Ross T. A Content Semantics for Quantified BRADY

Relevant Logics. x I

Abstract. We present an algebraic-style of semantics, which we call a "content semantics", for quantified relevant logics based on the weak system BBQ. We show soundness and completeness for all quantificational logics extending BBQ and also treat reduced modelling for all systems containing BB a Q. The key idea of content semantics is that true entailments A ~ B are represented under int6rpretation I as content containments, i.e. I (A) ~< 1(/3) (or, the content of A contains that of B). This is opposed to the truth-functional way which represents true entailments as truth-preservations over all set-ups (or worlds), i.e. ('Ca ~ K) Of I (A, a) = T then I(B, a) = T).

I present what I call a content semantics for a broad range of quantified relevant logics, in fact, for all those logics stronger than the base system BBQ

(see 1 below). I also present a "reduced modelling" of this semantics for all

logics containing the system BBd Q (see A content semantics for quantified

relevant logics, part II, to apper in Studia Logica). 2 A content semantics is essentially an algebraic-style semantics without

those algebraic operators which take the semantics towards more or less conventional algebraic theories and away from being a more or less direct semantics of the logical concepts. The main algebraic operator to be omitted from the sentential part of the semantics is the fusion operator 'o'.

This operator was introduced by Dunn in I-7] and later in [-1]. It was also employed by Routley and Meyer in [21] and in their sentential algebraic semantics in [16] and [17]. The point of the fusion operator was to bring out familiar algebraic properties like associativity, commutativity and idempo- tence, which would enable the algebraic semantics concerned to be brought into comparison with some familiar algebraic systems.

The problem with the fusion operator is that, unlike the other algebraic operators that one encounters in algebraic semantics of relevant logics, it does not correspond to a connective which has a suitable English reading. Such a fusion connective can be defined in stronger systems such as R as follows: A o B =~r ,,. (A -~ ,.~ B). (c.f. [1], p. 344-5, where fusion is called 'contenability'). However, for weaker systems such as D J (c.f. my [4] and 1-6]), this definition no longer applies and the fusion connective can here be characterized by the two-way rule: A ~ .B ~ C c e . A o B ~ C. The above definition of fusion does

1 This paper Was presented to the Australasian Association for Logic Conference, held at the University of Auckland from 9-12th July, 1986. I wish to thank those present for some helpful comments. I also wish to thank Kit Fine for some useful discussion on some topics of this paper.

2 The sentenfial logic BB was introduced by Layers in [15].

112 R.T. Brady

provide a rather awkward reading, for the systems concerned, but there is point in Concentrating on those weaker systems which yield a simply consistent (or non-trivial) naive set theory (c.f. my [2], 1-5] and [6].) and for which the tWo-way rule for fusion would apply. In such a case, fusion would be a technical device for grouping iterated premises, but for which there is no standard English reading. Also, the addition of fusion would cause triviality in naive set theories which would otherwise be simply consistent (or non-trivial). (The argument for this is due to Meyer and Slaney and appears in Chapter 4 of [6,1.) So, in conclusion, there is good reason not to include the fusion connective and hence it would be pointless to include the corresponding fusion operator in an algebraic-style semantics for relevant logics in general. Also, Lavers in 1-15], Chapter 3, omits fusion in his algebraic semantics of the sentential logic BB and its extensions, but for a different reason. In fact, as we shall see, our semantics is essentially a quantificational extension of Lavers' sentential semantics, which in turn leans heavily on Meyer and Routley's [17].

Now that we have argued for a content semantics, as introduced above, we now consider why we use the term 'content' for such a semantics. Well, we use the term in the same way as Priest does in 1-19], p. 416, that is, for the sense of a sentence. Priest, on p. 422, introduces true entailments of B from A as equivalent to the containment of the sense of B in the sense of A. The term 'content' is then seen as an extension of the use of the containment metaphor.

A more precise term is 'logical content ' because the contents (or senses) do depend on the logic concerned. Take any logically equivalent sentences, say, p and p v (p & q). They will then have the same contents. So, a certain circularity develops, in that the logic itself is used to determine the notion of logical content, which in turn is used to characterize the logic. Firstly, though, one would not expect the mere notion of content to help us much in pinning down a particular logic. And, secondly, there are a number of ways of helping to pin down a suitable logic or range of logics which not only yield simply consistent naive set theories but also satisfy other features as well (as would be required for a non-ad hoc solution to the set-theoretic paradoxes, anyway). Such features are given in Chapter 1 of [6-1, one of which is the satisfaction of the Depth Relevance Condition, which appears in my [3].

One of the main distinguishing features of algebraic-style semantics, and of content semantics in particular, is that true entailments are represented as containments of algebraic objects (contents). This is expressed in the form: I (A ~ B) E Tiff I (A) ~< I (B), for formulae A, B, interpretations I, where T is the truth filter, and it follows from our semantic postulate p7a, to follow. This can then be contrasted with the representation of true entailments in truth-functio- nal semantics, whether relational or operational. Generally what happens here is that true entailments are represented as truth-preservation over all set-ups, i.e. 1 (A --. B, T) = Tiff, for all set-ups a e K, if 1 (A, a) = T then I (B, a) = T, for formulae A, B, interpretations I, where K is the set of all set-ups and T is the base set-up. For unreduced Routley-Meyer relational semantics, however, the

A content s eman t i c s . . . 113

above equivalence is only satisfied at the level of validity in the semantics, i.e. A ~ B is valid in the semantics iff, for all model structures ~ ' , for all interpretations I in J/', for all set-ups a e K, if ! (A, a) = T then I (B, a) = T. Perhaps, this is a good reason for favouring the reduced semantics.

Added to this contrast between the content and truth-Tfinctional represen- tations of true entailments are the relative degrees of difficulty in obtaining a suitable semantics of each of the two types. For the content semantics this paper provides a fairly straight-forward intuitive semantics, neatly extending the corresponding sentential semantics. However, for the truth-functional semantics, Fine has established in [9] a negative result, showing that RQ and some other relevant logics in its vicinity are incomplete with respect to their constant domain semantics, which is a simple straight-forward extension of the Routley-Meyer sentential (relational) semantics. Additionally, I have searched for other constant domain semantics and also for variable domain semantics, which are extensions of the Routley-Meyer sentential semantics and all that I have tried, which have largely been of the more straight-forward type, have been abortive. However, Fine, in {10], has proved soundness and completeness for RQ with respect to a stratified domain semantics which extends his sententiat operational semantics in [8]. This is quite a complex semantics using a new interpretation of quantification which involves arbitrary objects. So, I think it is still fair to say that the more simple and straight-forward extensions of the Routley-Meyer sentential semantics do not seem possible for the characterization of systems at least in the vicinity of RQ. As a result, I think there is some preference for the content semantics and its representation of true entailments as content containments, over the truth-functional semantics and its representation of true entailments as truth-preservation over all set-ups.

We now return to algebraic-style semantics for quantified relevant logics. The only work in this area is due to Meyer, Dunn, and Leblanc in [18]. In their algebraic semantics, for which they show soundness and completeness with respect to RQ, they employ a fusion operator with the usual fusion properties for R, and they use the substitution interpretation for quantifiers, but no semantic postulates are given for generalized meets and joins. We have dealt with fusion above. Further, they admit that it is mainly personal preference as to whether the substitutional interpretation for quantifiers is used or whether domains are used. However, the reason that no semantic postulates are added for generalized meets and joins is that their quantificational extension of the sentential semantics is a trivial one in that all interpretations I satisfy the trivial condition, I (A)~ T, for all sentential instances A of quantificational axioms, where T is the truth filter (c.f.p. 107 of [18]). This ensures soundness by putting a condition on interpretations, thus obviating the need for any quantificational extension of the sentential semantic postulates. This suits the purposes of Meyer, Dunn and Leblanc as they are essentially interested in establishing the admissibility of.the rule ~: A, ~ A v B =~ B, for the logic RQ. However, we are interested in providing a good content semantics for quantified relevant logics

114 R.T. Brady

and so we would be looking for an extension of sentential postulates involving the use of generalized meets and joins, which would represent the universal and existential quantiflers, respectively. We would expect the postulates involving these generalized meets and joins to reflect the corresponding sentential postulates for meets and joins. We will see this feature in our content semantics to follow.

Further, one might expect such a content semantics to reflect one of the types of algebraic semantics that have been used for the classical predicate calculus, since certainly our general semantics for quantificational logics will include the classical predicate calculus as a special case. However, the type of semantics presented here is different from the three types of classical algebraic semantics that have been used, viz. the Rasiowa and Sikorski semantics using unrestricted generalized meets and joins (see [20], pp. 225-6 and pp. 286-8.), the Halmos semantics using polyadic algebras (see [12], pp. 20-33), and the Henkin-Monk-Tarski semantics using cylindric algebras (see [14], pp. 162-3 and p. 168). Whilst i t may be an interesting exercise to use these three types of algebraic semantics to capture the range of quantificational logics considered in this paper, our content semantics has some advantages over each of these types.

The main difference in our semantics is that n-place functions from D" to C are used to interpret n-place predicates, where D is the domain of individuals and C is the set of contents. The use of such functions is extended to the interpretation of n-place formulae by the use of closure conditions on the set of functions. The set H n of n-place functions is taken as primitive and this, together with the closure conditions, actually puts a restriction on the set of functions that can be used to interpret n-place formulae. Such a restriction is used in restricting the generalized meets and joins in such a way that they match up with the universal and existential quantifiers.

The idea of using primitive sets of n-place functions to interpret n-place predicates is similar to that used by Henkin in his truth-functional semantics of higher-order predicate calculus (see [13], p. 52 and pp. 55-6.) He shows that by expanding the set of models in this way, completeness for higher-order predicate calculus can be proved where incompleteness would otherwise be proved had n-place predicates been interpreted in the set of all such n-place functions. This means that Henkin's use of primitive sets of such n-place functions is essential to the obtaining of completeness. This is similar to our case where use of arbitrary sets of functions would prevent our completeness proof from proceeding.

We now consider advantages of our semantics over its main contenders, the Rasiowa and Sikorski type, the Halmos type, and the Henkin-Monk-Tarski type. One advantage over all of these is that none of them have any domains for predicates, thus making these semantics unsuitable as they stand for an extension to second-order quantificational logics.

The main problem with Rasiowa and Sikorski's type of semantics is that it

A content semantics... 115

uses complete Boolean algebras which have generalized meets and joins which are unrestricted. It is clear that many infinite meets and joins do not correspond to quantification since the interpretation of quantification is restricted to x-variants of an interpretation of a formula, for some formula and some variable x. This interpretation of quantification is applicable in Rasiowa and Sikorski's semantics, in particular (see p. 226 of [20]), and thus their use of complete Boolean algebras introduces superfluous generalized meets and joins.

The main advantage our semantics has over the Halmos and Henkin- Monk-Tarski types is that it is more perspicuous in that the contents are basically composed of n-place functions applied to a sequence of n domain elements, for some n ~> 0. The use of polyadic or cylindric algebras, on the other hand, makes the semantics more distant from the quantificational logic that is being captured, in that it is more algebraic, in character. Thus, our semantics is more in keeping with the idea of content semantics and our preference fo r restricted generalized meets and joins, ' ( ] ' and '[.)', over the more algebraic operators, '3' of polyadic algebras and 'c' of cylindrical algebras ties in with our dropping of the fusion operator 'o'. Further, the advantage of a content semantics is that it captures logical concepts more directly and relates them to the key semantical concepts of truth and validity.

We now proceed with our content semantics, firstly the unreduced semantics for BBQ, then the extension of this semantics to all extensions of BBQ, followed by the reduced semantics for BBaQ and all its extensions.

1. Unreduced content semantics for BBQ

We first axiomatize BBQ by adding quantificational axioms and rules to an equivalent of Layers' axiomatization of BB in [15].

Primitives

Sentential variables: p, q, r , . . . Predicate var iables ' f g, h, ... (Superscripts are sometimes added to indicate the number of argument places). Individual variables: x 1, x2, x 3 . . . . Connectives and Quantifier: ~ , &, -% V.

Definitions

A v B =as ~ (~ A & ,,, B). A~--~B =ai(A --, B)&(B--, A). (3x) A = as "~ (Vx) ~ A.

Axioms

A1. A ~ A A2. A & B - - . A A3. A & B ~ B A4. A & ( B v C ) ~ ( A & B ) v (A&C)

116 R. T. Brady

A5. .-~ -.~ A ~ A A6. (Vxk)A ~ A~l/x~, where x t is free for x k in A. A7. (VXk)(A v B) ~ A v (VXk)B, where x k is not free in A.

Rules

R1. A, A ~ B ~ B . R2. A, B ~ A & B . R3. A ~ B , C - * D ~ B ~ C ~ . A ~ D . R4. A--* ,,~ B:----* B ~ ,.~ A. R5. A ~ B , A ~ C ~ A ~ B & C . R6. A =~ (VXk) A. R7. (Vxk) (A ~ B)=~ A ~ (VXk)B, where x k is not free in A.

The (unreduced) content semantics for BBQ consists of BBQ model structures (BBQ m.s.) which are composed of the following concepts: (T, C, D, ~-, ~<, , , ca, ~ , ~ , ('], U ) , where C is a non-empty set of contents, T is a non-empty subset of C (considered as the true contents), D is a non-empty domain of individuals, '~<" is a 2-place relation on C, ' . ' is a 1-place function on C, 'n ' , 'w' and '=,.' are 2-place functions on C, ' 0 ' and 'U ' are 1,place functions from V (defined below) to C, and 3 r is the disjoint union of ~-o, ~ l . . . . . ~'", .,. (for all n/> 0), where each of the ~" ' s is non-empty, ~:o __ C and ~r, (n >i 1) is a set of functions from/Y' to C. f f is subject to closure conditions and all these concepts are subject to semantic postulates, to follow after some preliminary definitions. The above set ~-n of functions will provide interpretations for the n-place predicates and indeed n-place formulae, for each n. Formulae with no free individual variables will be interpreted in ~-o, ' ( ] ' and 'U ' are the generalized meet and join, which will represent the universal and existential quantifiers, respectively. The set V, defined below, has the effect of restricting their application so that they are in fact meets and joins of x-variants of interpretations of formulae, for some x.

We proceed with the preliminary definitions. We consider the set S of all functions s: N + --, D, where N + is the set of positive integers. So, for F n ~ TM, s ~ S, F" (s (il), -.., s (i,)) ~ C, for il, . . . , i, ~ N +. The functions s: N + ~ D repre- sent denumerable sequences of domain elements, such sequences .being used in a similar manner to sequences in Tarski's Semantics of Predicate C~dculus, i.e. each sequence represents an assignment in D to each of the individual variables, x l , . . . , x . . . . . . taken in order.

For s e S , beD, k e N +, we define the function sb/k: N + --, D, as follows:

F o r i r k, sb/k (i) = s (i). s /k (k) = b. sb/k is essentially the k-variant of s which assigns b to the k-th position in the seq,uence given by s.

F, or f " e ~ ~ and seS , we define the function F~s: (N+)" _. C, as follows:

For n>/1 , F"s( i l , . . . , i~)=V"(s( i t ) , . . . , s( i ,~)) , for all i~ . . . . . i~eN +. For n-~0, F ~ ~

A content semant ics . . . 117

It is convenient to use these functions for the interpretation of formulae, allowing one to separate off the n argument places.

For F" E ~-", s E S and il, . . . , i,, k E N*, we define k-constancy for expres- sions of the form F n S ( i l , . . . , i,) as follows: For n i> 1, F n s ( i l . . . . . i,) is k-constant iff kq~{i x . . . . . in}. For n = 0, for F ~ s ( = F ~ is k-constant, for all k e N +.

[Hence, if F n s (i x . . . . , in) is k-constant then F n sb / k ( i l , . . . , in) = F ~ s (i 1 . . . . . in)']

The k-constancy of expressions F n s (i x, . . . , in) is used to semantically represent the idea of the variable x k not occurring free in the formula represented by this expression.

For {Jx,-. ",Jra} ~- {1 . . . . . n}, a I . . . . , an-m, b e D , we define b/{jx, . . . , jm}(ax,. . . �9 . . , an-m) as the sequence of n elements of D formed by putting b in the Jx . . . . . jm-positions and putting a x . . . . . a n - s in the remaining n - m positions in order. In particular, for m = 0, we define b / O ( a l , . . . , a,) as the sequence (al, . . . , a,).

Further, we define the (n-m)-place function F n b / { j t , . . . , j , " } such that F " o / { j x . . . . . jm} (ax , . . . , a , _ " ) = Fn[b / { j x . . . . . jmJ(a~, . . . , a,_m)], for all a x . . . . . . . , a , - m ~ D . It is useful to have these functions so that generalized meets and joins of them can be easily formed (see below).

V = d y {{Fnb/{ j I . . . . . j"}(a x . . . . . a,,_,,): b e D } : m, n E N , n ) m , F n E . ~ "n, a t . . . .

. . . , a n - r e e D , { ja , " " , i s } ~-- {1, . . . , n}}. This is the domain of the generalized meet and join functions and so we have O { F n b / { j l . . . . , j m } ( a l , . . . , an-m): b e O } e C a n d U {Fn*/{Jx . . . . . . ] , , } (al . . . . . an-,.): b e D I e C . For FmE~-" and F n, G " E ~ " , we define the following functions:

The n-place function F"* such that F"*(a x . . . . , a n ) = F" ( a~ , . . . , an)*, for all a 1, . . . , a n E O .

The ( r e + n ) place function F m n G n such that F " c~ G"(ax , . . . , a, , , b x, .=., bn) =

= F m ( a x , . . . , am) c ~ G " ( b x, . . . , bn), for all a x . . . . , am, b l , . . . , b n ~ D .

The (m + n)-place function F m u G n such that F," u G n (ax, . . . , am, b x . . . . . bn) =

= F m ( a x , . . . , am) U G ' . ' ( b l , . . . , bn), for all ax, . . . , a " , b I . . . . , bnED.

The (m+n)-place function F " = ~ G n such that Fm=: ,Gn(ax . . . . . a , " , b t . . . .

. . . . b,) = F " ( a x , . . . , am)=: ,Gn(bx , . . . , b,), for all a 1 . . . . . a,", b l , . . . , b n ~ D .

Let {Jx . . . . . j,,} ___ {1, . . . , n}, for the following two definitions.

The (n-m)-place function (] {F"O/{j x . . . . ,Jm}: b e D } such that 0 {F"b/{Jx . . . . . . . . j,.}: b E O } ( a x . . . . . an- , . ) = 0 {Fnb/(Jx, . . . , j , , } ( a l . . . . . a , _ " ) : b E D } for all a l , . . . , a n - r e e D .

The ( n - m)-place function ~ {F" b/{j~ . . . . , Jm}: b E D ) such that U {F" b/{j x . . . .

. . . . J,,}: b e D } ( a x . . . . , an- , . ) = U { p b / { j , . . . . . j , . } ( a x, . . . , a n - , . ) : b E O } , for all a 1, . . . , a n - , . E D .

The functions F ~', F , . c ~ G n, F , . u G n, F , . = ~ G n, ( ' ] { F " b / { j x , . . . , j , ,}: b e D }

and U { F " b / { j x . . . . . j,.}: b e D } are all members of ~ ' , as determined by the

3 -- Studia Logica 2/88

l 18 R.T. Brady

closure conditions on ~ , given below. These functions enable n-place formulae to be interpreted in ~ .

Closure Conditions on ,~ cl. If F"~.~ TM then F"*E~ n. c2. If F m ~ .~,n and G n ~ ~ n then F" n G n, F" w G n, and F m ~ G n E .~'rn + n. C3. If Fn~Y" and {J l , . . . , J , , } - - -{1 , . . . ,n} then (-]{Fnb/{jl , . .... ira}: b~

e D } ~ n-" and U{F"b / { j l , . . . , jm} : b ~ D } e ~ n-re.

Semantic Postulates For c, d, e e C , F n E ~ n, G m e ~ m, i i . . . . , in, Jl . . . . ,Jm, k ~ N + : pl (a) c ~ c .

(b) If c<~d and d~<c then c = d . (c) If c~<d and d<~e then c~<e.

p2 (a) c n d ~ < c . (b) c n d ~ d . (c) If c~<d and c~<e then c ~ < d ~ e .

p3 (a) c ~ < c u d . (b) d < ~ c u d . (c) If c~<e and d<~e then c u d ~ < e .

p4 c n (d w e) ~< (c n d) u (c n e). p5 (a) e * * = c .

(b) If c ~< d then d* ~< c*. p6 (a) If c e T and c~<d then deT.

(b) If c e T and d e T then c n d e T . p7 (a) c = ~ d e T iff c~<d.

(b) If c~<d then d = . - e ~ < c ~ e . (c) I f c ~ < d then e ~ c ~ < e = ~ d .

p8 (a) 0 {Fnsb/k(ia . . . . . i,): b e D } <~ Fnsb/k(i~, . . . , i,), for all b e D . (b) If F"s(i~ . . . . . in) <<. G"sb/k(jx, . . . , j , , ) , for all b e D , then F"s( i l , . . .

. . . , i n )<-O{Gmsb /k (J l , . . . , Jm) : beD} , where Fns(il . . . . ,in) is k- -constant.

p9 (a) Fnsb/k(ia . . . . , i,) <~ 0 {Fnsb/k(i~ . . . . , in): beD} , for all beD. (b) If F nsb/k(ia . . . . . i , )<<.G"s( j j , . . . , j , , ) , for all beD, then ~ { F "

sb/k( i l , . . . , in): b e D } <~ GmS(jl . . . . ,j,,), where G m s ( j l , . . . , j m ) is k-constant.

pl0 ~ {( F n w Gm)sb/k(i l . . . . , i~,Jl, "",Jm): b e D } <~ F"s(i 1 . . . . . in) u {Gmsb/k(Jl, "'',Jm): beD} , where F"s(i~, . . . , in) is k-constant.

p l l (a) If Fnsb/k(i~ . . . . , i n ) e T , for all beD, then ~{Fnsb /k ( i~ , . . . , i n ) : b e D } e T .

One can see how each of the postulates p8-11 extend appropriate postulates concerning meets and joins of contents to corresponding properties of restricted generalized meets and joins. E.g. compare p4 and pl0.

We now define interpretations for the basic kinds of variables and then extend such interpretations to all formulae. An interpretation I in a BBQ m.s. is

A content semantics ... 119

an assignment to each sentential variable, predicate variable and individual variable, as follows:

(i) I (p )e ~ o (c_ C), for each individual variable p. (ii) I ( f " ) e ~ " , for each n-place predicate variable f", for n ~> 1.

(iii) I (Xk)e D, for each individual variable x k, for k >~ 1. Interpretations I are each extended to all formulae, inductively as follows:

(iv) I ( f" xil ... xi. ) = I ( f")(I (xil) . . . . . I (xin)). (v) I (--~ A) = I (A)*.

(vi) I (A & B) = I (A) n I (B). (vii) I (A ~ B) = I (A) =~ I (B).

(viii) I ( ( V X k ) A ) = N { I b / x k ( A ) : beD} , where Ib/x k is the xk-variant of I assigning b to Xk,

i.e. Ib/Xk(P)= I(p), for all sentential variables p, Ib /Xk( f f )= I~" ) , for all predicate variables f " (n/> 1), Ib/xk (Xt) = I (Xt), for all l ~ k, and Ib/Xk (Xk) = b.

In order to ensure that ~ {Ib/xk(A): b e D } is properly defined, i.e. to show that Ib/Xk (A) has the form F" b/{jl , . . . , Jm} (al . . . . . a,_,,), we need the following inductive definition of F~a, the definition of S I and Lemmas 2 and 3 below. Let A be a formula with n argument places (i.e. with n free individual variable occurrences) and let I be an interpretation. Then, we define the n-place function F~,x inductively as follows:

Fp~ =asI(p) , for all sentential variables p. F~%xq ...... ,,a = e l i ( f " ) , for all atomic formulae of form f " x a ... x i .

FZ Ad = dl F~,I*. f / ? l + n - - ?1~ /1

A & B , I - - d f FAd ~ FBa. f l q ~ + ll - - ?n tl A-*Bd -- df FAd =~ FBd. F'~w;Aa=ai(~{F"a,lb/{j l , . . . , j , ,} : beD}, where x k occurs free in just the Jl, . . . , j , , -a rgument places of A. [-Note: {J l , - . - , J , ,}-~ {1 . . . . . n}.] Lemmas 1 and 2 assist in the proof of the important Lemma 3.

LEMMA 1. For formula A with n argument places and for interpretations I, FnA,I e ~ n .

PROOF. By induction using the closure conditions cl, c2 and c3.

We define the function S F N + ~ D , for interpretation I, as follows:

s,(i) = I (x3 , for all i e N +.

LEMMA 2. For formulae A with n argument places, for interpretations I, for k e N + and b e D ,

F~a,mx~ = F~a and s m ~ = s b! k.

PROOF. The proof of F]a~/~ ~ = F"aa is by induction using the definitions of F ~ j and lb/xk . s ~ / ~ = S~k is also proved using definitions of its components.

120 R.T. Brady

LEMMA 3. For formulae A with n free individual variable occurrences xil . . . . . . . . xi., taken in order, and for interpretations I,

I (A) = F"aa sl (il . . . . , i,).

PROOF. By induction using the definitions of I(A), Faa, st and using Lemmas 1 and 2. We set out the case for (VXk)A, where A has n free individual variable occurrences, xl, . . . . . xi., in order.

l ( (VXk)a ) = (-] {Ib/xk(a): b e D } ,

= ( ) S F . b a. i /~smx~( i l . . . . , i.): b e D } , by induction hypothesis.

= (']{F"aaSlb/k(ia . . . . . i.): beD} , by Lemma 2

= 0 {Vna,' b/{Jl, "",Jm}(I(xikl), "", I(xik._m)): b e D } , where {j, . . . . . j.,} __ {1 . . . . , n}, k occurring in the j , , ...,jm-pOsi tions of (q . . . . , i.), leaving the remaining positions k l , . . . , kn-m, in order. [This means that ('] {Ib/xk(A): b e D } is now defined.] _N,F. D} - ~ a,, w l , "",Jm}: b e (I(xik) , . . . , l ( X i k n _ m ) )

- - f n-m (Vxk)A,I 1" - - (Vxk)a,lsi(ik~ . . . . , ik._~. ), by definition of F "-m s

We continue with further lemmas which are needed to assist in the proof of soundness of BBQ with respect to the content semantics.

LEMMA 4. For formulae A with n free individual variable occurrences xq , . . . . . . . xi., in order, and for interpretations I, (i) for n > O, I (A) is k-constant for all k r 1 . . . . . i,}, a n d

(ii) for n = O, I (A) is k-constant for all k.

PROOF. By Lemma 3 and the definition of k-constancy. Note that, for n = 0, I(A) = F ~ by Lemma 3. A,I ,

LEMMA 5. For F " e ~ " , s eS , i~, . . . , i., k e N +,

(i) I f F"s(i x, . . . , i,) is k-constant then F"*s(i 1 . . . . . i,) is k-constant. (ii) ~ {F"b/ {j~, . . . . Jm}: b e D } s(ik~ . . . . . ik._.. ) is k-constant, where { j l ,

. . . . j ,} _ {1, . . . , n} and k occurs in the Jl , . . . , j ,-positions of {i 1 . . . . , i,}, with the ka, . . . , k,_,,-positions remainin9.

(iii) U {F" b / { J l , ' " , Jm}: b e D} s (ik, . . . . , ik. _..) is k-constant, with conditions as for (ii).

PROOF. Trivial, by definition of k-constancy.

LEMMA 6. For formulae A with n aroument places, with x I free for x k in A, and for intepretations I,

I~a~,#,~,1 = F"aa.

PROOF. By induction using definitions.

We extend the interpretations I to deal with the defined connect ive ' v ' and the defined quantifier '=r. We need interpretations for ' v ' for the soundness proof and we also need to tie in ' v ' and 'B' with ' u ' and 'U' , respectively, to

A" c o n t e n t s e m a n t i c s . . . 121

motivate the introduction of 'w' and '(,3' into the BBQ model structures. Such relationships can be established for both intepretations I (A) and functions F" A , I ,

as follows:

LEMMA 7. For formulae A with m argument places and B with n argument places, and for interpretations I, (i) (a) I ( A v B ) = I ( A ) w I ( B ) ,

(b) ~,m+. r~ --A v e,I = Faa w F~ a, (ii) (a) I ( (Sx~)A)= U{Ib/xk(A): beD} ,

(b) - ' F~x,)A,, = U {F],, b/{J,, "" , J,}: beD} , where x k occurs free in just the Jl . . . . , j fargument places of A.

PROOF.

(i) (a) I (A u B) = (I(A)* c~ I (B)*)*, by applying definitions, and so to prove (i) (a) we need: c w d = (c* nd*)*, for c, d e C . This can be established using the following semantic postulates: plb , 2a, 2b, 2c, 3a, 3b, 3c, 5a and 5b.

(i) (b) --A~m+~v Z,I = (FAam* n *B,V~"* ~* and so we need: F m w G" = (F"* c~ G"*)*, for F " e ~ - " and G"e~'" . This follows readily from definitions using c u d = ( c * n d)* for c, d e C (see (i) (a)).

(ii) (a) I ( (3Xk)A)=(( ' ]{ Ib /xk(A) *" beD})* and so we derive U{Fmsb /k ( i , , . . . , i.): b e D } =((-]{Fmsb/k( i l , . . . , in)*: beD})*, for F" e ~ " and s e S, and then apply Lemmas 2 and 3. In the proof the following semantic postulates are used: p lb , 5a, 5b, 8a, 8b, 9a and 9b. We also require (('] {F"s b/k(ix, . . . , i,,)*: beD})* and (U { Fmsb/k(il , . . . , i,,): beD})* to be k-constant. This can be shown using Lemma 5 by re-expressing these expressions in the forms ((] {F m*b/{Jl, . . . �9 ..,Jr}: beV})*S( ik l , ' . . , i km- , ) and ( U { F ' b / { j l . . . . ,Jr}: beD})* s ( ik , , . . . , ik~,_,), respectively, where k occurs in the j l , . . . , J,,-i places of (i~ . . . . . i,.) and i k . . . . . . ik~_ , constitute the remainder, in order.

(ii) (b) , ,-t ~,,,* b/ �9 beD})*, x k F(3xu )a , ! - - - - ( ( ~ I " A,I {Jl , �9 �9 �9 , Jl} where occurs free in just the j a , . . . , j f a r g u m e n t places of A. We need to show: U{Frab'{j l . . . . ,Jl}: b e D } =((-]{Fm*~ . . . . ,jr}: beD})*, for Free e ~-" and {Jl . . . . . j~} _ {l . . . . . m}. This follows from definitions using U {Fmsb/k(ix . . . . . i.): beD} = ( A {Fmsb/k(il . . . . . i.) ~-- beD})*, for F " e ~-" and s e S (see (ii) (a)).

We proceed to define truth and validity in the BBQ content semantics.

A formula A is true for interpretation 1 in a BBQ m.s. ~t' iff I ( A ) e T.

A formula A is valid in the BBQ m.s. d/l iff A is true for all intepretations I in Jg.

A formula A is valid in the BBQ content semantics iff A is valid in all its BBQ m.s.

We show in the following two theorems that the BBQ content semantics is sound and complete with respect to the axiomatization of BBQ.

122 R. T. Brady

THEOREM 1. (Soundness) For all formula A, if A is a theorem of BBQ then A is valid in the BBQ content semantics.

PROOF. We need to show that, for each axiom A, I ( A ) e T, for any interpretation I in BBQ m.s. d/g, and, for each rule, truth for I is preserved by the rule, for any I, or validity in ~ / i s preserved by the rule, for any / / / . For validity of formulae of the form A ~ B, it is sufficient to show that I (A) ~ I (B), for any I in J//, by p7a. We test a sample of axioms and rules.

(A6) I ( ( V x k ) A ) = O{FnAaSzb/k(il, . . . , i,): beD} , where A has the n free variable occurrences, xi, . . . . , xi,, in order, by Lemmas 2 and 3.

<~ F"~,I st b/k (ij . . . . , i,), for all b e D, by p8a. I(AX'/Xk) = F"AaS/(X')/k(il, . . . , i,), by Lemmas 3 and 6. Hence, I ((VXk) A) <~ I (A~'/Xk).

(A7) I ( ( V X k ) ( A v B ) ) = ( ' ] ,+m b - �9 {FAvB,ISI / k ( t l , "" , in, J1, "'',Jm): beD} , where A has the n free variable occurrences, x~l . . . . , x~,, in order, and B has the m free variable occur- rences, xs,, . . . , x j,,, in order, by Lemmas 2 and 3,

= ~ {(F~ta u F~,,) sib/k ( i , , . . . , i,, j , , . . . , j,,): b e D}, by Lemma 7(i) (b), F"Aas,(il . . . . , i,) u N {F"~,*sIb/k(Jl, "' ' , J,,): beD} , by pl0, since F"Aasi(il . . . . , i,) is k-constant,

---I(A) w I((VXk)B), by Lemmas 2 and 3, = I(A v (VXk)B), by Lemma 7(i) (a).

(R6) Let A be valid in BBQ m.s. ~ . Then Ib/Xk (A) e T, for all b e D, for any I in rig. Hence F"Aa sib/k (il . . . . , i ,)e T, for all b e O, by Lemmas 2 and 3, with specification of A as above. By p l l a , ~ {F"A,iSlb/k(il, . . . , i,): beD} e T and hence I ( (VXk)A ) e T. Then (VXk) A is valid in ~ .

(R7) Let I ((VXk) (A ~ B)) e T. Then ~ " + m b �9 {FA-~n,t Sx/k (q, . . . , i,, Jr, " ' , J~,): b e D} e T, with specification of A and B as above. By p8a and p6a, "+" b - " ... j , ,)eT, for all beD, FA~B,ISI / k ( t l , . . . , i , ,J1, , By definitions and the k-constancy of F"aasi(il, . . . , i,), F"A,isj(il, . . . . . . . i,) ~ Fr~,i sib/k (Jl, "", Jm) e T, for all b e D. By p7a and p8b, F"aast(i~, . . . , i,) < O{F~, ,szb/k( ja , . . . , j , , ): beD} . Then, I (A) < I ((VXk) B) and, by p7a, I (A ~ (VXk) B) e T.

Before proceeding with the Completeness Theorem, we need certain preliminaries which set up the canonical BBQ model structure. If A' is a non-theorem of BBQ, we need to construct a canonical BBQ m.s. Jgc with canonical interpretation Ic such that I~(A')r T~ in J/{c, where Ji'~ consists of (T~, Co, Oc, O~c, ~<, *, n , u , =*-, ('], U ) , where the last six concepts are defined on Co, Dc and o~ .

A c o n t e n t s e m a n t i c s . . . . 123

Let D~ be the denumerable set of individual constants, {a s, a2, . . .

. . . , ak, . . . . } ( k zN+) . Add these constants to BBQ to obtain BBQ', with the associated modification to A6:

A6'. (Vxk)A ~ At/Xk, where t is an individual constant or an individual variable which is free for x k in A.

We will be generally working with sentences of BSQ', i.e. formulae of BBQ' with no free individual variables.

We define the remaining concepts of ~/c, by forming the usual Lindenbaum algebra.

[A[ = as {C: C is a sentence of BBQ' and f-~Bo4A ~ C}, for sentences A of BBQ'.

[N.B. I--BBQ, A++C iff IAI = IBI, for sentences A, B.]

Cc =ds{IA[: A is a sentence of BBQ'}. T~ =ds{[A[: A is a sentence of BBQ' and t--Bse, A}.

[N.B.~BBQ, A iff [AIeT~, for sentences A.]

ff~ = d S ~ 0 W ~ ~ . . . . . . W ffc" U , where =~-~-" (n >1 1) is the set of all func- tions, [A]": Dc" ~ C~, for any n-place formula A of BBQ, where [A]" (ai,, . . . . . . . ai,) = [A(ai,, . . . . ai,)[, for i a, . . . , i , ~ N +.

Similarly, H ~ is the subset of all elements [A] ~ of Cc, for any closed formula A of BBQ, where [ A ] ~ IA].

B be sentences of BBQ'. aBQ' A ~ B.

Let A and

IAI IBI = ds +

IAI* =dsl~ AI. IAI n IBI =.slA &BI. IA[ u tBI = dl [ A v BI. [AI ~ IBI = as iA ~ B].

Let A be a formula of BBQ' with at most x k occurring free. ~) {[A]"b/{j~, . . . , j , , } (a , , , . . . , ai,_,,): br =aSI(VXk)AI, where A has n ar- gument places with x k occurring in the j~ . . . . ,ira-positions and ail, . . . , ai,_,. making up the remainder, in order.

U {[A]" b / { j~ , . . . , j,,} (a i , , . . . ' ai" ,,): b ~ D~} =dS J(3Xk) AI, with A as above.

Because of Substitution of Equivalents holding for BB@' and BBQ, it can easily be shown these definitions are independent of the particular formulae A and B chosen to represent IA[, ]B[ and [A]".

To complete the definitions, we define the canonical intepretation I~ on J/go as follows:

(i) /~(P) = ]P[ = [p]O, for all sentential variables p. (ii) Ic(f") = [f"]", for all n-place predicate variables f " (n ~> 1).

(iii) I~ (Xk) = a k, for all k ~ N +.

It is clear that Ic (p) s f ro , Ic (f") e ~" ~'~, and I~ (x,) ~ D c.

124 R.T. Brady

We now proceed to establish the required properties of J//c and Ic which enable the Completeness Theorem to be proved.

THEOREM 2. (Completeness). For all formulae A, if A is valid in the BBQ content semantics then A is a theorem of BBQ.

PROOF. As indicated before, we need to construct a canonical BBQ model structure J /c and interpretation I c such that I~ (A') ~ T~, for some predetermined non-theorem A' of BBQ. Jg~ has been defined above and we need to show that it is a BBQ m,s. Firstly, we check the immediate properties: T~ 4= O, C~ ~ O,

of ~ c, ~ c , . . . , . . . , o~" r O (all n), D c ~ O, T~ _c C~, ~-~ is a disjoint union ~-o 1 ~,~, ~-0 ~*~ ___C~.

In order to check the closure conditions, we first establish the following identities relating elements of H c.

(i) For [A]" e o~r [A]"* = [ ~ a]". (ii) For [A]" s~-~ and [B]"eo~", [A]mn [B]" = [ a &B] m+", [A]" u [B]" =

= [A v B] "+" and [A]" => [BIn = [A ~ B] m+". (iii) For [A]"eO~c", {ja, . . . , j , } _ {1, . . . , n}, ~ {[a]"b/{ j a . . . . ,Jm}: bSDc} =

= [(Vxk) A]" - ' , and ~ {[A]" b~ { j~ , . . . , j ,}: b e D~} = [(3xk) a ] " - ' , where x k occurs free in just the j~ . . . . , j,,-places of A, with the remaining places taken up by elements of D~ after substitution.

(i), (ii) and (iii) can be proved by using definitions for BBQ m.s. in general, as well as definitions applying to the canonical BBQ m.s.

As a result of these identities, (i), (ii) and (iii), the corresponding closure conditions cl, c2 and c3 are immediate.

We proceed to check a sample of semantic postulates.

p3c. Let ]AI, IBI, [CI, ~ C~, where A, B, and C are sentences of BBQ'. Let Ial ~< ICI and Inl ~< ICI. Then f--BnQ, A ~ C and b-BBo, B ~ C . By A1, A5, R1, R3, R4 and R5, ~-nnQ, A v B ~ C and hence IAIwlnl ~< ICI. p8a. Let A be an n-place formula of BBQ. Let s be a member of S, and let k, i~ , . . . , i, e N +. Let ij, . . . . , ij,, = k for some subset { j~ , . . . ,Jm} of {1 . . . . , n}. Let ik~, . . . , ik,_,, be the remaining members of i 1, . . . , i,, in order, after ih, . . . , ij,, are removed. Let b~, . . . , b,_,, be s (ik~) . . . . . S (ik,_,,), respectively. Insert into the n argument places of A, x k in the j~, . . . , j , , -posit ions and ba . . . . , b,_ m in the remainder, in order. By A6, ~BBQ,(VXk)A ~A~/xk , for any b eD~. Hence, I(Vxk)al ~< Iab%l (by definition of '~<') and (~{[A]"b/{j l . . . . . j , ,}(bx, . . . , b,_/) : beDc} ~ [A] "b/ {j~, . . . , j , , } (b~, . . . , b , - m ) ( b y definition of ' 0 ' and I-A]"), for any b eDc. Coverting to the s-notation,

N {[A]"sb/k(il . . . . . i,): beDc} <. [A]"sb /k ( i l , . . . , i,), for all beDc, as required.

p8b. Let [A]"s(i I . . . . , i,) ~ [B]msb/k(j l , . . . , jm), for all beD~, where A and B are respective n-place and m-place formulae of BBQ, s e S and k, i~, . . . , i,, Jl, . . . ,Jm e N + . Let [A] ' s ( i 1 . . . . . i,) be k-constant, i.e. kq~{il, . . . , i,}. Let the

A content semantics... 125

n-argument places of A contain s (it), . . . , s (i,), in order, and let the m argument places of B contain s ( j 0 . . . . , s ( j , ) , in order. Then IA I ~ [Bb/k[, for all b e D ~ , where b/k is taken to mean that b is substituted in just the k~, . . . , ki-positions such that Jk,, " " , J k , = k. Note that no such substitution is warranted for A because k r . . . , i,}. Hence, t--BBQ'A ~ B b / k and b-BBQ'(A- -*B)b /k , for all b e D c . Let b be distinct from s( i 0 . . . . . s( i , ) , s o t . . . . , s ( j , , ) . Replace 'b' in the proof by a variable z which is new to the proof. Then t-- abe" (A --, B y / k, where z is substituted for each of the positions Ja . . . . . j,, that are equal to k. Hence, by R6, t-- BBO.' (Vz) ((A --* B) ~/k) and, by R7, t--- BBO.' A ~ (Vz) B z/k, since z is not free in A. Then, I h l < ~ l ( V z ) n = / k l and [ A ] " ( s ( i a ) . . . . , s ( i , ) ) < ~ { [ B ] m b / { k ~ , . . .

�9 . . , k l } ( b l , . . . , b , , - l ) : b e O c } , where b t . . . . , bin_ i consist of s ( j t ) . . . . , S(jm) , with the s(k)'s removed. So [ A ] " s ( i t . . . . , i , ) ~ ( - ] { [ B ] m s b / k ( j a , "'',Jm): be e D~}, are required�9

pl la. Let [A]" s b/k (i t , . . . , i,) ~ T~, for all b s D c, where A is an n-place formula of B B Q , s e S and k, i t , . . . , i, e N +. Let the n argument places of A contain s( i O, . . . , s( i , ) , in order. Then, as for the proof of p8b, IAb /k I sT~ and I-BBO.' A b/k, for all b ~ D c. Let b be distinct from s (it) . . . . . s (i,) and replace 'b' in the proof by a variable z which is new to the proof. Then, t--BBQ,. A ~/k and, by R6, t--sso, (Vz)(A ~/k). Hence, I(Vz)(A z/k)[ ~ T~ and, using s-notation, 0 { [ A ] " s b / k ( i l , ' ' ' , i,): b ~ D c } ~ Tc, as-required.

Hence d//~ is a B B Q model structure. It remains to show that the interpretation I~, defined above, extends to all formulae of B B Q in such a way as to ensure that I c ( A ' ) r Tc, where A' is our non-theorem of B B Q . We prove:

( . ) i f , / X k x b,/ (A (xix, xi .)) [A (ai, , ,bx/a,_ �9 �9 �9 X k l �9 �9 �9 , = . . � 9 a i n ) ~ , , �9 �9 .

Proof is by induction on A.

(i) (ii)

(iii) (iv)

b~/ ak,i"

i f , ~ Xk ~ . . . b,/ Xk ' (p) = IPl = IP b~/ ak~ . . . b,/ ak,i" lcb i / Xk 1 � 9 b d X k , f f n x i 1 . . . X i n ) =

= I~ ( f " ) ( i f a Xk ~ .. �9 b,/ Xk ' (Xi) ' . . . . �9 I f , / X k ~ be Xk ' ( X J )

= I f ] ( a i , , . . . , ai,) b'/ak . . . n n b , /ak ~ (these substitutions being made into the respective argument places of [f"]")

n a xbx/a,, bl/akz[" - - I f ( i x , , - - " ' � 9 a i . ) , , x " ' "

The cases for .-~ A, A &B and A ~ B are straight-forward. i f , / X k , . " . b,/ Xk ' ( ( y x k ) ( A (xi~ . . . . . x, .))) =

= ( - ] { Ib , /Xk ' . . . b , /Xkb /Xk(A(x ix ' . . . , Xi.)): b e D ~ } =

= N {[A (a,x, . . . . . . , ai .)b,/ak x b,, ak ,b/akl: b e De}, where ,bj/ak j, is removed in all cases in which k = k j,

= ~ { [ A ] , (aix ' . . . , t * i , ! ~ " l b / a t . k b a / a , . ~ k x " " " b f f a k : b ~ D c } , where the substitutions b/akbx/ak x . . . b,/ak ' are made into (ai, . . . . , ai.) as appropriate,

- (~ { [ A ] " b / { j x , jm} (a i~ . . . . , , b l / , b'/ak" b e D ~ } , where a k - - " �9 " ' a i h n _ m J ~ k l " " "

occurs in the Jr, .-., Jm-P laces and ihx , . . . , ihn_m Consist of it, . . . , i, with Jr , . . . , Jm removed, _ ~b,/a . b H ak~[ ' given that x k occurs free in the - [(VXk) A (ai~, . . . . . ai~._.., k~ "

126 R. T. Brady

Jl, .- . , jm-places of A. Note that we can re-insert those ,bj/ak S in which k = kj since such substitution is now vacuous.

Continuing with the completeness proof, as a result of (*), we have: Ic (A ' (x i l . . . . . Xi,,) ) = M'(ail , . . . , ai,)l, where xi,, . . . , xin are the free variable occurrences of A'.

In order to show that I~ (A') ~ T~, it suffices then to show that A' (ai, , . . . , ai,) is a non-theorem of BBQ', given that A' (x~l, . . . , xf.) is a non-theorem of BBQ. By reductio, let I--Bso, A' (ail , . . . , ai. ). Replace all distinct individual constants in the proof by distinct individual variables new to the proof. Hence, I-sBe A'(zl, . . . , z,), for such variables zl, . . . , z,, since there are no individual constants in the proof. Then by R6, A6, and R1, t--~BQA'(x h . . . . ,xi . ) , contradicting our assumption.

So, I~ (A')r T~, A' is not valid in ~/~ and A' is invalid in the BBQ content semantics. Thus, the Completeness Theorem follows..

References

[1] A. R. ANDERSON and N. D. BELNAP, JR., Entailment, The Logic of Relevance and Necessity, Princeton U.P., 1975.

[2] R. T. BRADY, The simple consistency of a set theory based on the logic CSQ, Notre Dame Journal of Formal Logic, Vol. 24 (1983), pp. 431-449.

[3] R.T. BRADY, Depth relevance of some paraconsistent logics, Studia Logica, Vol. 43 (1984), pp. 63-73.

[4] R. T. BRADY, Natural deduction systems for some quantified relevant logics, Logique et Analyse, 27 Anfiee, 1984, pp. 355-377.

[5] R. T. BRADY, The non-wiviality of dialectical set theory, in G. Priest, R. Routley and J. Norman (eds.), Paraeonsistent Logic, Essays on the Inconsistent, Philosophia Verlag, forthcoming in 1989.

[6] R. T. BRADY, Universal Logic, in preparation. [7] J. M. DUNN, The Algebra of lntensional Logics, Ph.D. Thesis, Uni. of Pittsburgh, 1966. [8] K. FINE, Models.for entailment, Journal of Philosophical Logic, Vol. 3 (1974), pp. 347 372. [9] K. FINE, Incompleteness of quantified relevance logics, in R. Sylvan (ed.), Directions in

Relevant Logics, forthcoming in 1988. [10] K. FINE, Semantics for quantified relevance logic, typescript, 1985. [11] D. GABBAY, On 2nd order intuitionistie propositional calculus with full comprehension, Archly

fiir Mathematische Logik, Vol. 16 (1974), pp. 177-186. [12] P. HALMOS, Algebraic Logic, Chelsea, New York, 1962. [13] L. HENKIN, The completeness of the first-order functional calculus, in J. Hintikka (ed.), The

Philosophy of Mathematics, Oxford U.P., 1969, pp. 51-63. [14] L. HENKrN, J. D. MONK and A. TARSKI, Cylindrical Algebras, North-Holland, 1971. [15] P. S. LAYERS, Generating lntensionai Logics, M. A. Thesis, University of Adelaide, 1985. [16] R.K. MEYER and R. ROUTLEY, Algebraic analysis of Entailment I, Logique et Analyse, Vol. 15

(1972), pp. 407-428. [17] R. K. MEYER and R. ROUTLEY, Algebraic analysis of Entailment II, typescript, 1973. [18] R. K. MEYER, J. M. DUNN and H. LE]~LANC, Completeness of relevant quantification theories,

Notre Dame Journal of Formal Logic, Vol. 15 (1974), pp. 97-121. [19] G. PRIEST, Sense, Entailment and Modus Porens, Journal of Philosophical Logic, Vol. 9

(1980), pp. 415-435. [-20] H. RASIOWA and R. SIKORSKI, The Mathematics of Metamathematics, Pafistwowe Wydaw-

nictwo Naukowe, Warsaw, 1963.

A content semantics... 127

[21] R. ROUTLEY and R. K. MEYER, The semantics of entailment, in H. Leblanc (ed.), Truth, Syntax and Modality, North-Holland, 1973, pp. 199-243.

[-22] R. ROUTLEY, R. K. MEYER, V. PLUMWOOD and R. T. BRADY, Relevant Logics and their Rivals, Vot. 1, Ridgeview, California, 1982.

LA TROBE UNIVERSITY BUNDGORA, VICTORIA AUSTRALIA

Received April l, 1987

Studia Logica XLVII, 2