The Lambek-Grishin calculus for unary connectives

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1 The Lambek-Grishin calculus for unary connectives Anna Chernilovskaya Utrecht Institute of Linguistics OTS, Utrecht University, the Netherlands Introduction In traditional Lambek style categorial grammar, derivable objects are sequents of the form Γ C (where Γ in the antecedent could be considered as a product formulae context, whereas a succedent C is a single formula). The absence of a structural connective in the succedent of a sequent makes the whole system asymmetric. This asymmetry is overcome with the introduction of the plus connective, which is symmetric to the product Lambek connective. Symmetry here means that the plus connective is also used to build a context, but in the succedent of a sequent. Both context connectives (product and plus) come together with their residuals, therefore there are two families of binary connectives, and they are linked to each other with structure-preserving interaction postulates. This then results in the Lambek- Grishin calculus first mentioned in (Grishin, 1983), which is a symmetric version of Lambekstyle categorial grammar. The construction of the Lambek calculus can be enriched with different types of unary connectives. For example, (Bernardi, 2002) considers a pair of residuated and a pair of galois unary connectives. (Bernardi and Szabolcsi, 2007) use two distinct pairs of residuated and a pair of galois connectives to model the quantifier scope and negative polarity licensing in Hungarian. In this paper, I propose a unary analogue of the (binary) Lambek-Grishin calculus. It is based on the minimal calculus built with two families of residuated unary connectives, which is then extended with some postulates expressing interaction between the two families. The structure of the paper is as follows: In section 1 I formulate the unary Lambek-Grishin calculus comprising two families of residuated unary connectives, and introduce some postulates linking these two families together. In section 2, the Kripke semantics for the suggested calculus is studied. Finally, in section 3 I discuss potential linguistic applications of the calculus. 1 Logic of unary connectives 1.1 Pure theory of unary logical connectives First let me remind the algebraic notions of (dual) residuated and (dual) galois pairs suggested in (Dunn, 1991). Given two (in general, distinct) partially ordered sets (X, X ) and (Y, Y ), two functions of one argument f: X Y, g: Y X and x X, y Y, consider a pair of I thank Michael Moortgat for fruitful discussion of the paper and the workshop reviewers for their valuable remarks. I also thank Markus Egg for his helpful comments. 1

2 functions (f, g). The table below specifies a relation between these two functions. iff x X gy gy X x fx Y y rp dgc y Y fx gc drp In the table, (d)rp and (d)gc stand for (dual) residuated pair and (dual) galois pair, respectively. Note that (f, g) is a residuated pair if and only if (g, f) is a dual residuated pair. Consequently, it suffices to consider residuated pairs of functions, which I will do in the remainder of this paper. In the inference rule format, the definition of a residuated pair (f, g) is double line means that the rule can be applied in both directions. fx Y y x X gy, where a I define a minimal calculus called LG 1, whose set of formulae is defined inductively as: F 1 = a, b, c,... elements of the set Atom, i.e. atomic formulae 1 F, 1 F the first residuated family of unary connectives 2 F, 2 F the second residuated family of unary connectives. Derivable objects of the calculus are sequents of the form A B, where A, B F 1 and the arrow is called a derivability relation. The rules of the system are as follows: an axiom scheme (reflexivity of the derivability relation): A A; a transitivity rule (transitivity of the derivability relation): if A B and B C then A C; residuation rules: i {1, 2} i A B iff A i B. Here two ordered sets coincide: (X, X ) = (Y, Y ) = (F 1, ). I have defined a system with two families of residuated unary connectives. Though the rules for two pairs are precisely the same, we would see a semantic difference between them in the section about Kripke semantics. The system above exhibits a symmetry denoted as, which is a function : F 1 F 1 acting as identity on the set Atom and defined with the following table for complex formulae: 1A 1 A 2 A 2 A The following lemma about the symmetry holds: Lemma. LG 1 A B iff LG 1 B A. The lemma can be proven by induction on the length of the derivation. It shows that the symmetry is arrow-reversing. Thus, two pairs of unary connectives are related to each other via. 1 and 2 play a role of context connectives on the left- and right-hand side, analogously to and, respectively. However, I have not yet considered any derivational postulate that would express communication between the unary families. 2

3 1.2 Grishin interaction postulates in the Lambek Grishin calculus The interaction postulates between the unary families that I propose here were inspired by the (binary) Lambek-Grishin calculus based on generalisation (and symmetrisation) of the (non-associative) Lambek calculus studied in (Grishin, 1983). Let me repeat its construction here. The Lambek-Grishin calculus is based on the minimal calculus, denoted here as LG 2, extended with additional postulates. The set of LG 2 formulae is defined below: F 2 = a, b, c,... atomic formulae A B, A \ B, B / A product, left and right division A B, A B, B A plus, left and right difference. The minimal Lambek Grishin calculus consists of: reflexivity and transitivity of the derivability relation; residuation rules for binary connectives: A C / B iff A B C iff B A \ C B C A iff C B A iff C A B. This system exhibits two kinds of symmetry: the one mentioned above ( ) and a new one ( ). Both are functions, : F 2 F 2 acting as identity on the set of atomic formulae and defined for the binary connectives with the tables below: C / D A B B C D C D \ C B A A B C D These symmetries act in different directions : Lemma. LG 2 A B iff LG 2 B A iff LG 2 A B. C / B A B A \ C B C B A C A. This claim can also be proven by induction on the length of the derivation. Intuitively, the lemma states that the symmetry is arrow-reversing, whereas the symmetry is arrowpreserving. The main idea of Grishin is not only to introduce the second residuated triple of connectives, which makes the (non-associative) Lambek calculus symmetric, but also to consider postulates of interaction between the two families. Grishin classifies them in two groups (so-called group I and group IV), each comprising four representatives. Because of the rules of the basic calculus LG 2, the additional postulates of every representative come in groups of six that are mutually interderivable in LG 2. As an example, consider six postulates taken from one representative of both groups: Group I Group IV (1) (B C) A B (C A) (1) (B \ C) A B \(C A) (2) A (C / B) (A C) / B (2) B \(C A) (B \ C) A (3) A (C B) (A C) B (3) A (C B) (A C) B (4) (A B) \ C B \(A C) (4) (A \ C) B C (A B) (5) (C B) A C (A / B) (5) (A B) / C A /(C B) (6) A (C \ B) (A C) B (6) A (B C) (C / A) \ B 3

4 Postulates of the group IV are identified as pertinent to linguistic phenomena in (Moortgat, 2007). However, from a logical perspective, both groups of interaction postulates are interesting. In principle, one could add to the minimal system both groups of postulates. We abbreviate LG 2 extended with postulates of the group I as LG 2 + I (analogously, there are LG 2 + IV and LG 2 + I + IV ). All calculi mentioned exhibit the same two symmetries / with the same properties of arrow preservation/reversion, respectively. 1.3 Grishin interaction postulates for the unary connectives I suggest extending the minimal unary Lambek-Grishin calculus LG 1 with some postulates that provide interaction between the two unary families. Analogously to the binary case, postulates come in two groups, each consisting of several postulates mutually interderivable in LG 1. In the unary case, each group has the only representative containing three interderivable postulates. Consider the following table: Group I Group IV (1) 1 2 A 2 1 A (1) 2 1 A 1 2 A (2) 2 1 A 1 2 A (2) 1 2 A 2 1 A (3) 2 1 A 1 2 A (3) 1 2 A 2 1 A One can see a connection between the unary and the binary postulates. Let me explain the informal idea. Suppose that the set of LG 2 formulae is extended with two units 1 and 1 with properties: A 1 1 A A 1 1 A 1 A A 1 1 A A 1 Then 1 \ A A / 1 and 1 A A 1, where abbreviates that sequents in both directions are derivable. If one then defines a partial function f: F 2 F 1 as A 1 1 A; 1 A 1 A A 1 2 A; 1 A 2 A A / 1 1 A ; 1 \ A 1 A A 1 2 A; 1 A 2 A one can translate the first three binary postulates into three unary ones having replaced two of three letters A, B, C with units. The last three postulates could not be translated. I introduce the notation LG 1 + I, LG 1 + IV and LG 1 + I just like for the case of LG 2. Note however that the postulates of the group IV are of precisely the same shape as the one of group I, and exchanging two families would exchange two groups. To be consistent with literature on the Lambek-Grishin calculus, a term unary Lambek-Grishin calculus is used for LG 1 + IV. 2 Kripke semantics for unary connectives This section provides the Kripke semantics for LG 1 enriched with interaction postulates, extending the work in (Chernilovskaya, 2007) and (Kurtonina and Moortgat, 2007). First 4

5 I consider models for LG 1, then I discuss the impact of the interaction postulates on LG 1 models. 2.1 The case of LG 1 A Kripke frame for LG 1 is a triple F = W, R 1, R 2, where W is a non-empty set of worlds and R i W W for i {1, 2} are binary accessibility relations. A Kripke model for LG 1 is a pair M = F,, where F is a frame defined above and (a forcing relation) is a binary relation between W and the set of atomic formulae Atom (it determines which atomic formulae are assumed to be true in a given world). Let me extend the forcing relation to determine truth conditions for the unary connectives in such a way that a unary connective i ( i, respectively), where i {1, 2}, is interpreted as an existential (universal) modality with respect to the accessibility relation R i. For all x, y W and all A F 1 the following conditions are defined: x 1 A iff y (R 1 xy and y A) y 1 A iff x (R 1 xy implies x A) x 2 A iff y (R 2 yx and y A) y 2 A iff x (R 2 yx implies x A) There are two kinds of symmetry that arise in the truth conditions: (1) the one within each family of residuated connectives: a connective i is interpreted as a corresponding universal modality for the rotation of R i (the rotation is understood as a reversal of the argument order); (2) the symmetry between two families of residuated connectives: the accessibility relation R 2 is the rotation of R 1 (together with the change of the relation name). The same symmetries arise for binary families of connectives, as one can see from the binary truth conditions discussed in (Kurtonina and Moortgat, 2007). However, for the time being, the two accessiblity relations are not connected to each other. The minimal calculus LG 1 is sound and complete with respect to the class of models described above: Soundness and completeness theorem. For all formulae A, B F 1 LG 1 A B iff A B (here A B means that for all frames F and all forcing relations defined for atomic formulae the set of worlds where A is true is a subset of the set of worlds where B is true). Proof. Correctness is proved trivially by induction on the length of the derivation. To establish completeness, one can build a canonical model based on a Henkin construction. The canonical model construction. In the canonical setting, worlds are weak filters. A set X of LG 1 formulae is called a weak filter if for all formulae A and B if LG 1 A B and A X then B X. Weak filters will be denoted with capital letters X, Y, Z. A canonical model for LG 1 is a tuple Wc, R c 1, R c 2,. In order to define canonical binary relations, I introduce two existential operations ( 1 and 2 ) on the set of weak filters. universal operations are defined analogously: X W c i {1, 2} i X = {B F 1 A F 1 (A X and LG 1 ia B)}; i X = {B F 1 A F 1 (LG 1 B ia implies A X))}. Respective 5

6 First, let us check that i X and i X are indeed weak filters. Take B i X and C F 1 such that LG 1 B C. By the definition of i, there exists a formula A X with the property LG 1 ia B. To show that C is in i X, one needs to find a formula A such that A X and LG 1 ia C. Take A = A. The fact that LG 1 ia C follows from transitivity of the derivability relation. The proof for i X proceeds in the same fashion: take B i X and LG 1 B C. Then for all formulae A LG 1 B ia implies that A X. To prove that C i X, take any formulae A such that LG 1 C ia. Then, by transitivity, LG 1 B ia, which yields A X, as desired. The following lemma shows that unary operators on weak filters act under the same residuation laws as unary connectives on formulae: Lemma. i {1, 2} the structure W c, i, i is a residuated algebra, i.e. for all weak filters X and Y i X Y iff X i Y. Proof. To prove this lemma, I need an additional fact: Fact. i {1, 2} A F 1 for all weak filters X (1) i A X iff A i X; (2) i A X iff A i X Proof of the Fact. (1) Suppose i A X. To prove that A i X, one needs to find a formula C X such that LG 1 ic A. Take C = i A. For the converse direction, suppose that there is such a formula C X with the property LG 1 ic A. To prove that i A X, note that LG 1 ic A iff LG 1 C ia. Since X is a weak filter and C X, one gets i A X. (2) Suppose i A X. Let us check the definition of i X. Take a formula C such that LG 1 A ic. The latter implies LG 1 ia C, which yields C X by the definition of the weak filter X. For the converse direction, assume that for any C if LG 1 A ic then C X. Taking C = i A, one gets i A X, as necessary. Now let me come back to the proof of the lemma. ( ) Suppose that i X Y. In order to prove that X i Y, take A X. By the definition of the operation i, this implies that i A i X, therefore, i A Y. By the fact proved above, this means that A i Y, q.e.d. ( ) Absolutely analogously, assume that X i Y and take A i X. This implies that i A X, i.e. i A i Y. By the definition of the operation i, one gets A Y, as desired. The notation for these weak filter operations is not accidentally chosen since there is a connection between a filter operation and the respective logical connective. I will formulate it in 6

7 terms of two special sorts of weak filters a principal filter generated by a formula A ( A ) and the set-theoretic complement of a principal ideal generated by A ( A ): A = {B F 1 LG 1 A B}; A = F 1 \ A = {B F 1 LG 1 B A}. Thus, the connection is formulated as follows: Lemma. i {1, 2} (1) i A = i A ; (2) i A = i A. Proof. (1) Consider a sequence of equivalent statements: a formula B i A if and only if i B A (by the fact proven above), which means that LG 1 A ib. This is if and only if LG 1 ia B. Having i A i A, the later is equivalent to B i A, as necessary. (2) Let me first prove the left inclusion: i A i A. Take B i A. To show that B i A, one needs to prove that LG 1 B ia. Suppose the opposite, i.e. that LG 1 B ia. But then, by the definition of the weak filter i A, i A i A, which is not true because having i A i A as an axiom of LG 1, one would deduce that A A. The latter is false by the definition of A. To prove the converse inclusion, take B i A, which means that LG 1 B ia. To show that B i A, consider a formula C such that LG 1 B ic. Then C must be in A, because otherwise assuming that LG 1 C A, one gets a contradiction with the fact that LG 1 B ia. Indeed: B i C i C i C i i C C C A i i C A i C i A B i A Thus, logical connectives are lifted to corresponding operations on the set of weak filters. Let us return to the construction of the canonical model for LG 1. The canonical model M c = W c, R c 1, R c 2, is built in the following way: W c is a set of all weak filters in LG 1, R c 1 XY holds iff 1 Y X and R c 2 XY holds iff 2 X Y. The order of worlds here reflects the symmetry number (2) mentioned with respect to the truth conditions. The forcing relation is defined on the set of atomic formulae as follows: for every weak filter X and every atom a X a iff a X. Let us prove that the canonical model is indeed a model of LG 1, i.e. if for some formulae A, B LG 1 A B then Mc A B. The latter means that for all worlds X such that X A holds that X B. Consider the following lemma: 7

8 Truth lemma. X W c A F 1 : X A iff A X. Having proven this lemma, one gets that the condition X A is equivalent to A X. Since X is a weak filter and LG 1 A B, one gets B X, which is equivalent to X B and thus, is what is necessary to prove. Proof of the Truth lemma. I prove the truth lemma by induction on the complexity of the LG 1 formula. For atoms, the claim of the lemma follows directly from the definition of the forcing relation. Let me show the proof for formulae 1 A and 1 A. The cases of 2 A and 2 A are absolutely analogous. 1) If X 1 A, by the truth conditions this means that Y W c such that R c 1 XY and Y A. It is necessary to prove that 1 A X, which is equivalent to A 1 X. By the definition of the canonical accessibility relation R c 1, R c 1 XY 1 Y X. As was shown above, this means that Y 1 X. By the induction hypothesis, from Y A one gets A Y, therefore, A 1 X, as desired. If 1 A X, i.e. A 1 X, to show X A it is enough to provide a weak filter Y such that R c 1 XY and Y A. Take Y = A. The second condition is then trivial. To show that 1 A X, remember that this is equivalent to A 1 X and, taking any B A would yield B 1 X since 1 X is a weak filter and A 1 X. 2) X 1 A by truth conditions means that for all weak filters Y, if R c 1 Y X then Y A. Take Y = 1 X. Then the first condition reduces to 1 X 1 X, thus, by induction hypothesis, A 1 X, which is equivalent to 1 A X, as necessary. Take now 1 A X and any weak filter Y satisfying the condition R c 1 Y X, i.e. 1 X Y. This is iff X 1 Y, then 1 A 1 Y, i.e. 1 1 A Y. But LG A A and Y is a weak filter, therefore A Y, q.e.d. The proof of the completeness part of the soundness and completeness theorem for LG 1 is now simple. Suppose, by contraposition, that for A, B F 1 LG 1 A B. Then Mc A B. Indeed, take a canonical world X = A. Obviously, A X, which, by the truth lemma, is equivalent to X A. However, it cannot be true that B X since LG 1 A B. By this, the soundness and completeness theorem is proved. 2.2 Extension of LG 1 with interaction principles I proceed by adapting the group I / IV interaction principles to the construction built above. These principles impose restrictions on the accessibility relations, therefore R 1 and R 2 would not be distinct any more. It is enough to choose one postulate in a group since they are interderivable. I consider here an example of extending LG 1 with an interaction principle 1 2 A 2 1 A taken from the group IV. If necessary, any other principle could be adapted analogously. 8

9 The frame constraint imposed by the postulate is X, Y, Z W ( (R 1 XY and R 2 ZY ) Y W (R 2 Y X and R 1 Y Z) ). Now one can establish soundness and completeness for the class of frames satisfying the frame condition above: Soundness and completeness theorem for the unary Lambek-Grishin calculus. A, B F 1 LG 1 + IV A B iff A B. The same theorem can be formulated for the calculi LG 1 + I and LG 1 + I + IV with respect to the appropriate classes of frames. Proof. Soundness is trivial because the frame condition is composed directly from the truth conditions for the postulate 1 2 A 2 1 A. As for completeness, one should show that in the canonical model described above this condition holds. Let me give a graphical illustration. if 1 X Y 2 Z A then Y : X Z A 2 1 Y On the left, I depict X 1 2 A with Z A, on the right X 2 1 A with Z A. Arrows with numbers represent respective accessibility relations. It is necessary to show that in the canonical model for all weak filters X, Y, Z constructed as on the left, there exists a fresh weak filter Y connected to X and Z as shown on the right. On the left part of the picture R c 1 XY holds iff 1 Y X and R c 2 ZY iff 2 Z Y. Take Y = 1 Z. One needs to check that R c 2 Y X and R c 1 Y Z. The second condition obviously holds since 1 Z 1 Z. Let us check that 2 1 Z X. As W c, 1, 1 is a residuated algebra, 1 y X Y 1 X, then 2 Z Y 1 X. Again, by residuation: 1 2 Z X. Now take any formula A 2 1 Z, which is equivalent to 1 2 A Z. One needs to show that A X. In the group IV there is a postulate 1 2 A 2 1 A, therefore, using that Z is a weak filter, one gets 2 1 A Z, which means that A 1 2 Z, i.e. A X, q.e.d. 2.3 Enrichment of the binary Lambek Grishin calculus with unary connectives LG 1 +I / IV can be merged with LG 2 +I / IV with the help of interaction postulates between unary and binary families of connectives. Analogously to the Grishin postulates taken from the group IV, I suggest the following postulates: 2 A B 2 (A B) and A 2 B 2 (A B) (1) They reflect the communication between the second family of unary connectives and the product family of binary connectives. These postulates bring the same intuition as initial 9

10 Grishin postulates number 3 from the group IV taking into account the analogy discussed in section 1.3. Take the images of these postulates, i.e. 1 (B A) B 1 A and 1 (B A) 1 B A. They should also be postulated since they are not equivalent to (1). However, they are analogous to binary postulates number 2. Turning to Kripke semantics, these new postulates would provide the connection between relations R and R 2 (R and R 1, respectively), thus a more restricted class of frames would be considered. 3 Linguistic application In this section I suggest possible linguistic applications of the two unary families introduced above. There are two groups of rules composing unary connectives: residuation rules, which work apart for every family, and interaction postulates, which link them to each other. I do not speak here about postulates bringing together binary and unary connectives. 3.1 Application of two distinct families (Areces et al., 2001) show how to use a pair of residuated and a pair of galois unary connectives to model scope relations between polarity items of the generalised quantifier type and negation. They assign a type denoted as q(a, B, C) to a generalised quantifier and use a certain logical rule to capture scope behaviour. In categorial grammars logical formulae serve as linguistic categories for words. Basic categories that are used here are np standing for noun phrase and s for sentence. There is a well known question of characterisation of quantifier scope. For example, consider a sentence Everyone likes someone. It has two different readings informally described below: 1) x y likes(x, y) > 2) y x likes(x, y) > The first reading is called local, the second one non-local. The sign > denotes which quantifier takes scope over which. Scope ambiguity corresponds to several derivations in categorial grammar. A generalised quantifier (GQ) acting locally like a category A in the context of the category B and binding at the level C is prescribed a category that we denote as q(a, B, C). The following rule captures the behaviour described: [A] B Γ[C] D Γ[ [q(a, B, C)]] D (q) (Γ and are -contexts). Therefore, sentential-level quantifiers acting locally like noun phrases (np) in the sentence (s) get a category q(np, s, s). What a categorial grammarian has to do is to indicate which category q(a, B, C) denotes so that the rule (q) becomes a derivable rule of a categorial grammar. In the Lambek calculus the following GQ category 10

11 was suggested: s /(np \ s). However, it works correctly only for a local reading of sentences analogous to the example above. In the (binary) Lambek Grishin calculus this category constructor looks like q(np, s, s) = (s s) np. The rule (q) is derivable, as necessary. From this category the known Lambek GQ category s /(np \ s), which provides only local quantifier scope, is derivable, but with the new category assignment non-local GQ scope could be gotten as well. Further, I will use q(a, B, C) meaning the Lambek-Grishin category. The problem that we will concentrate on here is the behaviour of so-called polarity items of GQ type and negation. The basic idea of what I present is completely the same as in (Areces et al., 2001), but the concrete realisation is slightly different. Namely, I suggest using in a category construction in a positive position a combination of residuated connectives instead of a combination of galois connectives, as it is done in the paper mentioned. Consider a tiny lexicon: John : np like : (np \ s) / np doesn t : ((s /(np \ s)) \ s) /(np \ 2 2 s) anything : q(np, 2 2 s, 2 2 s) something : q(np, 1 1 s, 1 1 s) A combination of a GQ ( anything, something ) with negation (contained in doesn t ) denoted as Neg provides different readings depending on GQ. For example, a sentence John doesn t like anything has the only reading Neg>GQ, whereas John doesn t like something only GQ>Neg. In categorial grammar, these two possibilities come from two different derivations. They are schematically spelled out in Figures 1 and 2. More generally, if a GQ has a type q(np, s 1, s 2 ), a certain reading is provided iff all of derivabilies hold: Neg > GQ : s s 1 GQ > Neg : s s 1 s s s 2 2 s s 1 1 s s s Note that the equations s i i s for i {1, 2} are satisfied independently of s 1, s 2 values. np np s s 1 np np np \ s np \ s 1 (np \ s) / np (np \ s 1 ) / np np ((np \ s) / np np) s 1 s s np ((np \ s) / np q(np, s 1, s 2 )) 2 2 s (q) np s /(np \ s) s 1 1 s (np \ s) / np q(np, s 1, s 2 ) np \ 2 2 s (s /(np \ s)) \ s np \ 1 1 s ((s /(np \ s)) \ s) /(np \ 2 2 s) (np \ 1 1 s) /((np \ s) / np q(np, s 1, s 2 )) np (((s /(np \ s)) \ s) /(np \ 2 2 s) ((np \ s) / np q(np, s 1, s 2 ))) 1 1 s Figure 1. Neg > GQ 11

12 s 2 2 s np np np \ s np \ 2 2 s np s /(np \ s) s s 1 (s /(np \ s)) \ s np \ s 1 ((s /(np \ s)) \ s) /(np / 2 2 s) (np \ s 1 ) /(np \ s) np np np \ s (((s /(np \ s)) \ s) /(np \ 2 2 s)) \(np \ s 1 ) (np \ s) / np (((s /(np \ s)) \ s) /(np \ 2 2 s)) \(np \ s 1 )) / np np (((s /(np \ s)) \ s) /(np \ 2 2 s) ((np \ s) / np np)) s 1 s s np (((s /(np \ s)) \ s) /(np \ 2 2 s) ((np \ s) / np q(np, s 1, s 2 ))) 1 1 s (q) Figure 2. GQ > Neg Substituting s 1 = s 2 = 2 2 s (the case of anything ), we satisfy only the first set of derivabilities, getting only the reading Neg > GQ, whereas s 1 = s 2 = 1 1 s (the case of something ) provides only the reading GQ > Neg. In fact, in the examples above I modified type assignments suggested in (Areces et al., 2001) and discussed in Chapter 7 of (Bernardi, 2002). My suggestion was to introduce the second pair of residuated unary connectives, which replaced the galois connectives. I make use of the fact that for i {1, 2} i i A A i i A, but 1 1 A 2 2 A. As I have shown, just like (Areces et al., 2001), I get only the reading Neg > GQ for the negative polarity item anything, and only the reading GQ > Neg for the positive something. 3.2 Application of two connected families (Bernardi and Szabolcsi, 2007) make use of partially ordered set of categories, illustrating their theory on Hungarian quantifiers. In this setting, two distinct families of residuated unary connectives are used to provide a partial ordering. Unary galois connectives are present as well. As the result, the underlying derivability scheme with unary connectives is very finegrained. I do not discuss unary galois connectives here, but consider the calculus LG 1 + I, i.e. LG 1 enriched with a postulate 2 1 A 1 2 A. I sketch another possible derivability scheme, which could be extended, if necessary s 1 1 s 2 2 s s 1 1 s 2 2 s s 12

13 Conclusion This paper presents the unary Lambek-Grishin calculus, gives its Kripke semantics and outlines potential linguistic applications of the new system. Further steps could be done in several directions. First of all, one could be interested in downward monotonic galois connectives and their influence on (one of the versions of) the Lambek-Grishin calculus. Here interaction postulates are to be formulated and Kripke semantics is to be studied. Moreover, a gap is left in linguistic applications. It could be interesting to study more natural language data and to try to accomodate the variation of derivability patterns of the unary Lambek-Grishin calculus. References Areces, C., Bernardi, R., and Moortgat, M. (2001). Galois connections in categorial type logic. In Proceedings FG/MOL, volume 47 of Electronic notes in theoretical computer science. Elsevier. Bernardi, R. (2002). Reasoning with polarities in categorial type logic. Ph.D. thesis, Utrecht Institute of Linguistics OTS, Utrecht University. Bernardi, R. and Szabolcsi, A. (2007). Partially ordered categories: optionality, scope, and licensing. In C. Barker and A. Szabolcsi, editors, New directions for proof theory in linguistics, course notes ESSLLI 2007, Dublin. Chernilovskaya, A. (2007). Completeness of the Lambek-Grishin calculus with unary modalities. In R. Bernardi and M. Moortgat, editors, Symmetric Categorial Grammar, course notes ESSLLI 2007, Dublin. Dunn, J. (1991). An abstraction of Galois connections and residuation with application to negation and various logical operations. In Proceedings of the European Workshop on Logics in Artificial Intelligence. Grishin, V. (1983). On a generalization of the Ajdukiewicz-Lambek system. In R. Bernardi and M. Moortgat, editors, Symmetric Categorial Grammar, course notes ESSLLI 2007, Dublin. Kurtonina, N. and Moortgat, M. (2007). Relational semantics for the Lambek-Grishin calculus. In R. Bernardi and M. Moortgat, editors, Symmetric Categorial Grammar, course notes ESSLLI 2007, Dublin. Moortgat, M. (2007). Symmetries in NL syntax and semantics: the Lambek-Grishin calculus. In R. Bernardi and M. Moortgat, editors, Symmetric Categorial Grammar, course notes ESSLLI 2007, Dublin. 13