ROSS T. BRADY

HIERARCHICAL SEMANTICS FOR RELEVANT

LOGICS*

(i) I’RELIMIKARIES

If we are going to represent entailments in a suitable relevant logic, we will need to supply intuitive characteristics that such logics arc expcctcd to have. The Relevance Condition on lhc sentential logic L, [viz. for all formulae A and B, if k,.A -+ B then A and B share a variable] prOvidcs a weak condition ensuring that antecedents and consequcnts of entailments contain some common content. To better characterize entailments we need to strengthen or add to this con- dition to bring out further features of entaihncnt. In an earlier paper Brady [I], the concept of depth relevance was introduced and used to define the Depth Rclcvance Condition. a stronger relevance condition that picks out a certain group of weaker relevant logics from system B through to Dp> including D W and DJ. This syntactical condition has a certain intuitive appeal and the object of this paper is to develop a semantics which is based on this condition and which accentuates its intuitive features. This semantics: which we will call hierarchical semantics. will overlay the Rowley-Mcycr semantics (cf. [8]? pp. 298-302)* which is used to capture systems satisfying the Relevance Condition.

To reslatc this earlier work. wc give the concept of depth? define the Depth Relevance Condition: and prcscnt the key logics in the group from B to Dp. WC consider formulae A with conncctivcs -, &. v , and +, and inductively deline the depth of occurrcnccs of sub- formulae within ,4.

(i) The subformula A of the formula A is of depth 0 in A. (ii) If -B is a subformula occurrence of depth d in A, then this

occurrcncc of B is of depth d in A. (iii) If B & C is a subformula occurrence of depth d in fgJ then

these occurrences of B and C arc of depth d in A.

358 ROSS T. BRADY

(iv) If I3 v C is a subformula occurrcncc of depth d in ,4, then thcsc occurrences of B and C arc of depth d in A.

(v) If ZI -+ C is a subformula occurrcncc of depth d in A, then these occurrences of lI and C are of depth d + I in A. That is, the depth of an occurrence of a subformula in a formula A is the number of nested ‘+“s rcquircd to reach the subformula, starting with A. WC will introduce the expression ‘d(B, A) to represent the depth of B in A, whcrc B is a specific subformula occurrence within A.

The Depth Relevance Condition for a sentential logic L, with for- mulae constructed as above, can be stated as follows:

For all formulae A and B, if kLA + B then A and B share a variabk at the same depth in A + B, i.c. there is some variable p with occurrences in both A and B such that d(p, A) = d(p, B) for these occurrences.

The systems B - DR* are made up of combinations of axioms and rules chosen from the following:

Primitives: - , &, v , +. Axioms.

Al. A + A.

A2. A&B+A.

A3. A&B-B.

A4, (A + B) & (A + C) -+. A + B & C.

A5. A-+AvB.

A6 B+AvB.

A7. (A+C)&(B+C)+.A v B+C.

A8. /I 8~ (B v C) + (A & B) v (A & C).

A9. --A +A.

AIO. A+ -B-+.B+ -A.

All. (A + B) & (B + C) +. A + C.

Al2. Av -A.

HIERARCHICAL SEMAKTICS FOR REI~FVAXT LOGICS 359

RI. A,A + B-B.

R2. A, B e. A & B.

u3. A -+ B. C + D a B + C +, 24 + D.

R4. A-+ wB=c-B+hA.

R5. J4 = -(A --P -A).

Mmi.-id.

MRI. If A =S B then C v A 3 C v B.

Spcc~Jk sJwi~ls.

B = Al-9. Rl-4.

DW = B + .AlO - R4.

DJ = DW + All.

DK = DJ + Al2.

DR = DK -r R5.

Bd = 13 + MRl.

DWli = DW + MRI.

DJd = DJ 7 MRI.

DK’ = DK + MRl.

DRd = DR + MRl.

B is the basic system of the Routley-Meyer semantics, as dewloped in their book [g]. in Ghaptcr 4. IIW’ is motivated in Slaney [IO] and Gcntzenixd in Brady [4]. DJ” is being advocated in Brady [5] as the most suitable entailment logic. It is shown there that naive set theory based on the logic DJd is simply consistent. DK is suggested in Routley [6] as a paraconsistent logic and is used as the logical base of a non-trivial dialectical (i.e. inconsistent) set theory in Brady [3]. DR” is the same as the logic shown to be depth relevant in Brady [l].

360 ROSS -I-. BRADY

although there it is called DR and set out with disjunctive forms of primitive rules rdthcr than the meta-rule. The disjunctive systems, coded with ‘d’, have a reduced Routlcy-Meyer semantics (cf. 0rady [5] for DJd) though some of them have the same set of theorems as their non-disjunctive counterparts, without the ‘d’ (cf. Slaney [9], p. 405).

In Brady ([I], p. 65), it is suggested that the Depth Relevance Con- dition distinguishes levels of entailment, occurring within the antecc- dent A and the consequent B of entailments ,4 + B, which can be defined as follows:

If a subformula occurrence C + D is of depth d in a formula A then this occurrence of ‘-+’ is of level d +- 1 in A, as is C and D.

In a logic that satisfies the Depth Relevance Condition, the com- mon variable(s) in the antecedent ,4 and consequent B occur as parts of entailments of the same depth in A and B. It is these common variables that are key components that ensure the validity of the cntaiIment A -+ B.

It is also suggested that these levels can be semantically rcprescnted in hierarchical model structures of the sorts used in [3] and [I], and that thcsc levels constitute remnants of the hierarchies of Russell and Tarski, which are eliminated respectively by naive set theory and the corresponding solution to the semantic parddoxes (cf. RoutIey [6] and Brady [3] and [5]). This would then provide a unified semantics whose levels could bc taken to represent the levels of an entailment which is meta-linguistically construed, semantically if not syntactically.

Tdking the idea of using hierarchical semantics to provide a seman- tical representation of logics satisfying the Depth Relcvancc Con- dition, it seems natural to interpret the levels difference between an entailment A --+ B and its antecedent A or its consequent B in succes- sivc levels of a hierarchical semantics, where each level would be used to represent the various depths of subformulae occurring in a for- mula. This would then mean, in particular, that formulae would be evaluated at levels higher than or equal to their degree, the degree being the maximum depth of any subformula occurring in the for- mula. phough it is possible to give an inductive definition for the degree of a formula, this will suffice as a definition.] In order to ensure that subformulae are evaluated at a level corresponding to

IIIERARCHICAL SEMAVTICS FOR RELEVANT LOGICS 361

their depth, wc would need to place appropriate level restrictions on their valuations and in the case of subformulae of the form A + B we would use the values of A and B from one level lower than that of A -+ B. There would also need to be some level at which all formulae could be evaluated for the purposes of determining the validity of arbitrary formulae.

(ii) HIERARCHIC.AL SEMANTICS

We proceed to set out such a hierarchical semantics based on the Routlcy-Meyer semantics and prove soundness and completeness for it with respect to the systems B through to DRd. WC will also show that there seems little prospect for the extension of hierarchical semantics to other relevant logics which contain the contraction principles, (A +. A + B) +.A + B,A -+ -A + -,4or,4& (A -+ B) 4 B or which contain the hypothetical syllogism principles, A-+B+.B+C+.A-+CorA+B+,C+A-+.C-+B. Further5 we will argue that there is little prospect for various other styles of hierarchical semantics. Our hierarchical semantics will thus provide a specialized semantics which will counter the sometimes quoted objection that the Routley-Meyer semantics. by itself, embraces too many logics.

We will use natural numbers to represent the levels of the hicr- archy, with the additional level, at which all formulae can be evaluated: to bc represented by (11. ti will bc considered to be numerically greater than any natural number. We will graft the levels i onto the set-ups 0 of the Routlcy-Meyer semantics to form set-ups 0: of the (unique) level L We will follow the account of their semantics given in [8]. Chapter 4. since our main treatment will bc for unreduced modelling and we will be concerned with systems that extend the basic system B. We prefer this to their original work [7] which provides a reduced semantics for the system R, with many of the arguments being simi- lar. The 3-place relation R, used to evaluate entailments, will apply to indexed set-ups (q, hI, c~) where i andj satisfy the relation 2 = ~1) or j c i - 1’.

We also need the overall restriction that any subformula D of a formula C can only be evzdluated at levels greater than or equal to the

362 ROSS T. BRADY

degree of C minus the depth of D in C. This will require intcrpre- tations to be given only to subfornnna occurrences D of such a for- mula C, rather than the usual Routley-Meyer one of interpreting arbitrary formulae built up from atoms. This restriction is then enforced by further restricting the above levclsj at which A and B can be evaluated, as will k seen in the valuation condition for A + B given klow.

(x) NON-DISJUNCTIVE LOGICS

Let L be a non-disjunctive logic between B and DR, obtained by adding any subset of the axioms AlO- and the rule R5 to the system B. We now prcscnt our hierarchical semantics for L. An L hierarchical modal structure (L h.m.s.) consists of the 5 concepts: Tw, Oc,, K, R and *, where K is the disjoint union of the non-empty sets, K,, K,, . . . , K, . . . , and KC:,, Ow is a non-empty subset of Ku, TCo E OU, R is a 3-place relation which is a subset of the union of Ku x Ki x Ki (Vi, 0 < i < m) and Ki x K, , x K, , (Vi, 0 -C 0 < m), and * is a I-place function on K, with a* G Ki if u E K,, for each i, 0 < i < u, such that certain of the following semantic postulates hold, given the following definition:

For all i, j, and all a,, bi E K, and c,, 4 E K,,

dl. ai < b, =a @-o, E Oc,Vk,,d,.

PI. q < (4

P2. if a, < b, and Rbicjd, then Ra,c]d,.

P3. a. z a?* t .

P4. if a, < b, then b,? G a?.

P5. if Raih,cj then Ra(c,?hy. [replacing p4]

Ph. if Ruibjcj then (3xj E .Jj)(Raibjxj and Ra,x,c,).

P7. if x”, E OU then x2 < x0.

Pg. if xc0 E OU then (!IJJ~,, zcO E OCs)Rx~ywz~.’

IIIERARCHIC.41~ Sl<MANTlCS FOR RELEV.4NT LOGlCS 363

For E h.m.s.? we have the postulates pl-p4. For extensions of I?, we add p5 if axiom Al0 occurs~ p6 if axiom .411 occurs, p7 if axiom Al2 occurs and p8 if rule R5 occurs.

A vuhutiorz TJ on the L h.m.s. .%I is a function which assigns a value T or F to each sentential variable JJ at each set-up U: E Ki, for all i, subject to the following condition: For Us? /I~ E K!. for all i,

if u, < !I~ and ZI(~? 0;) = T then U( 1~. b:) = T. (Hereditary Condition)

Whilst valuations are defined generally so that they can apply to any formula, their cxtcnsions to interpretations are limited to subformula occurrences of a given formula, which would normally be the formula for which validity is being dctcrmincd, but this formula could be chosen for other semantic information.

For any formula C, each valuation TJ on M can bc uniquely extended to an interprctatiot~ Z on M. for all subformulae occurrences 4 of C under the restriction that 4 is only evaluated at levels i greater than or equal to the degree of C minus the depth of ,4 in C (i.e. 01 > r’ 2 deg(C) - &I, C)):

For any i, for any a, E KJ.

W I(p. a) = c(p. u,).

(ii) Z(-,4, u,) = Tifff(,4, ~7) = F.

(iii) Z(A & B$ u!) = T iff 1(,4? u;) = T and f(& u,) = T.

(iv) Z(A v fl, u,) = Tiff 1(.4, 0,) = T or T(B. u,) = T.

~~~~~~ f(‘4 + B, lZ,$) = T iff: for all i such that i 2 d&C) - d(,4 + By C) - I, for all b,, ci E K!. if Ra,:,h!c, and 1(,4. b,) = T then f(II, c,) = 7’.

tvJ(bl For 0 < i < CJ, f(,d -+ fi, 0, , ,) = T iff, for aIt l7;. cS E K,? if Ra:+,b,c, and Z(A, bi) = T then f(B. cJ = T.

Note that in (v)(a) the condition ;j > de&C) - 0’(,4 + Bz C) - 1’ can be replaced by :i > deg(C) - c/(4, C)’ or yj > deg(C) - d(B. C).. In this casc.j 2 max(dcg(,4)z dcg(B)) follows. [Note that ‘deg. abbreviates ‘degree’ and ‘max’ abbreviates maximum..]

364 ROSS T. BRADY

The following lemrrm will provide some uniformity to the valuation procedure by ensuring that a formula, whichever formula it is a sub- formula of, will always take the same value in a particular set-up q, provided that i is firme and satisfies the appropriate restrictions for each such evaluation. There is also a more specialized result for Us.

CONSTANCY LEMMA. Let the formula A be u .&formula occurr- ence in formulae C, and Cl. This includes the case where A is either C, or C1.

(i) Le/ a< G K,, where co > i > max(dcg(C,) - d(A, C,), deg(C1) - d(A, Cl)). Then I(A, ai) in the evaluation of C, takes the same vaiue as I(A, a;) in the evaluation of Cz. This still applies q’ C, and CT are the same formula and A occurs at dtJterent depths within it.

(ii) Let a(” E KU. Then I(A, aa) in the evaluation of C, takes the same value as I(A, au) in the evaluation of C1, provided &>g(C,) - d(A, C,) = deg(Cz) - d(A, Cz).

Proof. Proof is by induction on the valuation procedure. For for- mulae of the form A + B in (ii), we need the induction hypothesis for A and I3, from both (i) and (ii).

A formula A is valid in the L h.m.s. A4 iff Z(A, T(,>) = T for all interpretations I on A4, and A is invalid in M otherwise. A is valid in the hierarchica/ L semantics (L-h-valid) iff A is valid in all L h.m.s.

HEREDITARY LEMMA. For all interpretations I un an L h.m.s., for all i 2 deg(C) - d(A, C), for all ai, bi E Ki, ifai < bi and I(A, ai) = T then Z(A, b;) = T.

Proof As in [7l, p 208 or [8], p 302, with level indices and restrictions.

ENTAILMENT LEMMA. (1) For ah interpretations I on a L h.m.s., $ for all i 2 deg(C) - d(A + B, C) - l,.fbr all a; E K,, tfI(A, ai) = T then I(B, ai) = T, then I(A -+ B, T,“) = T.

(2) for al1 L h.m.s. M, $ jar all interpretations I on M, for ail i 2 deg(C) - d(A + B, C) - I, for all ai E K,, tfI(A, ai) = T then l(B, ai) = T, then A -+ B is v&id in M.

(3) (For all L h.m..s. M, jar all interpretations I on M, for all i 2 a’eg(C) - d(A + B, C) - 1, for ail a, E KiT tfZ(A, a;) = T then I(B, a) = T) $I A -+ B is L-h-valid.

HlER.ARCHlC~4L SEMs4NTlCS FOR RELEVANT LOGICS 365

J%xI$ As in [S], pp 302-3, with lcvcl indices and restrictions.

SOUNDNESS THEOREM. For aii formulae A> if A is a theorem qf L then A is L-h-valid.

Proof. Each axiom-scheme and theorem-scheme is evaluated as a whole formula and its subformulae are evaluated as such. When evaluating such schemes of the form A .+ B at 7’,9z due to the Entail- ment Lemma. WC just cheek that it preserves truth at q. for all i such that 0~ > i 2 deg(A 4 B) - 1. We proceed as in [8], pp 303-5, with level indices and restrictions. Use is made of semantic postulates p 5-8 when checking axioms AlO- and rule R5, respectively. Note that in checking R3 we need the following: dcg(A 4 B) - 1 = max(deg(A), deg(B)) and dcg(B -+ c -+. A + II) - 2 = max(deg(A). deg(B), deg(CA degCD)h

COMPLETENESS THEOREM. For ull,formuhze A, i/A ix L-h-v&i then A is u theorem oj. L.

Proof. We follow the proof given in [8], Chapter 4, but we place an index i (0 < i < w) on sets Si of formulae in such a way that the degree restriction dcg(A) < i < cq holds. Appropriate indexing and corresponding degree restrictions are used throughout the completeness proof. We indicate in what follows the key points in the proof given in [8] and show? where the indexing and degre restrictions apply.

We introduce a canonical model structure (T,.,<,,. OL,(,,, U, KL.{. RL. *), with the following definitions of these semantic primitives. Wc dctinc KL,,O as the set of all prime L(,,-theories with each theory indexed by o), and K,,,)(O < i < ~0) as the set of all prime Li-theories composed only of formulae of degree < i with each such theory indexed by i. Here, an indexed set lz, of formulae is an L:-theory iff: for all formulae A, B of dcgrcc < i. if F,.A -+ B and A E a: then B G l;,, and if A E ai and B G u, then A & B E q. An L!-theory u, is prime iff. for all for- mulaeA;Bofdegree <i:ifA v Bea,thenAEu,orBeu!.

O,,,(,, is the set of all prime regular Lc,-theories, where an Lc9-theory is regzkzr iff it contains all the theorems of L. TL.u is an arbitrary element of Or,cs.

TL,C, and O,,.cI have no degree restriction but R,, and 07 have dcgrcc restrictions as follows:

366 ROSS 7.. BRADY

For any pair i, j such that i = CO or j = i - 1 holds, &a,hlcj is defined as, for all formulae A and B of degree Gj, if A + B E ai and A E bj then B E cj. Clearly deg(A -+ B) < i. Note that only the degree restrictions apply and that there is no mention of depth of subfor- mulac here.

a: = [,4 : deg(,4) < i and m A $ a,}. & is the set of all L2-theories ai composed of formulae of degree

< i. The notions of L-derivable and L-maximal are both defined for for-

mulae of degree < i. We use notation S for a set of formulae indexed by i such that the degrees of all its formulae are less than or equal to

A set T of formulae is L,-derivabfe from a set Si of formulae iff, for some A,, . . . , & in Si and some B,, . . . , B” in 7;, kLA, & . . . &

472 + B, v . . . v B,,. Note that the notion of L-derivability is based on provability of

formulae of the form ,4 -+ LI in L. The tact that such proofs of A -+ B may contain formulae of degree > i does not affect our argument.

A pair (S, T.) of sets of formulae is Li-maximal iff 7; is not Li- derivable from S, and Si u 7, is the set of all formulae of degree Gi.

It then follows that Si is a prime L,-theory. The Extension Lemma is stated as follows:

Let the sets S; and c be such thut T, is not L,-derivable from S,. Then there is un Li-muximaf pair <S,‘, T/ ) sucf~ thut S, c S,’ and T G T{. Thus, S; is a prime Li-theory.

In the proof we start by cnumemting the set of all formulae of degree <i, and inductively dctermincd from this the Li-maximal pzdir (S,‘, 7:.‘) of sets of formulae.

lk Priming Lemma is as follows:

k>t Y& f>e an L,-theory und Vi u set of jiiwmulue (oj’dqree < i) di.voint from T, und closed under distinct disjunction, i.e. for distinct A und B, if A E q und B E U, then A v B E Ui. Then there is a prime L,-theory r such thut T, G T,’ and T: is dibjoint,from U,‘.

lllER~4RCHICAL SEMANTICS FOR RELEVANT LOGICS 347

The following Corollaries to the Priming Lemma are required for the range of logics we are dealing with:

COROLLARY 1. [j’A is a non-theorem of L then there is a prime regular LO>-theory TL, swh thut A $ TO,.

COROLLARY 2. For a: E I? ,.,,, bl E l?I,,j und q, G KL,j, lj” l?,.alb,c.i then. ,for some xi G K[,,;! b, s .v, and l&a,x,ci. [I& is RI, applied to L-theories, possibly degree-restricted. which are not necessarily prime.]

COROLLARY 3. For uj E XL,,. b, E I?[.,, and cj E KL,j. [f &a: bjcJ then, for some xi E KL,, ~ a) s xi and &xi b, cl .

COROLLARY 5. For a: EZ?~,,$ b, E & und c, E I&], if& a) b,ci and C # cl then, for some xj E KL,] , &a,bjxj and C $ xj.

The indexing and degree-restriction cxtcnds to the proofs of these cor- ollaries. e.g. in the proof of Corollary 2? we define L$ as {A : deg(A) < j and (gB)(deg(B) < j, A + B E a! and B $ cj)j. Corollary 7. which is additional to those given in [8], p 309, is needed for p8 and can be found in [2]. p 239. In its proof, we put L; as {B: deg(B) < j and (3A)(dcg(A) < .j> A + h B E a, and A E b;);.

In checking that the canonical model structure satisfies the semantic postulates, we make use of the above corollaries. Degree restrictions already occur in the definitions of RLu,b,c, and CI: < b, and are just , reused as appropriatq but WC need to introduce them in existential constructions? e.g. for the construction of + for the postulate p6. There, define fj as {C: deg(C) < j and (%?)(deg(B) < j, B --+ C E q and B E bj)j and then use Corollary 2 to extend it to x,;. For ~8~ WC apply Corollaries 7 and 2 to extend the set of theorems of L.

The canonical valuation q. satisfies: z~~(p, a;) = T iITp E ai.

368 ROSS ‘I-. BRADY

To prove that this extends to all formulae, we need to show induc- tively that Z(L), u,) = Tiff D E q. We look in particular at the case of A -+ B, where i > deg(C) -- d(A + B, C).

(a) Let A -+ B E aw. To prove I(A + B, q,)) = T, WC let i 2 deg(C) - d(A -+ B, C) - 1, R,,awbic, and A E b,. BE c, then follows from the definition of RLa,;>bici and max(deg(A), deg(B)) < dcg(C) - d(A + B: C) - 1.

(b) Let A -+ 8 G a,+,. To prove Z(A + B, a,+,) = T, WC let RLa, , , b,lsi and A E b,. B E ci then follows.

(a) For the converse case, let A + ll +! a,,,. To prove Z(A -+ B, (I”,) = F, wc let i = deg(C) - d(A + Bz C) - 1. 5) and F, arc then constructed as in [8], with degree restrictions, as follows:

6, = {D: 1, A -+ D and deg(D) < i}

P, = {D: dcg(D) < i and (SlE)(deg(E) < i,

E + D E a”, and ,F E !I,)}.

bi and 2, are then Li-theories satisfying R,.a&Zi, B E fii and C $ ti, and are extended, using Corollary 6 to the Priming Lemma, to hi and c,, respectively.

(b) For A -+ B I$ uz .-, , we prove l(A + B, ui 4 ,) = Fin a similar way.

We arc then in a position to show that the non-theorem A of L, mentioned in Corollary 1 to the Priming Lemma, is f&c under the canonical interpretation I, evaluated at ‘f:,,, i.e. l(A, Tc,) = F. Thus A is invalid in the hierarchical semantics.

(/j) DISJUNCTIVE LOGICS

WC now proceed to present the hierarchical semantics for the disjunc- tive logics L between Bd and DRd, obtained by adding any subset of the axioms AlO- and the rule R5 to the system Bd. These systems have reduced semantics, with the following simplifications to the above semantics:

q < bi is defined as RT”,u,b,, for all i.

In the Entaihnent Lemma, the converses of (1) and (2) are provable.

For soundness, we need only check M R 1. Wc prove that if A * B then, if 1(A$ c,,) = T then /(B, TU) = Tz by induction on the proof procedure for .4 * B in L> using the converse of (1) in the Entailment Lemma.

For complctcncss we use D-L(,)-theories (c.f. the JL-theories in [S]. pp 337-8) to obtain the base theory, TL,<,, and then we use it to dcfinc T-Li-thcorics (cf. [8]. p. 306)? for all the non-regular theories. A more complete account of this completeness proof for the reduced seman- tics will appear in [5]. We proceed to set out the concepts of D-L(,,- theory? D-LP2-derivability and D-Lo)-maximality to clarify them for our purpose.

D-LU-f/rwr& q>, satisfy the following conditions:

(9 If 1~4 -+ B and ‘4 l a(,: then B 6 a(,,.

(ii) If A E (I~, and 11 G Us, then A & B E au.

(iii) If A + B E u,,; and A E ucX then 19 E qti.

(iv) If A -+ B E atd and C -+ D E u‘, then

B + c -+. A + IlEa,>.

CV) If A + - B e u* then B -+ - 4 g q,,. [if not redundant]

(vi) If A E q, then - (A + - .4) G q,,. [if needed]

(vii) If ‘4 + B and C v .A E agO then C v B E q>.

We deline the related concepts of derivability and maximality as follows:

A ,jiuwdu 11 is l~-I~c,-derivabie frotn ~t,fiwm~lu A> written A a,, B. iff B is derivable from A by successive application of the following rules:

(11 ,4 G- B, where -LA + B

09 A, B =s A & B.

(1~~1 A> A -+ B =s B.

370 ROSS T. BRADY

w A+B,C+D=PB+C-+.A--+D.

09 A--B-B+ -A. [if not redundant]

WV A*m(A+ -A). [if needed]

vu C v A =+ C v B, where A =s B.

A set bu of form&w is D-I+,-derivable from a set aTz, of fhmulae, written au =+ bcO, iff, for some A,, . . . , An, c aw and some B,, . . . , Bn E bc3, A, & . . . & Am =+ B, v . . v B,l.

A pair of s& of formulae (a”, bw) is D-Lw-muximal iff (1) bw is not D-Lu-derivable from acO, and (2) acOUbm = the set of all formulae.

The Extension and Priming Lemmas can then be proved with thcsc concepts replacing the usual ones. By Corollary 1 to the Priming Lemma, we establish the regular D-Lu-theory Tr,,, which is then used to defmc T-Lj-theories ai thus.

A T-L&zcor~ ai is a set of formulae with degree < i satisfying:

61 If A -+ B G T,,.,, and A E ai then B E a,.

(ii) IfAEu,and Beqthen A&BEu;.

These T-L,-theories form the set-ups of the canonical model structure, at the various levels. In the remainder of the proof, the important point to note is that TL.*, contains all the theorems of L, is closed under the rules of L and is a T-Lw-theory.

(iii) PROSPECTS FOR EXTENSIONS TO TllE LOGICS

IIaving shown soundness and completeness for systems in the range B - DR’ with respect to hierarchical semantics, it is worth seeing if this can be extended to other logics. We will examine some axioms in turn that may be used to extend DR’.

To facilitate this discussion, we introduce a relation Zij which regulates the vahrations of entailments A -+ B at ui by providing the levels j at which A and B arc to be evaluated. In our hierarchical semantics Iij can be defined as ‘i = m orj = i - I’ which then allows the valuation condition for A -+ B to become:

I(A -+ B, al) = Tiff, for allj such that Tij and j 2 dcg(C) - d(A + B, C) - I, for all bj, cj IZ Kj, if Raibjc, and /(A, b,) = T then I(B, q) = T.

HIERARCHICAL SEMANTICS FOR RELEVAXT LOGICS 371

The problem with the two forms of hypothetical syllogism, ,4 + B +.B-+C+.A+CandA+B+.C -+A +.C+ B,isthedif- fcrence in depth of the occurrences of A and B in the antecedents as compared to those in the consequents. In the soundness argument we evaluate the two formulae at the level U, and hence the antecedent A’s and B’s are evaluated down to level d-2 whilst the consequent A-s and B’s are evaluated down to level d-3: where d is the degree of the axioms.

The problem with the three forms of contraction (.4 -+. A + B) +.A + B,A + -A + - A and A & (4 + B) -+ B. lies essentially in the completeness argument. In order to salvage them for sound- ness. one needs to allow the relation I? to apply to triples (ui, bj, c,), where i = j, i.c, where fii would hold. I<j would become .i = w or, I’ # w and j = i or i - I’, In the completcncss argument. WC would need to detme R,.ufhic! as (V4, B of degree <i - 1) (if A -+ B E a, and A G b, then B G c!), with the degree restriction on A and B to enable deg(A -+ B) < i. A clash develops between the dcgrcc restric- tions on L-theories and those applying to RLqhiq. The probkm arises when a prime theory hi needs to be established by use of the Extension Lemma from a theory 2: , ~ obtained such as to satisfy an RL relation. Such a theory .?,-, cannot be shov,n to bc a theory of level i, because of the differing ranges of quantification over formulae.

The axiom. A +. A + B + B, cannot be captured in the hierarchi- cal semantics because its semantic postulate, if Rabc then Rhc, can- not be consistently indexed, though there are also problems here with degree restrictions. Similarly: the semantic postulate, (!Ix E 0)Ruxa. for the rule> A * A -P B + BF cannot be consistently indcxcd.

There are also problems with trying to vary the form of the hier- archical semantics. Firstly, let us consider dropping the depth rcquire- ments in the valuation procedure and using just the restrictions pertaining to the dcgrcc of the formula that one is evaluating. This would mean that there is no need to specify the whole formula of which a particular subformula, that one is evaluating, is a part. The valuation condition for A -+ B would then read:

For i > deg(A -+ B). [(A -+ B, a,) = Tiff, for all j such that Zij and ,j 2 deg(A + B) - 1, for all bj, cj E K,. if Ra, b.(c, and Z(A, bj) = Y-then I(B, c,,) = T. Note that deg(A -+ B) - 1 = max(deg(A), deg(B)).

372 ROSS T. BRADY

The problem in this case lies with the soundness argument for R3: A+B,C+D=-B-+C-+.A + D. B and C could have a high dcgrce in relation to A and D. WC riced to prove that I(A + D, ai) = ‘r, given f(B + C, ui) = T, for any i > max(dcg(B + C), deg(A -+ D)). So, WC let Bj,j 2 max(deg(A), deg(D)), bj, cj e Kj, Ru$,c, and [(A, hj) = T, with the aim of proving Z(D, t;) = T. If we put i = (0, A and D would have to bc evaluated at the level, max(deg(A), deg(D)), in contrast to the lcvcls, max(deg(A), deg(B)) and max(deg(C), deg(D)), available from the premises of the rule. lf rmx(deg(A), deg(B)) and max(dcg(C), deg(D)) are higher than max(dcg(A), deg(D)), we will not be able to USC Z(A, hj) = 7’ to obtain I(B, e,) = T, f(C, cj) = Y’, and hence l(D, cj) = T.

Secondly, WC consider hierarchical semantics with the removal of degree restrictions. This would leave Iij as the sole control over lcvcls in the evaluation of entailments. A + B. Ry varying the relation Bj (e.g. putting it as ‘i = co orj < i -c 01’ or as ‘i = co orj < i -c m’), WC can obtain soundness results for systems as strong as T, but the completencs argument would break down due to the lack of suitable subsets of formulae to rcprcscnt the various levels. The point here is that closure under -, & and v for the subsets of formulae occurs in a number of places in the completeness argument and so the degrees seem to provide the obvious way of delineating the lcvcls. Further, if WC dispense with the idea of using subsets of formulae to represent lcvcls, it is not clear how the lcvcls would be defined nor, if they wcrc defined, how the Extension Lemma would enable extensions to be contained within a given lcvcl.

One peculiar feature of the current approach is that the soundness argument for some of the rules cannot be directly carried through. In particular, if one tries to prove that A + B, B + C =+. A -+ C prc- serves validity by assuming the validity of A + B and B -+ C and using the Entailment Lemma, the argument fails in the case where B has a higher degree than A or C. [Even though it is not common for B to have a higher degree than A and C, B’s degree can be artificially rdised due to the equivalence, 1 B ++ B & (B v D).] The point is that the preservation of truth from A to B and from B to C will be estab- lished for set-ups of level i > max(deg(A), deg(B)) and i 2 max(deg(B), deg(C)), respectively, whilst WC need to show that truth

is prcscrwd from A to C for set-ups of level i 2 max(dcg(A). de&C)).

I-Iowwcr, WC have already proved that the rule prcscrvcs validity and Lhis provides an indirect method of proof. sincq by RX 11 + lI +. d4 -+ C is valid and, by RI. A + C’ is valid. The Entailment Lemma is still used for R3, but the validity of .4 + C is cstablishcd by RI at the w level. Another indirect method of proof for the rule is to use All. (A + B) & (I3 -+ C) +. ,I + C; R2 and RI, for systems with this axiom.

In conclusion. we have motivated hierarchical semantics as a semantical rendition of the Depth RelelJance Condition and WC arc now in a position to see the close relationship that exists bctwcn thcsc two. We have seen that the hierarchical semantics does not appear to bc modifiable in ways that remove the depth considerations from it. Also. the logics that arc depth rclcvant very closely corre- spond to those that have hierarchical scmamics and it does not appear to be possible to cxk!nd the semantics to familiar systems that arc not depth relevant nor is it possible 10 extend depth rclwancc to sy-stems that do not seem to have a hierarchical semantics (cf. [I]. p 72).

* This paper was prescntcd to the 1990 meetins of the .4ust1~lasian Associatiw for Logic held at the Lnivcrsity of Sydney. and I thank those present for helpful commcn~s made. ’ I ha\:e rwli& sinw the pul&xtion of 181 that this more finely tuned semantic pas- tulatc suffices for R5. in cuntradiction to my statement on p 346. in fw~notc 2. Sound- ness and complctencss arguments for it are indicated in the current paper.

374 ROSS T. BRADY

[S] R. T. Brady. “Universal Logic”, in preparation. [6] R. Routlcy, *‘Ultralogic as Universal?“. Relevunce t.ogic Ntwslc~~cr, Vol. 2 (1977).

pp. 50-X9 and 13% 175, and also in “Exploring Meinong’s Jungle and Beyond”, A.N.U., 19gO.

[7] R. Routley and R. K. Meyer, “The Semantics of Entailment I”, in Tru/!r, Synfux und Mod&y, cd. by 1.1. Leblanc, North Holland, 1972, pp. 199-243.

[g] R. Routlcy, R. K. Meyer, V. Plumwood and R. T. Brady, Rekvani L.ugics und Their RiwLs, Vol. 1, Ridgcview, 1982.

[9] J. K. Slancy, “Reduced Models for Relevant Logics Without WI”, No& Kane Journal of Formal Logic 2S (1987). pp. 395S407.

[IO] J, K. Slaney, “General Logic”, Auwulasian Journal oj Philosophy 68 (l990), pp. 7488.

La Trobe University, Bundooru, Victor& Austrulia.