Categorial grammar, modalities and algebraic semantics 
Koen Versmissen 
Onderzoeksinstituut voor Taal en Spraak 
Universiteit Utrecht 
Trails 10 
3512 JK Utrecht 
Netherlands 
koen. versmissen@let, ruu. nl 
Abstract 
This paper contributes to the theory of 
substructural logics .that are of interest to 
categorial grammarians. Combining se- 
mantic ideas of Hepple \[1990\] and Mor- 
rill \[1990\], proof-theoretic ideas of Venema 
\[1993b; 1993a\] and the theory of equational 
specifications, a class of resource-preserving 
logics is defined, for which decidability and 
completeness theorems are established. 
1 Introduction 
The last decade has seen a keen revival of investi- 
gations into the suitability of using categorial gram- 
mars as theories of natural language syntax and se- 
mantics. Initially, this research was for the larger 
part confined to the classical categorial calculi of Aj- 
dukiewicz \[1935\] and Bar-Hillel \[1953\], and, in partic- 
ular, the Lambek cMculus L \[Lambek, 1958\], \[Moort- 
gat, 1988\] and some of its close relatives. 
Although it turned out to be easily applicable to 
fairly large sets of linguistic data, one couldn't real- 
istically expect the Lambek calculus to be able to ac- 
count for all aspects of grammar. The reason for this 
is the diversity of the constructions found in natural 
language. The Lambek calculus is good at reflect- 
ing surface phrase structure, but runs into problems 
when other linguistic phenomena are to be described. 
Consequently, recent work in categorial grammar has 
shown a trend towards diversification of the ways in 
which the linguistic algebra is structured, with an 
accompanying ramification of proof theory. 
One of the main innovations of the past few years 
has been the introduction of unary type connectives, 
usually termed modalities, that are used to reflect 
certain special features linguistic entities may pos- 
sess. This strand of research originates with Morrill 
\[1990\], who adds to L a unary connective O with the 
following proof rules: 
F,B,F'FA \[mL\] OF~-A \[oR\] 
F, OB, F ~ b A OF b DA 
OF here denotes a sequence of types all of which 
have O as their main connective. The S4-1ike modal- 
ity o is introduced with the aim of providing an ap- 
propriate means of dealing with certain intensional 
phenomena. Consequently, O inherits Kripke's pos- 
sible world semantics for modal logic. The proof sys- 
tem which arises from adding Morrill's left and right 
rules for \[\] to the Lambek calculus L will be called 
Lb. 
Hepple \[1990\] presents a detailed investigation into 
the possibilities of using the calculus L• to account 
for purely syntactic phenomena, notably the well- 
known Island Constraints of Ross \[1967\]. Starting 
from the usual interpretation of the Lambek calculus 
in semigroups L, where types are taken to denote 
subsets of L, he proposes to let D refer to a fixed 
subsemigroup Lo of L, which leads to the following 
definition of its semantics: 
\[oAf = \[A\]n Lo 
As we have shown elsewhere \[Versmissen, 1992\] 1 
the calculus LD is sound with respect to this seman- 
tics, but not complete. This can be remedied by 
1This paper discusses semigroup semmatics for L and 
LO in detail, and is well-suited as an easy-going in- 
troduction to the ideas presented here. It is available 
by anonymous ftp from ftp.let.ruu.nl in directory 
/pub/ots/papexs/versmissen, files adding.dvi.Z and adding, 
ps. Z. 
377 
replacing the rule \[OR\] with the following stronger 
version: 
FlbOB1 ... Fo~-OB, F1,...,FnbA 
rl,..., F, ~- raA \[oR'\] 
Hepple \[1990\] also investigates the benefits of us- 
ing the so-called structural modalities originally pro- 
posed in \[Morrill et al., 1990\], for the description 
of certain discontinuity 'and dislocality phenomena. 
The idea here is that such modalities allow a limited 
access to certain structural rules. Thus, we could for 
example have a permutation modality rap with the 
following proof rule (in addition to \[rapL\] and \[OpR'\] 
as before): 
r\[oeA, B\] ~ C 
r\[8, opA\] ~- C 
The symbol ~ here indicates that the inference is 
valid in both directions. The interpretation of OR 
would then be taken care of by a subsemigroup Lop 
of L having the property that x • y = y • x whenever 
z•Lnpory•Lop. 
Alternatively, one could require all types in such 
an inference to be boxed: 
F\[rapA, DpB\] I- C I 
r\[opB, OpA\] ~- C 
In this case, Lop would have to be such that z. y = 
y- x whenever z, y • Lop. 
Closely related to the use of structural modalities 
is the trend of considering different kinds of prod- 
uct connectives, sometimes combined into a single 
system. For example, Moortgat & Morrill \[1992\] 
present an account of dependency structure in terms 
of headed prosodic trees, using a calculus that pos- 
sesses two product operators instead of just one. On 
the basis of this, Moortgat \[1992\] sketches a land- 
scape of substructural logics parametrized by prop- 
erties such as commutativity, associativity and de- 
pendency. He then goes on to show how structural 
modalities can be used to locally enhance or con- 
strain the possibilities of type combination. Morrill 
\[1992\] has a non-associative prosodic calculus, and 
uses a structural modality to reintroduce associativ- 
ity at certain points. 
The picture that emerges is the following. Instead 
of the single product operator of L, one considers a 
range of different product operators, reflecting differ- 
ent modes of linguistic structuring. This results in 
a landscape of substructural logics, which are ulti- 
mately to be combined into a single system. Specific 
linguistic phenomena are given an account in terms 
of type constructors that are specially tailored for 
their description. On certain occasions it is necessary 
for entities to 'escape' the rules of the type construc- 
tor that governs their behaviour. This is achieved by 
means of structural modalities, which license con- 
trolled travel through the substructural landscape. 
Venema \[1993a\] proves a completeness theorem, 
with respect to the mentioned algebraic interpreta- 
tion, for the Lambek calculus extended with a per- 
mutation modality. He modifies the proof system by 
introducing a type constant Q which refers explicitily 
to the subalgebra Lo. This proof system is adapted 
to cover a whole range of substructural logics in \[Ve- 
nema, 1993b\]. However, the semantics given in that 
paper, which is adopted from Dogen \[1988; 1989\], dif- 
fers in several respects from the one discussed above. 
Most importantly, models are required to possess a 
partial order with a well-behaved interaction with 
the product operation. In the remainder of this pa- 
per we will give a fairly general definition of the no- 
tion of a resource-preserving logic. The proof theory 
of these logics is based on that of Venema, while their 
semantics, with respect to which a completeness the- 
orem will be established, is similar to that of Hepple 
and Morrill. 
2 Resource-preserving logics with 
structural modalities 
2.1 Syntax 
The languages of the logics that will be considered 
here are specified by the following parameters: 
t~ Three finite, disjoint index sets Z, J and/C; 
A finite set B of basic types. 
Given these, we define the following sets of expres- 
sions: 
The set of binary type connectives 
c = {/i, \0~z; 
Two sets of unary type connectives 
M~ = {Aj}je.~ and M v = {~Tk}~¢Jc; 
~, The set of type constants q = {Qj}j~ u {Qk}kE~; 
The set of types T, being the inductive closure 
of B U Q under C U Mz~ U M e; 
The set of structural connectives SC = {oi}iez; 
The set of slructures S, being the inductive clo- 
sure of T under SC; 
c, The set of sequents {F b A I r • S,A • T}. 
The division of the unary type connectives into two 
sets Ma and M v reflects the alternatives mentioned 
in Section 1. Modalities/Xj are those whose struc- 
tural rules only apply when all types involved are 
prefixed with them, whereas only a single type pre- 
fixed with XTk needs to be involved in order for the 
accompanying structural rules to be applicable. 
2.2 Equational specifications 
We will use equational specifications to describe the 
structural behaviour of connectives and modalities, 
as well as the algebraic structures in which these are 
interpreted. To start with, we recall several impor- 
tant definitions and results. 
378 
A signature E is a collection of function symbols, 
each of which has a fixed arity. Let V be a countably 
infinite set of variables. The term algebra T(E, 1)) is 
defined as the inductive closure of l; under ~. An 
equational specification is a pair (~,,~) where ~ is a 
signature and E is a set of equations s = t of terms 
s,t E T(~,12). A ~-algebra .4 is a set A together 
with functions F A : A" --* A for all n-ary function 
symbols F E ~. A E-algebra .4 is a model for a set 
of equations E over T(~, N), written as .4 ~ £, if 
every equation of ~ holds in A. A (E, g)-algebra is 
a ~-algebra that is a model for £. 
Let E be an equational specification. Then we de- 
fine Ezxi to be the equational specification obtained 
from E by prefixing each variable occurrence with 
A~. The equational specification Ev~ is defined as 
follows (where V(F = G) denotes the set of variables 
occurring in F = G): 
(F=G)lx*'-Vkz\] mD FI~*-vkxl=G\[x*"Vkx\] 
(F=G)v/, =O UzCV(F=o) (F=G)\[z*-Vk*\] 
£vk --D UEE~r Ev~ 
To give a concrete example of these definitions, let E 
consist of the following two equations: 
x+y = y+x 
x+(y+z) = (x+y)+z 
Then ~ contains these two: 
Ajz+Ajy = Ajy + Ajx 
A~x+(A~y+Aiz ) = (Ajx+A~y)+A~z 
whereas gw is comprised of five equations in all: 
Vkz+Y = y+~7kz 
z+Wky = VkY+X 
w~+(y+z) = (wx+y)+z 
x+(Vky+z) = (x+Wy)+z 
x+(y+Vkz) = (x+y)+Vkz 
We will call a term equation resource-preserving if 
each variable occurs the same number of times on 
both sides of the equality sign. An equational spec- 
ification is resource-preserving if all of its member 
equations are. Note that this definition encompasses 
the important cases of commutativity and associa- 
tivity. On the other hand, well-known rules such 
as weakening and contraction can't be modelled by 
resource-preserving equations. Not only do they fail 
to be resource-preserving in the strict sense intro- 
duced here, but also they are one-way rules that 
would have to be described by means of rewrite rules 
rather than equations. 
2.3 Resource-preserving logics 
A resource-preserving logic is determined by the fol- 
lowing: 
Instantiation of the language parameters B, Z, 
,7 and K; 
t, An equational specification E over the signature 
{+~}iEz; 
Two sets of indices {ij}j¢,7, {ik}~er C_ Z; 
t> Two sets of equational specifications {Ej}jej 
and {Ek}ke/c, where Et is specified over the sig- 
nature {+i, } (I E ,7 U K). 
Of course, all equational specifications occurring in 
the above list are required to be resource-preserving. 
The operator + is intended as a generic one, which 
is to be replaced by a specific connective of the lan- 
guage on each separate occasion. We will write £* for 
the equational specification obtained by substituting 
• for + in E, but will drop this superscript when it 
is clear from the context. (Ej)zxi will be abbreviated 
as £~j, and (£k)Vk as £W" 
Henceforth, we assume that we are dealing with a 
fixed resource-preserving logic £. 
2.4 Proof system 
For £ we have the following rules of inference: 
AFA 
FI-A A(B) I- fi roiAt- B 
A\[(BIiA) ol r\] ~ c \[/,L\] r k aliA \[/,R\] 
FI-A A(B) I- C Aoirl- B \[\~L\] \[~iR\] 
A\[r ol (A\iB)\] ~- C r I- A\iB 
F, FQt r~l-Ot r\[Q,\]l-A \[Qd 
r\[rl oi, r2\] ~- A 
r\[Al i- B 
F\[AjA\] F B 
rtQjl ~ e 
r\[A/A\] k B \[~jL2\] 
r\[A\] F B r\[o,\] F a 
r\[VkA\] I- B \[vkLq r\[VkA\] k- B \[vkL2l 
FI-A FI-Q/ \[A#R\] FFA rl--Ok \[VkR\] 
r b AjA r b VkA 
rI-A rI-A r,t-Q, ... r,F-0al\[~d 
A ~ A ! \[E\] 'A k A 
rI-A A\[A\] I- B \[Caq 
A\[F\] b B 
In these rules i, j and k range over I, `7 and JC, 
respectively, and 1 ranges over `7 U/U. As before, a 
I indicates that we have a two-way inference rule. 
The \[£(0\]-rule schemata are subject to the following 
condition: there exist an equation s = t E E(' 0 and a 
substitution a : V -- T such that A can be obtained 
from r by replacing a substructure s ~ of r with ft. 
On \[Ell we put the further restriction that the ri's 
are exactly the elementary substructures of s a. For 
example, for gj = {x + y = y + z} we would obtain 
the following rule: 
r~ k oi r~ k Oi r\[r, % r2\] k A I 
\[6\] r\[r2 % r,\] f- A 
379 
NP I- NP 
NPI-NP SI-S \[\L\] 
NP\S I- NP\S NP, NP\S I- S 
NP, NP\S, (NP\S)\(NP\S) I- S \[~L\] \[/LI 
NP, NP\S/NP, NP, (NP\S)\(NP\S) I- S 
NP, NP\SINP, vpNP, (NP\S)\(NP\S) I- S \[VpLI\] 
NP, NP\S/NP, (NP\S)\(NP\S), VpNP I- S \[evp\] \[IR\] 
NP, NP\S/NP, (NP\S)\(NP\S) I- Sl V~' NP REL I- REL 
RELI(S/VP NP), NP, NP\S/NP, (NP\S)\(NP\S) I- REL 
Figure 1 
I/L\] 
NI-N NPI-NP \[/L\] 
(NP/N) o N I- NP 
NI-N NPI-NP I/L\] 
NI-N (NPIN) oNI-NP \[~L\] 
(NPIN) o (N o (N\N)) I- NP \[vALll 
(NP/N) o (N o VA(N\N)) F NP \[E~A\] 
((NP/N) o N) o VA(N\N) I- NP \[IR\] 
(NP/N) o N I- NP/VA (N\N) NP I- NP \[\L\] 
((NP/N) o N) o ((NP/VA (N\N))\NP) I- NP \[ILl 
((NP/N) o N) o ((((NP/VA (N\N))\NP)/NP) o ((NP/N) o N)) F NP 
Figure 2 
2.5 Some sample applications 
We will address the logical aspects of the calculi de- 
fined in the last section shortly, but first we pause for 
a brief intermezzo, illustrating how they are applied 
in linguistic practice. 
As our first example we look at how the Lambek 
calculus deals with extraction. Suppose we have the 
following type assignments: 
John, Mary : NP 
loves : NP\S/NP 
madly : (NP\S)\(NP\S) 
We would like to find type assignments to who 
such that we can derive type REL for the following 
phrases: 
1. who John loves 
2. who loves Mary 
3. who John loves madly 
As is easily seen, assignment of REL/(S/NP) to who 
works for the first sentence, while REL/(N P\S) is the 
appropriate type to assign to who to get the second 
case right. However, the third case can't he done 
in the Lambek calculus, since we have no way of 
referring to gaps occuring inside larger constituents; 
we only have access to the periphery. This can be 
handled by adding a permutation modality VP and 
assigning to who the type REL/(S/VP NP) to who. 
This single type assignment works for all three cases. 
For the third sentence, this is worked out in Figure 1. 
As a second example, consider the following noun 
phrase: 
the man at the desk 
For the nouns and the determiner we make the usual 
type assignments: 
the : NP/N 
man, desk : N 
From a prosodic point of view, at should be assigned 
type (N\N)/NP. However, semantically at combines 
not just with the noun it modifies, but with the en- 
tire noun phrase headed by that noun. Moortgat & 
Morrill \[1992\] show how both these desiderata can 
be fulfilled. First, the type assignment to at is lifted 
to ((NP/(N\N))\NP)/NP in order to force the re- 
quired semantic combination. This is not the end 
of the story, because due to the non-associativity of 
the prosodic algebra we still can't derive a type NP 
for the man at the desk. To enable this, they add a 
structural modality VA to the type assignment for 
at to make it ((NP/VA (N\N))\NP)/NP, after which 
things work out nicely, as is shown by the derivation 
in Figure 2. 
2.6 Cut-elimination and the subformula 
property 
Before turning to the semantics of/~ we will prove 
the Cut-elimination theorem and subformula prop- 
erty for it, since the latter is essential for the com- 
pleteness proof, and a corollary to the former. 
380 
As we remarked earlier, our proof rules are 
adapted from \[Venema, 1993b\]. Therefore, we can 
refer the reader to that paper for most of the Cut- 
elimination proof. The only notable difference be- 
tween both systems lies in the structural rules they 
allow. Note that resource-preservation implies that 
for any \[E(j)\]-inference we have the following two sim- 
ple but important properties (where the complexity 
of a type is defined as the number of connectives oc- 
curring in it): 
1. Each type occurring in r occurs also in A, and 
vice versa; 
2. The complexity of r equals that of A. 
Therefore, in the case of an \[C(0\]-inference, we can 
always move \[Cut\] upwards like this is done in Ve- 
nema's paper, and thus obtain an application of \[Cut\] 
of lower degree. Hence, \[Cut\] is eliminable from £. 
The subformula property says that any provable 
sequent has a proof in which only subformulas of that 
sequent occur. Under the proviso that Qj is consid- 
ered a subtype of AiA, and QI, of wkA, the subfor- 
mula property follows from Cut-elimination, since in 
each inference rule other than \[Cut\], the premises are 
made up of subformulas of the conclusion. 
Let £. be the logic obtained from £ by adding a 
set of product connectives {*i}iez to the language, 
and the following inference rules to the proof system: 
roiAI-A \[,~L\] rFh AFB \[.~a\] 
r.i A P A roi A F A*i B 
Like £, the system £, enjoys Cut-elimination and 
the subformula property. Note that this implies that 
if an £-sequent is/:.-derivable, then it is £-derivable. 
This property will be used several times in the course 
of the completeness proof. 
Now consider a naive top-down 2 proof search strat- 
egy. At every step, we have a finite choice of possi- 
ble applications of an inference rule, and every such 
application either removes a connective occurence, 
thus diminishing the complexity of the sequent to 
he proved, or rewrites the sequent's antecedent to a 
term of equal complexity. Therefore, if we make sure 
that a search path is relinquished whenever a sequent 
reappears on it (which prevents the procedure from 
entering into an infinite loop), the proof search tree 
will be finite. This implies that the calculus is decid- 
able. 
2.7 Semantics 
The basis for any model of £ is a (E, C)-algebra ,4, 
where I\] = {+i}iex and the product operation in- 
terpreting oi is denoted as "i. We say that 3 C ,4 
is an Fd-subalgebra of ,4 if it is closed under .~j, and 
2Note that we use the term top-down in the usual 
sense, i.e. for a proof search procedure that works back 
from the goal to the axioms. Visually, top-down proofs 
actually proceed bottom-up! 
s ° = t ¢ whenever s = t E gj and a : V --, 8. An easy 
Ck-subalgebra of`4 is a subset of ,4 that is closed un- 
der "ik, and such that s ° = t ° whenever s = t E gk 
and a : V --* ,4 assigns an element of $ to at last 
one of the variables occurring in the equation. A 
model for £ is a 4-tuple (,4, {,4j}jeJ, {,4k}ke~:, i.I) 
such that: 
t> ,4 is a (~, C)-algebra; 
Aj is an Ci-subalgebra of `4 (j E if); 
t> `4k is an easy gk-subalgebra of`4 (k E/C); 
t, \[.\] is a function B --* 7)(`4). 
Here, :P(,4) denotes the set of all subsets of,4. The 
interpretation function \[.\] is extended to arbitrary 
types and structures as follows: 
\[Od = ,4t (l e y u Ic) 
t> IB/,A\] = {c e ,4 I Va e \[A\]: c., a e \[\[3\]} 
> \[A\iB\] = {c E ,4 I Va e \[A\] : a "i c E \[13\]} 
z> EAoiB\]--{cE,4\[~aE\[A\],bE\[Bl:c=a.+b} 
A sequent F k A is said to be valid with respect to 
a given model, if ir\] g \[A\]. A sequent is gene~lly 
valid if it is valid in all models. The proof system 
is said to be sound with respect to the semantics if 
all derivable sequents are generally valid. It is com- 
plete if the converse holds, i.e. if all generally valid 
sequents are derivable. 
2.8 Soundness and completeness 
As usual, the soundness proof boils down to a 
straightforward induction on the length of a deriva- 
tion, and we omit it. 
For completeness, we start by defining the canon- 
ical model .A4. Its carrier is the set S/--, where 
= is the equivalence relation defined by r _-- A iff 
VA : r F A ¢~ A F A. The ----equivalence class con- 
taining F will be denoted as \[r\]. On the set S/_= 
we define products "i (i E 27) by stipulating that 
\[r\] .i \[A\] = \[r oi A\]. We need to prove that this 
is well-defined. So suppose r - r', A - A' and 
r oi A F A. For a structure O, let O* be the £.-type 
obtained from O by replacing each oi with oi. The 
sequent O* \[- A can be derived from O ~- A by a 
sequence of \[.L\]-rules. By definition of -- we know 
that r ' F" r* and A ~ }" A*. Now, r' ol A' I- A by the 
derivation below: 
r ol A I- A \[.L\]* r' r" r ° oi A ° I- A t- 
r' oi A" }- A \[Cut\] A' I- A" 
r' ol A' I- A \[Cut\] 
Evidently, .A4 = (S/=, {.i}icz) is a (E, ~)-algebra. 
Next, we define ¢~41 = {IF\] \[ F ~- Qz} (! e ,\] u/C). 
It must be shown that these have the desired prop- 
erties. Since it would be notationally awkward to 
have to refer to an arbitrary equational specifica- 
tion, we do this by means of an example. Let 
381 
rl oi# r~ • A 
Q# • Q# r~ oi# r~ I- A \[.L\] 
AjF~,2 • Q# \[A#L2\] Air~ oi# AiF~ t- A \[A#L1\] 
AjF~ oi~ A#F~ • A \[t:Aj\] 
r~ oi# A#F~ b A 
r~ • zx#rt \[Z~#R\] \[Cut\] 
r2 oi# rl • A 
r2 F r~ F2 F Qj 
r2 F Air** \[AIR\] 
\[c.t\] 
Figure 3 
ga# = {Aim +i~ Ajy = Ajy +ij Ajx}. Sup- 
posing that \[rl\], \[r2\] • .N4Aj we must prove that 
\[rl\] "ij IF2\] = \[r2\] "ij IF1\], i.e. that VA : r, % F2 F 
A ,## r~. oij rl F A. This follows from the derivation 
in Figure 3. The proof for A4Vk is similar. 
Finally, we set \[B l - {\[r\]l r e B} for B • 
B, which completes our definition of the canonical 
model. 
We proceed to prove the so-called canonical 
lemma: 
Lemma 
IT\] = {\[r\] I r F T} for all T • T. 
Proof 
We prove this by induction on the complexity of the 
type T. 
~, For basic types T it is true by the definition of 
\[.\]; 
~, For Qt (1 • 3" U/C) it is true by the definition of 
A4a; 
~, For T = B/iA: 
1. First, suppose \[r\] • ~"\]\] -- ~'B/ia\]. Then 
for any \[A\] • \[A\] we have that \[F\]., \[A\] = 
\[r oi A\] • \[B\]. By the induction hypothesis 
we deduce from this that r oi A I- B. In 
particular, since \[A\] • \[\[A\], we have that 
r ol A I- B, whence, by \[/iR\], it follows that 
r I- B/iA. 
2. Conversely, suppose that r F B/iA, and let 
\[A\] • ~A\]. Then, by the induction hypoth- 
esis, A I- A. We now have the following 
derivation: 
AI-A BI-B \[/,L\] 
r ~ B/,A (B/,A) o, A I- B 
A F A r oi A I- B \[Cut\] 
r oi A F B \[Cut\] 
From this we conclude by the induction hy- 
pothesis that IF oi A\] = \[r\] .i \[A\] • \[B\] for 
all \[A\] • \[A\]. That is, \[F\] • \[B/IA\]I, and 
we're done. 
For the other binary connectives, the proof is 
similar. 
t> For T = AjA: 
I. First, suppose \[r\] • \[AjA\] = \[A\]n A41. 
Then, by the induction hypothesis, r F A. 
Also, by the definition of A4~, I" t- Qj. 
Applying the \[Aj R\]-rule two these two se- 
quents, we find that I" I- AjA. 
2. Conversely, suppose r I- AjA. Then r I- A: 
A k A \[A~L1\] FFAjA AjA I- A 
r F A \[cut\] 
From this we conclude by the induction hy- 
pothesis that \[F\] • \[Al. Also, r 
\[&#L2\] FFA~A AjA I- Qj 
r F Q~ \[Cut\] 
From this we find by the definition of .A4j 
that \[r\] • \[Qj\] = .A4j. So \[r\] • lAin 
\[Qfll = IAjA\]. 
For ~7k, the proof is similar. 
Now suppose that the sequent r I- A is not derivable. 
Then in the canonical model we have, by the lemma 
we just proved, that \[r\] ¢ \[\[A\]. Since IF\] • \[r\], this 
implies that IF\] ~ \[\[A\]. That is, r I- A is not valid 
in the canonical model, and hence is not generally 
valid. \[\] 
3 Further research 
It will not have escaped the reader's attention that 
we have failed to include the set of product con- 
nectives {.i}iEz in the language of the resource- 
preserving logics. The reason for this is that a com- 
pleteness proof along the above lines runs into prob- 
lems for such extended logics. This is already the 
case for the full Lambek calculus. Buszkowski \[1986\] 
presents a rather complicated completeness proof for 
that logic. It remains to be seen whether his ap- 
proach also works in the present setting. 
Although we've tried to give a liberal definition 
of what constitutes a resource-preserving logic, some 
choices had to be made in order to keep things man- 
ageable. There is room for alternative definitions, 
especially concerning the interaction of the modali- 
ties with the different product operators. It would 
seem to be worthwile to study some of the systems 
that have occurred in practice in detail on the basis 
of the ideas presented in this paper. 
382 
Finally, it is important to realize that we limited 
ourselves to resource-preserving logics in order to ob- 
tain relatively easy proofs of Cut-elimination and 
decidability. Since such results tend also to hold 
for many systems with rules that are not resource- 
preserving, such as weakening and contraction, it is 
probably possible to characterize a larger class of 
equational theories for which these properties can be 
proved. We hope to address this point on a later 
occassion. 
Acknowledgements 
The task of preparing this paper was alleviated con- 
siderably thanks to enlightening discussions with, 
and comments on earlier versions by Kees Ver- 
meulen, Yde Venema, Erik Aarts, Marco Hollenberg 
and Michael Moortgat. 
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