KOSTA DOSEK

MODAL TRANSLATIONS

IN SUBSTRUCTURAL LOGICS

ABSTRACT. Substructural logics are logics obtained from a sequent formulation of intuitionistic or classical logic by rejecting some structural rules. The substructural logics considered here are linear logic, relevant logic and BCK logic, It is proved that first-order variants of these logics with an intuitionistic negation cm be embedded by modal translations into M-type extensions of these logics with a classical, involutive, negation. Related embeddings via translations like the double-negation translation are also considered. Embeddings into analogues of S4 are obtained with the help of cut elimination for sequent formulations of our substructural logics and their modal exten- sions. These results are proved for systems with equality too.

We shall show that the embeddings via modal and double-negation translations between systems based on Heyting’s and classical logic have analogues among embeddings between systems based on logics weaker than Heyting’s and classical. These weaker logics are obtained by rejecting some structural rules in a sequent formulation of Heyt- ing’s or classical logic, whereas the rules for logical constants are kept unchanged. We call such logics substructural, which may convey that something has been subtracted in the structure, but should also con- vey, according to the standard meaning of the word “substructural”, that these logics are in the foundations, in the supporting structures, of Heyting’s and classical logic.

We shall consider three of these substructural logics, displayed in the following diagram:

BCW BCK (lacks Thinning) , / (lacks Contraction)

BC (lacks Thinning and Contraction)

The names of these logics are derived from the corresponding combi- nators. The logic BCW is a variant of r&vunf logic, and BC is related to hear logic; the logic BCK is already known by this

Journal of PhiIosophical Magic 21: 283-336, 1992. ICI I992 Kluwr Acudemic Publishers. Printed in the Xe~herlands

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name. Some historical remarks and references on these substructural logics will be given in Section 1 below (a survey with additional refer- ences may be found in Ono 1990, Dosen 1988 and 1990).

By sheer volume, the literature on relevant logic overwhelms the literature on all the others, but this literature does not seem to be very widely read. This is why a number of things have been redis- covered when linear logic, and to a lesser degree BCK logic have attracted attention in the last few years. One of the reasons that have prevented easy access to relevant logic may be the failure of its main proponents to introduce this logic in a standard Gentzen format. Instead, a Fitch-style natural deduction technique was preferred, whereas sequent systems suffered because the leading experts on rel- evant logic insisted on having distribution of conjunction over dis- junction in their systems. If one wants to consider variants of relevant logic that lack distribution, as we do here, then one obtains sequent systems for them by a simple rejection of Thinning (as in Kripke 1959, where rejecting Contraction is also envisaged). This makes clear the leading idea of relevant logic. The failure of decidability and inter- polation in distributive propositional relevant logics, which was proved with much ingenuity by Urquhart (1984 and 1993) seems to be mainly due to the presence of distribution (but, according to Brady 1990, we should also blame Contraction). Apart from philosophical reasons, the best justification for having distribution in relevant logic is probably the elegance of the “first degree entailments” of Anderson and Belnap (1975, chapter III). However, sometimes one has the feel- ing that Anderson, Belnap and their pupils refuse to consider relevant logic without distribution because they think that would make life too easy (cf. Dunn 1986, 4 3.9, p. 174).

A fourth substructural logic has attracted attention not so much in logic proper, but in the related areas of mathematical linguistics and category theory (see Lambek 1988 and Szabo 1978). This logic could be called Lumbek’s logic, because its essential part is the Lambek calculus of syntactic categories of (1958). In our diagram, Lambek’s logic would be below BC, since, in addition to Thinning and Contraction, it lacks the structural rule of Permutation (also called Interchange). Lambek’s logic has an associative variant and a

MODAL TRANSLATIONS IX SLBSTRUCTURAL LOGICS 285

nonassociative variant, whose names in the notation with combina- tors, would be B and I respectively (actually, the combinator I could be included in the names of all our substructural logics). Nonassocia- tive Lambek’s logic I, which stems from (l961), is the minimal sub- structural logic, and our complete diagram would look as follows:

Bcw\ / BcK BC

In this paper, we focus our attention just on BC, BCW and BCK, and disregard the logics below BC. Our reason for doing that is that below BC we have problems with negation that is like classical negation, as will be explained at the end of Section 4.

Lath of our substructural logics comes in a version with an intuitionistic negation and in a version with a negation that is like classical negation, because it is involutive and satisfies De Morgan’s laws. Heyting’s logic, which we call H, yields classical logic with classical negation. The relation between BC with intuitionistic negation and BC with classical negatibn is analogous to the relation between H and classical logic. The same holds when BC is replaced by BCW or BCK. We will show here that this analogy is kept in particular when we consider embeddings via modal translations and double-negation translations (as we will see below in Section 2, the latter translations may be considered a special case of modal translations).

In Section 1, we introduce Hilbert systems for our substructural first-order predicate logics without equality.

In Section 2, we introduce modal extensions of these Hilbert systems and we consider embeddings via modal translations where the nonmodal logic embedded is not weaker than the underlying non- modal logic of the modal system. Namely, if the first logic has an intuitionistic negation, the second logic has an intuitionistic negation,

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whereas, if the first logic has a classical negation, the second logic may have an intuitionistic or classical negation. This type of embed- ding is rather easy to prove.

A type of embedding more difficult to prove is when the embedded logic has an intuitionistic negation and the underlying nonmodal logic has a classical negation. Let us call such embeddings provision- ally dificuff embeddings. A difficult embedding is the very well known embedding of H into S4. The analogous embedding of a variant of BCW with intuitionistic negation into an S4-type extension of BCW with classical negation was foreshadowed already by Orlov in (1928) (see Do&en 1992). We will also deal with embeddings that resemble the embedding of H into linear logic considered by Girard in (1987). However, this last embedding is not exactly of the same type as the difficult embeddings that are our main concern here.

In Sections 3 and 4, we develop proof-theoretical tools that we will use to establish our difficult embeddings. In Section 3, we formulate Gentzen systems, i.e. sequent systems, for our logics and prove cut elimination. Sequents in our Gentzen systems will be based on multi- sets of formulae. In Section 4, we establish the equivalence between the Hilbert and Gentzen formulation of our logics. At the end of this section, we explain why in this paper we don’t deal with Lambek’s logic.

With our proof-theoretical tools, we establish in Section 5 our dif- ficult embeddings of intuitionistic versions of BC, BCW and BCK into %&type extensions of classical versions of BC, BCW and BCK.

In Section 6, we extend our difficult embeddings to first-order variants of BC, BCW and BCK with equality. The nondifficult embeddings with equality may trivially be obtained for all our logics, but the difficult ones require a special adaptation of our proof- theoretical tools. Namely, with equality, our Gentzen systems will have a restricted form of Contraction involving equality even for logics that otherwise lack Contraction. We prove cut elimination for these Gentzen systems, which will enable us to get the desired difficult embeddings.

In a brief concluding section, we make a few remarks on trans- lations of proofs from hypotheses.

MODAL TRANSLATIONS IN SUBSTRUCTUR.4L LOGICS 287

I. HILBERT SYSTEMS

Let L be a first-order language with the following logical vocabulary:

binary connectives: +, l , A , v

wary connective: 7

@t-order quant$ers: Vx, 3x.

The connective l is called~i~o~, as in relevant logic. We assume L has infinitely many individual variables. We don’t have different sym- bols for free and bound variables. We use the following schematic letters:

individual variables: x, y, z, . . ,

indivihal terms: a, b, c, . . .

formulae: A, B, C, . . .

possibly with subscripts or superscripts. A schema of the form A; will stand for the formula obtained from A by substituting the term a for every free occurrence of x, provided we have satisfied the usual proviso for substitution that prevents free variables being captured by quantifiers; i.e., a is free for x in A. As usual, A ++ B is defined as (A + B) A (B + A).

Our basic Hilbert system in L, called BC, has the following axiom- schemata and primitive rules:

(1) A-A

@9 (A + B) 4 ((C -+ A) + (C + 23))

w (A + (B + C)) + (B + (A + C))

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

( 0 intr) A + (B + (A l B))

(Aelim) (AAB)+A, (AAB)+B

(A intr) ((C-A) A (CdB))+(C+(A A B))

(v elim) ((A 4 C) A (B -+ C)) * ((A v B) + C)

(v intr) A + (A v B), B + (A v B)

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OwJ (A -+lB)-P (B-+1/I)

(V elim) VXA + Ai

(V intr) Vx(C + A) + (C + VxA), provided x is not free in C

(!I elim) Vx(A -P C) + (3x,4 + C), provided x is not free in C

(Zl intr) A: + 3xA

(modus pmmzs) A A-B

B

A B (udjuncrion) -

AAB

A (generalization) -

VxA

SomeGmes, ( l elim) is called imporrarion, and its converse, which can replace ( l intr), is then called exportation (importation and ex- portation are related respectively to modus ponens and the deduction theorem). The converse of (v elim) can replace the axiom-schemata (v intr), and if we assume the rules:

AAB AAB ~ ~ A B

the converse of (A intr) can replace the axiom-schemata (A elim). Similarly, if we assume the rule:

VxA

4

the converse of (V intr) can replace (V eiim) and the converse of (El eiim) can replace (3 ink). The axiom-schema (neg) can be replaced by the following two schemata:

(A + B) + (1B + 1A)

(cf. Dogen 1988 $2.10, pp. 382-383).

MODAL TRAVSLATIONS IN SLBSTRUCTURAL LOGICS 289

When the axiomatization of BC is extended with the axiom-schema:

(39 (A + (A + B)) + (A 4 B)

we obtain the system BCW, and when the axiomatizdtion of BC is extended with the axiom-schemata:

W-J A * (B + A)

WRl 1‘4 + (x4 -+ B)

we obtain the system BCK. The system where (W), (KL) and (KR) are all added to the axiomatization of BC is the Heyting first-order predicate calculus, which we denote by H.

It is easily demonstrated by induction on the complexity of C that all our systems are closed under replacement of equivalents:

A-B

C - C[A/B]

where C[A/B] is obtained from C by replacing zero or more occur- rences of A by B. In general, all the Hilbert systems of this paper will be closed under replacement of equivalents. Since in H we can prove (A l B) * (PI A B), this means that in this system the connectives l

and A are synonymous; i.e., wc can always, even nonuniformlyF replace one by the other so that provability is preserved. In BCW. we can prove (A A B) -+ (A l B), but not the converse, whereas, in BCK, we can prove (A l B) + (A A B), but not the converse.

The system BC is related to hear logic (see Girard 1987 and Avron 1988; cf. Szabo 1978$ chapter 8. Meyer and McRobbie 1982, Slaney 1984, Giambrone 1985> Komori 1986, Giambrone and Kron 1987, Bull 1987 and Brady 1990) whereas BCW is a variant of the relevant logic R (see Anderson and Belnap 1975 and Dunn 1986). Both BC and BCW have an intuitionistic minimal negation. We may also call this negation purely implicarionul, because the schema (neg) is an instance of (C) when 1A is defined as A + 1, where 1 is an arbitrary propositional letter. The system BCW differs from the usual version of R in one more important respect. In BCW. we cannot prove the distribution of conjunction over disjunction; i.e., we lack:

(distr) (A A (B v C)) + ((A A B) v (A A c)).

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However, the implicational fragment of R, which is, to use a term of Anderson and Belnap (1975, $1, p. 3), the “heart.’ of this logic, is identical with the implicational fragment of BCW. Relatives without (distr) of BCW have been considered by Meyer (1966; see also Dunn 1986, $3.9, p. 174, Smimov 1972, chapter 6, Anderson and Belnap 197.5, $25.1, p. 299, Bo% 1983, McRobbie and Belnap 1984, Thistle- Waite er uZ. 1988, Dosen 1981 and 1985). In Ono and Komori (1985) one finds mainly relatives of BCK. Relatives of all our systems are also surveyed in Do&en (1988 and 1990).

From now on, X will stand for BC, BCW, BCK or H. When the axiomatization of X is extended with the axiom-schema:

(double neg) TTA+A

we obtain the system X, the subscript c standing for “classical’*. Of course, Hc is the classical first-order predicate calculus. The systems BCWc and BCc are closer to the systems originally considered by Anderson and Belnap (1975) and Girard (1987) since these systems also have a negation that satisfies (double neg). The implication- negation fragment of R, which was first axiomatized by Orlov in 1928 (see Dosen 1992), is identical with the implication-negation fragment of BCWc. However, BCWc, as well as BCc and BC&, still lacks (distr). The logic BCKc and its algebraic semantics were investigated by Grishin (1974 and 1981). A system with the same inspiration as BCK has been considered previously by Fitch (1936). Something similar may also be found in the many-valued logics of Lukasiewicz. There is a considerable literature on algebras related to BCK logic (see references in Ono and Komori 1985 and Dosen 1988; in Palasin- ski and Wrotiski 1986 one may also find references about the related implicational calculus).

2. MODAL TRANSLATIONS

Let now LIJ be the first-order language obtained from L by adding the unary connective 0 to the logical vocabulary of L. The system XD4 is the system in LCI obtained by extending the axiomatization of X with the following axiom-schemata and primitive rule:

CW q l(A + I?) 4 (CIA + q IB)

(04) ElA+LlClA

MODAL TRANSL.4TIONS IN SUBSTRLCTUR.4L LOGICS 291

A (necemirdon) -

CIA

The system Xtriv is the system in LIJ obtained by extending the axiomatization of X with the axiom-schema:

(&iv) ElA++A.

We say that a system S, is a subsyslem of a system S, or, in short- hand, S, z S2, iff every theorem of S, is a theorem of S2 and S2 is closed under the primitive rules of S,. It is easy to see that, for every X3 we may have XD4 G Xtriv. The system Xtriv and, a firtiori. XD4 are consewdtive extensions of X in the language L (in a proof of A of L in Xtriv, we just delete all boxes and, after an eventual omission of empty steps> obtain a proof in X).

Consider now the trunsfazion, i.e. one-one mapping? m from L into LCI defined by the following recursive clauses:

m(A) = CIA, if A is atomic

m(AzB) = El(m(A) z m(B)), if z is -+. l , A or v

m(/3A) = q pm(A), ifD is 1. V.x or 3x.

In other words, nr(,4) is the result of prefixing •i to every subformula of a formula A of L. The translation m and analogous translations are called modal transIations.

Let r be a translation from the language of a system S, into the language of a system S2. We say that S, can be embedded by r in S2, or9 in shorthand, S, +’ S-,, iff, for every formula A of the language of S, , we have:

A is provable in S, iff 7(A) is provable in S.

Then we can easily prove the following proposition:

PROPOSITIO3 1. Zf XD4 G S G Xtriv, then X -+“’ S. Proof. Suppose XD4 G S G Xtriv. If A is provable in X, we show

by induction on the length of proof of A that m(A) is provable in XD4; u forriori, it is provable in S.

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If m(A) is provable in S, then it is provable in Xtriv, and, hence, by replacement of equivalents, ,4 is provable in Xtiiv. Since Xtriv is a conservative extension of X in L, it follows that ,4 is provable in X.

q.e.d.

This proposition could be strengthened by using a system with modal postulates weaker than those of XD4. However, XD4 has the advantage that it is axiomatized with customary modal postulates. Other strengthenings obtained by replacing XD4 in Proposition 1 by a weaker system follow from Proposition 2 below.

With some of the systems S of Proposition 1, we could use for the embedding of X a more economical translation than m. Since in BCD4 we can prove:

q (UA -+ CD) -+ (CIA + q B), (CIA l cm) + cl{clA l cm)

q (UA A lx) -+ (CIA A CM), (CIA v ml) + q (UA v cm) q hllA -+ 1lIlA

q lVxCiA -+ VxOA, 3xUA + q l3xlZlA

whenever one of the converse implications is also provable in S, by replacement of equivalents, in m(A) we can omit Cl in front of the corresponding connective in order to obtain an equivalent formula. We perform this omitting systematically, starting from the outermost occurrences of the corresponding connective (formally, we make an induction on the complexity of A to show that m(A) and the formula obtained from m(A) by omitting 0 are equivalent). So, if in S we can prove all instances of the schema:

UW ClAdA

we have the converses of the implications in the right column, which yield what we shall call ouzer right equivalences (outer, because an outer box is omitted). With these equivalences, in m(A) we can omit Cl in front of every occurrence of l , v and 3x, and still obtain our embedding. Since l and A are synonymous in HD4, in our transla- tion we can omit 0 in front of l or A for this system. Dually, if in S we can prove all instances of the schema converse to (UT):

(UT-‘) ‘4 -+ DA

MODAL TRANSL.ATIONS I!V SUBSTRUCTL-RAL LOGlCS 293

we have the converses of the implications in the left column, which yield OUIU fefi equivalences. With these equivalences, in n?(,4) we can omit Ll in front of every occurrence of -+? A. 1 and ‘d.q and still obtain our embedding.

A different kind of economy is obtained as follows. Since in BCD4 we can prove:

zl(clA + B) + q (UA + q B) q (A + UB) 4 q (CA 4 Ill?)

cll/4-+cllrJ4

q (UA l UB) + q (A l B)

U(A A I?) -+ q (UA A CM) U(UA v ml) -+ U(A v I?) q lVxA -+ q lVxClA q lxEA -+ IlxA

whenever one of the converse implications is provable in S, by replacement of equivalents, in m(A) we can omit J at the appropriate places within the scope of the corresponding connective in order to obtain an equivalent formula. We perform this omitting systematic- ally* starting from the innermost occurrences of the corresponding connective (formally3 as before, we make an induction on the com- plexity of A to show that m(A) and the formula obtained from m(A) by omitting ?I are equivalent). If in S we can prove all instances of (UT), we can also prove the converses of the implications in the left column, which yield inner left equivalences. Dually, if in S we can prove all instances of (UT ‘). we can also prove the converses of the implications in the right column which yield inner right equivalences. We can use inner left or inner right equivalences to obtain embed- dings with more economical translations.

The tirst inner left equivalence can replace (04) and (EIT) in the presence of (OK): (necessilation) and the rule converse to (necessitation):

LIA A

If S is closed under this last rule, which is guaranteed in the presence of (UT)? then. without spoiling our embedding> we can also omit the main Ll in our translation; i.e.> we prefix Cl only to proper subformulae.

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Such a translation m’ from L into LCI is defined by the following recursive clauses:

m’(A) = A, if A is atomic

m’(A a B) = Urn’(A) a Urn’(B), if a is -+, l , A or v

m’(j?A) = blIlm’(A), if /3 is 1, VX or 3~.

Since Elm’(A) is m(A), we have that m(A) is provable in S iff m’(A) is provable in S. However, if in S we can prove (UT), we can use the three inner left equivalences to obtain a further economy in m’. Let the translation m” from L into LCi be defined as follows:

m”(A) = A, if A is atomic

m”(A -+ B) = Elm”(A) + m”(B)

m”(A l B) = Elm”(A) l Elm”(B)

m”(A A B) = m”(A) A m”(B)

rn#(A v B) = Elm”(A) v Urn”(B)

m”(iA) = 1 Elm”(A)

m”(VxA) = Vx m”(A)

m”(3xA) = 3xlJm”(A).

Then q lm”(A) is equivalent in S with Elm’(A), and hence m”(A) is provable in S iff m(A) is provable in S. An advantage of m’ and m” over m may be that with them we translate a connective into a con- nective; for example, v is translated by the connective vO defined by:

A vu B =df CIA v LIB.

However, m’ and m” are not suitable to translate proofs from hypoth- eses (see the remarks at the very end of this paper).

Consider now the following modal schemata, which are obtained by a modal translation of, respectively, (W), (KL) and (KR):

wa El(Cl(ElA + q l(ClA + q B)) + q l(ElA 4 q IB))

WJU q l(ClA + lIl(OB + CIA))

WW El(~llIlA + q (iZlA + UB)).

MODAL TRANSLATIONS IN SUBSTRUCTURAL LOGICS 295

With the same simple technique we have used for Proposition I> we can prove the following proposition:

PROPOSITION 2. BCW +“’ BCD4 + (WLI);

BCK -wm BCD4 + (KEIL), (KCIR);

H -Pi BCD4 + (WEI), (KOL), (KiElR);

H +“’ BCWD4 + (KCIL), (KEIR);

H -,“r BCKD4 + (WCI).

The embeddings of this proposition are like the modal embedding of Girard (1987, IV, 4 5. l), because on the right-hand side we have a weaker nonmodal logic extended with modal axioms that mimic the missing nonmodal axioms.

The system XDS is obtained by extending the axiomatization of XD4 with the following axiom-schema, which is obtained by a modal translation of (double neg):

a3 q (cllcllzlA + FM).

The system X=triv is obtained by extending the axiomatization of Xc with (triv). The following version of Proposition 1 is also straight- forward to prove with our simple technique:

PROPOSITION 3. ZfXD5 c S s Xtriv, then Xc 4’” S.

We can also easily demonstrate a version of Proposition 2 where on the left-hand side we have Xc instead of X and on the right-hand side we have an extension of BCD5, BCWD5 or BCKD5 with (WEI); or (KqL) and (KCIR), in the appropriate combination. The system on the right-hand side may also have (double neg).

The most interesting corollary of Proposition 3 is when S is the system X011, which is obtained by extending the axiomatization of X with:

It is easy to show that XD5 is a subsystem of X011, and X011 is of course, a subsystem of Xctriv. xote that XD5 is a proper subsystem of X011, since in XD5: which is included in the modal

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logic S5, we don’t have A -+ q A. Consider now the translation n from L into L such that n(A) is the result of prefixing 11 to every subformula of A. From the fact that Xc +m XCITI, it follows easily that Xc -+n X. As a special case, we have Hc +” H, which is the well- known embedding of classical logic into Heyting’s logic first con- sidered by Kolmogorov.

We could also use a more economical translation that R to embed X into X. Since in BC we can prove A -+ 1-1 A, which corresponds to (UT-‘), we can prove equivalences obtained from the outer left equivalences above by replacing El with II. For negation, we have also:

TTIA++IA.

So, in n(A) we can omit II in front of +, A, 1 and Vx. A dif- ferent kind of economy can be achieved by using the equivalences obtained from the inner right equivalences above by replacing Cl with IT. These equivalences, which involve 4, l , v and J.Y, are prov- able in BC. Of course, analogous economies could be obtained for the embedding of Xc into XUII via m.

The economy obtained by omitting the main 1-1 in n(A), which is analogous to what we had with m’ and m”, is not available for em- bedding BCWc into BCW and Hc into H. Otherwise, since in BCWc and Hc we have A v 1 A, we could prove II A v I A in BCW and H. This also shows that BCW and H are not closed under the rule:

IIA

A

However, BC and BCK are closed under this rule (see the remarks after the proof of Proposition 5 in Section 4 below). Hence, to embed BC‘ into BC and BCK into BCK we can omit the main 11 in n(A) or in one of the more economical translations of the preceding para- graph. It follows that the (= , I, v , 3) fragments of BC and BC coincide, and analogously for BCK and BCKc.

AI1 the embeddings considered up to now were rather trivial to prove. We come to a type of embedding more difficuh to prove when on the left-hand side we have X and on the right-hand side we have a modal extension of Xc. Metaphorically speaking, the modal operators

MODAL TRANSLATIONS IX SUBSTRUCTURAL LOGICS 297

work now to cancel the effect of (double neg); they subdue classical negation and transform it into an intuitionistic negation. A very well- known embedding of this type is the embedding of H via m. or an equivalent translation, into S4, which was first considered by Godel. We obtain S4 when we extend the axiomatization of He with the modal postulates (OK), (04) (UT) and (~ece.G~u~~~~). To show that if A is provable in H, then m(A) is provable in S4, it is enough to take the first half of the proof of Proposition 1 and appeal to the fact that HD4 is a subsystem of $3. But to prove the converse is a more difficult matter than before. In the following sections, we develop proof-theoretical techniques that will enable us to obtain this more difficult type of embedding for our substructural logics BC, BCW and BCK.

3. GEYTZEN SYSTEMS

Consider first the set F of finite sequences of formulae of L in which formulae may be repeated; the empty sequence also belongs to F. Over F, we define an equivalence relation by saying that, for every S. l E Fz we have s = i iff s can be obtained by permuting members of l or s is equal to t. A finite multi.~et of formulae of L is defined as a set X of sequences from F such that, for some s E & we have X = {t E F: t = s}; i.e., X is an equivalence class of sequences from F. Let M be the set of all finite multisets of formulae of L, and let us use the letters X, Y, Z, . . . , possibly with subscripts or superscripts for elements of M. For every X, YE M, let:

XY = {t E F: (for some s, E X) (for some s2 E Y) t = s, .sz}

where s,.s? is the sequence obtained by concatenating S, and .r?. It is clear that the operation so defined on M is associative and commuta- tive. To every sequence (,4,, . . . , ,4”) from F, where /r 2 O> there corresponds a unique X from M such that (A,, . . . > A,,) E X. So, we can designate the multiset X by any (A,, . . . , A,,) from X (of course, two different sequences may designate the same multiset). However, to make it clear that we are designating a multiset, and not a sequence, we write [A,. . . . , AJ instead of <A!: . . . , A,,). The

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multiset designated by [A] is a singleton multiset, and [ ] designates the empty multiset, i.e. the equivalence class of the empty sequence (formally speaking, the empty multiset has one element, viz. the empty sequence). The expression [A,, . . . , A,,] where n = 0 stands, of course, for [ 1. We say informally that a formula is from a multiset X iff it is a member of a sequence from X.

A Gentzen system is a subrelation of M x M. Every pair (X, Y), which we write X F Y, is called a sequenr. A Gentzen system G is specified, as usual, by giving a set of basic sequents of G and a list of rules of G. An application of a unary rule is a pair of sequents such that the first is a premise and the second the conchsion; and similarly with rules of greater arity. A derivation of X I- Y in G is, as usual, a tree made of applications of rules of G that, at the top, has basic sequents of G and, at the bottom, X k Y. A sequent is in G iff it has a derivation in G.

Note that this notion of Gentzen system differs from the standard notion because a sequent X F Y is not a string of symbols of the language L enlarged with some auxiliary symbols. If X and Y are finite sequences of formulae of L, we may take a wquent X F Y to be a string of symbols of the language L enlarged with the comma to separate formulae within X and Y, and the turnstile to xparate X from Y (we have something analogous when X and Y are terms made of formulae of L with a bmdry comma, as in Dogen 1988). It is prob- ably the wish to make sequents strings of symbols, i.e. purely syntac- tic objects, that motivated Gentzen in (1935) to base his sequents on finite sequences of formulae. This may be better adapted to his purposes and more in the spirit of Hilbert’s school. Otherwise, since Gentzen deals only with classical and intuitionistic logic, he could have had sequents X F Y where X and Y are finite sets or for- mulae. However, many things in the standard approach can straight- forwardly be adapted to the approach based on multisets, or sets, of formulae, and we shah not bother to introduce everything anew. The previous two paragraphs serve just to fix not&ion and basic terminoi- ogy, and not to bring forward anything new. (Both our notion of sequent and the standard one differ from Lambek’s sequents, called arrows in Lambek and Scott, 1986, $1.1, which are labelled by a term recording the history of their derivation; in Lambek’s sense, for a

MODAL TRAXSLATIONS IN SUBSTRUCTURAL LOGICS 299

given X and Y, we mdy have more than one sequent X k Y with dif- ferent labels.)

Our first Gentzen system, called G(BC), is given by the following basic sequents and rules, whose archetype is in Gentzen (1935):

Basic sequents:

Ml I- [Al Structural rule:

KW x I- Y[A] [A]Z I- w

xzt YW

Rules for logical constants: -.

x t Y[A] [E]Z t w (+ R) Lw 1 WI [A -+ E]XZ I- YW x I- Y[A + E]

[A. E]X t Y

[ A l E ] X t Y (0 RI

x t Y[A] z !- W[E] x-z I - YW[A . E ]

[A]X I- Y [E]X t- Y [A A B]X I- Y [A A B]X’ k Y

(,,, Rj ,x’k WI ,YI- WI A’ + Y[A A B]

In the rules (VR) and (3L), “proviso $’ means, as usual, that there is no formula from the multisets of the conclusion in which the eigen- variable y occurs free (of course, y is also restricted by the usual proviso that y is free for x in PI. which goes with the notation AC).

When to the rules of G(BC) we add the structural rule:

(Contr) zzxl- YWW

zxl- YW

300 KOSTA DOSEN

we obtain the Gentzen system G(BCWJ, and, when to the rules of G(BC) we add the structural rule:

(Thin) xl- Y

zxl- YW we obtain the Gentzen system G(BCKJ. With both (Contr) and (Thin), we obtain the Genzen system G(h). For all these new Gentzen systems, basic sequents are unchanged.

The rules (Contr) and (Thin) are usually given in the form where one of Z and W is the empty multiset, and, moreover, the remaining one is a singleton multiset. If we don’t want to investigate systems that have Contraction or Thinning on one side, but not on the other, this is just a stylistic matter, and it is clear that the effect of our (Contr) and (Thin) is obtained by repeated use of (Contr) and (Thin) in these restricted forms Our form of (Contr) and (Thin) enables us to achieve a certain economy of presentation. It will be clear from the context, and sometimes we shall explicitly note, whether Contraction or Thinning on the left or the right is essential. However, with the modal rules below and in Section 6, we will consider restricted forms of Contraction and Thinning on the left.

We call a sequent X 1 Y where Y is a singleton multiset or the empty multiset a single-c04z&un sequent; otherwise it is a muftiple- conclusion sequent. In the spirit of Gentzen, we obtain the Gentzen system G(X) from the Gentzen system G(X) by introducing for all rules the following Single-Conclusion Restriction:

conclusions must be single-conclusion sequents. Since basic sequents are single-conclusion sequents, this has the effect of allowing only single-conclusion sequents in derivations. We can formulate G(X) by adding to (1 R) the proviso that Y is the empty multiset, and to (Thin) the proviso:

1

if Y is not the empty multiset, then W is the empty multiset;

if Y is the empty multiset, then W is a singleton multiset or the empty multiset.

Basic sequents and all other rules may be taken as for G(X), though with the Single-Conclusion Restriction some of these rules can be for- mulated mom economically.

MODAL TRANSLATIONS 13 SLBSTRUCTURAL LOGICS 301

Analogues of the Gentzen systems G(BCW), G(BCWJ, G(H) and G(HJ could be formulated with sequents x k Y where X and Y are finite sets of formulae of L. This would require a reinterpretation of our rules (in particular, we would have problems with l in the absence of Thinning)> whereas Contraction could not be formulated any more; it would be incorporated in the notion of sequent! as the structural rule of Permutation (or Interchange) is incorporated in our notion of sequent based on finite multisets of formulae (cf. Dogen 1985).

We shall now introduce Gentzen systems with sequents based on finite multisets of formulae of the language LCi instead of L. A multi- set x of formulae LEl will be called moduiized iff every formula from x is of the form LIZ3 (the empty multiset is modalized too). We use A-y Y=, z:, . . . for modalized finite multisets of formulae of LO. For the Gentzen system G(XcS4) we assume whatever tie have assumed for G(Xc) plus closure under the following additional rules, which stem from Curry’s (1950) and (1952), and whose analogues are in many other places:

We will also consider Gentzen systems with the following restricted forms of (Contr) and (Thin):

(Contr II) ZLZOXk Y

ZUXk Y

(Thin IIL) Xk Y

zn‘Yi- Y

Note that (Thin OR) involves multiple Thinnings on both the left and right (to formulate this rule. our style of presentation, where we have arbitrary multisets Z and IV rather than singleton multisets? is particularly handy).

302 KOSTA DOSEN

We need the following terminology for the cut-elimination result below. We will say that a formula A of L or LO is regular iff no variable occurs both free and bound in A; otherwise, it is irregular. For every formula A of L, there is a regular formula A’ of L such that in all our Hilbert systems in L we can prove A - A’; the same holds when L is replaced by LO. The regular A’ is obtained from an irregular A by renaming some bound variables (see Kleene 1952, (i 33). Note that if a formula A of L is regular, then m(A) is regular too. A variable occurs free (respectively bound) in a sequent iff it occurs free (respectively bound) in a formula from either the left or right multiset of that sequent. Then, a sequent is called regular iff no variable occurs both free and bound in that sequent (see Kleene 1952, 678).

We also need the following terminology for our proofs below. A formula A is said to be involved in an application of a rule (*) iff A is from a multiset in the conclusion and the main connective or quanti- fier of A is introduced by this application of (*), or if (*) is (Contr), (Thin) or (Thin OR), then A is from the Z or W of this application of (*), or if (*) is (Contr 0) or (Thin q L), then A is from the Z” of this application of (*).

We can now demonstrate the following cut elimination for our Gentzen systems:

PROPOSITION 4. If’s regular sequent is in one of the following Gentzen systems.

G&J, G(X), WW),

G(BC,S4) + (Contr q ),

G(BC,S4) + (Thin q L), (Thin OR),

G(BC,S4) + (Contr q ), (Thin EIL), (Thin EIR),

G(BCW,S4) + (Thin q L), (Thin OR),

G(BCK,S4) + (Contr El),

then is has a derivation in the same Gentzen system without (Cut). Proof: Since this proof is only an exercise in standard Gentzen

techniques, we will only give a sketch of it with comments.

MODAL TRANSLATIONS IN SUBSTRUCTURAL LOGICS 303

First, note that the restriction to regular sequents in the formu- lation of our proposition is essential, because an irregular sequent may be derivable only with (Cut). For example, p-~tlx xRy] I- [.yRx] cannot be derived without (Cut) (cf. Kleene 1952, 078: Example 4). Instead of this restriction, Gentzen in (1935) and many authors fol- lowing him, have different symbols for free and bound variables, which makes the language accept only regular formulae and regular sequents.

Consider now the derivation of a regular sequent. As a preliminary step, by renaming free and bound variables, we can transform this derivation to ensure two things. First, we ensure that the derivation be reguZurt i.e. that no variable occur both free and bound in it (a variable occurs free, respectively bound, in a derivation iff it occurs free, respectively bound, in a sequent in this derivation). Second, we ensure that the eigenvariable 1’ of an application of (VR) or (3L) occur nowhere else in the derivation except in sequents above the con- clusion of this application of (VR) or (3L). A derivation in which these two things are ensured will be called disciplined (Kleene says that it has the pure variable property; see 1952, 8 78, Lemmata 37 and 38).

Then we show that, for a disciplined derivation of a regular X k Y with a single application of (Cut), there is a disciplined derivation of X t Y with no application of (Cut) or with one or two applications of (Cut) of strictly smaller complexity.

(1) There is nothing new in cases we have to consider for G(BC,) and G(BC), and we can copy the appropriate parts of the proof in Klecne (1952, 0 78, Theorem 48).

For systems with (Contr) or (Thin), we have additional cases, when the left or right premise of our application of (Cut) is the conclusion of an application of (Contr) or (Thin):

(2.1) If the cut-formuir, i.e. the A in the premises of (Cut) elimi- nated by applying (Cut), is not involved in this application of (Contr) or (Thin). then we show that we can first apply (Cut) and then (Contr) or (Thin), i.e. that we can push (Contr) or (Thin) below (Cut).

(2.2) If the cut-formula is involved in an application of (Contr) above the left or right premise of our application of (Cut), then we

304 KOSTA DOSEN

can try pushing (Contr) below two applications of (Cut); i.e., the left figure would be replaced by the right figure:

u I- V[A] L4ZL4Z~ I- yww (Contr)

LWX k ycut) uzxt VYW

u t V[A] [A]Z[A]ZX I- YWW u t V[A] UZ[A]ZX t VYWW (Cut)

uzuzx I- VVYWW (Cut)

uzxt VYW (Contr)

and we would proceed analogously when the left premise of our application of (Cut) is the conclusion of (Contr).

Formally, the cut-elimination procedure consists in a double induc- tion on degree and rank as in Gentzen (1935) or Kleene (1952). Then the upper application of (Cut) in the right figure above has a lower rank than the application of (Cut) in the left figure, and, by the induction hypothesis, it can be eliminated. However, after this elimi- nation is made, we are still left with the lower application of (Cut), whose degree is not lower and for which it is not clear that it will have lower rank (in Szabo 1978, appendix C, pp. 229-243, this is not proved). To forestall these difficulties, in Gentzen systems with (Contr) we should replace (Cut) by the following version of Gentzen’s mix rule:

(Mix) x t Y[A]” [A]” z t w

xzt YW where n, m 2 1 and [A]” and [A]” abbreviate [A] repeated n, respec- tively m, times. The cut-elimination procedure consists then in a mix- elimination procedure, which imposes certain adjustments in the pro- cedure we are sketching.

If the cut-formula is involved in an application of (Thin) above the left or right premise of our application of (Cut), then our application of (Cut) will disappear after replacing the left figure by the right figure:

(izx" L fyw (Thin)

MODAL TRANSLATIONS IN SUBSTRUCTURAL LOGICS 305

and we proceed analogously when the left premise of our application of (Cut) is the conclusion of (Thin).

This establishes our proposition for all our systems G(X,) and G(X).

It remains to consider systems with the modal rules. We have the following additional cases:

(3.1) If the left or right premise of our application of (Cut) is the conclusion of an application of (*), which is one of the rules (JL), (Contr Cl) and (Thin ZL), and the cut-formula is not involved in this application of (*) above the right premise, then we push (*) below (Cut) as in case (2.1) above.

(3.2) So, the left premise of our application of (Cut) is the con- clusion of an application of either (OR) or (Thin q R). With (JR), we have:

z” L [A] X’ t Y’ y; Z” t [OA] [ciA]X I- Y (*)

z-xl-y

Then suppose (*) is (Contr 3) or (Thin OL) in whose application the cut-formula 3A is involved. We can push (Contr 0) below two applications of (Cut), or our application of (Cut) will disappear with (Thin IlL), as in case (2.2) above. Formally, in Gentzen systems with (Contr Cl), we should then consider the elimination of a restricted version of (Mix) where the mix-formula A in the premises is of the form 3B and n = 1.

If (*) is (OL) and X’ t Y’ is [A]X t Y, then we have:

(Cut> Z’-; L [A] [A]X t Y

z’;‘xL Y

where A is of strictly smaller complexity than EA. With our restricted version of (Mix), we may have the following

figure:

(OR) (Mix) Z”Xk Y

306 KOSTA DOSEN

This is replaced by the figure:

(OR) ZD k [A]

zn t [EM] [A][OA]mX 1 Y zo I- [A] IA]ZDX t Y

(Mix)

ZGZUXk Y

We have in the new figure an application of restricted (Mix) above an application of (Cut). However, in Gentzen’s terminology, this appli- cation of (Mix) has lower rank than the application of (Mix) in the old figure; so, by the induction hypottiesis, it can be eliminated. After this elimination, we are left with the application of (Cut) in the new figure, whose cut-formula is of strictly smaller complexity than the mix-formula of the application of (Mix) in the old figure.

If (*) is (OR) and X’ t Y’ is hence [OAJP I [B], where Xis X0 and Y is [LIB], then we push (OR) below (Cut) as follows:

(Cut) z” k [DA] [UAJX~ I- [B]

(‘JR) ZOP k [B]

zoxo k [ml]

If (*) is (Thin OR), then we have two cases depending on whether the cut-formula DA is in X’ or not. If it is in x’, then X’ t- Y’ is [OA]X” t [ 1, and we push (Thin q R) below (Cut) as follows:

((&) Zn t u-1 D‘wn I- [ 1

(Thin UR) ZOX”k[ ]

ZOXk Y

If the cut-formula CIA is not in X’, then we get Z’X t Y from x’ I- Y’ by a simple application of (Thin OR). There are no possibili- ties left for (*).

(3.3) If the left premise of our application of (Cut) is the conclusion of an application of (Thin OR), then we have:

x0 11 I (Thin q R) zxn t W[A]

VW [‘4]U I- Y

zx”ut WY

MODAL TRANSLATIONS IN SUBSTRUCTURAL LOGICS 307

and we get ZX” U 1 WV from X0 t [ ] by a simple application of (Thin iIlR). This concludes the proof of Proposition 4. y.e.d.

4. EQUIVALENCE BETWEEN HILBERT SYSTEMS AND

GENTZEN SYSTEMS

We shall now establish to which Hilbert systems our Gentzen systems correspond. For that, we want to assign to each sequent some for- mulae of L or LO. To describe this assignment, WC will find it useful to define in L and L3 the following binary connective:

A + B =dT l/i + B.

This connective (which may be found in Orlov 1928, Anderson and Belnap 1975, $27.1.4, p. 314, Grishin 1981 and Girard 1987) is dual to l in the systems X,, since in these systems we can prove:

(A l B) h -I(A -+ -IB).

As a matter of fact, in BC,, BCW, and BCK,: each of +, l and + could serve to define the remaining two connectives with the help of -I. Likewise, with the help of 1 we could give De Morgan-like defi- nitions of v in terms of A I and of 3 in terms of V, and vice versa. So, a sufficient base for BC,, BCW, and BCK, could be, for example, (-+,lr A,V)or(*, 1: v , 3). We have noted at the end of Section 2 above that the ( l : 1, v , 3) fragments of BC and BC, coincide, and analogously for BCK and BCK,. So, BC and BCK include as a fragment the whole systems BC, and BCK, respectively. This is anal- gous to the fact that the Heyting propositional calculus includes as its ( A . 1) fragment the whole classical propositional calculus.

Both l and + are associative and commutative in X,, and, in H,: they are also idempotent. Jn H,, the connective + is synonymous with v . The connectives + and l could be characterized in all the Gentzen systems GO(,) and G(X) by the double-line rules:

[A]X i Y[B] [A, B]X i- Y

X k Y[A + B] [A . B]X 1 Y

where the double line means that WC have in each case two rules, one going downwards and the other upwards. Analogously, in G(X,), the

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connective + could be characterized by:

x t Y[A, B]

x I- Y[A + B]

This makes it clear that to a sequent [A,, . _ . . A, ] I- [II,, . . . , B,,,], where n, m 2 1, we should assign any of the formulae:

(A;. . . . . A;) + (B; + . . . + B;)

where the bracketing in the antecedent and consequent is arbitrary, and the sequences (A;, . . . AA) and (B;, . . . , Bi) are permutations of, respectively, (A,, . . . A,,) and (B, , . . . , B,). To a sequent [ I t- P,, . . . 3 B,], where m 2 1, we assign just B; + . . . + II;, and to [A,, . . . , A,] k [ 1, where n 2 1, we assign i(A; l . . . l A;), which, for n > 2, is equivalent in BC with (A; l . . . l A;. ,) + 1A; (we don’t assign anything to the sequent [ ] 1 [ 1. which is not impor- tant to us here). For a given sequent X t- Y. we designate by h(X t- Y) any of the formulae of L or LIZ assigned to X k Y (the letter h stands for “Hilbert”).

Let now X,S4 be the Hilbert system in Lc7 obtained by adding to the axiomatization of X, the modal postulates (OK), (Cl4), (UT) and (necessitcrtion). The system H,S4 is the first-order predicate calculus S4. We will also consider Hilbert systems obtained by adding, for some X, to the axiomatization of X,S4 the axiom-schemata:

(WO) (OA + (DA + B)) -+ (DA --* B)

(knL) A + (OB -+ A)

(k q R) q I~CIA 4 (OA -+ B)

(with I primitive, (kOR) would amount to 01 + B; for (kOL), see Anderson and Belnap 1975, 5 29.1, pp. 39 l-392). Since in XS4 we can prove:

fl(oA + q B) - fJ(ClA + B)

and since X,S4 is closed under the rules:

A

DA

MODAL TRAKSLATIONS IN SUBSTRUCTURAL LOGICS 309

adding to the axiomatization of X,S4 the axiom-schema on the left- hand side has the same effect as adding the axiom-schema on the right-hand side:

w3 amounts to q (nA --f (CIA -+ B)) + (OA + B),

(KOL) amounts to CIA --) (38 -, A).

WOW amounts to KliClA + (OA -+ B), i.e. (kClR).

So. (KOR) has exactly the same power as (kElR). On the other hand: it is clear that (~0) and (kDL) yield respectively (WO) and (KZL) (it is not important to us here that the converse need not hold).

We can now establish the following proposition:

PROPOSITION 5. The sequent [ ] t [A] is in a Gentzen system in the left column i# A is provable in the Hilhert system in the sume line in the right column:

WQ

G(X)

G(X, S4)

G(BC,S4) + (Contr 0)

G(BC,S4) + (Thin 3L), (Thin OR)

G(BC,S4) + (Contr U), (Thin CIL), (Thin 3R)

XC X

x34

BC,S4 + (wU)

BC,S4 + (kOL). (k3R)

BC,S4 + (wEI), (kCIL), &OR)

G(BCW,S4) + (Thin CIL), (Thin q IR) BCW,S4 + (kOL). (kOR)

G(BCK,S4) + (Contr 0) BCK,S4 + (id)

Proof. We show first by induction on the length of derivation that if X t Y is in a Gcntzen system. then all the formulae /1(X k Y) are provable in the corresponding Hilbert system. This yields one direc- tion of our proposition.

For the other direction, we derive in the Gentzen system [ ] k [A] for every axiom A of the corresponding Hilbert system, and we show closure under the primitive rules of this Hilbert system. For example,

310 KOSTA DOSEN

if [ ] 1 [A] and [ ] t [A + B] are in the Gentzen system, then, by using [A, A + B] 1 [B] and (Cut), we obtain that [ ] 1 [B] is in our Gentzen system. q.e.d.

As a consequence of Proposition 5, we can show that BC and BCK are closed under the rule:

1lA A

By analyzing how [ ] t [I-I A] can be obtained in G(BC) or G(BCK) without (Cut), we see that we must also have [ ] k [A]. We can simi- larly show that BC, and BCK, have the disjunction property (if A v B is provable, then either A or B is provable) and the existence property (if 3xA is provable, then, for some term a, A,” is provable). The intuitionistic systems BC, BCW, BCK and H also have the disjunction and existence properties, but, there, these properties are expected, whereas for the classical systems BC, and BCK they may seem sur- prising. For all Hilbert systems of Proposition 5 whose corresponding Gentzen system lacks Contraction on the right, we can show that, in addition to having the disjunction and existence properties, they are closed under the rules:

A.B A.B - - A B

For closure under the rule displayed at the beginning of this para- graph, we used the fact that the corresponding Gentzen systems lack Contraction on both sides.

Note that BCW,S4 differs from the system Rn of Anderson and Belnap (1975, 927.1.3, pp. 343-344) not only in lacking (distr), but also in lacking:

(OA A q B) -+ q (A A B).

Instead, we have only the converse of this implication and ( q A l Cl B) + q l(A l B), which are provable already in BCD4. As a related schema, in BCW,S4, we also have q l(ClA A q IB) cf q l(A A B), which is one of the inner left equivalences of Section 2.

MODAL TRANSLATIONS IN SUBSTRUCTURAL LOGICS 311

(Modal extensions of relevant logic are studied in Routley and Meyer 1972, Belnap ef al. 1980, Boiic 1983, and Fuhrmann 1990.)

We have taken as our basic Hilbert system BC rather than a weaker system in whose Gentzen formulation we would lack the structural rules of Permutation or Association (see DoSen 1988). We could easily prove for such weaker systems embeddings of the type of Propositions l-3 (cf. DoSen 1990. $6). However, to prove embeddings of the more difficult type, which are our main concern here, it seems we cannot use Gentzen systems like those we are working with since the last section. Namely, speaking of sequents based on finite sequences of formulae, with either of the four pairs of sequents:

A,-lA I- , tA,lA

A,-lAL, tlA,A

lA,A I- , rA,1A

lA,Ab , 1lA.A

we obtain Permutation. For example, with the first pair? we have:

(Cut)

kA,lA B,A l-B*‘4

(Cut) BbB*A,lA A.lAt

A,BbB.A A*Bl-B.A

(with the second pair, we also start with B, A t B l A, whereas. with the remaining two, we start with A, B t- A l B). With the double-line rule for +, we can analogously obtain the commutativity of + : i.e. Permutation on the right. Similarly, with sequents based on a non- associative and noncommutative binary comma (see DoSen 1988), we could obtain the structural rule of Association and Permutation with either of our four pairs (provided we have rules to cancel the empty G-term 0, as in DoSen 1988, $1.3). And it seems we must have at least one of these pairs if our systems are to obey rules like those we have @ven for -I in Gentzen systems without the Single-Conclusion Restriction. So, (double neg) brings in Association and Permutation in its natural Gentzen formulation. This does not mean that Gentzen systems of another type could not accommodate (double neg) without

312 KOSTA DOSEN

bringing in Association and Permutation. We could try using sequents based on signed formulae, or, analogously, ordinary sequents that, in addition to rules like those above, have for every constant separate rules to introduce the negated constant at the left and right of the turnstile (see Zaslavskii 1978, Bull 1987, and Brady 1990). However, it seems better to give this matter a separate treatment and to concen- trate first, as we do here, on what can be treated with more standard tools.

(Hilbert systems corresponding to Gentzen systems without Associ- ation and Permutation, which have an intuitionistic negation, in par- ticular purely implicational negation, have been treated in Dozen 1988, Q 2.10. What we have been saying in the paragraph above does not mean that adding (double neg) to one of these weak Hilbert systems necessarily yields the associativity and commutativity of the fusion connective.)

5. MODAL TRANSLATIONS IN ANALOGUES OF S4

Armed with our Gentzen systems and cut elimination, we now pro- ceed with the proof of embeddings analogous to the embedding of H in S4 via the modal translation m or an equivalent translation.

We must however introduce some additional terminology. In appli- cations of the rules for our Gentzen systems, the multisets denoted by X, Y, Z, W, Z” and X0 in our formulation of these rules are called parametric multisets. Let [A,, . _ . , A,], where n 3 1, be a parametric multiset, and let f bc a one-one mapping of the set of occurrences of formulae {A,, . . . , A,) onto itself such that f(Ai) and Ai are occur- rences of the same formula. Note that this mapping is not unique. We will say that an occurrence of a formula from a parametric multiset in the conclusion of an application of a rule is f-clustered with an occur- rence of the same formula from the same parametric multiset in a premise iff f assigns one occurrence to the other. Let us associate with every nonempty parametric multiset in an application of a rule in a derivation a single mappingf, save for applications of ( A R), ( v L), (Contr) and (Contr Cl), where we associate two, one for each of the two occurrences of the parametric multiset in the premises. Let us call a set of such mappings Q. Starting from a particular occurrence of A

MODAL TRANSLATIONS IN SUBSTRCCTURAL LOGICS 313

from a multiset in a sequent in a derivation, and a particular set Q for this derivation, we obtain a tree of occurrences of A by adding successively all the occurrences of A that for some f E Q are f-clustered with those occurrences already in the tree. We call such a tree an A-cluster (Gentzen in 1938, 0 3.41, introduces a slightly different notion of cluster; a notion analogous to ours is in Maehara 1954, $2.621).

Note that, starting from a particular occurrence of A, we may obtain several different A-clusters, because 0 need not be unique. For example, an A-cluster obtained from an A in [A + A, A, A] below:

[Al I- [Al (Thin) [A, A] t [A]

(Thin) iA + Al A, A] I- [A]

may be a chain with two or three nodes. All occurrences of A in an A-cluster are always on the same side of k. Binary branching in an A-cluster may occur when we have applications of (A R), ( v L), (Contr) or (Contr 0); with (A R) and (v L), branches go to different premises, whereas, with (Contr) and (Contr Cl), they go to the same premise. (If linear logic is deemed linear because, due to the lack of (Contr), it eschews branching in clusters, this holds good only for the fragment without A and v , since these connectives introduce another branching in clusters. The rule (Contr Cl), when added to linear logic, also brings in branching. Of course, the logic BCK is linear to the same extent.)

In the absence of (Thin) and (Thin OR), the rule (1R) is the only rule that can have a single-conclusion sequent as a premise and a multiple-conclusion sequent as a conclusion. If the conclusion of an application of (-rR) is the premise of an application of (OR): this conclusion cannot be a multiple-conclusion sequent, because the premise of (OR) must be a single-conclusion sequent. We call such applications of (1 R) passive, and we prove the following auxiliary proposition:

PROPOSITION 6. For every regular formula A qf!fL, if [ ] !- [m(A)] is one of the modal Gentzen systems of Proposition 4, then it has a derivation in the same Gentzen system in kvhich every application of (1 R) is passive.

314 KOSTA DOSEN

Proof: If [ ] t [m(A)] is regular, by Proposition 4, it has a deri- vation without applications of (Cut). If in this derivation we have the figure on the left-hand side, we replace it by the figure on the right- hand side:

tww I- y y x I- Y[lrn(B)]

bmlX I- y

(1 R) (*I [m(B)]X’ I- Y’

X’ I- Y’[lrn(B)] X’ k Y’[lrn(B)]

where (*) is (+ W, (0 U ( A L), ( v RI, (1 J.4, (1 R), VU (VW, (3L), (3R), (CIL), (Contr) in whose application -rm(B) is not in- volved, (Thin), (Contr Cl) or (Thin q L). We proceed similarly if (1 R) is followed immediately by (- L) or ( l R). Note that, since we are within a derivation of [ ] 1 [m(A)] without applications of (Cut), an application of (1 R) cannot be followed immediately by an appli- cation of a rule that introduces a connective or quantifier such that the formula involved is of the form: Cx lm(B) or lm(B)aC, where u is +, 0, A or v; or i-~m(B); or VxC, where Ct is im(B); or ZixC, where C,X is lm(B). Also, (Thin OR) cannot follow immediately (1 R). So, we have only four cases left: (1 R) followed immediately by (A R), or ( v L), or (Contr) in whose application im(B) is involved, or (OR).

Suppose we have the following figure:

[m(B), m(C)lX I- Y s i: Fi [m(C)IX t Y[-W)l WW’ 1 WW91

[m(C) v mW)lx I- W m(B)1 where 6 is the derivation that has at the bottom the right premise of our application of ( v L). Then consider a lm(B)-cluster in 6 obtained by starting from the occurrence of urn indicated in [m(D)]X I- Y[lm(B)]. At the top of this lm(B)-cluster, we replace the figure in the left column by the figure in the same line in the right column:

(basic sequent) [T m( B)] I- [1 m( B)] [W91 1 bW1 (l L, [-W% m(B)1 k [ 1

(-, R) bW)lZ t W Z I- W[lm(B)]

b@W t- W

MODAL TRANSLATIONS IN SUBSTRUCTURAL LOGICS 315

(Thin) ut v

zu t VW[lrn(B)]

ut v (Thin) [m(B)]ZU t VW

(Thin OR) x2 t [ ] xu t [ ]

Z‘P t W[lrn(B)] (Thin q R) [m(B)]ZX~ t w and we replace in 6 our TM(B)-cluster on the right of t by an iso- morphic tree made of occurrences of m(B) on the left of I. Then, by applying the same rules as in 6, in the same order, we obtain a deri- vation 6’ of [m(B), m(D)]X t Y, which yields:

(VL) [W), MCW k Y tm% ww I- y

[m(B), m(C) v m(D>]X i- Y (l R, [m(C) v m(D)]X t Y[lrn(B)]

We proceed analogously when (1 R) is above the right premise of (v L). We could proceed analogously with (A R) that follows immedi- ately (1 R), but this case is anyway excluded because we cannot have [lm(B), m(C) A m(D)] on the right of !- in a derivation without (Cut) of [ ] t [m(A)] (we cannot “transport” lm(B) or m(C) A m(D) to the left of t, with the help of (- L) or (-IL), before prefixing it with Cl, and we cannot prefix either of these formulae with 0, with the help of (OR), because of the presence of the other). For the same reasons, we cannot have that one of (- R); ( l R), (v R), (VR) and (SR), or (1 R) where the formula involved is different from -urn, follows immediatley our application of (-I R).

Suppose we have the following figure:

[m(B)]ZZX t ~W[l?n(B)1 w &T; zzx !- YW[-lm(B)lW[1 wo1

zx t YW[-lm(B)]

Then we proceed as above to transform the derivation 6 into a deri- vation 6’ of [m(B)]Z[m(B)]ZX t YWW, which yields:

-, (Contr)

[m(B)]Z[m(BfiZX !- Y ww

(1 R) [m(B)lZX ’ yw zx t YW[1nr(B)]

316 KOSTA DOSEN

Since our transformation of 6 into 8, in both cases considered above, did not perturb the order of the rules applied in 6, it does not matter from which application of (-IR) in our derivation we will start pushing (-tR) below other rules. And, since we are within a deri- vation of [ ] I- [m(A)], the rule (-tR) cannot be the last rule applied in this derivation. So, every application of (-IR) must be followed immediately by an application of (OR).

To be quite precise, we should recast the sketch of our proof above as a formal induction. For this induction, we introduce for our deri- vations a complexity measure equal to the sum of the lengths of all paths starting at the conclusion of an application of (1 R) and termi- nating at the premise of the corresponding application of (OR) below. The length of a path is the number of sequents in it minus one, and the premise of the corresponding application of (OR) below is at the bottom of all lm(B)-clusters obtained from the lm(B) involved in the application of (1 R). Pushing (-I R) below (1 R) with different formulae -tm(B) involved in the two applications of this rule is excluded, as we have mentioned above; with the same formula lm(B), involved, this is an empty step, in which nothing is done. With every pushing of (-tR) below another rule, this complexity measure becomes strictly smaller, until it reaches zero. q.e.d.

To state the next auxiliary proposition, we shall use the following terminology. An application of ( + L) is called passive iff its left pre- mise is a basic sequent or the conclusion of an application of (OR). Analogously, an application of (1L) is passive iff its premise is a basic sequent or the conclusion of an application of (OR). For example, the applications of (+ L) and (-IL) below are passive:

Z” t [A] (OR) zn t- [DA]. [ElB]Y k w (- L, [CIA

PI I- [Al + q B]zDYt w (lL) [1A, A] t [ ]

We are interested in passive applications of these rules because, in the absence of (Cut) and (Contr), no application of a rule save a non- passive application of (+ L) or (-I L) can have .a multiple-conclusion

MODAL TRANSLATIONS IN SUBSTRUCTURAL LOGICS 317

sequent as a premise and a single-conclusion sequent as a conclusion. Now, we prove the following auxiliary proposition:

PROPOSITION 7. For every regular formula A of L. if [ ] I- [m(A)] is one qf the modal Gentzen systems of Proposition 4. then it has a deri- vation in the same Gentzen system without (Cut) in which every uppli- cation of (+ L) and (1 L) is passive.

ProojI If [ ] I- [m(A)] is regular, by the proof of Proposition 4, it has a disciplined derivation without applications of (Cut). Then we show that, if the left premise of an application of (- L) in this deri- vation is the conclusion of an application of a rule (*). we can always push (-+ L) above (*), provided (*) is not (OR). Namely, we replace the left figure by the right figure:

(*I X’ k Y’[m(B)] [m(C)]2 I- W (+ L, [m(B) + m(C)]X’Z k Y’W

(- L) X k Y[m(B)] [m(C)]2 I- W

[m(B) + m(C)]XZ - YW

(*I [m(B) + m(C)]X’Z t Y’W

where (*I is ( - R), ( l Lh ( A L), ( v RI, (1 I-1, (1 RI, O’U 0%

(3L), (3R): (OL), (Contr) in whose application m(B) is not involved, (Thin) in whose application m(B) is not involved, (Contr 0) or (Thin 17L). In case (*) is (VR) or (3L), the proviso for the eigen- variable will be satisfied because our derivation was disciplined (it remains disciplined after the replacement of the left figure by the right figure). We can similarly push (- L) above a binary rule, i.e. (- L), (o R), ( A R) or (v L): by which the left premise of our application of (- L) was obtained. With (A R), we will obtain two applications of (+ L) that have as conclusions the premises of an application of (A R), and analogously with (v L). Note that, in the figures above, the formula introduced by (+ L) must be of the form m(B) + m(C) because it is a subformula of HZ(A).

We must also show that in case we push (+ L) above (-+ L) whose application was passive, the corresponding application after the

318 KOSTA DOSEN

replacement will remain passive, as when the upper figure is replaced by the lower figure:

x I- bwl bm1z I- ww91 + m(E)]XZ I- W[m(B)] [m(C)]U I- V

[m(B) --t m(C), m(D) -. m(E)]XZU 1 WV

bQ.W b WNOI b4C)lu t P’ (+ Lj

(- L) X I- [m(D)] [m(B) + m(C), m(E)]ZU I- WV

[m(B) -P m(C), m(D) -+ m(E)]XZU I- WV

For the remaining cases, we replace the left figures by the right figures immediately below:

(Contr) ZZX t YW[m(B)] W[m(B)]

ZX k YW[m(B)] (+ L, [m(B)

PWW 1 V + m(C)]ZXU t YWV

ZZX t- YWW91 WmU91 MWJ t V + m(C)jZZXU I- YW[m(B)] WV [m(C)]U t V

+ m(C)]z[m(B) -+ m(C)]ZXUU I- YWWVV

[m(B) 4 (C)]ZXU I- YWV

Xl Y [m(C)]U I- V

[m(B) + m(C)$ZXU t- YWV

min) [m(B) xl- Y

--) m(CJZXU t YWV

xD I- [I Crhin q R) ZXn t w[m(B)]

(-’ L, [m(B) [m(C)]U k V

4 m(C)]ZXnU t WV

x0 I- 1 I uhin q R) [m(B) + m(C)]ZX” U I- WV

The fact that, in the first case, with (Contr), we have a single appli- cation of (+ L) in the left figure and two applications of (+ L) in the right figure immediately below will cause problems when we try to recast

MODAL TRANSLATIONS IN SUBSTRUCTURAL LOGICS 319

the sketch of our proof as a formal induction (see the end of the proof below). Then in Gentzen systems with (Contr) (actually, with Contraction on the right, where Win (Contr) need not be the empty multiset) we should replace (- L) by the following rule:

x t Y[A]” [B]Z I- w [A + B]XZ I- YW

where n > 1 and [A]” abbreviates [A] repeated n times. We should similarly replace (1L) by:

x I- Y[A] [lA]Xb Y

This is analogous to what we had with (Cut) preceded by (Contr), which required the introduction of (Mix), in case (2.2) of the proof of Proposition 4. In general, the present proof is similar to a cut- elimination procedure (note, by the way, that (Cut) is obtained from:

x b Y[A] [A]Z I- w (- L) [A +A]XZt YW

by omitting [A -+ A]). Of course, we don’t have these complications with (Contr) if our Gentzen system is one of the modal Gentzen sys- tems of Proposition 4 that lacks (Contr), like G(BC,S4) and G( BCK, S4).

So, the only remaining possibility for the left premise of our appli- cation of (- L) is that it is obtained by (OR) or that it is a basic sequent. We proceed analogously to show that applications of (1 L) can be confined to passive applications. We must also show that, when we push (1L) above (- L) whose application was passive, the corresponding application of ( --t L) after the replacement will remain passive, as in the case above where we were pushing (+ L) above (+ L) whose application was passive. The left premise of an appli- cation of (+ L) or the premise of an application of (1 L) cannot be the conclusion of a passive application of (-IL).

Actually, we can push applications of (- L) and (1 L) above applications of (Cut), but, since we are working within a cut-free deri- vation, we don’t need that.

320 KOSTA DOSEN

To be quite precise, we should recast the sketch of our proof above as a formal induction. In analogy with Gentzen’s left rank, we define the rank of an application of (- L) introducing A + B on the left as the largest number of consecutive sequents in a path such that the lowest of these sequents is the left premise of our application of (-+ L) and each of the sequents has A in the right multiset. We define analogously the rank of an application of (1 L) introducing 1 A on the left. In the basis of the induction, we show that every application of (+ L) or (-rL) of rank I is either passive or can be eliminated (this elimination occurs in cases with (Thin) and (Thin OR)). In the induction step, we show that every application of (- L) or (1 L) of rank greater than 1 such that every application of (- L) or (1 L) above it is passive can be replaced by one or two applications of (+ L) or (1 L) of strictly lower rank. A passive application of (+ L) or (1 L) above the replaced application of (- L) or (1 L) remains passive after the replacement. q.e.d.

Let us sum up the import of the auxiliary propositions 6 and 7. The rule (1R) can produce the following pattern of sequents in derivations:

single-conclusion multiple-conclusion

which violates the Single-Conclusion Restriction, whereas (- L) and (1 L) can produce the converse pattern:

multiple-conclusion single-conclusion

Passive applications of these rules cannot do that any more. To reduce applications of (1 R) to passive ones, we have pushed them down in the proof of Proposition 6, whereas to reduce applications of (-+ L) and (1L) to passive ones, we have pushed them up in the proof of Proposition 7. Proposition 6 is effective in the absence of Thinning on the right, in the sense that the first pattern above will never occur. Proposition 7 is effective in the absence of Cut and Con- traction on the right, in the sense that the second pattern will never occur. Of course, we are not obliged to reduce ail applications of

MODAL TRANSLATIONS IN SUBSTRUCTURAL LOGICS 321

(1 R): (+ L) and (1 L) to passive ones, as we have done, but only those that produce the first pattern for the first rule and those that produce the second pattern for the latter two rules. In the absence of the first -pattern? derivations must be made exclusively off single- conclusion sequents, provided basic sequents are single-conclusion sequents, as they are in all our Gentzen systems. In the absence of the second pattern, derivations of single-conclusion sequents must be made exclusively of single-conclusion sequents.

We can now prove the following proposition:

PROPOSITION 8. For every regukur formula A ofL: if[ ] I- [m(A)] is in G(BC,S4), then [ ] k [A] is in G(BC).

Proof. If the antecedent holds, then: by Proposition 6, we have a derivation of [ ] f- [m(A)] in G(BC,S4) in which all sequents are single-conclusion sequents. This is because (Thin) and (Thin 3R) are absent from G(BC,S4). Then we delete all the unary connectives 3 from this derivation and, after the omission of empty steps, obtain a derivation of [ ] t- [A] in G(BC). q.e.d.

An alternative way to prove this proposition is to use Proposition 7 and the fact that (Contr) is absent from G(BC,S4). By imitating the proof of Proposition 8. we can prove the following proposition:

PROPOSITION 9. For every regular formulu A oSL, $[ ] I- [m(A)] is in G(BCW,S4), then [ ] t [A] is in G(BCW).

Note that, now, there is no alternative proof via Proposition 7, since (Contr) is present in G(BCW,S4). We need Proposition 7 to prove the following proposition:

PROPOSITION 10. For every regular formula A of L, if[ ] 7 [m(A)] is in G(BCK,S4), then [ ] i [A] in in G(BCK).

Proof. If the antecedent holds, then, by Proposition 7, we have a derivation of [ ] t [m(A)] in G(BCK,S4) in which all sequents are single-conclusion scqucnts. This is because there are no applications of (Cut) in our derivation, (Contr) is absent from G(BCK,S4), and

322 KOSTA DOSEN

[ ] k [m(A)] is a single-conclusion sequent. Then, as before, we delete all the unary connectives q from our derivation and, after the omission of empty steps, obtain a derivation of [ ] k [A] in G(BCK).

q.e.d.

Note that, now, there is no alternative proof via Proposition 6, since (Thin) is present G(BCK,S4). Note also that we apply Prop- osition 7 only for G(BCK,S4), and perhaps G(BC,S4); in both of these cases, Proposition 7 can be proved without the complications with (Contr).

We can finally prove our main embedding results. First, we have the following proposition:

PROPOSITTON Il. Zf BCD4 c S s BC,S4, then BC -+m S. Proof. If A is provable in BC, then, by Proposition 1, m(A) is prov-

able in BCD4, and, hence, also in S. Suppose now that m(A) is prov- able in S. Hence, it is provable in BC,S4. Then, there is a regular formula A’ of L such that A f--) A’ is provable in BC and m(A) cf m(A’) in BC,S4. So, m(A’) is provable in BC,S4. Then, by Prop- osition 5, [ ] t [m(A’)] is in G(BC,S4), and, by Proposition 8, [ ] k [A’j is in G(BC). Then, again by Proposition 5, A’ is provable in BC, and, hence, A is provable in BC. q.e.d.

We proceed similarly using Propositions 2, 5 and 9 to obtain the following embeddings:

PROPOSITION 12. ZfBCD4 + (WO) E S c BCW,S4, then BCW -+m S.

Among the systems S of this proposition, we find in particular BC,S4 + (WEI), whose Gentzen formulation is G(BC,S4) + (Contr Cl).

From Propositions 2, 5 and 10, we obtain the following proposition:

PROPOSITION 13. vBCD4 + (KOL), (KOR) s S sBCK,S4, then BCK +m S.

MODAL TRANSLATIONS IN SUBSTRUCTURAL LOGICS 323

Among the systems S of this proposition, we find in particular BC,S4 + (kcIL). (kClR), whose Gentzen formulation is G(BC,S4) + (Thin q L), (Thin OR).

Finally, we can state the following proposition:

PROPOSITION 14. rfBCD4 + (WO), (KUL), (KJR) G S c H,S4, then H -.“I S.

This is an immediate consequence of Proposition 2 and the well- known fact that H +m H,S4. Among the systems S of this prop- osition, we find in particular the last three systems listed together with their Gentzen formulations in Proposition 5. The embedding of H into a system G related to BC,S4 + (wEI). (kOL) considered by Girard in (1987, IV, 5.1) (cf. Schellinx 1991 and Troelstra 1990: 9 5) is not exactly of the same type as the embedding of H via ~71 into BC,S4 + (WEI), (kElL), (knR), which is covered by Proposition 14. First, Girard has a translation more economical than m (we have envisaged such a translation with m” in Section 2 above), but this is not an essential difference. A more essential difference should be that Girard does not translate the intuitionistic 1 of H in terms of 0 and the classical 1 of G (which in Girard’s notation is written as a super- scribed I), but in terms of q and an intuitionistic negation of G, defined with the help of the linear implication of G and a prop- ositional constant (for which Girard uses respectively the signs -c and 0). For this intuitionistic negation of G, we have (neg) and (kElR).

Our auxiliary propositions 6 and 7 are not sufficient by themselves to prove that H can be embedded by m into H,S4, i.e. S4, because the application of the first proposition is blocked by the presence of (Thin) and the application of the second by the presence of (Contr) in G(H,S4). For this proof we need something else, or something more. Proofs of this embedding using Gentzen-style techniques were given by Maehara (1954), Schiitte (1968, chapter IV) and Prawitz and Malmnas (1968) (Hacking 1963 and Zeman 1973. pp. 206-212, pre- sent such proofs for the propositional case). Maehara’s proof of (1954) is similar to our proof via Proposition 6. In an analogue of Proposition 6 (Sac 3 of 1954), he pushes down applications of (1 R)

324 KOSTA DOSEN

that violate the Single-Conclusion Restriction in order to reduce them to passive applications. His style of proof differs slightly from ours. He replaces at once a -urn-cluster on the right of t above the premise of:

(ORI ZG 1 [lrn(B)]

zn t [cllrn(B)]

by a m(B)-cluster on the left of t, in order to obtain:

Our proof of Proposition 6 relied on such replacements of clusters only in the presence of (v L) and Contraction on the right, which introduce branching in the lm(B)-cluster. Otherwise, we have pushed down (1 R) step by step. However, Maehara pushes down not only applications of (1 R), but also applications of (- R) and (VR) that violate the Single-Conclusion Restriction in order to reduce them to passive applications, i.e. applications immediately above applications of (UR). As a matter of fact, we could similarly push down appli- cations of (A R) too (we have to replace a derivation with a (m(B) A m(C))-cluster on the right of I- by two’derivations, one with a m(B)- cluster on the right of t and the other with a m(C)-cluster on the right of b). Maehara works with a multiple-conclusion sequent for- mulation of Heyting’s logic obtained from G(H,) by having only (- R), (1R) and (VR) fall under the Single-Conclusion Restriction. For this formulation, [A,, . _ _ , A,] t [B,, . . _ , I$,,], where n, m > 1, is interpreted as (A, A . . . A A,) + (B, v . . . v B,). In the absence of Thinning on the right, such multiple-conclusion sequent formulations of our intuitionistic systems BC and BCW cannot be introduced in the same way, but they also need not be introduced. It is enough to reduce applications of (1R) to passive ones. Maehara’s proof can be adapted to prove that BCK +m BCK,S4 by working with a multiple-conclusion sequent formulation of BCK. This formu- lation, in which [A,, . . . , A,] t [B,, . . . , B,,,], where n, m > 1, is interpreted as (A, l . . . l A,) + (B, v . . . v B,), is obtained from

MODAL TRANSLATIOTiS IN SUBSTRUCTURAL LOGICS 325

G(BCK,) by having only (+ R), (1 R), (VR) and (A R) fall under the Single-Conclusion Restriction. In the analogue of Proposition 6, we reduce applications of all these rules to passive ones. However, our proof via Proposition 7 takes better advantage of the absence of Con- traction on the right.

In Propositions 4-7, we have proved more than is strictly needed to obtain Propositions 8-10 and the embeddings of Propositions 1 l-13. We could have concentrated only.on the Gentzen systems G(X) and G(X,S4) where X is BC, BCW or BCK (cut elimination for these Gentzen systems is a rather standard affair). However, the other Gentzen system2 are interesting in their own right, and, at the cost of considering a few additional cases, we have preferred to state more general results.

6. SYSTEMS WITH EQUALITY

Suppose now that L and L3 have in their logical vocabulary the binary predicate = of equality. When the axiomatization of one of our Hilbert systems S is extended with the axiom-schemata:

(= elim) a = b -+ (AT, + Ai)

(= intr) a=a

we obtain the Hilbert system S=. It is clear that, if we write t for a = a. we can prove t -+ (A -+ A) and t in S= (cf. Anderson and Belnap 1975. (3 27.1.2, pp. 342-343; in the notation of Girard 1987, t would be written 1).

It might be more in the spirit of systems lacking (KL) and (KR) to have (= elim) with the proviso that x occurs free at least once in A, so that a occurs at least once in A: and h at least once in Al. We could have had something similar already in L without =, where we could require that V’xA and 3xA be well-formed only if x occurs free at least once in A. However, here we shall assume (= elim) without this proviso. What we want to say about modal translations does not depend on whether we have the proviso or not (other things may depend; for example, with the proviso, we cannot always prove t -+ (A -+ A)).

326 KOSTA DOSEN

The schemata (= elim) and (= intr) can be replaced by either of the following two schemata:

A;trvx(x=a+A)

A; ++ 3x(x = a. A)

(to obtain (= intr), we let A be x = a in the right-to-left direction of the first schema, and x = a + a = a in the left-to-right direction of the second schema; the remaining direction yields (= elim)).

It is an easy matter to obtain embeddings like those of Propositions l-3 for the corresponding systems with equality. However, for embed- dings like those of Propositions 1 l-14, the matter is more complex.

Problems arise with equality in the absence of (W). Namely, with- out some form of (W), we cannot assume (= elim) only for atomic formulae A, and then prove (= elim) for every formula A by induc- tion on the complexity of A. For example, from the two formulae:

we can pass to:

a = b -+ (a = b + ((C; . &) + (C; l Q)));

but then we need an instance of (W) to obtain:

a = b + ((C-,x. x) + (C; l G)).

However, the instance of (W) we need is provable in BC= , since from the following instance of (= elim):

a = b + ((a = b -+ (a = b --t B)) -+ (b = h + (b = b + B)))

with the help of(C), (= intr) and (modus ponens), we obtain:

(W=) (u=b-+(u=b+B))-+(u=b+B).

As we shall see below, related problems involving Contraction arise when we want to give a cut-free Gentzen formulation of a system with equality that lacks (W).

Let now S be one of the Hilbert systems of Proposition 5. To for- mulate the Gentzen system G(S=), which corresponds to S= in the sense of Proposition 5, we use the following notation. If X is

MODAL TRANSLATIONS IN SCBSTRUCTURAL LOGICS 327

[A,:. . . , A,], where n b 0, let Xl be [(A,);, . . . , (A,,):]. Then G(S=) is obtained by adding to G(S) the rules:

(= L) z;xt w:r ZiXk w;r

[a = b]Z,“X k w; Y [h = u]ZiX t w; Y

the basic sequents:

(=R) []!-[~=a]

and the following restricted form of (Contr):

(Conlr =) z=z=xk Y

Z”Xl- Y

where Z= is a finite multiset such that every formula from it is of the form a = h. Of course: (Contr =) is superfluous if we already have (Contr), but, in the absence of (Contr), we have to assume this new rule, which corresponds to (W =), in order to prove cut elimination.

If G(S) is a modal Gentzen‘system, then we introduce a modifi- cation in the definition of modal&d multisets. Namely, for G(S”), a multiset X’ is modulized iff every formula from XZ is of the form q IB or a = h. The modal rules are now understood in terms of this new definition of modalized multisets. This modification is needed because from the following instance of (= elim):

a = h + (O(u = u) + cl(a = 6))

with the help of (C), (= intr), (necessitation) and (modus ponens), we obtain a = h -+ q (u = b). The converse implication is an instance of (UT). The equivalence a = h ++ q (u = 6) enables us to make one additional economy in our modal translations. With the new defini- tion of modalized multisets, the rule (Contr =) is a special case of (Contr Cl).

The rules (= L) can equivalently be formulated with X and Y being the empty multiset. It is clear that in the presence of (Contr =), but not in its absence, the two rules (= L) can be replaced by the four rules:

[A;]X i Y X I- Y[A;]

[a = 6, A;]X I- Y [u = h]X + Y[A;]

[A;]X L Y Xl- Y[A;] [h = a, A;]X t Y [h = u]X - Y[A;]

328 KOSTA DOSEN

Cut elimination with equality is considered by Takeuti in (1975, chapter 1, 0 7) but not with (= L) and (= R). Postulates essentially identical with (= L) and (= R) may be found in Kanger (1963) and Lifschitz (1968), as well as in a number of notes, contemporaneous with the latter paper, by Mints, Orevkov and their collaborators, published mainly in the Seminar Notes of the Leningrad Mathematical Institute. However, I have been unable to find a published proof of cut elimination with these postulates (Lifschitz 1968 announces such a proof for a continuation, which does not seem to have appeared). So, it might be worth sketching the proof of the following proposition:

PROPOSITION 15. If‘ a regular sequent is in the Gentzen systems G(S=), then it has a derivation in G(S=) without (Cut).

Proof. As in the proof of Proposition 4, we first transform the deri- vation of a regular sequent into a disciplined derivation. Then we have one more preliminary transformation to ensure that in our deri- vation none of the rules of our Gentzen systems, apart from (= L), ever precede an application of (= L); i.e., we push every rule except (= L) below (= L). The cases that require an essential use of (Contr =) arise with the rules ( + L), (0 R) and (Cut). We take the case with (- L) for our example, in which the left figure is replaced by the right figure:

x t Y[&] (- L) [A;

[B,“]Z 1 w

(= L) + B,“]XZ t YW

[a = b, A: + l$]X’Z’ t- Y’W

x t Y[A;] [B,“]Z t w [a = b]X’ I- Y’[Ai] [a = 6, &]Z’ k W’ (= L,

All the other cases are rather straightforward. We remark only that, for (VR) and (3L), we use the fact that our derivation is disciplined, whereas, for (OR) and (Thin OR), we use the new definition of modalized multisets.

MODAL TRANSLATIONS IN SUBSTRUCTCRAL LOGICS 329

Before we continue with the proof. we introduce the following ter- minology. In an application of (= L), the formulae CI = h and b = u in the conclusion are said to be principalIy involved, and a formula Ai from the multiset Zi or Wdr in the conclusion is said to be suhordi- nately involved, provided Ai is different from AZ in the premise. A formula that is either principally or subordinately involved is said to be involved. In an application of (Contr =), a formula from Z’ is said to be involved.

Now, as in the proof of Proposition 4, we eliminate applications of (Cut) starting from the top. In addition to cases covered by that proof, we must consider the following cases:

(1) If the left premise of our application of (Cut) is a basic sequent as in the following figure:

(=R) []k[h=b] [b=b]XtY Xl- Y (Cut)

then we have to show that X t Y can be derived without (Cut). If the right premise is a basic sequent, or the conclusion of an appli- cation of a rule that is neither (= L) nor (Contr =), or the con- clusion of an application of (= L) or (Contr =) in which the cut- formula b = b is not involved, then we proceed as in the proof of Proposition 4.

If the cut-formula b = b is principally involved in the application of (= L) whose conclusion is [b = b]X t Y, we clearly have a deri- vation of X I- Y without (Cut). If it is subordinately involved in this application, then [b = b]X I- Y is like the first line of the following figure:

(= I-1 [a = b, b = b]Z;C: I- W;;‘V

[a = b, a = b: a = b]Z,” U I- Wdr V twice (Contr =) [a = b]Z,“U I- wd’v

whose last line is X i Y. If the cut-formula b = b is involved in the application of (Contr =)

whose conclusion is [b = b]Z= c’ F Y. where Z’ C: is X, as in the upper

330 KOSTA DOSEN

figure, then we can replace this figure by the lower figure:

[b = b]Z-[b = b]Z”U k Y

1 1 I- [b = 4 (Contr =)

[b = b]Z” U t- Y ccutj z=ub Y

[]k[b=b] [b=b]Z=[b=b]Z=UtY

1 1 I- Ib = bl Z= [b = b]Z= U t Y (Cut) (Cut)

Z’Z’UI- Y Z’UF Y

(Contr =)

(we have an analogous case with (Contr El) instead of (Contr =), where Z= is replaced by ZO). Formally, as with (Contr) and (Contr 0) in cases (2.2) and (3.2) of the proof of Proposition 4, we should then consider the elimination of the following restricted ver- sion of (Mix):

X k Y[a = b] [a = b]“Z k W xzt YW

where m 2 1. (2) If the left premise of our application of (Cut) is the conclusion

of an application of (= L) in which the cut-formula is not involved, we push (Cut) above (= L) as in the proof of Proposition 4. If the left premise of our application of (Cut) is the conclusion of an appli- cation of (= L) in which the cut-formula is involved (it can be so only subordinately), as in the left figure, we replace this figure by the right figure:

(Cut)

[&]U t v Z,xX t W: Y[A,“] [a = b, A:]U t ?’ (= L)

( = L) [a = b]Zb’XU t W; YV

(Contr =) la = b’ a = b]Z,xXU I- W,x YV

[a = b]Z,“XU I- Wd; YV

MODAL TRANSLATIONS IN SUBSTRUCTURAL LOGICS 331

Since, due to the preliminary transformation, above the application of (= L) in the left figure, we may have only applications of (= L), after finitely many such replacements, the left premise of our appli- cation of (Cut) will be a basic sequent.

(3) Let the left premise of our application of (Cut) be the con- clusion of an application of a rule (*) that is not (= L), and let the cut-formula be involved in this application of (*) by having a connec- tive or quantifier introduced by (*) (all the other cases are dealt with as in the proof of Proposition 4). Then; considering how the right premise of our application of (Cut) was obtained, the only case not covered by the proof of Proposition 4 is when the right premise is the conclusion of an application of (= L) in which the cut-formula is subordinately involved (it cannot be principally involved because it is not of the form u = b). Then we shift (= L) from the right premise to the left premise, as in case (2) we shifted it in the opposite direc- tion. Namely, we replace the left figure by the right figure:

ly’ !- v (*) u 1 V[A;]

[Af]Z,:X I- w,x Y

(Cut) [a = b, A;]Z;X k W: Y (= L,

[a = b]KZ;X I- VW;Y

(*) U’ I- V’

(= L) u I- V[&]

(Cut) [a = b]U I V[A:] [A:]Z,“X t Wd; Y (= L) [a = b]UZ,“X 1 VW; Y

(Contr =) b ; a

“b;u~~~~;~w~~ Y b b”

Since, due to the preliminary transformation, above the application of (= L) in the left figure, we may have only applications of (= L), after finitely many such replacements, the right premise of our appli- cation of (Cut) will be a basic sequent of the form [A] 1 [A]. q.e.d.

It is an easy matter to prove that every Gentzen system G(S’) is equivalent in the sense of Proposition 5 with the Hilbert system S= . Note that, as we have shown at the beginning of this section, in the systems S=, we can prove (W =), which corresponds to (Contr =).

332 KOSTA DOSEN

We can easily extend the proofs of the auxiliary propositions 6 and 7 to obtain analogues of these propositions for modal Gentzen sys- tems G(S=). However, for the proof of the analogue of Proposition 6, we have to modify our definition of an A-cluster. The multisets Z,X, W” Z” and w” in applications of the rule (= L) are also considered 07 h h as parametric multisets, and an A: from, for example, Z: may be fclustered with A$ from Zi. So, an A-cluster need not be made any more only with occurrences of A, but can be made also with occur- rences of substitution instances of A that correspond to each other according to (= L).

We can then prove for every regular formula A of L that, if [ ] I- [m(A)] is in G(X,!%=), where X is BC, BCW or BCK, then [ J t [A] is in G(X =), from which we obtain the following analogues of Propositions I l-13:

l$BCD4= 5 S E BC,S4=, then BC= +m S;

tfBCD4= + (WO) E S c BCW,S4’, then BCW = +m S;

ifBCD4’ + (KCIL), (KCIR) s S c BCK,S4=, then BCK= +“’ S.

The following analogue of Proposition 14:

ifBCD4= + (WO), (KOL), (KOR) c S & H,S4’, then HE +m S

follows From Hz +m H,S4=. I don’t know of any published Gentzen- style proof of the fact that Hz -+m H,!U’, but an elegant syntactic proof, inspired by algebra, may be found in Flagg and Friedman (1986). Maehara’s proof of (1954), mentioned at the end of Section 5 above, could be adapted with the help of (= L) and (= R) to yield a proof of this embedding.

7. TRANSLATIONS OF PROOFS FROM HYPOTHESES

The embeddings considered in this paper were of the form:

(1) A is provable in S, iff r(A) is provable in S,.

It is often considered that one has proved more if for a relation IF of syntactic consequence (i.e. deducibility from hypotheses) between sets

MODAL TRANSLATIONS IN SUBSTRUCTURAL LOGICS 333

of formulae I and formulae one has established:

(II) I It A holds for S, iff {r(B): B E I’} It r(A) holds for S?.

However, if our relation of syntactic consequence is compact (i.e., I IF A iff, for some finite I’ E I, I’ It A), and we have modus ponens and a deduction theorem so that:

{B,, . . . , B,,t It A holds for S iff B, -+ (. . . -+ (B, + A) . . . ) is provable in S

then (I) and (II) need not differ essentially, and, for many translations z, one can easily obtain (II) from (I). In any case. with substructural logics, it may not always be clear what is the appropriate relation of syntactic consequence (cf. Do&n 1990). If It is assimilated to the t of our Gentzen systems, then we can easily infer from what we have already estabished that, for example:

(II’) [B,, . . . . B,] k [A] is in G(X) iff

LtiB,),..., m(B,,)] 1 [m(A)] is in G(X,S4)

(we just use the fact that, in X,S4. we can prove m(C) iff we can prove the formula obtained from m(C) by omitting the main box and boxes prefixed to fusions). The equivalence (II’) holds also when X is replaced by X= and X,S4 by X,S4-. So, behind our results of form (I), it may not bc difficult to find apparently stronger results of form (II). Note that in (II’) we can replace m by more economical translations, but not by m’ or m”, since in G(X,S4) we may lack [A. El.,4 + q lB] k [B] or [A, q A + B] t [B] (cf. Prawitz and Malmnas 1968, pp. 219, 222).

ACKNOWLEDGEMENTS

I would like to thank the Science Fund of Serbia (Grant 04OlA) and the Alexander von Humboldt Foundation for supporting my work on this paper. I would also like to thank the Universities of Constance and Tubingen, and, in particularl Peter Schroeder-Heister and Andre Fuhrmann, for their hospitality. I am grateful to Johan van Benthem and the Dyana project for enabling me to present my results in a talk at the University of Amsterdam. I am also grateful to Andreja

334 KOSTA DOSEN

Prijatelj, Harold Schellinx, Dirk Roorda and Maarten de Rijke for their hospitality and willingness to discuss matters related to these results. In particular, Andreja Prijatelj has drawn my attention to an infelicity in the exposition in the first version of the paper.

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