A Kripke-style semantics for R-Mingle using a binary

J. YICHAEL DIYNN

A Kripke-Style Semantics for R-Mingle Using a Binary Accessibility Relation*

1. Introduction The sentential calculus Ir (R.Y) is o b t a i n e d b y adding t h e axiom scheme A - + ( A ~ A ) t o t h e B e l n a p - A n d e r s o n s y s t e m 1~ of R e l e v a n t I m p l i c a t i o n . W e shall here p r e s u m e t h a t R M has been explicitly axiomatized as in [2], a n d shall cite various t h e o r e m s of R M listed t h e r e b y name, e.g., T r a n s i t i v i t y . ICY has a m o d e l t h e o r y in t e r m s of Sugihara matrices, as was discovered b y ) I e y e r [9] 1. Here, however, is p r e s e n t e d a model t h e o r y using K r i p k e ' s device of model s t r u c t u r e s with a b i n a r y accessibility relation. The principal p o i n t of d e p a r t u r e f r o m K r i p k e is t h e consideration of models which allow sentences to b e s i m u l t a n e o u s l y b o t h " t r u e " a n d "false". W h e n t h e basic results here were first o b t a i n e d b a c k in 19692 t h e r e was no need to explicitly m e n t i o n t h a t t h e semantics presented u s e d a binary accessibility relation. T h a t was surely a p a r t of t h e ordinary ~teaning of t h e phrase " K r i p k e - s t y l e semantics". B u t since t h e n t h e w o r k of l~outley a n d M e y e r (cf. [13]) a n d others has shown h o w to e x t e n d K r i p k e - s t y l e semantics to allow for t e r n a r y accessibility relations. This has b e e n p a r t i c u l a r l y f r u i t f u l in t h e case of t h e relevance logics, and [13] its successors contains completeness results for various of these logics, including R, a n d ICY (cf. also [4]). I t m u s t b e f r a n k l y confessed t h a t t h e ingenuity a n d t h e p o w e r of t h e t e r n a r y semantics were so o v e r w h e l m i n g to this a u t h o r t h e y caused delay until n o w of full p u b l i c a t i o n of t h e b i n a r y semantics for RM. There h a d been at first t h e v a i n h o p e t h a t t h e b i n a r y semantics could b e e x t e n d e d to t h e other relevance logics, a n d t h e n with t h e success of t h e t e r n a r y a p p r o a c h t h e b i n a r y semantics b e c a m e to a p p e a r old-fashioned a n d special. * Grateful acknowledgement is made for partial support to the bTational Science Foundation, U.S.A., grant GS-33708. 1 Cf. [1] for an extended presentation of Meyer's results (originally obtained in 1966), and also [2] for a generalizatioll using algebraic methods. 2 The author first arLno~mced these results in passing in a talk (mimeo privately circulated) ia the joint American Philosophical Association- Association for Symbolic Logic Symposium, iVatural language versus formal language, New York, December, 1969, and actually presented them in A Kripke-style semantics for R-mingle, contributed to the Tarski Symposium, University of California, Berkeley, JuDe, 1971 (abstracts of contributed papers were distributed to participants).

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While all of that still remains true, nonetheless there are reasons why the binary semantics for ICM is interesting. After all a binary accessibility relation is less complex than a t e r n a r y one. This lea.ds on the philosophical level as it turns out to a rather familiar glossing of the semantics for RIV[ in terms of the flow of time increasing information, whereas there has been as yet it seems no completely satisfactory glossing of the t e r n a r y semantics. On the mathematical level it happens t h a t one can use the method of filtration straightforwardly on the binary semantics to show :RM decidable, whereas the method of filtration seems to have at least no straightforward application to the t e r n a r y semantics. 2. The binary semantics We define an (l~.l~f) model structure to be an ordered triple (G, K, R), where K is a set, G~ K, and R is a (weak) linear ordering of K, i.e., a reflexive, anti-symmetric, transitive, connected relation. I t further does not h u r t to require t h a t G be the least element of K 1ruder R, as the reader can check for himself as he works trough the proofs. An (:RM) nwdel on such a model structure is a function 9(_P, H), where P ranges over sentential variables and H ranges over members of K. We specify t h a t the possible values of ~ are ST~t j, {F}, and {T, F}, and also require t h e ~:]E]~DITAI~Y CO=NDITIO~: :[or all H , H ' e K , _~ ~ ( P , H').

if H R H ' then ~(P,H)___

We then extend q to complex sentences inductively as follows: (~.T) TEr iff F e ~ ( A , H ) ; (~F) FEr iff T E ~ ( A , H ) ; (&T) T ~ ( A & B ) , H ) iff T E ~ ( A , H ) and T E ~ ( B , H ) ; (&F) F E q ( A & B , H ) iff Fe~v(A,H) or F E ~ ( B , H ) ; (vT) T E ~ ( A v B , H) iff T e q ( A , H ) or T E q ( B , H); (vF) F~q~(AvB, H) iff F r and F r (-->T) Tr q~(A--+.B, H) iff for all H'E K such that H R H ' , (i)TE ~(A, H') only if T e ~ ( B , H'), and (if) F r H') only if Fr H'); (->F) Fr iff either (i) T r H) or (if) T E q ( A , H) and F ~ ( B , H). INote that an easy induction shows that tp(A, H) is always non-empty. V~e next define a sentence A to be a logical consequence of a set of sentences S in I~M (in symbols S kR~ A) iff for all models ~ on all :R.M model structures ( G , K , R), if for all B ~ S , T e ~ ( B , G), then T t q ( A , G ) . The chief result of this paper is t h a t the semantic notion of logical consequence in RM is coincident with the syntactic notion of derivability in RM. 3. Informal /nterpretation In [5] a semantics was presented for the first-degree entailments (no nesting) of Anderson and Belnap using the idea that an adequate modeling

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of a s y s t e m of beliefs w o u l d p e r m i t t h e assignment to a given sentence of b o t h or neither of t h e t r u t h values T a n d F (as well as of course t h e u s u a l assignments of e x a c t l y one). Pa,ins were t a k e n to stress that such modelings were regarded as epistemologicalIy rather t h a n ontologically based. One is sometimes told (whether b y informants, nature, t h e o r y , intuition, w h a t e v e r ) t h a t A is both t r u e a n d false, a n d other times one h~s no information at all regarding A's t r u t h or falsity. A n d y e t in fact p r e s u m a b l y A is precisely t r u e or precisely false. I t was r e m a r k e d t h a t one was able to c a p t u r e t h e fh'st-degree implications of R M b y basically considering only t h o s e " a m b i v a l e n t " models in which sentences were always assigned at least one t r u t h value. ~re here e~tend this observation, a.rguing in effect t h a t :RM is t h e logic a p p r o p r i a t e to reasoning in a situation of complete b u t n o t necessarily consistent information. I n [5] these a m b i v n l e n t models were essentially " s t a t i c " . :No a c c o u n t w.~s t a k e n of ch~nge in i n f o r m a t i o n concerning A over time. :Now K r i p k e [8] nnd Grzegorczyk [7] i n d e p e n d e n t l y d e v e l o p e d a semantics for intuitionistic logic which can be t h o u g h t of as " d y n a m i c " . Since w e are using K r i p k e ' s model structures, it is n a t u r a l to talk in t e r m s of t h e m , b u t sometinms b o r r o w i n g a p a r t i c u l a r l y vivid image from Grzegorczyk's motivations. The rough idea is that, t h e m e m b e r s of K are evidential situations (G b e i n g t h e a c t u a l one) a n d t h a t t h e accessibility relation R is to be u n d e r s t o o d as t h e relation of possible extension of one evidential situation so as to obtain another. The K r i p k e model s t r u c t u r e s for intuitionism requh'e R to be reflexive a n d transitive, b u t n o t necessarily c o n n e c t e d nor nnti-symmetric. The last requh'ement could h a v e b e e n m a d e with no h a r m ; however, connectedness w o u l d give rise to a semantics for D m n m e t ' s L C , un extension of t h e intuitionist logic (cf. [14]). This is p l e a s a n t since LC can b e t r a n s l a t e d into I~.M [cf. [6]), a n d t h e r e m u s t be some connection there. Once R is c o n n e c t e d t h e r e seems no reason not to t h i n k of it simply as t h e relation of t e m p o r a l priority. There is in t h e K r i p k e - G r z e g o r c z y k semantics an a s s y m m e t r i c ~ l t r e a t m e n t of t r u t h a n d falsity. Thus K r i p k e [8, p. 98] says: " B u t ~ ( A , I-I) = F does not m e a n t h a t A has been p r o v e d false at H. I t simply is not (yet) p r o v e d at H, b u t m a y be established later". Grzegorczyk, who seems to h a v e at t h e b a s e of his n~otivations t h e idea t h a t t h e atomic sentences are something like o b s e r v a t i o n sentences, rings a more philosophical n o t e w h e n he says [7, p. 596]: " T h e c o m p o u n d sentences are n o t a p r o d u c t of experiment, t h e y arise from reasoning. This concerns also n e g a t i o n s : we see t h a t t h e lemon is yellow, we do not see t h a t it is n o t blue". N o w t h e r e need u l t i m a t e l y be nothing w r o n g with such a preferred t r e a t m e n t of t r u t h , and indeed it seems consonant with t h e original motiv a t i o n s of intuitionism. B u t t h e semantics we are presenting here is more even b a n d e d in its t r e a t m e n t of t r u t h and falsity. I t takes a more " p o s i t i v e "

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stance t o w a r d lalsity. P e r h a p s (contra Grzegorczyk) we do a f t e r all see t h a t t h e lemon is n o t blue. I n this it is quite similar to T h o m a s o n ' s s t u d y of constructible falsity [15] 3. The K r i p k e - G r z e g o r c z y k semantics makes its prejudice in favor of t r u t h f o r m a l l y explicit in t h a t it requires t h a t once a sentence is t r u e in an evidential s i t u a t i o n t h a t it r e m a i n s t r u e in all later evidential situations, b u t t h e corresponding r e q u i r e m e n t is n o t m a d e for falsity. T h o m a s o n does m a k e t h e same r e q u i r e m e n t for f a l s i t y as t r u t h , a n d we do so also. I t is obvious t h u t this c~nnot be done while w o r k i n g with models t h a t give each sentence precisely one t r u t h v a l u e (as do K r i p k e ' s models) w i t h o u t t h e models d e g e n e r a t i n g into w h a t are in effect static models, for all t h e evidential situations would be indistinguishable in t e r m s of which sentences t h e y established. T h o m a s o n works with models in which some sentences h a v e no t r u t h value, whereas we are w o r k i n g t h e other side of t h e street.

4. Semantical soundness W e define an R M - t h e o r y T to be a n y set of sentences of R ~ which is closed u n d e r M o d u s ponens a n d a d j u n c t i o n . W h e n convenient we r e g a r d tOM itself as an R ~ - t h e o r y b y i d e n t i f y i n g it with t h e set of its theorems. W e n o t e t h a t an R M - t h e o r y is a f o r t i o r i an l~-theory in t h e sense of [10], a n d t h u s t h a t we can bring over various definitions a n d results f r o m [10]. Thus, e.g., where T is an l~M-theory a n d A e T, we w r i t e ~T A. A n d we say t h a t A is derivable f r o m a set S of sentences in t h e l~M-theory T (in symbols S br A) iff A is a m e m b e r of e v e r y P~M-theory which includes SuT. W e can now s t a t e t h e ~:EI~IAiNTICALSOI_T:ND1N'ESS Tlt-EO~E~[.

If

S

F-R~~ A , then S I~R.~~ A .

P~ooF. The only rules of R ~ are modus ponens a n d a d j u n c t i o n . B o t h of these r a t h e r obviously preserve t r u t h . [For t h e f o r m e r look at ( ~ T ) a n d recall t h a t R is reflexive, for the la.tter j u s t look at (&T).] Thus t h e proof reduces to verification of axioms. This is even more tedious t h a n usual because of t h e " d o u b l e e n t r y b o o k k e e p i n g " needed because of clause (ii) in (-~T). s After th6 body of the present paper was written, Routley's [11] was published. His semantical analysis cleverly employs a "star operation" (also used in the ternary semantics, ef. [13]) in place of the Thomasoll device of truth value gaps. It appears that the star operation could be similarly employed in place of our truth value gluts. Cf. [12], which also came to the author's attention after the paper was written, for a binary sema~ltics usiug the star operatiou for auother systt~ln r(datcd to the A~ldcrsonBelnap systems. Cf. also [5] for tlfis author's doubts about the philosophical (as opposed to mathematical) sense of the s~ar operation.

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W e v e r i f y fh'st t h e characteristic 1%1~ a x i o m A-+(A-+A). I n checking for TE ?(A--.(A--.A), G) it suffices t o show t h a t if G R H , t h e n (i) (ii)

Tr only if T E ~ ( A - + A , H ) , F E g ( A - + A , H ) only if F E ~ ( A , H ) .

and

h~ow TE ~(A--.A, H) can easily b e seen to hold since it boils d o w n to t h e t a u t o l o g y t h a t TE ~ ( A , H') only if Te ~ ( A , H'), and t h e same thing for F. So (i) is trivially t r u e b y v i r t u e of a t r u e consequent 4. As for (ii) it is easy to see t h a t since always TEq~(A~A,H), if F E ~ ( A - + A , H ) this m u s t b e b e c a u s e of (-+F, ii). So we h a v e F e ~ ( A , H ) 5. The verification of t h e other axioms is left to t h e reader. The following will b e e x t r e m e l y useful for t h a t p u r p o s e : I-iE~EDITAICY L v , ~ A .

For any sentence A, if HRH', then ~ ( A , H )

c q~(A, H'). P ~ o o F is b y s t r a i g h t f o r w a r d induction on t h e length of A. The only case to give ~ny p a u s e is w h e n A ----B--.C and Feq~(A, H). There are 3 subcases (note t h a t in 2 a n d 3 we use linearity of R). S u b c a s e 1. ]H~ so H R H , a n d T~ ~ ( B , H~) a n d Tide(C, H~). E i t h e r H ' R H ~ or H ~ R H ' . If t h e first, t h e n ( b y ->T) clearly Tr q~(B~C, H') a n d so ( b y -+F(i)) F~ ~(B--->C, H'). If t h e second, t h e n T ( ~ ( B , H') b y inductive hypothesis, since T~ ~ ( B , H~). Also since Tr ~(C, H~), t h e n F e ~(C, H~), ~nd, again b y i n d u c t i v e h y p o t h e s i s , F~ ~(C, H'). B u t since T~ ~ ( B , H') a n d F~ ~(C, H'), t h e n ( b y -+F(ii)) F e r It'). S u b c a s e 2. 3H~ so H R t I ~ a n d F ~ ( C , s y m m e t r i c a l l y to subcase 1.

Its) and F i b ( B , I-I~). A r g u e d

S u b c a s e 3. Te ~ ( B , II) a n d F~ ~(C, It). Then b y i n d u c t i v e hypothesis T e ~ ( B , It') a n d F e ~ ( C , II'), a n d so (by -+F(ii)) Fe~o(B-+C, tI').

5. Semantical completeness W e first define some requisite n o t i o n s . An l%lK-theory T is prime iff w h e n e v e r ~-TAVB, t h e n either FTA or ~TB. W h e r e T is a prime g ~ t h e o r y we define t h e canonical model str~cture determined by T to be (GT, KT, RT) , w h e r e G T = - T , K T ---{T': T' is a p r i m e l%M-theory and T _ T'}, a n d T ' R T T " iff T' c T " . The canonical model determined by T, q~T, is t h e n defined on this model s t r u c t u r e so t h a t (i) Teq~T(_P , T') if:[ P e T', a n d (ii) FE 00T(ip, T') iff ,'~/ge T'. One not used to relevance logic might be surprised at the author's delicious sense of forbidden pier, sure derived from the use of the material conditional here and throughout the metalauguage. 5 Note that neither the line.~rity of the accessibility refection nor the ~ssumption that e~ch sentence is either true or false w,~s used in the verif cation of the characteristic R1K ,~xiom. Aud yet adding th.~t axiom to R produces ~R~(A---*B)v(B--~A), verification of which seems to require both assumptions.

J. .Michael Du.~

168 W e n e x t prove a series of L:ESL~IAS.

LESDIA 1 (LINDE~]3AUSI'SLF.MSIA). Let T be an RM-theory and suppose not kTA. Then there is an RM-theory To such that (i) T ~ To (ii) To is prime, a~d (iii) not kr0A. P~O0F is s t a n d a r d (Cf. T h e o r e m 3 of [10]). LE~r~A 2. Let To be a prime RM-tl~eory a,nd let T1 and T2 be RM-theories such that To =- T~, To. Then either T~ =_ T2 or T2 =- T~. PROOF. Suppose ~-T1Aand not [-T2A, while kT.,B a n d not k ~ B . :Now k~I(A--->B)v(B~A) [2]; hence ~ToA~,B or k T o B ~ A . If kToA-+B t h e n ~lso k ~ A ~ B , and hence, since }-T1A, we would have kT1B , c o n t r a d i c t i n g our supposition t h a t not k T B . A c o n t r a d i c t i o n follows ,~imflarly on t h e a l t e r n a t i v e t h a t ~T B ~ A . L E ~ A 3. .Let T be an l~M-theory. Tlten FTA--->B iff both A 5T B a n d - . ~ B ~T ~ A . P~ooF. The implication f r o m left to right is obvious since RM-theories are closed u n d e r b o t h Modus ponens and (hence b y Contraposition) Modus tollens. I n proving t h e converse implication we find it convenient to suppose t h n t RM has been enriched with t h e primitive sentential consta.nt f a n d its negation t =dl f_+f6. We n o t e fh~st t h a t as a substitution instance of t h e characteristic t~M axiom scheme we h a v e f - > ( f ~ f ) , i.e., f-+t. Al.~o t is useful in stating t h e following, which follows i m m e d i a t e l y f r o m t h e corresponding result for P~ in [10]. D~DUCTI0~- TIm0~E~[ Fol~ R ~ . Let S be a set of scnte,~ces and T be an RM-theory. Then if S, A ~ r B , then S ~ A & t ~ B . The following outlines t h e basic moves in proving t h e converse iraplication Assumption. 1. A kT• Assumption. 2. ,~B }-T~-,A 1, D e d u c t i o n Theorem. 3. kT A & t ~ B 2, D e d u c t i o n Theorem. 4. ~r ~ B & t--->~ A 4, Contraposition, Double :Negation, De 5. kTA---~Bvf )Iorgan, Disjunction, Transitivity. 5, kR~~ f-+t, I d e n t i t y , Disjunction, 6. k~ A - + B v f Transitivity. 6, I d e n t i t y , Conjunction. 7. kT A->A & (B vt) 7, Distribution, Transitivity. 8. kT A->(A & B ) v ( A & t ) 8, Simplification, 3, Disjunction, Tra, n9. t-T A-+B sitivity. 6 That f can bo added consorv,~tively to R or RM with the axiom ~A*-~(A-*f) is well-known.

A Kripke-style semantics fo, R-mi~gle... LE~'~A 4.

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I-f. A-->B iff for all prime RM-theories T' such that T ~_ T', ~-T"B, and ~'T".~B only "if ~ . . ~ A .

P~.ooF. Left-to-right is immediate. For right-to-left we argue contrapositively, assuming not F~. A--->B. Then by L:E~IA 3 we know either not A ~TB or not -~B FT --.A. Let TH-A be the smallest l~M-theory including Tw{A}, and similarly for T § Quite clearly either B is not 9~ theorem of T ~-A or .~A is not a theorem of T § ~.B. Using LEhI~L~_1, either T-J- A can be extended to a prime RM-theory T' so t h a t not ~T' B, or T-J-,B to a prime R~[-theory T " so that not ~.. ~.A, thus completing the proof. LE~[.~ 5. _Let q~ be the cano~ical model determined by some prime RM-theory To. Then for all sentences A of R M , and all prime RM-theories T _ To, (i) T e ~ ( A , T) iff FTA, and (ii) F ~ 0 ( A , T) iff kr .~,A. P~ooF. By induction on the length of A. C a s e 0. A is a sentential variable. The result holds by definition. Case 1. A = ~ B . (i) T e ~ ( ~ B , T ) iff FE~p(B,T) iff (inductive hypothesis) -..Be T. (ii) FE ~( .~B, T) iff TE ~(B, T) iff (inductive hypothesis) FT B iff (double negation) ~T .--.~B. C a s e 2. A ----B&C. (i) T E ~ ( B & C , T ) iff both T E ~ ( B , T ) and TE r T) iff (inductive hypothesis) both ~TB and ~TC iff (Simplifica.tion, Adjunction) ~ r B & C. (ii) F E ~ ( B & C, T) iff F e g ( B , T) or Fe cp(C, T) iff (inductive hypothesis) FT N B or FT ~.C iff (Addition, primeness) FT .~Bv---C iff (DeBIorgan) Vr ~ ( B & C). C a s e 3. A = B v C . Similnr to case 2 (alternatively dispense with v as primitive, setting A v B = a ~ . ( . - . B & .~C)). C a s e 4. A = B ~ C . Upon invoking the clause for -> in the inductive definition of ~ and applying the inductive hypothesis, (i) is just LES.~A 3. For (ii), we divide the equivalence into halves. Ja'guing left-to-right, let F~ 9(B--->C, T). Then either Tr162 T) or else T ~ ( B , T) while F~ ~(C, T). In the first case not VT B--->C (by (i), just established). But then ~ .~(B-~C) (Excluded Middle, primeness). I n the second case, ~T B and VT ~.C (inductive hypothesis). But then again ~T --.(B-~C) (since ~1r B & .~C-->.~(B--->C)). For right-to-left, suppose VT .-.(B--->C). We are to show t h a t either Tiq~(B-~C,T), or else T e g ( B , T ) while F e ~ ( C , T ) . We have ah'eady argued (i) so the first ~Iternative amounts to not VT B ~ C , whereas by inductive hypothesis the second alternative just amounts to ~ B and VT -~C. I t thus suffices to further nssume V~.B--->Cfor the purpose of showing ~TB and V~. ~-C. Since ~Ir (T" of [2]), by Modus ponens ~T .~BvC-~.B-->C. Then by Mod~.s tollens, VT .~(.-.BvC). But then by DeB'Iorgan, Simplific.~tion, and Double ~Neg~tion, ~T B and ~-~ ....C, as desired.

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J. .Michael Dun~ We can now p r o v e the S:E~'IA:NTICAL COMPLETENESS TKE0:I~:EM. ] f

S

~l~SIA, then S ~r~ A

I~OOF precedes contrapositively. If not S FRM A, t h e n b y LE~I)fA ] there is a prime ICM-theory To with S ~ To and not bye A (letting th( T of the Lv,~r~A be the smallest l~M-theory including S). Consider th( canonical model qr0" L~)fM~ 2 then contains the only non-evident in. formation needed to see t h a t the c~nonical model determined b y a prim( R~Vf-theory is indeed an E1VLmodel, and L E ~ A 5 assures t h a t for al B~ S, TeqT0(B , To), whereas T iq~ro(A , To).

6. Decidability by filtration That I~M h~s the finite model property and is decidable was first show~ by Meyer [9] using matrix methods (cf. note 1). I t is none the less o: some minimal interest t h a t the method of filtration (cf. [14]) nmy b~ applied to the binary Kripke-style semantics. P a r t of the interest lie: in the fact t h a t the t e r n a r y semantics of Routley and Meyer has so fa~ resisted ~iltration (at least for the stronger systems like E, 1~, and P,~[) The 1)~00F precedes by straightforward modifications of the argument: of [14], and the interested reader can work it out for himself. To mak, sure he gets started off on the right track it m a y be well to define th, essential equivalence relation. Where 9v is an RS[ model on an R ~ 1node structure (G, K, R) and where ~ is ~ set of sentences closed under subfor mulas, define for H , H ' E K , H - - , H ' iff ~ ( B , H ) = ~ ( B , H ' ) for al Be v2. Thus the reader should be sensitized to the fact t h a t H a n d H being indiscernible with respect to which sentences in W t h e y make tru, does not suffice (as it does in [14]) t u support the appropriate equivalence They must also be indiscernible with respect to which sentences in ~ the: make false.

7. RM models and Sugihara matrices I t turns out t h a t there are connections between the present mode theory for ]~M and the earlier model t h e o r y in terms of Sugihara matrices Due to limitations of space we merely sketch the main outlines of thes, connections leaving m a n y details of proof and even statement to the in terested reader. We make use of definitions relating to Sugihar~ matrices f r o m [2] but first take this opportunity to correct an error there. On p. 2 it w a intended t h a t a Sugihara matrix be a chain with involution-, i.e., t h a t be a 1-1 order inverting mapping of period two : of the chain onto itself with the designated elements and operations specified as indicated. 7 The italicized words were i a a d v o r t a n t l y omitted, although they wore ii~ t h a b s t r a c t [3]. Aeknowledgem6nt is expressed to M. Tokarz for p o i n t i n g out this error

A Kripke.style semantics for R-mingle...

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I t would be quite well-known were we working on K r i p k e models for t h e intuitionist sentential calculus to t r a d e a K r i p k e model w ( A , H ) for its equivalent v a l u a t i o n IIAII, = {HE K : ~0(A, H) ----T}. W e w a n t to do s o m e t h i n g similar, b u t because of t h e a m b i v a l e n t n a t u r e of I~M models we m u s t t h i n k of t h e equivalent v a l u a t i o n as IIAII~ =