we can alter our canonical model of Section 4 above to stipulate that PL consist only of normal theories. The proof, with the altered definition of & can still be shown to be an Lnis. Thus, we call add t.he following condition to our definition of an Lms, without compromising completeness:

P15. P ( Q ) + a = a*.

We call P15 the normality condition and Lnis that satisfy it. n o m u l niodels. I n t,he present section, we show that normal niodels have some very nice propert.ies.

Following KRIPKE [9], a Dms is an ordered pair P = (P,T), where P is a 11011-

empty set and T is a binary relation on P such that for all a E P there is some b E P such that Tab. A Dms is a Tms if T is reflesive; it is a Brim if T is also symmetrical; and i t an S4ms if T is transitive and reflesive. Moreover, an S4ms is an S h s if i t is also a Brms. Now, let X be an n-place relation on a set Y and Z he a subset. of Y. Then, following standard practice, we let S f 2 = X fl 2" (S I 2 is called the restriction of X to 2). Moreover, let P be the set of a E Ii such that. P ( a ) . The following lemmas are easily proved:

L e m m a 6.1. Let K = (I<, P, R , S , * ) be a normal RDms. Then P = ( P I T I P ) is a Dms.

L e m m a 6.2. Let K = (I<, P, R, S, *) be a noi.n~.al R T m . s . Then. P = ( P . T I P ) is a Tms.

L e m m a 6.3. Let K = ( I i ' ,P ,R,S ,*) be a normal RBnns. Then P = ( P , T 1 P ) is a Brms.

L e m m a 6.4. Let K = (I<, P, R , S , * ) be a normal R4ms. Then P = ( P , T I P ) is an S4ms.

L e m m a 6.5. Let K = (Ii, P, R , S , *) be a normal R5rns. Then P = ( P , T I P ) is an S m s .

For a normal RDnis (RTms, RBrms, R4ms, R5nis) K = (I<, P, R. S, *), we say that P = ( P , T f P ) is the Dms (Tms, Brms, S4ms, SSms) determin.ed b y K.

NOW we define a classical m.odal interpretation.' For L E {D,TIBr,S4,S5} and an Lms P = (P,T), a valuation v is a fuiiction from ordered pairs of propositioiial variables and members of P into {T,F}. A formula A is said to be in t.he motlal vocabulary or a modal formula iff the only logical connectives that occur in it are A , -, and 0. A classical modal interpretation V (a function from modal foriiiulas and worlds into {T, F}) associated with a valuation 2' is defined as follows:

V ( p , a) = v(p , a) , for each propositional variable p , V ( A A B, a) = T iff V(-4, a) = T and V ( B , a) = T, V ( - A , a ) = T iff V(A,a) = F, V(OA, a ) = T iff Vt(Taz

Vp. VA. v-. VO. V ( A , +) = T).

As the following lemma shows, classical interpretat,ions hear a tidy rela.tion t.0 the interpretations on relevant modal structures defined in Section 2 above.

176 Edwiii D. Mares

L e m m a 6.6. Let K = (KIP, R,S,*) be a nonnal RDnis (RTms, R B r m s , R4ni.s, R5ms) and P be the Dms (Tms, Brms, S m s , S m s ) deiermined b.y K. Let u be a valuation on K and v be a valuation on P such that for all a E P and all proposiiioii.al variables p , u ( p , a ) = v ( p , a ) . Moreoz~er, let I be the RD (RT, RBr, R4, R5) interpretation associated with u and I’ be th.e S4 interpretation associnicd with. 11.

Th.en, for all modal fornaulas A an.d all a E P , I ( A , a ) = V(A, a). P r o o f . Case 1. A = p . Follows by hypothesis. Case 2. A = B A C!. Follows

by VA, TA, and the inductive hypothesis. Case 3. A =- B. I ( - B , u ) = T iff I ( B , a’) = F. a E P, by the normality condition, u = a*, whence I ( -B, a ) = T iff I ( B , a ) = F. Suppose I(-B,a) = T. Then, by what. we have just said, I ( B , a ) = F. So, by the inductive hypothesis, V ( B , a ) = F, whence, by V-, V ( - B , a ) = T. Now suppose that V(-B, a ) = T. By the saiiie reasoning, I(-B, a ) = T. Case 4. A = o B . Suppose I ( O B , a ) = T. Then, for all b E P such that T u b , I ( B , b ) = T. By the inductive hypothesis, V(B ,b ) = T, so, by VO, V ( O B , a ) = T. For the converse, suppose that V ( O B , a) = T. Then, for all b E P, if Tab, then V ( B , b ) = T. By the inductive hypothesis, I (B , b ) = T. By TO’, I ( O B , a) = T, concluding t.he lemma.

The above leminas show that the following corollary holds: c o r o 11 a r y 6.7. If A i s a modal formula, then ~ - R D A i f f kD A. P r o o f . Suppose A is a modal formula. Assume that it is a theorem of D. Then,

by the soundness theorem for D of KRIPKE [g], A is verified in all Dms. In pa.rt,icular, it is true in all Dms determined by RDms. Thus, by Leiiima 6.6, it is verified on all RDms. For the converse, formulate D in the full vocabulary (i.e., for1nulat.e it using A, 7 0 , and 4). Clearly, every theorem of RD is also a theorem of D SO

formulated. In particular, every modal forinula that is a theorem of RD is a theorem of D, concluding the lemma.

In the same way we can prove the following: C o r o 11 a r y 6.8. If A i s a modal formula, th.en

(a) FRTA i f f F T A , (b) F R B ~ A i f f F B ~ A , (c) F R 4 A *ff FS4A9 (d) F a 5 A i f f I-SSA-

Therefore, each of our systems is classically complete, as we claimed a.t. the out.set. of the paper.

References

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ANDERSON, A. L., and N. D. BELNAP JR, Entailment: The Logic of Relevance and Necessity, Volume 1. Princeton University Press, Princeton N. J. 1975. BACON, J., Being and Existence: Two Ways of Forinal Ontology. Yale Universi1.y PhD Dissertation, New Haven 1969. D O ~ E N , K., The first axiomatization of a relevant logic. Zentrum Philosophie und Wissenschaftstheorie Bericht, Konstanz 1990, pp. 9-90.

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(Received: July 10, 1991)

classically complete modal relevant logics

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