Lambek Calculus and its relational semantics: Completeness and incompleteness

Lambek Calculus and its Relational Semantics: Completeness and Incompleteness

HAJNAL ANDRt~KA* and SZABOLCS MIKULftS* Mathematical Institute of the Hungarian Academy of Sciences Budapest, Pf 127, H-1364, Hungary

(Received 13 November 1992; in final form 2 June 1993)

Abstract. The problem of whether Lambek Calculus is complete with respect to (w.r.t.) relational semantics, has been raised several times, cf. van Benthem (1989a) and van Benthem (1991). In this paper, we show that the answer is in the affirmative. More precisely, we will prove that that version of the Lambek Calculus which does not use the empty sequence is strongly complete w.r.t, those relational Kripke-models where the set of possible worlds, W, is a transitive binary relation, while that version of the Lambek Calculus where we admit the empty sequence as the antecedent of a sequent is strongly complete w.r.t, those relational models where W = U x U for some set U. We will also look into extendability of this completeness result to various fragments of Girard's Linear Logic as suggested in van Benthem (1991), p. 235, and investigate the connection between the Lambek Calculus and language models.

Key words: Lambek Calculus, relational semantics, language models, algebraic logic, residuated semigroups, representation theorems.

1. INTRODUCTION

Lambek Calculus was introduced (Lambek, 1958) with both linguistic and logical motivations. It has been strongly investigated since then, e.g., because it is an example of substructural logics. Van Benthem (1991) relates this inves- tigation well. See also Buszkowski (1986), Dogen (1990), Roorda (1991), Gabbay (1992), Pentus (1993).

The so-called relational semantics for Lambek Calculus was suggested by Johan van Benthem around 1988, van Benthem (1989b), where plenty of motivation is given for this semantics. Clearly, relational semantics is strongly motivated by dynamic semantics for natural languages. We quote the "slogan" behind relational semantics: "Natural language is a programming language for effecting cognitive transitions between information states of its users." It is proved by van Benthem (1991) that Lambek Calculus is sound w.r.t. relational semantics, and it was asked whether it was also complete. In this

* Supported by Hungarian National Foundation for Scientific Research grant Nos. 1911, 2258, and by TEMPUS JEP No. 1941-92/2.

The first report on the first part of this work was in the Banach Centre, Warsaw, 17th October 1991.

Journal of Logic, Language, and Information 3: 1-37, 1994. (~) 1994 Kluwer Academic Publishers. Printed in the Netherlands.

2 HAJNAL ANDRI~KA AND SZABOLCS MIKULAS

paper we prove (Thm.2.1) that Lambek Calculus is indeed complete w.r.t. relational semantics.

In order to prove this completeness, we had to allow "relativized" relational models, because the original Lambek Calculus is not complete w.r.t. "unrelativized", or "absolute" relational models. The question of what strengthening of Lambek Calculus would make it complete w.r.t, the more natural unrelativized relational semantics, naturally arises. It turns out that Lambek Calcu- lus can be modified in two very natural ways, both modifications making it complete w.r.t, the stronger relational semantics. (See Thm.3.1.)

All the above concerned strong completeness. Given an arbitrary inference system, we call it strongly complete if it models the consequence relation between sets of formulae and formulae, and we call it weakly complete if it reflects only tautologies (or theorems). In more detail, an inference system is strongly complete if for every set r of formulae, and for every formula ~, cp can be derived (proved) from [' whenever ~p is a semantical consequence of 1-'. The inference system is weakly complete if ~ can be derived whenever cp is a valid formula.

We also investigate a connection with another kind of semantics for Lam- bek Calculus, the so-called language-model semantics. More precisely, we will concentrate on language models where we admit languages with the empty word. We show that Lambek Calculus is not weakly complete and that there is no strengthening of Lambek Calculus which is sound w.r.t, relational semantics and would be strongly complete w.r.t, the language models of the above kind. Weak completeness of Lambek Calculus w.r.t, language models without the empty word had long been an intriguing open problem. Interesting results in this line can be found, e.g., in Buszkowski (1986) and in Gabbay (1992). Recently Pentus (1993) gave a positive solution.

We also investigate what happens if we introduce new connectives in Lambek Calculus, moving towards Linear Logic. We find that structural (or additive, or static, or Boolean) conjunction does not cause any problem, however, "structural" disjunction makes a strongly complete strengthening impossible. (Weak completeness is still possible.)

2. STRONG COMPLETENESS OF LAMBEK CALCULUS W.R.T. RELATIONAL SEMANTICS

2.1. Definition of the Lambek Calculus

First of all, let us define the language of Lambek Calculus, LC. Given a denumerable set P of primitive symbols, we let the set FormLc of formulae be the smallest set containing every primitive symbol and closed under \ , / ,

LAMBEK CALCULUS AND ITS RELATIONAL SEMANTICS 3

and e, i.e., if A, B E FormLc, then A \ B , A / B , A �9 B C FormLo The set of sequents is the set of all expressions of the form A 1 , . . . , An =~ A0 where n is a positive integer and Ai E FormLc for each i _< n.

LC is given by the following axiom and rules of inference, where A, B, C stand for formulae and x, y, z stand for finite sequences of formulae including the empty sequence unless the contrary is asserted.

Axiom: (LCO) A ~ A.

Rules of inference:

(LC1) x ~ A A ~ B

x=~ B x non-empty

x ~ A y ~ B ( L C �9 r) x, y non-empty

x , y ~ A o B

(LC o l) x , A , B , y ~ C

x, A o B, y ~ C

(LC\r ) A , x => B

x=> A \ B x non-empty

(LC\O x => A y , B , z ~ C

y, x, A \ B , z ~ C x non-empty

(LC/r ) x, A ~ B x ~ B / A x non-empty

x ~ A y , B , z =:> C (LC/I) x non-empty.

y , B / A , x , z ~ C

A theorem of LC is a sequent deducible in LC @-LC), i.e., by the usual recursive definition, a sequent is a theorem iff it is an instance of (LCO), or it is given by some rule of inference from some theorem(s). More generally, let I" be a set of sequents and ~ be a sequent. We say that ~ is LC-deducible from F, F }-LC ~, iff

1. 9~ 6 F o r 2. 9~ is an instance of (LCO) or 3. there is a set A of sequents each of whose elements is LC-deducible from

A is an instance of this rule. F and there is an inference rule such that Note that if we consider only derivations from the empty set, then, in the definition of LC, (LC1) is superfluous, a result of Lambek (1958). ((LC1) is

4 HAJNAL ANDRt~KA AND SZABOLCS MIKULfi~S

really the cut-rule, and the result of Lambek is a cut-elimination theorem.) On the other hand, if we want to have strong completeness, i.e., we are dealing with derivations of the form F ~-LC ~ (P arbitrary set of sequents), then (LC1) is needed. (Indeed, let A, B, C E P and {A ~ B, B ~ C} = P. Then, as we will see soon, A ~ C is a semantical consequence of r . Since each rule but (LC1) introduces a new connective in the sequent to be derived, r [-LC A ~ C uses (LCI).)

REMARK 2.1. If the set of primitive symbols is the set of basic types, then the formulae are types and, roughly speaking, ~ of LC corresponds to the derivability relation of a version of Categorial Grammar. At the same time, if P is considered as a set of propositional variables, then LC is a Gentzen type inference system, and hence it is a fragment of Linear Logic. II

2.2. Relational semantics for the Lambek Calculus

The thought of giving a natural semantics, hopefully characterizing LC, arises naturally. The so-called relational semantics motivated by dynamic semantics for natural languages was proposed by Johan van Benthem, cf., e.g. van Benthem [1989b, 1991]. In van Benthem (1991) he showed that this semantics is sound for LC and he asked whether it was complete, too. In section 2.3 we will show that this semantics does indeed characterize LC, i.e., it is not only sound but also complete for it. We note that Brown and Gurr (1991) contains another completeness theorem for LC via relational quantales. That semantics is close to relational semantics, the difference is that the interpretation of the implications (i.e., the slashes) is different.

In this section we give the definition of relational semantics.

DEFINITION 2.1. [RelSem] By a relational model for LC we mean an ordered pair <W, v) where W is a transitive binary relation and v is a mapping of the set P of primitive symbols into the powerset P(W) of W.

Next we define the satisfaction relation. Let W = (W, v) be a relational model for LC, and let A,B, Ao, A1,.. . ,An E FormLc for any n > 0. Let x E W be arbitrary. We define W , x IF ~ for formulae and sequents cp


]d

A I \ b / X

i A . b N a > b

/ \ \ /

AI /~7 \ B A \ ~ I b I \ \ I

A \ b .V -VVZ a ) b

Fig. 1.

f

i 9 "x

,A f

i " \

Fig. 2.

inductively. See Figure 1. Let x = (a, b).

W , x IF p W , (a, b) IF A �9 B

W , (a, b)IF A\B

W,(a,b) IF A/B

W , x IF (A1, . . . ,A~ =~ Ao)

iff xEv(p), f o r p E P . iff there is c such that (a, c), (c, b) E W

a n d W , (a,c)IF A, W,(c,b)IF B. iff for all c such that (c, a) r W,

W , (c, a) IF A implies W , (c, b) IF B. iff for all c such that (b, c) C W,

W, (b,c)IF B implies W , (a,c)IF A. iff (W,x IF ((A1 * A2). . . ,,A~)

implies W , x IF A0).

We say that a sequent ~p of LC is true in the model W , in symbols W ~ cp, iff W , x IF ~ for all x 6 W. A sequent is valid with respect to RelSem iff it is true in every relational model. We denote this by ~R ~. We say that ~ is a (RelSem) consequence of F, in symbols I" ~R ~, iff, for every relational model W of LC, W ~ I' implies W ~ ~, where W ~ I" abbreviates that, for every ~ E 1-', W ~ ~. II

For instance, the intuitive meaning of the / is: A/B is true at a pair (a, b) iff whenever we "add" a B-pair (b, c), the formula A is true at the "resulting" pair (a, c). See Figure 2.

REMARK 2.2. In the above definition, FormLc can be thought of as the set of formulae of a modal logic with three binary modali t ies . , \ , / , where �9

6 HAJNALANDREKA ANDSZABOLCS MIKULAS

is a possibility type modality, and the residuations are, in a certain sense, dual and conjugate modalities of �9 More precisely, we define Kripke-models with one ternary accessibility relation C on which all three modalities are based: by a Kripke-model for LC we mean a triple W = (W, C, v) where W is a set, C C_ W x W x W, and v is an evaluation of the propositional variables, i.e., v �9 P ~ 79(W). Truth is defined as follows. Let x E W, and let A, B E FormLc. Then

W, x IF p W, xlF A � 9

W, x IF A\B

W, x IF A/B

W, x tF ( A 1 , . . . , An :=~ AO)

iff xCv(p), f o r p E P , iff there are y, z E W such that Cyzx and

W, y I F A , W, z I F B , iff for all y, z E W such that Cyxz and W, y iF A,

we have W, z IF B, iff for all y, z E W such that Cxyz and W, y IF B,

we have W, z IF A, iff (W,x IF ((A1 �9 A 2 ) . . . . A n )

implies W, x IF Ao).

Indeed, the modality \ is related to the modality �9 in a similar fashion as the temporal modality Always-in-the-past, denoted as [P], is related to Sometime-in-the-future, denoted as (F), in, e.g. Andr6ka et aI. (1991) and Goldblatt (1987). In Andr6ka et al. (1991), (P) is called the conjugate of (F) and [P] is the dual of (P). Then \ is a conjugate of a dual of .. It is instructive to meditate over the two steps leading to \ from �9

We obtain a conjugate modality in temporal logic by reversing the accessibility relation, this corresponds to permuting the arguments of the ternary accessibility relation C, and we obtain a dual modality by passing from an existential quantifier to a universal one (more precisely, by replacing the arguments with their negations and then negating the whole expression). So the obvious dual of �9 would be [] defined as

W, x I~- A [] B iff for all y, z such that Cyzx, either W, y IF A or W, z IF B.

We can get another dual [] of �9 by fixing (i.e., not negating) the first argument

as W, x IF A [] B iff for all y, z such that Cyzx,

W, y IF A implies W, z IF B.

We then get a conjugate [] by interchanging the second and third arguments of C obtaining

---+

W, x IF A[]B iff for all y, z such that Cyxz, W, y IF A implies W, z IF B.


. - +

Then one can see that [] is just \ , i.e., for all W, x, we have W, z IF A \ B

iff W, x IF- ANB. This is what we meant by saying that the slashes \ , / are certain duals of conjugates of |

In the above Kripke-frames, the elements x E W are "abstract pairs", called "arrows" in Arrow Logic, investigated by J. van Benthem [1991, 1992], and by Y. Venema (1992). More about the connection with arrow logic see, e.g. Marx et al. (1992), Marx (1992), Mikul~is (1992b), Simon (1992).

Now, in RelSem, a relational model is nothing but a Kripke-model (W, C, v) where the worlds are real pairs, i.e., where W is a (transitive) binary relation, and C is defined the natural way:

Cyzx iff y = (a,b) ,z = (b,c) ,x = (a,c) forsomea, b,c.

In Dogen (1990) these relational Kripke-frames are called, very suggestively, "two-dimensional ternary frames". II

2.3. Strong completeness theorem for Lambek Calculus with respect to relational semantics

Now, we can formulate our first theorem, which was first presented by Mikul~is (1991).

THEOREM 2.1. [Strong Completeness Theorem for LC w.r.t. RelSem] For any set F of sequents, and for any sequent ~,

REMARK 2.3. In the case of I" = ~), we have Weak Completeness Theorem w.r.t. RelSem. II

COROLLARY 2.1. [Compactness Theorem] For any set F of sequents and sequent qo, i fP ~R ~, then there is a finite A C_ F such that A ~R ~.

Proof of Corollary 2.1: By Theorem 2.1 it is enough to show that if 1-' ~-LC ~, then 2x ~-LC ~ for some finite subset A of F. And this is straightforward by the definition of LC-deduction. |

The proof of Theorem 2.1 will be based on an algebraic representation theorem (Thm.2.2). To state this, we need some definitions.

8 HAJNAL ANDRI~KA AND SZABOLCS MIKUL~,S

DEFINITION 2.2. [Representable relativized relational structure, RRS] Let W be a binary relation and let R, S C_ W be subrelations of W. The so-called left and right residuals relative to (or relativized to) W are defined as follows:

R \ W S de~--~-f {(x,y) e W ' V z ( ( z , x ) e R ""+ (z,y) e S)} R / W Sd*2 {(~,y) e W " Vz((y,z) e S - . (z,z) �9 R)}

and o denotes usual relation composition, i.e.,

n o s de=f {(~, V) " 3z ( (x , z) e R A (~, y) e S)}.

We will deal with ordered algebras whose elements are binary relations, whose operations are o, \ w , / w , and whose ordering is the set-theoretical inclusion relation. We will call such structures representable. In more detail, we call .A = (A, ,,, \ , / , C_) a representable relativized relational structure, an RRS, iff

- A is a set of binary relations, - o, \ , / are binary operations on A, - o, \ , / c o i n c i d e on A with o, \ w , / w , respectively, where

W = U A = { ( x , y ) . ( g R c A)(x ,y) c R}.

We note that W is a transitive relation because A is closed under o, which is not relativized to W. However, W is not necessarily reflexive or symmetric. Also, i f A = (A, o, \ v , / v , c_) is an algebra for an arbitrary transitive V such that A C_ 79(V), then V can be taken to be W = U A, i.e., for all R, S C A we have R \ v S = R \wS , R / v S = R / w S . We will often omit the index W from \ w , / w .

II

We note that the operations \ and / are highly dependent on W, i.e., R \ w S changes if we change W but leave R, S fixed. That is why we call the RRS's "relativized". This "relative behaviour" is inherent in \ , / j u s t as in Boolean complementation. However, later we will speak of "unrelativized" \ and / . By this we will understand that we choose W in a natural way (to be a Cartesian space).

DEFINITION 2.3. [Relational structure, RS, ordered residuated semigroup, ORS]

(i) We call an algebra with three binary operations and a binary relation on it a relational structure (RS). We usually denote the operations of an RS by o, \ , / and its relation by _<. Thus .4 = (A, ,,, \ , / , <_) E RS iff A is an


arbitrary non-empty set , . , \ , / are arbitrary binary operations on A, and _< is an arbitrary binary relation on A.

(ii) E denotes the following set (A1)-(A7) of formulae (in the first-order language with equality of NS), where x, y, z, u are variables:

(I) _< is an ordering, i.e., (A1) x _< x (A2) x _< y A y <_ z --* x <_ z

(A3) x _< y A y < x --* x = y .

a semigroup operation which is monotonic in both arguments w.r.t. (II) . is _<, i.e.,

(A4) ( x . y ) . z = x . ( y . z )

(A5) x _< y A z < u --~ x . z < y . u .

(III) \ and / are the so-called left and right residuals o f . , i.e., (A6) x . y < z ~-~ y < x \ z

(AT) x | y < z ~-~ x < z / y .

If `4 E RS, then `4 ~ E denotes that the set E of (open) formulas is valid in `4, i.e., if the universal closures of elements of E are valid in .4. For instance, .4 ~ x < x iff .4 ~ V x ( x < x ) . If .4 ~ E, then we call .4 an o r d e r e d r e s idua t ed s e m i g r o u p (see, e.g. Buszkowski (1986)), and ORS denotes the class of all ordered residuated semigroups. |

The operations \ , / of taking residuals in semigroups have long been investigated in semigroup theory. In algebraic logic, they correspond to some kinds of implications, see, e.g., Pratt (1985). Recently, they came into focus in several works, see, e.g., Pratt (1992), Jdnsson and Tsinakis (1992), Jipsen et aL (1992), Jipsen (1992).

We have defined two subclasses, ORS and RRS, of RS. As we shall see, ORS reflects very closely the syntactic derivations of LC, while RRS reflects very closely relational semantics of LC. Completeness of LC w.r.t, relational semantics will then be based on the algebraic representation theorem saying that ORS and RRS coincide, up to isomorphisms (see Theorem 2.2).

To make the above ideas more concrete, for any subclass K of RS we will define a semantics ~ K for LC, and then the overall idea of our completeness proof will be the following. For any set r of sequents, and for any sequent ~p, we prove

(1) I" FLC ~ iff P ~ O R S r (2) ORS = IRRS, (3) P ~RRS (t9 iff F ~R qo.

In the above, (1), (3) are more or less trivial (because ORS is "very close" to the definition of LC, while RRS is "very close" to the definition of relational

10 HAJNAL ANDRI~KA AND SZABOLCS MIKUL~,S

semantics), the hard part will be step (2) (which is Thm.2.2), saying that every ordered residuated semigroup is isomorphic to a representable relativized relational structure. IRRS denotes the class of all algebras isomorphic to a member of RRS.

THEOREM 2.2. ORS = IRRS, i.e., for every A C RS,

A ~ ~ iff A E IRRS.

We now elaborate the above steps (1), (3), and we will prove Theorem 2.2 in section 2.4.

First we define the Lindenbaum-Tarski formula algebras of LC and we show that they are ORS's. These are known facts, with easy proofs (see, e.g., Buszkowski (1986)). For completeness, however, we include here detailed proofs.

Let ~ = (FormLc, �9 \ , / ) be the "term-algebra", where 0, \ , / are the natural operations on FormLc. Let I' be any set of sequents. We define the relations _<r and ---r on Formbc as follows. For any A, B E FormLc,

A < r B iff F t-LC A :=> B A = r B iff ( A _ < r B and B < _ r A ) .

LEMMA 2.1. For any set I' o f sequents, =-r is a congruence relation on .~ and, for any A, B, A ~, B I such that A ---r At and B ==-r B ~, we have

A <_r B iff A' <_r B'.

Pro�9 of Lemma 2.1: _~r is reflexive and transitive by (LC0), (LC1), so = r is an equivalence relation. To show congruence, assume that A -=r X , B - r B ~. First we want to show A �9 B - r A t �9 B~. By F }--LC A ::> A t, P [-LC B ~ B ~ and (LC �9 r), we obtain F k-LC A, B ~ A t �9 B ~, from which we obtain F ~-LC A �9 B ~ A t �9 B ~ by using (LC �9 l), i.e., A �9 B <_r A t �9 B'. We obtain A I �9 B' <_r A �9 B similarly, so A �9 B --r A t �9 B ~ as desired. The proofs for \ , / a r e completely analogous, therefore we omit them. So - - r is a congruence relation,

Assume now further that A _<r B. Then A ~ _<r A and B _<r B ~, by A - : r A ~ and B - r B~, so by transitivity of _<r we obtain A t <_r B t. l

For A E FormLc, A / - v denotes the equivalence class of ~ r A is in.

DEFINITION 2.4. The Lindenbaum-Tarski algebra/21" of the LC is defined as

cr=(z,.,\,/,<_),

LAMBEK CALCULUS AND ITS RELATIONAL SEMANTICS ] |

where <L, . , \ , / ) is the factor-algebra 7 / - - r and _< is the image of _<r, i.e.,

L = F o r m L c / - - r = { A / - F : A E FormLc}, ( A / - - r ) �9 ( B / - r ) = (A �9 B)/=-r,

and similarly f o r / , \ , and

( A / = r ) < ( B / - - r ) iff A _<r B.

Clearly, s E lqS. II

LEMMA 2.2. /2i, is an ordered residuated semigroup, i.e., s ~ ~, for any set F of sequents.

Proof of L e m m a 2.2: <__r is an ordering on L because it is reflexive and transitive by (LCO), (LC1), and it is asymmetric because we factorized by

~ 1 ~.

Later in this proof we will use the following ( .) several times: For any A, B, C E FormLc,

( .) F bLC A, B :=> C iff F ~-LC A �9 B ~ C.

Indeed, the "only if" direction follows immediately by using (LC �9 l), while the other direction follows from (LCO), (LC �9 1), (LC1).

To show associativity of o, we get F ~-LC A, B, C ~ (A �9 B) �9 C by using (LCO), (LC �9 r), then we get F ~-LC A * (B * C) ~ (A * B) * C by using (LC �9 l) twice. The proof of F ~-LC (A * B) * C =~ A * (B * C) is similar.

Monotonicity of e: From F ~-LC A ~ B by (LCO) and (LC �9 r) we get F ~-LC A, C ~ B * C, from which F ~-LC A * C ~ B * C by (*). To show monotonicity in the other argument is similar.

Residual property: Assume F F-LC A �9 B ~ C. Then I" ~-LC A, B => C by (*), hence F F-LC B ~ A \ C by (LC\r). Assume now P F-LC B ~ A\C. Then P ~-LC A * B ~ A * (A\C) by monotonicity of . . By (LCO), (LC\I) it is easy to show F ~-LC A * (A\C) ~ C, so one application of (LC1) gives P ~-LC A * B ~ C. The proof for / is completely analogous. II

For any class K C RS we define a semantics ~ K for LC, as follows. A K-model for LC is a pair (9, v) where G E K and v �9 7 ..... ~ G is a homomorphism (we recall that 7 is the "word-algebra" of LC), i.e., if 9 = (G, e, \ , / , <) , then v : FormLc ~ G such that for any A, B E FormLc, v(A . B) : v(A) �9 v(B), v (A \B) = v(A)\v(B), v (A /B) = v(A)/v(B). M(K) denotes the class of all K-models of LC. Let qo be a sequent, say, cp is A1, . . . , AN =:~ Ao, and let AA = (6, v) E M(K). Then we define M ~ ~ iff


v ( ( A 1 �9 A 2 ) �9 �9 �9 �9 An)) <_ v(Ao) in G. Let F be a set of sequents, qa a sequent of LC. Then .A4 ~ F iff .A4 ~ 15 for all 15 E 1', and F ~/~ ~ iff AA ~ ~ for all .A4 E M(K) such that .A4 ~ F.

LEMMA 2.3. Let F be a set of sequents and let qo be a sequent of LC. Then (i), (ii) below hold.

(i) P ~ans ~ iff P ~n ~. (ii) F ~ORS qO iff F [-LC ~0.

Proof of Lemma 2.3: (i): Assume F ~R ~ and let A4 E M(RRS) be such that .s ~ F. We want to show M ~ ~. Let A,4 = (~, v / with G = (G, o, \ w , / w , C_) E RIq$, W = U G. Then W is a transitive relation, hence W = (W, v I P) is a relational model for LC. For any A E FormLc define

w(A) = {(a,b) E W : W , (a,b)It- A}.

Then by the definition of It- we immediately have that w(A �9 B) = w(A) o w(B), w ( A \ B ) = w ( A ) \ w w ( B ) , w ( A / B ) = w(A) /ww(B) . Thus w = v because w r P = v [ P and v is a homomorphism. Hence A4 ~ 15 iff W ~ 15, for any sequent lb of LC. Thus W ~ F by A/I ~ F, hence W ~ qo by F ~R ~o, hence A,4 ~ r by W ~ qo, and we are done. The proof of "F ~RRS qO implies P ~R ~o" is very similar. We omit it.

(ii): Assume F ~ o R s ~o, we want to show P t--LC ~o. Let v(B) = B~ - r for any B E ForrnLc. Then .A4 = (L:r, v) E M(OFIS) by Lemma 2.2. Let 15 be any sequent of LC of the form A1, . . . , An ~ A0, and let A -- ((A1 �9 A2). . . �9 An). By definition, AA ~ 15 iff ( A / - r < A o / - r ) in Z:r. By a generalization of (,) in the proof of Lemma 2.2, we also have that F [-LC A ~ A0 iff F }-LC 15. Thus for any sequent 15,

(.) .M ~ 15 iff 1-" ~-LC 15/).

Now .A4 # s by (*), since s FLC 15 for any 15 E s thus A4 # ~ by s ~ORS # and .h4 E M(ORS), and then P }-LC ~ by (*) again. The proof of "F FLC ~ implies F ~ o R s ~o" goes by an easy induction along the steps of the FLc-derivation. We omit that part. I

Proof of Theorem 2.1: Now Theorem 2.1 immediately follows from Lemma 2.3 and Theorem 2.2. (We note that ~ K is the same as ~ IK for any class K, where ~ IK denotes the consequence relation determined by the class consisting of all algebras isomorphic to a member of K, because if (~, v / ~ and h : G ~ ~' is a homomorphism, then (G', h o v) ~ qo.) 1


REMARK 2.4. In Buszkowski (1986), a semantics called GS-semantics, is introduced and completeness of LC w.r.t. GS-semantics is proved. Here we show that Thm.2.2 is a strengthening of this theorem. Namely, we show that RelSem is a kind of "subsemantics" of GS-semantics.

Let W be a transitive relation. We define a semigroup as follows. Let u be a new element, not a pair and not in W, and let W + = W U {u}. We define the binary operation, on W + as follows: For any x, y E W +,

(a,e) i f z = ( a , b ) , y = ( b , c ) forsomea, b,c x . y = u otherwise.

For any R _ W let us define h(R) = R U {u}. Then it is easy to check that h is an isomorphic embedding of the RRS (79(W), o, \ w , / w , c) into the r. semigroup spread over (W +, =, .). This shows that completeness w.r.t. GS- semantics follows from completeness w.r.t. RelSem. Since h also preserves intersection, completeness w.r.t. IGS-semantics in w (Buszkowski, 1986) also follows from our Thm.2.3 (which we shall state later). II

2.4. Proof of the representation theorem

In this section we prove that every ordered residuated semigroup is isomorphic to a representable relativized relational structure. Without residuations, representability of ordered semigroups was already studied, e.g., in Bredi- hin and Schein (1978). Representation theorems of this kind (representing abstract algebras as algebras whose elements are relations of some specified rank) are a major theme in algebraic logic, cf., e.g., Henkin et aL (1985), N6meti (1991), Marx (1992), Andr6ka (1991).

Proof of Theorem 2.2: The next proof will be similar to the proof of Lemma 3 (Andr6ka, 1991).

It is easy to check that E is valid in every RRS. For the other direction, let us assume that .A E RS and A ~ E. Step by

step we will build a directed graph G = (U, E, g) the edges (E) of which will be labelled (g) by the elements of our structure ~4. We will use this graph to define a representation function rep, which will be an isomorphism from A to a structure of binary relations on U.

In each step c~, we will define a directed graph G~ = (Us, Es , gs), where Us is the set of nodes, E~ _ Us x Us is the set of edges, gs : Es > A is the labelling function (A is the universe of A) such that

(I) /~s is irreflexive and transitive (II) (x, y ) , (y, z) E Es implies g~ (x, z) < g~ (x, y) | g~ (y, z) .

14 HAJNAL ANDRI~KA AND SZABOLCSMIKULAS

Choose an infinite cardinal ~ such that JA I < ~. Let V be a set ofcardinality t~, and let cr �9 ~ ~ 3A x 2V x 3 be such that

(V (g, b, c, x, y, i) e 3A x 2V x 3)(V/~ < a ) (3u < ~) [,~ _< u A cr(u + 1) = (a ,b ,c ,x ,y , i ) ] .

To see that there is such a function o-, let f : t~ ~ 3A x 2V • 3 x t~ be a bijection. If we fix a, b, c, x, y, i, then, for t~ many ordinals 7, f('Y) = (a, b, c, x, y, i, 5) for some 5 < t~. So, for each A < t~, there is v /> )~ such that f ( v ) = (a, b, c, x, y, i, 5') for some 5' < ~. Let 9 : 3A X 2V x 3 x ~; 3A x 2V x 3 with 9 ( a , b , c , x , y , i , A) = ( a , b , c , x , y , i } for each,~ < e;. If we define o-(v + 1) = 9 ( f ( v ) ) and (r(v) arbitrary for limit v < ~, then ~r meets the requirements.

On the intuition for a: We want the final graph, G, have the following additional properties:

(HI) (Va e A)(Vx �9 U)(~u E U)g {u,x) = a (IV) (Va e A)(Vy e U)(~v e U)g,(y,v} = a (V) (W, b,~ e g)(Vx, y e U)[(x,y} e E A e ( ~ , y ) = ~ / ~ < ~ . b ~

(3z e U)(~ (x,~} = ~ A e (z,V} = b)].

When building G step by step, we will "maintain" properties (I), (II) and will "bring-about" (III)-(V) by putting in appropriate points. We will build G in t~ steps, V will be a universe from which we will choose our new elements to put into U, and cr will be the "scheduling" for building the graph: or(A) is the "task" to take care of in the Ath step. The condition on cr is that each "task" recurs after each step (this will be needed because we will care for a task only if its conditions are already met). If or(A) = (a, b, c, x, y, i), then in the Ath step we will examine the edge (x, y) from the point of view of labels a, b, c and i indicates the "type of the activity" to be carried through.

Oth step. For each element c of A, we choose two different elements from V, say uc and vc such that uc, vc are all different for different c's. Let U0 = {u~,v~ : c C A}. We can assume that I V \ U0l = ~. Let E0 = {(u~, vc) : c E A} and ~0 (Uc, vc) = c. Clearly, (I) and (II) hold.

c~ + 1st step. Let cr(c~ + 1) = ( c , a , b , x , y , i ) . If (x ,y} r E~ or ga (x, y) r c, then go to a + 2nd step. Otherwise we have three subcases according to the value of i.

i = 0. See Figure 3. Choose an element from V \ Us, say, u. Let


% /

i

X J d!y / q , d

I l.,l ~ . i / ~.qoC

r., ~ '1

Fig. 3.

i

\ d,a \

Fig. 4,

\ t j ;

Uo~-t-1 = UoL u {~}

E.+I = E~u{<~ ,p ) . ( x , ; ) e E . } u { ( ~ , x ) }

i = 1. See F igu re 4. C h o o s e an e l e m e n t f r o m V \ U<~, say, v. Le t

u .+ i = u . u {v} Ec~+l : E a u { ( q , v ) " (q,y) e E a } O { ( y , v ) } e~+~ : e~ u {(<q,v> , ~ <q,v> �9 a>. <q,v> e E~} u {<<V,~> ,~>}.

i - - 2. See F igure 5. I f c f a �9 b, then go to nex t step. O the rwi se let z E V ' . . U~ and


f-771Z ~..

.X, / , C �9 \

T - d / d ' q b.o. ~.

f

Fig. 5.

u,~+~ = u,~ u {~} E,,+~ = E ~ u { ( r , z } : (~,x} e E,~}u

{(z, ~) : (V,~} e E,~} u { (x ,z ) , (z,y)} e,~+~ = e ~ u { ( ( ~ , z ) , a ) , ( ( z , y ) , b ) } u

{ ( ( r , z } , e ,~ ( , ' , x ) . a> : <r,~> e E~,}U {<<z,~> ,b.e,~(y,~>> : <~,~) e ~,~}.

It is easy to check that property (I) is preserved in the c~ + 1 st step. We also have to prove that the new transitive triangles constructed in the

oz + 1st step have property (II). We have to check only the new triangles, i.e., triangles in which new edges occur. We have three cases according to the value of i above.

i = 0. The new edges are { (u ,p ) : p = x or ( x , p ) E E~} . The typical situation is represented on Figure 6, where the two kinds of new triangles are the ones determined by u, x , p l , and by u , p l , p 2 . We have to show that a4 _< a �9 al and a5 _< a4 �9 a3. By the construction of the graph we have a4 = a �9 a l , a5 = a �9 a2; and by our induction hypothesis that (II) holds for E~ we have that a2 _< al �9 a3. Thus a4 < a �9 al by (AI) (reflexivity of _<),

and a5 = a �9 a2 _< a �9 ( a l �9 a3) ---- (a �9 a l ) �9 a3 = a4 �9 a3 by (A5), (A1), (A4) (i.e., monotonicity and associativity of .) .

i = 1. This case is completely analogous to the case i = 0. The typical situation is represented on Figure 7, we omit the computation.

i = 2. The new edges are {(r,z} : r = x or ( r ,x) C E~} U {(z ,s ) : s = y or (y, s) E E~}, and the typical situation is represented on Figure 8.

The new triangles to be checked are the ones determined by the following triples of points: r l x z , r z r l z , x z y , x z s l , r l z y , r lZSl , z y s l , ZSlS2. Checking these is very similar to the previous cases. As an example, we


0, t

OA I

i / XL,." i /o) , , J i / l

t

t ~ I t

g/

Fig. 6.

\ - '~ cl

. T I % \ % . . . . . . ~" q'z

Fig. 7.

check the triangle rlZS 1. We show that a3 ~ a5 # a6. By the construction

of the graph we have a5 = al * a, a6 = b �9 a2, c < a * l) and by our induction hypothesis on E a we have a3 _< a4 �9 a2 and a4 ~ al * c. So

a3 <_ aa,a2 <_ (al , c ) , a 2 ~_ (al , ( a o b ) ) , a 2 = (al ,a ) | = as ,a6 by monotonici ty and associativity of o.

Thus Ga+l satisfies (1I) as well.

18 HAJNAL ANDRt~KA AND SZABOLCS MIKUL,/~S

qt

...~. _Z

I % I Q, \ \ ~6 I Q~ / / / O ~ l)++, + / / l k N q > J / os " >~s,

/ \

T// \.~ r~ s2 Fig. 8.

Limit step. If o~ is a limit ordinal, then let Us = U/~<~ U/~, E~ =

U/~<~ E/~ and g~ = U/~<c~ g~.

Let G = G~, i.e.,

~;=Uuo, E=U~o and e=Ueo. Clearly, G satisfies (I) and (II).

Now, we are ready to define the representation function rep. For every c E A, let

rep(c) = {(u, v)" g (u, v) _< c}.

We have to show that rep is an isomorphism from A to a structure whose elements are binary relations on the set of nodes of our graph. Clearly, rep(c) is a binary relation on U for any c E A.

We prove that rep is an isomorphism w.r.t. <, i.e.,

a _< b iff rep(a) _C rep(b).

Indeed, if g (u, v) _< a, then, by transitivity of <, g (u, v) <_ b, so (u, v) E rep(a) implies (u, v) E rep(b). Ifrep(a) C_ rep(b), then for every (u, v) E E, if g (u, v) _< a, then g (u, v) < b. Since g (ua, va) = a (see the 0th step), we have a <_ b.


Z ~

ta " - / o

Fig. 9.

Now we show that rep is one-one, i.e.,

a r b implies rep(a) r rep(b).

Assume rep(a) = rep(b). Then rep(a) C_ rep(b) and rep(b) C_C_ rep(a), so a < b and b < a by the previous proof, thus a = b by (A3) (antisymmetry of _<).

We check that rep preserves the operations too.

Checking the operation |

rep(a .b) = {(u,v) : g(u,v) <_a.b} (~=) =, {(~, ~} : ?z(e (u, z} < a A e (z, ~} < b)} = = { ( u , z } : g(u,z) <_ a} o { ( z , v ) : g(z,v} <_ b} = = rep(a) o rep(b).

(i): (C_): Let c = g (u, v). Then, for some a + l , c r (a+ l ) = (c, a, b, u, v,'2). So in the a + 1st step we put a z into the graph such that ~ (u, z) = a/rod e ( z , v ) = b .

(~.): By properties (I) and (II), by the transitivity of _<, and by (A5).

Checking the operation \."

repia\b ) = { (u, v)" ~ (u, v} _ a\b} (ii) { (u, v) �9 a �9 ~ (u, v} <_ b} (ii_~)

= {<~,~>. v z ( e ( z , ~ ) < ~ ~ e ( z , ~ ) s b ) } = = r e p ( a ) \ r e p ( b ) .

(ii): c < a\b iff a �9 c < b by (A6). (iii): (C__): By monotonicity of . . (~_): The triangle in Figure 9 is in the graph. (iv): (C_): By properties (I) and (II).

20 HAJNAL ANDRt~KA AND SZABOLCS MIKULAS

Z ~

Fig. 10.

(D): Figure 9 is in G, so if g(z,u} < a, then t ( z , u ) �9 s _< a �9 e (~, ~) = e (z', ~ / � 9 e (~, ~) = e (z', ~) <__ b.

Checking the operation/:

rep(b/a) = { (n ,v ) " e (u , v } <_ b/a} (~) {(u ,v} " t ( u , v ) . a < b} (v=i)

= { (~ ,~ } . w (e (~ ,~ )_< a ~ e ( ~ , , , ) , e(~,,~)<__ b)} (~2)

= r e p ( b ) / r e p ( a ) .

(v): By (A7). (vi): (C_): By monotonicity of . . (D): Figure 10 is in G. (vii): (C_): By properties (I) and (II). (_D): By Figure 10 g (u, v) �9 a _< b.

Thus rep is the desired isomorphism, since the image of A,

({rep(a)" a E A } , \ , / , o , C_),

is in RRS. Thus Theorem 2.2 is proved. I

2.5. Adding new connectives lq, U to Lambek Calculus.

In this section we investigate what happens if we add "static conjunction" M, and "static disjunction" U to LC.

Let LCC denote Lambek Calculus with static conjunction. This means the following: in the language of LCC we have one more connective, i.e., A M B E FormLcc whenever A, B E FormLcc. Otherwise, FormLcc and the sequents are defined the same way as in section 2.1. The models for LCC are those for LC. Let W be a relational model for LC, x E W and let


A, B E FormLcc. Then

W , x tF A F1 B iff (W, x I~- A and W , x IF- B).

This conjunction is sometimes called structural, or Boolean (van Benthem, 1991), or additive (Roorda, 1991) versus multiplicative. We adopted the term "static" (versus "dynamic") from Pratt (1992).

The axioms and rules of LCC are those of LC together with the following two axioms, and rule:

(MI) A r q B ~ A A M B ~ B x ~ A x ~ B (nr)

x ~ A r q B

Otherwise everything is defined as in the case of LC.

THEOREM 2.3. [Strong completeness of LCC] For each sequent ~ of the language of LCC, and for each set F of sequents of LCC,

r ~-LCC ~ /ff P ~R ~.

Proof of Theo rem 2.3: The proof of Theorem 2.3 is based on the following representation theorem, Theorem 2.4, exactly the same way as the proof of Thm.2.1 was based on Thm.2.2. I

Let RRC denote the class of all RRS's endowed with the operation of taking intersection. That is, an ordered algebra (A, o, \ , / , N, C_) is in RRC iff (A, o, \ , / , _C) E R R S and A is closed under taking intersection, i.e., for all R, S C A we have R M S E A. Let E) be E together with the following axiom:

(A8) (z <_ x A z <_ y) +-+ z < x n y . Thus ,A ~ 6) means that ,A is a semilattice-ordered residuated semigroup.

THEOREM 2.4. Every semilattice-ordered residuated semigroup is isomorphic to a representable one, i.e., for any algebra ,,4 (of the right type),

A ~ ~) iff A C IRRC.

Proof of Theorem 2.4: We use the same construction that we used in the proof of Theorem 2.2. We only have to show in addition that the function rep defined there is an isomorphism w.r.t, the operation 73 as well. Indeed, by (A8),

rep(aMb) = {(u, v) " g (u, v) < a M b } = = ( (u ,v ) ' g (u ,v ) < a } n { ( u , v ) ' s < b } = = rep(a) r~ rep(b).

I

22 HAJNAL ANDRI~KA AND SZABOLCS MIKUL,~S

Now we turn to investigating adding "static disjunction" U to Lambek Calculus. In view of the above, the natural thing would be if adding U to LCC, we would get completeness by adding the following two axioms and rule:

(Ur) A =~ A U B B =~ A U B A ~ C B ~ C (ul)

A u B ~ C

This is not the case as, e.g., Theorem 2.5 below shows, where we prove that no finitely many axioms or rules can ensure strong completeness if we add U to the set of operations of LCC.

In relational semantics we interpret U as follows: Let W be any relational model for LC and let x E W. Then

W , x IF A U B iff (W, x IF A or W , x IF B).

THEOREM 2.5. Let Q denote any extension of LCC with a finite set o f axioms or sequent rules for U. Then Q cannot be sound and strongly complete w.r.t. relational semantics.

Proof of Theorem 2.5: Let Q be any extension of LCC with a finite set of axioms and sequent rules for U. We denote derivability in Q by FQ, and ~R denotes consequence in relational semantics as before. Assume that Q is sound, i.e., for any F and ~o, we have

I ~ [--Q ~ implies s ~R ~"

We will show that FQ is not strongly complete, i.e., there are s and ~ such that

r but r %

Let RRD denote the class of all RRC's endowed with the operation of taking union. That is, an ordered algebra (A, o, \ , / , A, U, C__) is in RRD iff (A, o, \ , / , N, C_) E RRC and A is closed under taking union. Then it is easy to see, just as before, that for any F and qo,

(1) r ~ R ~ iff F ~RRD eft.

The axioms and sequent rules of Q translate into a finite set A of equational implications (or, in other words, quasi-equations) in the language of R RD, by using the standard techniques in algebraic logic (N6meti, 1991). For example, the sequent rule (Ul) would translate into the quasi-equation

x <__ z A y <__ z ~ ( x U y ) < z.


Then A defines a class QRS of algebras which is analogous to O R S in that it reflects F-Q, i.e., for any F and ~,

(2) P }-Q ~ i f f f' ~QRS ~.

Soundness of Q implies that RRD ~ A , i.e., that RRD __ ORS. Next we show that results of Andr6ka (1991) imply that the quasi-

equational theory of RRD is not finitely axiomatizable, i.e., there is a quasi- equation q such that

( .) RRD ~= q and QRS g= q.

Indeed, in the proof of Theorem 4 in Andr6ka (1991), a sequence of algebras .An, and quasi-equations qn are defined for which the following hold:

- the operations of .An are U, [q, o, - , ~, 0, 1 ~ the first three being binary, the next two unary, and the last two are constants;

- qn contains only the operation symbols U, rq, | RRD ~ q,~o and .An qn;

- any non-principal ultraproduct of the .An's is isomorphic to a relation set algebra on some set U, i.e., to an algebra 13 = (B, U, N, % ,-~, - I , O, !d} where B is a set of binary relations on U, ~ denotes the operation of taking complement w.r.t. U x U, -1 denotes the operation of taking converse of a relation (i.e., R -1 = {{a, b) �9 (b, a) E R}), and Id is the identity relation on U (i.e., Id = { (u, u) �9 u E U}).

For any n, we define the ordered algebras .A~n as follows:

.A~= (An,* , \ , / , rq , U, <_l

where An is the universe of .An, U, rq, �9 are the original operations of .An, while \ , / , _< are defined from the original operations of.An as follows:

a/b = - ( ( - a ) �9 b ~)

a < b iff a ~ b = a .

Let /3 ~ denote the algebra we obtain f rom/3 likewise. Then/3t E RRD, /3~ is isomorphic to a non-principal ultraproduct of the .A~'s and .A~ ~= qn. By /3~ C RRD we have B ~ ~ A, and then by A being finite we have A~ ~ A for some n, i.e., .A~ E QRS. Let q --- qn. Then Q R S ~= q by .A~ ~ qn, but RRD ~= q.

We have seen that ( ,) holds for some quasi-equation q. Let q be such a quasi-equation. By [x = y iff (x _< y/x y _< x)], we may assume that q is of

24 HAJNAL ANDRI~KA AND SZABOLCS MIKULJ~S

the form ('O <_ al A . . . A ~-n _< crn) ~ "r0 _< o-0. Now, again by the standard translation techniques, Ti, cri translate into formulae Ai, Bi (by just replacing the variables with primitive symbols in P). Let P = {AI ~ B 1 , . . . , An =~ Bn}, (p = Ao =~ Bo. Then [qFID ~ q means that P ~RRD ~, while ORS ~= q means that 1-" ~=QRs qo. Together with (1), (2) these imply that F ~R ~ but I-' ~/Q % showing that Q is not strongly complete w.r.t, relational semantics. II

We note that Theorem 2.5 remains true if we want to add any connectives the relational semantics of which would correspond to an operation expressible (or definable) in relational set algebras (i.e., expressible from U, ,.o, o , -1 , Id as defined in the proof of Theorem 2.5.) We also note that Theorem 2.5 above remains true if we replace relational semantics in it with "unrelativized relational semantics" (which will be defined in the next chap- ter), because the ultraproduct/3 in the proof of Theorem 2.5 had U x U as biggest element.

3. LAMBEK CALCULUS WITH EMPTY SEQUENTS AND UNRELATIVIZED RELATIONAL SEMANTICS FOR LAMBEK CALCULUS

In the definition of relational models W = (W, v) for LC we required W to be transitive only. The question of what other nice properties of W in a relational model we can require naturally arises. In this section we show that if we require W to be reflexive, or a Cartesian space, then we loose completeness of LC. The other natural question that arises is what happens if we omit the conditions of x, y, z being non-empty in the definition of LC. We will show that if either we allow empty sequents in LC (i.e., delete the conditions on being non-empty in the definition of LC) or if we add four new rules to LC, then the so obtained stronger calculi will be complete w.r.t, the stronger (U x U type) semantics.

The question why we required W to be transitive at all also might arise. The reason is that if we do not require transitivity of W, then LC is not sound w.r.t. this weaker semantics. (Namely, both (LC �9 r) and (LC �9 l) fail to be sound.) An example is the following. Let W = {(0, 1}, (1, 2), (2, 3), (1, 3}, (0, 3)}, let A, B, D, E be primitive symbols and let v(A) = {(0,1)}, v(B) = {(1,2)}, v(D) = {(2,3}}, v(E) = 0, let W = (W,v). Then W (A, B, D ~ E) while W ~ (A, B �9 D ~ E).

We think it is possible to give a weaker version LC- of LC which would be sound and complete w.r.t, relational semantics where we do not require transitivity.


Let W = (W, v) be a relational model for LC. We say that W is a square model if W is a Cartesian space, i.e., W = U x U for some set U. Let RelSem + denote relational semantics for LC where we allow only square relational models. Let P be a set of sequents and let ~ be a sequent of LC. Then ~R+ ~ denotes that T is valid in all square relational models of LC, and similarly, F ~R+ ~ denotes that ~ is true in every square relational model of LC in which F is also true.

PROPOSITION 3.1. LC is not complete w. r.t. RelSem +.

Proof of Proposition 3.1: Let us consider the sequent qo = p ~ p �9 (p\p), where p is a propositional variable. We will show that ~R+ ~ while not ~R 79. Indeed, let W = (W, v) be any square model with W = U • U. Let I d = { (u ,u ) �9 u C U}. Then Id C_ v(p\p) , thus v(p) = v(p) o Id C_ v(p) o v(p \p) , showing W ~ ~. This shows ~R+ ~. On the other hand, let W = {(0,0) , (0, 1)}, let v(p) = {(0, 1)} andlet W = (W,v ) be a relational model for LC. (W is transitive.) Then it is easy to see that v(p) o v (p \p) = ~, thus W ~ ~p showing ~:R ~. By Theorem 2.1 we have that ~ is not LC- derivable (by ~:R ~), thus LC is not complete w.r.t. RelSem + (by ~R+ ~)" !1

The odd behaviour of the above sequent was already known in the literature Dogen (1990). By Thin.3.1 below we will have that (in this respect) these are the only "odd" sequents. (So this means that the second kind of odd sequents ( P \ P ) \ Q ~ Q, mentioned in Dogen (1990), can be derived from the first o n e s . )

Now we define two strengthenings, LC + and LC ~ of Lambek Calculus. Let LC + be LC together with the following four rules.

A ~ B A ~ B

C =~ C . ( A \ B ) C ~ ( A \ B ) | C

A ~ B A ~ B

C ~ C �9 ( B / A ) C ~ ( B / A ) �9 C

Now we introduce another strengthening, LC ~ of LC. Let A0, A 1 , . . . , An E FormLc. We call A1 , . . . , AN ::~ A0 a generalized sequent (or sequent in the wider sense), if n /> 0. That is, we allow n = 0 also. These sequents with n = 0 will be denoted by ~ A0. Let W = (W, v) be a relational model for LC and let (u, v) E W. Then we define satisfaction of the generalized sequent ~ A as

W , ( u , v ) l~- ~ A iff ( u = v implies W , ( u , v ) l~-A).

26 HAJNAL ANDREKA ANDSZABOLCS MIKULAS

(The motivation coming from dynamic semantics for natural languages is the following. If starting from a state u we did not move at all (n = 0), then this transition (i.e., (u, u)) is in A (van Benthem, 1989b).)

Let LC ~ denote the calculus we obtain from LC by omitting all the conditions stating non-emptyness in it, and where at the same time by sequent we mean sequent in the wider sense.

THEOREM 3.1. Both LC + and LC ~ are strongly complete w.r.t. RelSem +. That is, (i), (ii) below hold.

(i) Let ~ be a (non-generalized) sequent and F be a set of(non-generalized) sequents o f LC. Then

s [-LC + ~9 i f f s #R+ ~9.

(ii) Let qD be a generalized sequent and let [' be a set of generalized sequents. Then

I ~ FLCO ~9 i f f s ~R+ ~P.

The proof of Theorem 3.1 is based on algebraic representation theorems, just as in earlier cases. The proof of Thin.3. l(i) will be a close parallel to that of Thm.2.1, while the proof of Thm.3.1 (ii) will be a refined version of that.

Let RRS + be the "square" version of RRS: .4 = (A, . , \ , / , <) E R R S + iff there is a set U such that

- A is a set of binary relations on U, - ~ are binary operations on A coinciding with o , \ g x u , / U x V

(restricted to A), respectively, - < is a binary relation on A coinciding with C_ (restricted to A).

We note that in general W = U A r U x U, all we can know is that W is a reflexive, transitive relation on U (i.e., {(u, u) �9 u E U} c_ W and W o W c_ W) .Ye t , R \ w S = R \ u x u S foral lR, S E A, by R \ u x u S C_ W. Thus, RRS + c RRS.

Let C+ be C together with the following four formulae.

x < y -+ z < z �9 ( x \ y ) x <_ y --~ z < z ~ ( y / x )

x < _ (x \y) o z x <_ y -- , z < ( y / x ) �9 z

THEOREM 3.2. For every .4 E RS,

.,4 ~ ~+ iff -4 E IRRS +.

Proof o f T h e o r e m 3 . 2 : The "if" part is easy and omitted.


Assume that .4 ~ ~+. We will construct, as in the case of Theorem 2.2, a directed and labelled graph, and we will define the representation function using this graph.

Let G = {V, E, t) , where V is the set of nodes, E = V x V is the set of edges and g : E ) 79(A) is the labelling function. So one difference from the proof of Thm.2.2 is that G is a full graph, and another difference is that we label with sets of elements of A and not only with elements of A.

G will have the following five properties.

(I) (Vu, v, w C V)(Va, b)(a C g (u, w) A b e g (w, v) --+ ac(c _< a . b A c e g (~, v}))

(II) (Vu, v E V)(Va, b,c E A)(a <_ be c A a C g {u, v) (Bw e V)b e g iu, w) A c e g (w, v) )

(m) ( w ~ v ) (va c A)3~(a e g (~, ~) A (W C V)~ r ~ g{w,v) = { a * h ' h e t ( u , v ) } )

(w) (Vv ~ V)(Va ~ A)3w(a E g (v, w) A (Vu E V)u # v g(u,w) = { h . a " h e giu, v)})

(V) (Vu E V)g(u ,u ) _D I, whereZ = {a\b" a <_ b} U {b/a" a <_ b}

We will define G by recursion. Let a and o- be as in the proof of Theorem 2.2. We will use the following notation. If X, Y C_ A, then we let X �9 Y = { x e y ' x C X , y c Y } .

Oth step. Let V0 = {ua, va : a C A}, Eo = Vo x Vo and W = { (Uat Va) , (Ua, Ua) , (Va, Va): a C A} where ua, Va (a C A) are all different. Moreover, let Co {ua, va) = {a} and Co <u~, Ua) = CO {va, v~) = I, and let Co (u, v) = 0 if (u, v) E (V0 x V0) \ W.

(I) holds because of the new formulae in ~+, and (V) is satisfied as well.

a + l s t s t e p . Let o-(a + 1) = {a,b,c ,x ,y , i ) .Wehavethreesubcases according to the value of i.

i = 0. See Figure 11. Let z be a new point (z ~ V~), and let

~ + 1 E~+I g~+l

= G u { z }

= G + I x G + I = ~. u {((~,~),z), ((z,x), {a} �9 ~,~ (x,x) u {a})}U

{((z,p), {a} �9 G (x,p)) : p c G A p 4: x}u {((p, ~), ~) :p e G}.

28 HAJNALANDREKA AND SZABOLCSMIKULAS

i r

- Z -,~

{,:,iv I ({d" e, o,,.>)~

I

<j, -" . . ta}.rg~<,c,~,.-

r

Fig. 11.

i~,./q., t~.>. {,:,} ~ /

/

Fig. 12.

J f

q,,~ ~ t', ?>

" "~PO v )

Z~ .~

i [d" u

I

i = 1. See Figure 12. Let z be a new point, and let

V~-I- 1 Ec~+l

g~+l

= v,~ u {z} = v,~+l x v,~+i = ~ . u { ( ( z , z } , Z } , ( (> z} , ~,~ (y, y) o { a } u { a } ) } u

{ ( (q ,z ) ,ga (q,Y} " {a} ) : q e Vo~ Aq g: y}U {((z, q}, ~} : q e v~}.

i = 2. See Figure 13. If a ~ b �9 c, or a ~ g~ (x, y), then go to a + 2nd

step. Otherwise let z be a new point, and let

V~-t- 1 Ea+l t a + l

= y,~ u {z} = v~+ l x v~,+l = e ~ , u { ( ( z , z ) , { ~ } o e , ~ ( > x ) o { b } u I } } u

{((,-, ~ ) ,~ . (,-, ~ ) . {b}) : ,- e v . / , , - # x } u

{((:~, ~} ,e,~ (x,x} �9 {b} u {b})} u {((z, y) , {~} �9 ~. (v,v} u {~})} .


,~ .1" ....

\

, , / i " J

b

Fig. 13.

Limit step. If a is a limit ordinal, then let

/~<~ ~<~ /~<~

We note that if in the case i = 2 we have z = y, then ga+l (z, z) r (~, g~+l (z, x) r 0, hence we may not assume that G is directed in the sense that (Vu, v E G, u r v)[g (u, v) = 0 V g (v, u) = 0]. Because of this, in the case i = 2 we also may have g~+l (z, z) D I.

Let G = G,,. Clearly, G satisfies (V). G also satisfies (I), since in each step this property was preserved, checking this is a mechanical and tiresome calculation. As an example, we check one case. Assume we are in case i = 2, with the above notation, we want to check the triangle z z z . Assume d E g~ (y, x), we want to show that e _< b �9 (c �9 d) for some e E g~ (x, x). (See Figure 14.) Indeed, a E g,~ (x, y), thus e < a �9 d for some e E g~ (x, x) by our induction hypothesis, thus e < a �9 d _< (b �9 c) �9 d = b �9 (c �9 d) by a < b e c a n d E +.

Moreover, (II), (III) and (IV) hold for G by the construction. Let, for every a E A,

rep(a) = {<u,v> " (3h E g(u,v>)h <_ a}.

Then rep clearly preserves <, and is one-one because of the 0th step in the construction.

Now we show that rep is a homomorphism. First we show that

rep(a) o rep(b) = rep(a �9 b).

30 HAJNAL ANDREKA AND SZABOLCSMIKULAS

b Z

Fig. 14.

Indeed, if (u, v) E rep(a) o rep(b), then

3w((3h~ E ~.(u,w))h,~ < aA (~hb e e(w,v))hb <_ b)

and, by (I),

3w(3h E e(u,v))(3ha E ~.(u,w))(~hb E e(w,v))h < ha �9 hb <_ a o b,

i.e., (u, v) E rep(a ,, b). The other direction is a straightforward consequenc of (II).

We also have rep(a\b) C_ rep(a)\rep(b),

since if (u, v) E rep(a\b), then (3h E e (u, v))h < a\b, so, by (I),

Vw((3ha E e(w,u))ha < a---* (3h' E l (w ,v) )h ' <_ a . ( a \ b ) <_ b),

i.e., Vw (w, u) E rep(a) ~ (w, v) E rep(b) whence (u, v) E rep(a)\rep(b). To show that

rep(a)\rep(b) C_ rep(akb)

we have to distinguish two cases. In the first case, we assume that u r vanc (u,v) E rep(a)krep(b). Then

Vw((w, u) e rep(a) ~ (w, v) E rep(b)),

i.e., Yw((3ha E l (w,u))ha <_ a --+ (3hb E e(w,v))hb <_ b),

so, by (HI),

3w((3h E l Iu , vl)(3hb E l Iw, v))a,, h : hb < b).


Thus (3h E t (u, v) )a �9 h < b, so (3h E s (u, v) )h <_ a\b, i.e., (u, v) E rep(a\b).

Now we assume that u = v, i.e., (u, u) E rep(a)\rep(b). By the construction 3w(t(w, u) = {a} �9 u) U {a}), so we conclude that

3w(a <_ b V (~h E e (u, u) )(3hb E t (w, u) )a . h = hb <_ b).

Then, by (V), and because g(u, u) C_ I, (3h E e (u, u) )h < a\b, i.e., (u, u) E rep(a\b).

Similar argument, using (IV), shows that

rep(a/b) = rep(a)/rep(b).

Let B = {rep(a) : a E A}. Then B is a set of binary relations on V, by the definition of rep. Also, B is closed under the operations o, \ v • 2 1 5 because `4 is closed under o, \ , / and rep is a homomorphism w.r.t, these operations. (That is, we checked that rep(a\b) -- rep(a)\vxvrep(b) for all a, b E A). Thus B = (B, o, \ , / , C_) E RRS + and rep is an isomorphism between .4 and/3. Therefore Theorem 3.2 has been proved. I

In the representation theorem we will use to prove Thm.3.1(ii), we will have two additional constants e, 0 denoting the identity relation and the empty relation, respectively.

R R S ~ denotes the class of all RRS's expanded with Id, 0 as constants, i.e., .4 = (A, o, \, /, e, O, C_) E RRS ~ iff (A, o, \ , / , C_) E R R S + and e = { (u, u) : (3R E A) (u, u) E R} and 0 is the empty set.

RS ~ denotes the class of all ordered algebras expanded with two constants, i .e. , .4 = (A, . , \, /, e, O, <) E RS~ iff (A, . , \ , / , <) E RS a n d e , 0 E A.

Let E0 be E together with

e o x = x e e = x 0 . x = x . 0 = 0 0 < x .

Let A be the set of the following formulae

x e y = 0 ~ (x = 0 V y = 0) x o y <_ e ~ (x = 0 V y = 0 V x = y = e).

Note that A is not valid in RRS ~ (while E0 is). That is why, in the following theorem, only one direction is stated.

THEOREM 3.3. For every ,,4 E RS ~

A ~ ~o u A implies A E IRRS ~

32 HAJNAL ANDRt~KA AND SZABOLCS MIKUL,/~S

Proof of Theorem 3.3: We make essentially the same construction as in the proof of Theorem 3.2 with some modifications.

We will construct a directed and labelled graph, G = IV, E, ~), satisfying the following six properties. Propei'ties (I), (II) and (V) will be the same as in the proof of Theorem 3.2. We require properties (III) and (IV) only for a E A \ {e, 0}. The graph will have this feature too:

(VI) (V(u,v) E E ) 0 ~ g ( u , v ) A ( e E g ( u , v ) - - - + u - - - - v ) .

Let cr and ~; be as before. We define the graph by recursion using the original construction in the proof of Theorem 3.2.

Oth step. This is the same as before, we just choose v,~ and ua for a E A \ {e, 0} only.

c~+lststep. Let ~r(c~ + 1) = (a,b,c,x,y,i).Wehavethreesubcases according to the value of i again.

i = 0 or i = 1. Do the original construction, provided a ~ {0, e}. Otherwise go to next step.

i = 2. If a ~ b �9 c, or a ~ g.a(x,y), then go to next step. Otherwise, by property (VI), we have that 0 r {b, e}. Ifb = e or c = e, then go to next step. Otherwise, by A, b ~; e and c ~ e. In this case, do the original construction.

Limit step. Take the union as before.

Let G = G~. Then properties (I)-(V) are achieved. (This is an easy consequence Of the original construction.) Further, (VI) is clearly preserved in each step.

Let rep(a) = { (u ,v ) : (3h E g(u,v))h <_ a}.

It is easy to prove that rep(e) = { (u, u) : u E Y}, rep(0) = 0 and rep(0\y) = rep(x/0) = V • V. The other cases, in checking that rep is an isomorphism, are the same as in the proof of Theorem 3.2. I

DEFINITION 3.1. Let I" be an arbitrary set of generalized sequents. We define an analogue,/2~, of the formula algebra, Z;r, of LC.

Let e, 0, 1 be three new elements not in FormLc, and let T = FormLc U {e,0, 1}. Let

T=<T, \ , / , . ,<r , e ,O)

where the definitions of the operations and the relation < r go as follows. On A, B E FormLc these are defined as before, i.e., A <_r B iff P [-LC o A ~ B.


For every x E T and A C FormLc, let 0 --<r x _<r 1 and e _<r e, and let e _<r A iff P t'-LC0 ::::k A. Let 0 �9 x = x �9 0 = 0 and e �9 x = x * e = x, and if x r 0, then let 1 �9 x = x �9 1 = 1. Let 0 \ x = 1 and e\x = x, and if x r 0, then let x \ 0 --- 0. Moreover, if x ~ (e, 0}, then let x\e = 0 and x \ l = 1. Finally, if x ~ 1, then let 1 \ x = 0. The other s lash, / , can be defined in a similar way. Let

x - - r y iff (x<_ry and y < _ r x ) ,

and let C ~ = ( T / - r ) . I

LEMMA 3.1. c o b sOu A.

Proof of L e m m a 3.1: It is easy to check, using the definition above, that < r is a partial ordering, and that �9 is an associative operation which is monotonic w.r.t. < r . It is not difficult to show, by case distinction, that if a _<r b and c < r d, then a �9 c < r b �9 d. The rest of E0 is easy, by the definitions of the slashes, and A holds, by the definition o f . , as well. I

Proof of Theorem 3.1: The proof of (i) proceeds exactly as the proof of Thm.2.1, but now we use Thm.3.2 instead of Thm.2.2.

Proof of (ii): Soundness of LC ~ w.r.t. RelSem + is easy to check. To prove strong completeness, let F be a set of generalized sequents, ~o be a generalized sequent, and assume that F ~R§ T. We want to show that F ~-LC 0 ~.

Let us look at the Lindenbaum-Tarski algebra s We will tum this algebra into a square relational model W such that LCU-provability from I? will coincid.e with validity in this model. By Lemma 3.1 and Theorem 3.3, C ~ is isomorphic to an RRS ~ Let h �9 C ~ ~ A4 be such an isomorphism, where

F 0 34 = (M, o, \ , / , Id, ~, C_) E RRS . To turn A4 into a relational model, l e t W = U • U w h e r e U = {u : (u,u) E R f o r s o m e R C M } , a n d l e t v(p) = h(p/--r) for all p C P . Then W = (W, v) is a square relational model of LC. It is easy to check by induction that for all x E W and for all A C F o r m L c ,

(**) W , x I~- A iff x E h(A/_--r).

Let A, B E FormLo By the definition of C ~ we have that F t-LCO (A ~ B) iff ( A / - r ) _< ( B / - r ) in C~ and by h being an isomorphism the second statement holds iff h ( A / - r ) C h(B/--r). By (**) we have that h(A/--r) C_ h(B/=--r) i f f W IF (A ~ B).

We have seen that

F F-LC0 (A ~ B) iff W I~- (A ~ B),

34 HAJNAL ANDRI~KA AND SZABOLCS MIKUL.~S

Now look at the sequent ~ B. The above argument remains valid if we replace A with the empty sequence, and (A/=r) with e (of s in it. Thus we get

P [-LC o (=~ B) iff W IF (=~ B).

Let finally r be any sequent of form A 1 , . . . , An ~ B with n > l, and let A = ((Al * A2) * . . . * An). Then by (*) in the proof of Lemma 2.2 we get

P }-LCO if) iff P ~LC o (A :::> B),

and clearly W It- r iff W IF (A =v B), thus we obtained

(* * *) r [-LC o r iff W It- r

for any generalized sequent r Recall that F ~R+ (P. By ( , �9 , ) we have W It- F, hence W IF (p by

1-' ~R+ qo. Then applying ( , �9 , ) once more we get F ~-LC 0 ~, and we are done. I

4. COMPLETENESS W.R.T. LANGUAGE MODELS

Now we prove that LC is not weakly complete w.r.t, language models (LM) and that there is no extension of LC which is sound w.r.t. U • U type relational semantics and is strongly complete w.r.t. LM. First, we recall the definition of language models from van Benthem (1991), p. 189.

DEFINITION 4.1. [Language Model] A language is a set of finite, possibly empty, sequences. A family of languages is a set {L~ : i 6 I}, where Li is a set of finite sequences (words) over a finite alphabet.

A language model is a family of languages enriched with the following operations.

La �9 Lb def {xy " x E La, y E Lb}

La\Lb d_ef {x" (Vy E La)yX C Lb}

Lb/La def {x" (Vy e La)xy e Lb}

A sequent A1,. �9 An :=~ AO is true in a language model if

v(A1) * . . .* v(A,~) C_ v(Ao)

where v is the valuation function defined in the obvious way. The consequence relation ~LM is the usual as well. I

PROPOSITION 4.1. LC is not weakly complete w.r.t, language models.


Proof of Proposition 4.1: By the definition of \ the empty sequence is in L \ L for every language L. Thus x ~ x �9 ( x \ x ) is valid in every language model.

On the other hand, if W = a = { (0, 1) }, then afL a o (a\a) in the FIRS with universe W, so, by Theorem 2.1, x =~ x �9 ( x \ x ) is not deducible in LC. 1

PROPOSITION 4.2. There is no calculus containing LC which is sound w. r.t. RelSem + and strongly complete w.r.t, language models.

Proof of Proposition 4.2: We will show that there are a set F of sequents and a sequent ~ such that F ~R+ ~ but 1-' ~LM ~.

It is easy to check that

{x ~ x . x , y ~ x} V:R+ Y =~ x � 9

(let r e p ( x ) = {(1,0), (0,0)} and r e p ( y ) = {(1,0)}). On the other hand,

{x :::~ x �9 x, y ~ x} ~LM Y ::~ x �9 y

because of the following. Let Lx, L u be two languages and assume that Lx C_ Lx �9 Lx, Ly C_ L~. We want to show Ly C_ L~ �9 Ly. If Lx = O, then L u = 0 and we are done. So we can assume that Lz r 0. Let w E Lx. Then, since Lx C_ Lx �9 Lz, there are Ul, Vl E Lz such that w = UlVl. By the same argument, for each number i, there are Ui+l, Vi+l C Lx with ui = Ui+lV~+l. Sooner or later, since w is finite, either ui or vi is the empty sequence. Hence Ly C L~ �9 L u. 1

COROLLARY 4.1. LC ~ that version of the Lambek Calculus where we admit sequents with empty antecedent, is not strongly complete w.r.t. LM.

Proof of Corollary 4.1: LC ~ contains LC. 1

We note that S-semantics of Buszkowski (1986) differs from our LM- semantics in that in S-semantics the empty word is not allowed in any language. A version of Proposition 4.2 is proved by Buszkowski (1986), Lem- ma 11, for S-semantics instead of LM-semantics. We note that the sequent P ~ P �9 ( P \ P ) is LM-valid but not S-valid. (This shows also, by the results in Buszkowski (1986), that the calculus LSC1 of Buszkowski (1986) is not a conservative extension of LC, an interesting fact.)

36 HAJNAL ANDREKA AND SZABOLCSMIKULAS

A C K N O W L E D G E M E N T S

We are gra te fu l for interest and usefu l c o m m e n t s f rom Johan van Ben them,

Serge i A r t e m o v , Wojc i ech B u s z k o w s k i , D o v Gabbay , Istv~in N6met i , I ld ik6

Sain, and Andr~is S imon . We are also grateful to the two a n o n y m o u s re fe rees w h o s e c o m m e n t s and sugges t ions we found very useful .

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Lambek Calculus and its relational semantics: Completeness and incompleteness