Natural Language Semantics in Biproduct Dagger Categories

Transcription

1 Natural Language Semantics in Biproduct Dagger Categories Anne Preller To cite this version: Anne Preller. Natural Language Semantics in Biproduct Dagger Categories. Journal o Applied Logic, Elsevier, 2014, 12, pp < /j.jal >. <lirmm > HAL d: lirmm Submitted on 10 Sep 2013 HAL is a multi-disciplinary open access archive or the deposit and dissemination o scientiic research documents, whether they are published or not. The documents may come rom teaching and research institutions in France or abroad, or rom public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diusion de documents scientiiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche rançais ou étrangers, des laboratoires publics ou privés.

2 Natural Language Semantics in Biproduct Dagger Categories Anne Preller Abstract Biproduct dagger categories serve as models or natural language. n particular, the biproduct dagger category o inite dimensional vector spaces over the ield o real numbers accommodates both the extensional models o predicate calculus and the intensional models o quantum logic. The morphisms representing the extensional meanings o a grammatical string are translated to morphisms representing the intensional meanings such that truth is preserved. Pregroup grammars serve as the tool that transorms a grammatical string into a morphism. The chosen linguistic examples concern negation, relative noun phrases, comprehension and quantiiers. Keywords: Compositional semantics, biproduct bagger categories, compact closed categories, quantum logic, pregroup grammars, compact bilinear logic, proo graphs, two-sorted logic 1 ntroduction Biproduct dagger categories have been studied extensively in quantum logic, [Selinger, 2007], [Abramsky and Coecke, 2004], [Heunen and Jacobs, 2010]. They also constitute a natural candidate as a oundation o natural language semantics, because they ormalize count words (biproduct) and relative pronouns (dagger), two logical abstractions present in natural language with a ew exceptions - both are absent in the Amazonian Pirahã, [Everett, 2005]. These two operations are powerul enough to comprehend the structure o a compact closed category and with it the representation o morphisms by graphs that represent inormation low. normation low along morphisms handles among other things the grammatical notions o dependency and control. The noun phrase birds who ly in 5.2 illustrates how the graphical representation o meanings recuperates the dependency links. An example concerning control can be ound in 3.6 o [Preller and Prince, 2008]. The biproduct and dagger also make it possible to imbed predicate logic. n particular, the logical content o words like and, or, not, all, no, who, some can be captured by an explicit deinition and the notions o truth and logical LRMM, 161 rue Ada, Montpellier, France. preller@lirmm.r 1

3 1 NTRODUCTON consequence can be introduced. The property o the morphisms interpreting some, however, does not restrict to a unique morphism. Each occurrence o the word some in the text may by interpreted dierently. The biproduct and dagger also provide an abstract deinition o the inner product and thereore a geometrical representation o the linguistic notion o similarity by the inner product (cosine) and o the logical notion o negation by orthogonality. Thus, the geometrical operators on subspaces and projectors proposed in [Rijsbergen, 2004] and [Widdows, 2004] can be generalised rom FdVect R, the inite dimensional vector spaces over the ield o real numbers, to biproduct dagger categories. The semantic categories considered here are biproduct dagger categories withageneratingobject. TypicalsemanticcategoriesarethecategoryFdVect R and the category 2SF o inite sets and two-sorted unctions. The latter are models o two-sorted irst order predicate logic, which according to [Benthem and Doets, 1983] is equivalent to second order logic with general models. The mathematical tool or recognising grammatical strings o words and computing their meanings is that o pregroup grammars. The syntactical analysis is carried out in the ree quasi-pregroup C(B) generated by a partially ordered set B o basic types introduced in [Lambek, 1999]. t is a compact bicategory, that is to say a compact closed category. The pregroup dictionaries o [Lambek, 2008], which list pairs word : T consisting o a word and a type, lack semantics. Thereore, the pregroup lexicons proposed here consist o triples word : T :: word, where word is a ormal expression in the language o compact closed categories. These ormal morphisms play a role similar to that o lambda-terms in categorial grammars. The meanings o grammatical strings are computed in the lexical category C(B L), the ree compact closed category generated by the partially ordered set B and the set o basic morphisms L given by the pregroup lexicon. The interpretation o the grammatical string is mediated by a unctor rom the lexical category into an arbitrary biproduct dagger category with a generating object. the unctor preserves the compact closed structure, it guarantees compositionality, but it must satisy supplementary conditions to become a model or natural language. The compositional semantics o [Clark et al., 2008] and [Kartsaklis et al., 2013] or vector space models in FdVect R also use pregroup grammars and a unctorial method. This claim has to be taken with a caveat. The suggested unctor is partial, it is not deined or strings containing logical words, relative, determiners and so on. n act, such a partial unctor cannot be extended to the logical words expressing negation and implication, neither or classical logic nor or quantum logic, [Preller, 2013]. The categorical semantics in biproduct dagger categories proposed here is more general. One can not only deine vector space models in the abstract setting o an arbitrary semantic category, but also unctors that are deined on all o the lexical category and that also simulate truth and logical consequence. These are the truth-theoretical models. For every truth-theoretical model there is a canonical vector space model and an isomorphism rom the lattice o

4 2 BASC PROPERTES predicates in the ormer to the lattice o projectors introduced by quantum logic in the latter. This is due to the act that the projectors corresponding to words have a common basis o eigenvectors. The material o this article is organised as ollows. Section 2 presents the basic properties o biproduct dagger categories with an emphasis on the class o projectors called intrinsic, because their matrix representation is the same in any biproduct dagger category. They include the morphisms arising rom grammatical strings. Section 3 concentrates on biproduct dagger categories with a generating object and their important property o explicit deinitions. Section 4 establishes the equivalence between the quantum logic o intrinsic projectors and the logic o predicates. The essential characteristic o a predicate is that it assigns truth values both to individuals and sets o individuals. Section 5 starts with a cut ree axiomatisation o compact bilinear logic, [Lambek, 1993] and [Buszkowski, 2002], and shows how pregroup grammars construct syntactical analysis and semantical representation in the lexical categories based on the proo graphs o [Preller and Lambek, 2007]. The section concludes with a ew linguistic examples linking relative noun phrases and comprehension as well as quantiiers and negation. 2 Basic properties This section recalls deinitions and properties requently intervening in quantum logic, see or example [Abramsky and Coecke, 2004], [Heunen and Jacobs, 2010], [Selinger, 2007]. Only the emphasis on intrinsic morphisms is new. 2.1 Biproduct dagger categories A dagger category is a category C together with a contravariant involution unctor dagger : C C that is the identity on objects. This means that the ollowing equalities hold or any object V and morphisms : V W, g : W U V = V 1 V = 1 V (g ) = g : U V = : V W. Call the adjoint o. n a dagger category any coproduct o V and W with canonical injections q 1 and q 2 is also a product o V and W with canonical projections q 1 and q 2 and vice versa. ndeed, the dagger inverts the diagram expressing the universal property. Hence coproducts are biproducts in a dagger category. Similarly, an initial object 0 o a dagger category is also a terminal object. ndeed, i 0 V : 0 V is the unique morphism rom 0 to V then 0 V : V 0 is the unique morphism rom V to 0. Hence 0 is a zero object where 0 VW = 0 W 0 V : V W is the unique morphism that actors through 0. The subscripts may be dropped, context permitting.

5 2.1 Biproduct dagger categories 2 BASC PROPERTES Deinition 1. A biproduct dagger category is a dagger category C equipped with an initial object 0 and binary coproducts such that the canonical injections q 1 : V V W and q 2 : W V W satisy q i q i = 1,q j q i = 0 or i,j = 1,2, i j. (1) NotethatV 0 V. ndeed,q 1 : V V 0andq 1 : V 0 V areinverseo eachother, because(q 1 q 1 ) q 1 = q 1 = 1 V 0 q 1 and(q 1 q 1 ) q 2 = 0 = 1 V 0 q 2. Thereore q 1 q 1 = 1 V 0. Given g j : U V j, denote g 1,g 2 : U V 1 V 2 the unique morphism satisying q j g 1,g 2 = g j or j = 1,2. Similarly, or h i : W i E denote [h 1,h 2 ] : W 1 W 2 E the morphism determined by [h 1,h 2 ] q i = h i or i = 1,2. Finally, or i : V i W i, denote 1 2 : V 1 V 2 W 1 W 2 the unique morphism such that q i ( 1 2 ) q i = i and q i ( 1 2 ) q j = 0 VjW i, or i,j = 1,2, i j. We have or any g : U U and h : E E g 1,g 2 g = g 1 g,g 2 g, h [h 1,h 2 ] = [h h 1,h h 2 ] ( 1 2 ) g 1,g 2 = 1 g 1, 2 g 2 [h 1,h 2 ] ( 1 2 ) = [h 1 1,h 2 2 ] Any morphism : V 1 V 2 W 1 W 2 is uniquely determined by the our morphisms q i q j, or i,j = 1,2. These our morphisms may be displayed in the orm o a matrix ( q M = 1 q 1 q 1 q ) 2 q 2 q 1 q 2 q. 2 Proposition 1. The ollowing equalities hold in a biproduct dagger category 0 VW = 0 WV 1, 2 = [ 1, 2 ] ( 1 2 ) = 1 2. (2) Any biproduct category C is enriched over abelian monoids, i.e. the binary operation deined on each hom-set C(V,W) by = [1 W,1 W ] ( 1 2 ) 1 V,1 V, or 1, 2 : V W is associative and commutative with unit 0 VW.

6 2.1 Biproduct dagger categories 2 BASC PROPERTES Moreover, addition is bilinear h ( ) g = h 1 g +h 2 g, or g : V V,h : W W and ( ) = q 1 q 1 +q 2 q 2 = 1 V W. (3) t ollows that any biproduct category has a matrix calculus, i.e. the ollowing equalities hold M +g = M +M g and M g = M g M, (4) where the right-hand matrices are deined the usual way, entry by entry. For example, i kj are the entries o M and g ik those o M g the entry h ij o M g M satisies h i = g i1 1j +g i2 2j. Deine the n-ary biproduct with canonical injections q in : V i V 1... V n by induction thus V 1... V 0 := 0 V 1... V 1 := V 1 V 1... V n := (V 1... V n 1 ) V n or n 2, q 00 = 1 0 q 11 = 1 V1 q i2 = q i : V i V 1 V 2 or i = 1,2 q in = q 1 q i(n 1), or i = 1,...,n 1 and q nn = q 2, where q 1 : (V 1... V n 1 ) (V 1... V n 1 ) V n, q 2 : V n (V 1... V n 1 ) V n. context permits, write q i instead o q in. n the case where V i = V or all i = 1,...,n, write n V = V 1... V n 1. Adopt a similar convention or n, where : V W. Equalities (1) - (3) generalise to n-ary biproducts. Together, they constitute the generalised Dagger Biproduct Calculus. For example, the generalised version o (1) is q i q i = 1 Vi, q i q j = 0 VjV i, or i,j = 1,...,n,i j. Any morphism : V 1... V m W 1... W n is completely determined by the nm morphisms q i q j, or j = 1,...,m, i = 1,...,n. Hence, = g i and only i the ollowing Matrix Equalities hold q i g q j = q i q j, or j = 1,...,m, i = 1,...,n. (5) The nm morphisms q i q j may be displayed in the orm o an nm-matrix M = q 1 q 1 q n q 1... q 1 q m.... q n q m The equalities (2) - (4) then generalise to arbitrary biproducts. 1 The notation V n is reserved or the n-ary tensor product introduced in Section 3.3.

7 2.2 Examples o biproduct dagger categories 2 BASC PROPERTES The most interesting case is when V j = W i = or all indices j,i, where is any object non-isomorphic to 0. Then the entries o the matrix are endomorphisms o, i.e. q i q j :. every object is isomorphic to a inite coproduct o some distinguished object, say, then the entries o a matrix reer to the coproducts V = m and W = n. 2.2 Examples o biproduct dagger categories The categories H o linear maps and inite dimensional Hilbert spaces over the ields = R or = C are biproduct dagger categories. Every object is a inite biproduct o, identiied with a one-dimensional space. The adjoint o α is the conjugate o α. = R then α = α. The matrix o the adjoint o a linear map in H is the transpose o the conjugate matrix o. The category 2SF o two-sorted unctions Two-sorted irst order logic has two sorts o variables, one or elements x, and one or sets X. Besides an equality symbol or each sort, there is a binary symbol requiring elements on the let and sets on the right, x X. There are two sorts o quantiiers, x, X etc. Functional symbols accept both sorts as arguments. Models interpret every unction symbol by a two-sorted unction : A B satisying ({x}) = (x) or x A ( ) = (X Y) = (X) (Y) or X,Y A. The category 2SF o two-sorted unctions as morphisms and inite sets as objects is a biproduct dagger category. Any two-sorted unction is determined by its values on elements, because all sets are inite. The adjoint : B A o : A B is given by (b) = {a A : (a) = b or b (a)}. The biproduct is the disjoint union o sets, with as the zero object. With these deinitions, any object o 2SF is isomorphic to a inite biproduct o any arbitrarily ixed singleton set. Note that it has exactly two endomorphisms, namely the identity map and the zero map, which sends the unique element o to the empty set. Thereore, α = α or all endomorphisms α :. The sum o,g : A B is the set-theoretical union (+g)(x) = (x) g(x). The right-hand side o the last equality involves an abuse o notation: i (x) or g(x) is an element, we should have used the corresponding singleton set. English makes the same abuse. Compare apples and pears with the teacher and the student. The category R o semimodules over [0,1] Recall that the linear order on the real numbers in [0,1] induces a distributive

8 2.3 Generalising geometrical notions 2 BASC PROPERTES and implication-complemented lattice structure on [0, 1], namely α β = max{α,β} α β = min{α,β} α β = max{γ : α γ β} α = α 0. This lattice is not Boolean, because α = 1 α or 0 < α < 1. The lattice operations deine a semiring structure on = [0,1] with neutral element 0 and unit 1 by α+β = α β α β = α β. The category R o ree semi-modules over the real interval = [0,1], generated by a inite set is a biproduct dagger category. The zero-object is the semi-module reduced to a single element, {0}. The interval = [0,1] is a semimodule, generated by the singleton set {1}. The endomorphisms o identiy with the elements o. The biproduct o two semi-modules is generated by the disjoint union o the two semi-modules. Hence every object o R is a inite biproduct o. Every endomorphism/element o is by deinition, its own adjoint α = α. The adjoint : n m o a linear map : m n is the linear map deined by the transpose o the matrix o. 2.3 Generalising geometrical notions Several geometrical notions amiliar rom real or complex vector spaces can be generalised to arbitrary dagger biproduct categories. The letter C denotes an arbitrary biproduct dagger category in the ollowing. Deinition 2. Morphisms : U W and g : V W are said to be orthogonal in W i g = 0. A projector is an idempotent and sel-adjoint morphism p : V V, i.e. p p = p and p = p. Orthogonality is a symmetric relation. Every morphism is orthogonal to 0. n general, a morphism can have several distinct orthogonal morphisms. Projectors, as we shall see later, determine a maximal morphism that is orthogonal to them. This maximal orthogonal morphism is itsel a projector, the negation o the original projector. The rest o this section is an argument that iterated biproducts o any object V 0 internalise propositions and inite subsets. Projectors will play the role o propositions, the canonical injections q i : V n V the role o individuals. Note that the canonical injections q i and q j are distinct or i j, because 1 V 0 VV. Subsets o individuals are internalised as sums o distinct canonical injections. A morphism : V W is unitary i = 1 V. A unitary is necessarily monic and its adjoint is epic. n the case where is an isomorphism, is unitary i and only i = 1 W i and only i = 1. For example, a morphism o 2SF is unitary i and only i it maps dierent elements to nonempty disjoint subsets.

9 2.3 Generalising geometrical notions 2 BASC PROPERTES Proposition 2. Let V be any object o C. Assume K = {i 1,...,i k } and M = {l 1,...,l m } are disjoint subsets o {1,...,n}. Then q K = [q i1,...q ik ] : k V n V is unitary and orthogonal to q M = [q l1,...q lm ] : m V n V. The endomorphism p K = q K q K : n V n V is a projector and p K +p M = p K M. (6) Proo. First, recall that [q i1,...q ik ] = q i 1,...q i k. Hence, the Matrix Equalities (5) characterise p K as the unique morphism satisying { q i p 1 V i i = j and j K K q j =, or i,j = 1,...,n. (7) 0 VV else Next, use the Dagger Biproduct Calculus and the Matrix Equalities to show that q i 1,...q i k [q i1,...q ik ] =1 q i 1,...q (8) i k [q l1,...q lm ] =0. This proves that q K is unitary and orthogonal to q M. Finally, check that p K is sel-adjoint, via the equality recalled initially, and that it is idempotent, via the irst equality o (8). Equality (6) ollows rom the Matrix Equalities and bilinearity o addition. Corollary 1. V 0, the map K p K is a one-to-one correspondence between subsets K {1,...,n} and the projectors p K. Proo. Use the characterising equalities (7) and the act that 1 V 0 vv. Corollary 2. K M = and K M = {1...,n} then p K +p M = 1 n V = q 1 q 1 + +q n q n. Proo. The equality p K M = 1 n V is a special case o (8). Hence, p K +p M = 1 n V ollows by (6). Make V = and think o the canonical injections q i : n = N as individuals. Then every p K assimilates to a property o individuals, namely the property that is true or q i i p K q i = q i and alse or q i i p K q i = 0. Recall that a morphism g : U V is a kernel o : V W i it satisies g = 0 and is universal or this property. Universality means that or any h : X V with h = 0 there is a unique h : X U with h = g h. Proposition 3. Let K = {i 1,...,i k } be a subset o {1,...,n} = N. Then q N\K is a unitary kernel o p K and q K. Moreover, q K is the image o p K.

10 2.3 Generalising geometrical notions 2 BASC PROPERTES Proo. The equality p K q N\K = 0 is a particular case o (8). To prove the universality o q N\K, assume that g : U n V satisies p K g = 0. Let h := q N\K g : U (n k) V. Then g = (p K +p N\K ) g = p K g+q N\K q N\K g = q N\K q N\K g = q N\K h. This proves that ker(p K ) = q N\K. The equality ker(q K ) = q N\K ollows, because q K g = 0 implies p K g = 0. Finally, using the deinition im(p K ) := ker((ker(p K )) ) o [Heunen and Jacobs, 2010], compute im(p K ) = ker((ker(p K )) ) = ker(q N\K ) = q K. p K n V n V q q K K k V Note that v : W n V is invariant under composition with p K exactly when v actorises through q K. ndeed, v = p K v implies v = q K (q K v). Conversely, v = q K g implies v = q K (q K q K) g = p K v. Proposition 3 also implies that the orthogonal complement p K := ker(p K ) (ker(p K )) o the projector p K satisies p K = p N\K. Proposition 4. The projectors p K,p L satisy or any K,L {1,...,n} The relation given by p K p L = p L p K = p K L. p K p L p K p L = p K (9) is a partial order on the projectors p K inducing a Boolean Algebra structure such that p K p L = p K p L p K p L = p K L p K = p N\K = p K Under the assumption that V 0, the map K p K is an isomorphism rom the Boolean algebra o subsets o N into the projectors o n V. n particular, p K is the largest projector orthogonal to p K, i.e. p L p K i and only i p K p L = 0. Proo. Partition K L into the three disjoint subsets M = K L, M = K \(K L), and M = L\(K L). By Proposition (2), p K = p M +p M and

11 3 SEMANTC CATEGORES p L = p M +p M and the mixed terms p M p M, p M p M, p M p M are equal to 0. Thereore p K p L = (p M +p M ) (p M +p M ) = p M p M = p M. Similarly, p L p K = p M. This proves the irst assertion. The rest is now straight orward. Make V = and think o the canonical injections q i : n = N as individuals. Recall that p K is true or the individual q i i p K q i = q i and alse or q i i p K q i = 0. Then p K p L is true or q i i and only i both p K and p L are true or q i. Proposition 5. The partial order o the projectors p K is isomorphic to the partial order o their canonical images im(p K ) = q K. More precisely, or arbitrary subsets K = {i 1,...,i k } and M = {j 1,...,j m } o {1,...,n} the ollowing equivalence holds p K p M i and only i q K q M as subobjects. Proo. Recall that p K p M is equivalent to K M by (9). Assume that q K q M as subobjects and let g : k V m M be the morphism such that q K = q M g. Then q K = q M g implies q il = q K q l = q M g q l and thereore i l M, by Proposition 3, and this or l = 1,...,k. Hence, K M. Conversely, the inclusion K M provides an obvious actorisation q K = q M g. 3 Semantic categories This section mentions well known properties o biproduct dagger categories with a chosen generating object. The most important consequence or natural language semantics is the Property o Explicit Deinitions, Proposition 8. Deinition 3. Let C be a biproduct dagger category. An object o C is a generating object i the ollowing holds - α β = β α or all α,β : - or every object V there is an integer n 0 and a unitary isomorphism b V : n V. A semantic category is a biproduct dagger category that has a distinguished generating object 0. n the category 2SF o two-sorted unctions, is a distinguished singleton set. n the category R o semi-modules over the real interval [0,1], the distinguished object is this interval, = [0,1]. For real Hilbert spaces, = R, or complex Hilbert spaces, = C.

12 3.1 Spaces, vectors, scalars 3 SEMANTC CATEGORES 3.1 Spaces, vectors, scalars This subsections transers the terminology amiliar rom Hilbert spaces to an arbitrary semantic category. Hopeully, this will not conuse the reader. A space is an object V o C together with a unitary isomorphism b V : n V, called the base o the space. The integer n is the dimension o the space. A vector o V is a morphism rom to V. A scalar is an endomorphism o. The scalars orm a commutative semiring where multiplication is composition and addition is deined by Proposition 1. A scalar β is positive, i it has the orm β = α α. For example, 1 and 0 are positive. The positive scalars are closed under multiplication and addition, e.g.[selinger, 2007]. These notions have the usual meaning in Hilbert spaces. A space o 2SF is a set B and an enumeration o its elements B = {b 1,...,b n }, where b i = b V (q i ), or i = 1,...,n. The dimension is the cardinal o the set B. The vectors o B are the subsets o B. There are two scalars in 2SF, namely the map 0 that maps the unique element o to the empty set and the map 1 that maps the unique element to itsel. Both are positive. Scalar multiplication is deined or any scalar α : and vector v : V by αv = v α. Scalar multiplication is associative and commutes with addition (αβ)v = α(βv) and α(v +w) = αv +αw. For any morphism : V W and vector v : V, the value (v) o at v is (v) = v. All morphisms o C are linear, that is to say or : V W, v,w : V and α,β : (αv +βw) = α(v)+β(w). Assume that b V : m V is the base o V. The vectors a j = b V q j : V, j = 1,...,m are basis vectors o V and A = {a 1,...,a m } is the chosen basis o V. The basis vectors satisy a i a j = δ ij, or i,j = 1,...,m, (10) where δ ii = 1 and δ ij = 0 or i j. There are exactly m distinct basis vectors, because otherwise we would have 1 = 0, which contradicts 0. The equalities (10) mean that the basis vectors are unitary and pairwise orthogonal. Proposition 6. Every vector o V can be written uniquely as a linear combination o the chosen basis vectors. Proo. Let {a 1,...,a m } be the basis o V and v : V and α i = q i b V v, or i = 1,...,m.

13 3.2 Matrix calculus 3 SEMANTC CATEGORES Recall that q 1 q 1 + +q m q m = 1 m, by (6). Hence v = b V (q 1 q 1 + +q m q m) b V v = b V q 1 q 1 b V v + +b V q m q m b V v = a 1 α 1 + +a m α m = α 1 a 1 + +α m a m. This proves the existence. To see the unicity, assume v = a 1 β 1 + +a m β m. Multiplying both sides o the equality on the let by q i b V, we get q i b V v = β i, or i = 1,...,m. Corollary 3. Let K = {i 1,...,i k } {1,...,n}, v : k and j {1,...,n}. Then q K v = q j implies j K. Proo. Recall that q K = [q i1,...q ik ] : k n and thereore q K q l = q il or l = 1,...,k. Assume v : k and q K v = q j. Write v = k l=1 α lq l, where α l :. Then q j = q K ( k l=1 α lq l ) = k l=1 α l(q K q l ) = k l=1 α lq il. Coordinates are unique, thus j = i l and α l = 1 or some l k and α l = 0 or l l. Finally, q j = q il implies j = i l, which terminates the proo. Reer to the (unique) scalars α i, i = 1,...,m, such that v = α 1 a α m a m as the components o v. The notation V = V A expresses that A is the basis o the space V. the last element in a composition is a vector we may switch to set-theoretical notation to highlight the analogy between categorical and set-theoretical properties, e.g. p(v) = w instead o p v = w etc. 3.2 Matrix calculus The matrix calculus amiliar rom Hilbert spaces generalises to arbitrary biproduct dagger categories with a generating object. Proposition 7. Every morphism is uniquely determined by its values on the basis vectors. Proo. Let A = {a 1,...,a m }, B = {b 1,...,b n } and suppose that,g : V A W B coincide on the basis vectors a j = b V q j or j = 1,...,m. Then q i b W b V q j = q i b W g b V q j or i = 1,...,n,j = 1,...,m. Hence, b W b V = b W g b V, which implies = g. Proposition 7 has a converse Proposition 8 (Explicit Deinitions). Given vectors w 1,...,w m in W B, there is a unique morphism : V A W B satisying a j = w j, or j = 1,...,m. (11) Proo. The components o w j = φ 1j b 1 + +φ nj b n, or j = 1,...,m, deine a unique morphism g : m n such that q i g q j = φ ij, or i = 1,...,n, j = 1,...,m. Then = b W g b V satisies (11).

14 3.2 Matrix calculus 3 SEMANTC CATEGORES Proposition 8 can be rephrased by saying that semantic categories admit Explicit Deinitions. The morphism is explicitly deined by the values w j W B or the basis vectors a j V A in (11). By Proposition 8, every morphism : V A W B determines and is determined by the scalars φ ij = q i b W b V q j, or j = 1,...,m, i = 1,...,n. The scalars φ ij then determine with respect to A and B. The corresponding matrices are φ φ 1m φ φ n1 M =.. M =... φ n1... φ nm φ 1m... φ nm The Dirac notation can be introduced with its usual properties: Assign to any vector v = α 1 b 1 + +α n b n o V = V B a row matrix and a column matrix v = M v = ( α 1... α n), v = Mv = The inner product o v and w = β 1 b 1 + +β n b n : V is v w := M v M w = α 1 β 1 + +α nβ n, α 1.. α n. and the outer product o any vector u = γ 1 a 1 + +γ m a m o U = U A and w β 1 γ 1 w u := M w M u =. β n γ 1... β 1 γ m..... β n γ m Otherwise said, v w is the matrix o v w and w u is the matrix o b V w u b U : U V. The outer product o a basis vector b i = n k=1 δ kib k o V B and a basis vector a j = m l=1 δ jla l o U A is b i a j = (δ ij kl ), where δ ij ij = 1 and δij kl = 0 or k i or l j, k = 1,...,n, l = 1,...,m. ndeed, δ ij kl = δ kiδ jl = δ kiδ jl. n particular, the outer product b i b i is the matrix o the projector p {i}, or i = 1,...,n. Deinition 3 can now be reormulated or vectors in terms o the inner product. Vectorsareorthogonal i and only i theirinnerproductequals0. Avector is unitary i the inner product o the vector with itsel equals 1.

15 3.3 Compact closed categories 3 SEMANTC CATEGORES Examples o explicitly deined morphisms are the diagonal d V : V V V and the symmetry σ VW : V W W V deined or the basis vectors {a 1,...,a m } o V and {b 1,...,b n } o W thus d V (a j ) = a j a j σ VW (a j b i ) = b i a j, or j = 1,...,m, i = 1,...,n. Note that the morphism d V : V V V satisies { d V (a a i i i = j j a i ) = 0 i i j. Let K, L be subsets o the basis A o V, v K be the sum o basis vectors in K and similarly or v L. Then d V (v K v L ) = v K L = v k v L. (12) The morphisms d V and d V play the role o multiplication and comultiplication o the Frobenius algebra over the vector space V o FdVect R. This means that the compositional semantics o [Kartsaklis et al., 2013] can be generalised to an arbitrary semantic category. 3.3 Compact closed categories Recall that a monoidal category consists o a category C, a biunctor, a distinguishedobject andnaturalisomorphismsα VWU : (V W) U V (W U), λ V : V V and ρ V : V V subject to the coherence conditions o [Mac Lane, 1971]. t is a compact closed category i every object has a right and a let adjoint. A compact closed category is symmetric i there is a natural isomorphism σ VW : V W W V such that σ 1 VW = σ WV subject to the coherence conditions o [Mac Lane, 1971]. For notational convenience, the associativity isomorphisms α VWU and the unit isomorphisms λ V and ρ V are replaced by identities, e.g. (V W) V = V (W U), V = V and V. The tensor product is deinable in semantic categories. t plays the role o a bookkeeping device and internalises matrices as vectors o a tensor product space. Let b V : m V and b W : n W be spaces with chosen basis vectors a j = b V q j, b i = b W q i, where q j : m, q i : n, j = 1,...,m, i = 1,...,n, are the canonical injections. The tensor product o V and W and the dagger isomorphism b V W : n (m ) V W are V W := n V, b V W := n b V. Let q i : m n (m ), i = 1,...,n, be the canonical injections. The tensor product o a j and b i is the vector a j b i := b V W q i q j : V W.

16 3.3 Compact closed categories 3 SEMANTC CATEGORES Under these deinitions, the tensor product distributes over the dagger and the biproduct ( g) = g : W D V U V (W U) (V W) (V U). A compact closed category is a symmetric monoidal category C together with a contra-variant unctor and maps η V : V V and ǫ V : V V, called unit and counit respectively, such that (ǫ V 1 V ) (1 V η V ) = 1 V (ǫ V 1 V ) (1 V η V ) = 1 V. Proposition 9. Semantic categories are compact closed. Proo. (Outline) Follow the construction o the dual space in [Abramsky and Coecke, 2004] or the category o complex Hilbert spaces. First, introduce the dual scalar multiplication or v : V A and α : α v := v α = α v. This deinition creates a dual version o Proposition 6: Every vector can be written uniquely as the sum o dual scalar multiples o basis vectors. ndeed, let β i = α i or j = 1,...,m. Then m m m α i a i = α i a i = β i a i. i=1 i=1 The dual space V A is the space V A where vectors are given as sums o dual scalar multiples o basis vectors. n the case o 2SF, R and real Hilbert spaces, we have V A = V A, because α = α or all α :. Given : V A W B, use the principle o Explicit Deinitions to introduce the morphisms : V A W B and the dual : W B V A such that Then (a j ) = n i=1 φ ij b i = n i=1 φ ij b i, or j = 1,...,m (b i ) = m j=1 φ ij a j = m j=1 φ ija j, or i = 1,...,n. i=1 = = : W B V A. Note that the dual coincides with the dagger in the categories 2SF, R and FdVect R, which are the most requently considered categories in natural language semantics. The unit η V : V V and counit ǫ V : V V are the morphisms deined explicitly thus η V (1 ) = n i=1 a i a i ǫ V (a i a j ) = δ ij, or i,j = 1,...,n, where δ ij = 1 i i = j and δ ij = 0 i i j.

17 4 THE NTERNAL LOGC 4 The internal logic An internal logic o a category consists o a class o morphisms, the propositions, and a set o equalities expressing the truth o propositions. The internal logic o semantic categories ollows quantum logic in letting projectors stand or propositions. Logical connectives are deined in such a way that the projectors orm an ortho-complemented lattice with the identity as largest element. The approach to logic via predicates also is possible in semantic categories. The equivalence between the notions will be the subject o this section. n both cases, the basis vectors o the domain play the role o individuals. Basis vectors are generalised to Boolean vectors to capture the plurals o natural language. A vector v = α 1 b 1 + +α n b n is said to be Boolean i α i = 0 or α i = 1, or all i = 1,...,n. EveryBooleanvectorv : V B determinesauniquesubsetk = {i 1,...,i k } o N = {1,...,n} such that v = i K b i = v K. The propositional connectives are lited rom subsets K N to Boolean vectors such that v K v L = v K L, v K v L = v K L, etc. hold. Hence, the map K v K is a Boolean isomorphism. The Boolean vectors orm a Boolean algebra with largest element 1 = n i=1 b i = v N and smallest element 0 = v = 0 VB. Sometimes it is convenient to identiy a subset A = {b i1,...,b ik } B o basis vectors with the corresponding Boolean vector A = b i1 + +b ik = v K. 4.1 The logic o intrinsic projectors Projectors will stand or propositions in this subsection. The truth o a proposition p is expressed by the equality p = 1 V. A semantic category may have projectors that do not intervene in natural language semantics. For example, i all entries o a square matrix are equal to 1, it deines a projector in 2SF and in R, but not in a Hilbert space. We avoid the unwanted morphisms by limiting interpretations to intrinsic morphisms. Deinition 4 (ntrinsic morphism). A morphism : V A W B is intrinsic with respect to A and B i it sends every basis vector in A to a basis vector in B or to the zero vector 0. The identity 1 V and the diagonal d V : V V V, which maps any basis vector b o V to b b, are intrinsic. ntrinsic morphisms are closed under composition and tensor products. They are ubiquitous in natural language. Determiners, relative pronouns and verbs, to mention but a ew, are interpreted by intrinsic morphisms.

18 4.2 The two-sorted logic o predicates 4 THE NTERNAL LOGC Observe the ollowing properties, which are are straightorward except possibly (?), which is proved in [Preller, 2012]. Proposition 10. Let B = {b 1,...,b n } be the basis o space V B in C. Then - a projector p : V B V B is intrinsic i and only i p(b i ) = b i or p(b i ) = 0, or i = 1,...,n - the entries o the matrix (π ij ) ij o an intrinsic projector p satisy π ij = 1, i i = j and p(b i ) = b i π ij = 0, else, i,j = 1,...,n - intrinsic projectors map Boolean vectors to Boolean vectors - every intrinsic projector p has the orm b V p K b V, where K = {i : p(b i ) = b i, 1 i n} - the morphism b V p K b V is an intrinsic projector o V B or every K {1,...,n}. The unitary isomorphism b V lits the lattice operators deined on the canonical projectors p K o n to the intrinsic projectors o V B. Context permitting, we write p = p K instead o p = b V p K b V. Proposition 11. Let P B the set o projectors o V B that are intrinsic w.r.t. B. The map K p K is a negation preserving lattice isomorphism rom the Boolean algebra o subsets o N = {1,...,n} onto the lattice P B o intrinsic projectors o V B. Moreover, the ollowing properties hold or any K,L N - p K p L = 1 VB i and only i p K p L = p K - p K p L = p K i and only i p L v K = v K - p L v K = v L K. n particular, p L ( 1) = v L and p N v K = v K - p L v K = v K i and only i p L (b i ) = b i, or all i K - intrinsic projectors are monotone increasing on Boolean vectors. The restriction to P B makes the logic classical. V B is a Hilbert space then the elements o P B are exactly the projectors or which B is a common basis o eigenvectors. The internal logic proposed here uses only the classical part o quantum logic. 4.2 The two-sorted logic o predicates Predicates and two-sorted logic are deinable in an arbitrary semantic category. Let S be a ixed two-dimensional space with basis vectors = b S q 1 and = b S q 2. The two-sorted connectives introduced below are morphisms and as such they are determined by their values on the basis vectors.

19 4.2 The two-sorted logic o predicates 4 THE NTERNAL LOGC The two-sorted negation not S : S S is deined explicitly by not S ( ) =, not S ( ) =. Recall that the ull vector 1 o S satisies 1 = +. Then More generally, not S ( 1) = 1 and not S ( 0) = 0. not S (α +β ) = β +α. The two-sorted conjunction and S : S S S and two-sorted disjunction or S : S S S are deined explicitly on the our basis vectors o S S and S ( ) =, and S ( ) = and S ( ) = and S ( ) = or S ( ) =, or S ( ) = or S ( ) = or S ( ) =. Note that the two-sorted connectives are dierent orm the set-theoretical connectives introduced or Boolean vectors at the beginning o this section. Proposition 12. The two-sorted connectives deine a Boolean structure on the vectors o S. n particular, or arbitrary vectors v : S and w : S the ollowing holds not S not S v = v, not S and S (v w) = or S (not S v not S w). Proo. Show the property or basis vectors and use the act that morphisms commute with addition and scalar multiplication. A morphism p : V S is a predicate i it is intrinsic. t is a predicate on V i it maps basis vectors o V to basis vectors o S, i.e. p(x) = or p(x) =, or every basis vector x o V. By an n-ary predicate on E we mean a predicate on V = E... E. We may think o basis vectors as individuals o the universe o discourse (sort one ) and o Boolean vectors as subsets o individuals (sort two ). A predicate accepts individuals and sets o individuals as arguments. For an individual there areonlytwopossibletruthvalues, namely and. Forsetsoindividualsthere are at least two more. Proposition 13. Denote m = the m-old sum o the unit 1. Let p : V B S be a predicate on V B and A = {b i1,...,b ik } B a subset o k distinct basis vectors. Then there are non-negative integers k 1 k and k 2 k such that p( x Ax) = k 1 +k 2. (13) Proo. The predicate p partitions A into two disjoint subsets A 1 and A 2 such that p(x) = or all x A 1 and p(x) = or all x A 2. Let k i be the number o elements in A i.

20 4.2 The two-sorted logic o predicates 4 THE NTERNAL LOGC counting property n Hilbert spaces, a predicate counts the number o elements o a set or which it is true and also the number o elements or which it is alse. The integers k 1 and k 2 are unique and satisy k 1 +k 2 = k. undamental property in 2SF A two-sorted predicate inorms us whether it is always true, always alse or partly true and partly alse on a set A identiied with the Boolean vector x A x. p(a) = 0 p(x) = 0 or all x A A = p(a) = p(x) = or all x A and A p(a) = p(x) = or all x A and A p(a) = + p(x) = and p(y) = or some x,y A. (14) ndeed, the k-old sum o the unit o 1 is equal to 1 whenever k > 0. Hence, k =. The counting property and the undamental property give us a clue to deining truth in semantic catgories. truth-values Let p be a linear predicate on V A and X any vector o V A. We say that - p(x) is true i there is α 0 such that p(x) = α - p(x) is alse i there is α 0 such that p(x) = α - p(x) is mixed i p(x) = α +β or some α 0,β 0 - p(x) is mute i p(x) = 0. Proposition 14. Predicates are closed under composition with the two-sorted connectives. More precisely, assume that p : V B S and r : V B S are predicates on V = V B. Then the morphisms not S p, and S (p r), or S (p r) are again predicates on V B respectively on V B V B and satisy not S not S p = p not S and S (p r) = or S ((not S p) (not S r)). For any x B, A B p(x) = not S (p(x)) = p( x A x) = k 1 +k 2 not S (p( x A x)) = k 2 +k 1 (15) Whereas not S (p(x)) = is equivalent to p(x) or anyy basis vector x, this no longer holds or arbitrary vectors. For the counter example, let a and b be two distinct basis vectors such that p(a) = and p(b) =. The predicates and S (p r) and or S (p r) are predicates on V V. Composing them with the diagonal d V : V V V, we obtain the predicates and S (p r) d V and or S (p r) d V on V such that the equalities (15) still hold. Hence, the predicates on a given space orm a Boolean algebra.

21 4.3 ntrinsic projectors and predicates 4 THE NTERNAL LOGC 4.3 ntrinsic projectors and predicates LetC beanarbitrarysemanticcategoryandv = V B beanyspaceoc withbasis B. For every intrinsic projector p : V V, deine a predicate (p) : V S by the condition { i p(x) = x (p)(x) =, or all x B. else Conversely, given a predicate p : V S on V, deine an intrinsic projector J(p) : V V by { x i p(x) = J(p)(x) =, or all x B. (16) 0 else Proposition 15. The maps and J are inverse o each other and the ollowing holds or any intrinsic projectors p,r : V V ( p) = not S (p) (p r) = and S ((p) (r)) d V (p r) = or S ((p) (r)) d V (p r) = and S ((p) (r)). (17) Moreover, or any Boolean vector v L = i L b i and any intrinsic projector p : V V p(v L ) = v L i L ((p)(b i ) = p(v L ) = (18) 0 i L ((p)(b i ) =. n particular, i k is the number o basis vectors let invariant by p then (p)( n b i ) = k +(n k). i=1 Proo. t is suicient to veriy (17) or basis vectors, an easy exercise. The equalities (18) ollow rom (13). The switch between predicates and intrinsic projectors is common in natural language. Typically, an adjective in attributive position is interpreted as an intrinsic projector big : V B V B. The same adjective, when in predicative position, is interpreted by a predicate big : V B S such that big(x) = x big(x) =, or all x B. The isomorphism (16) transorming a predicate into a projector is related to the relative pronoun, see Section 5.3.

22 5 COMPOSTONAL SEMANTCS 5 Compositional semantics 5.1 The lexical category The description o the lexical category given below is a notational variant o the description in [Preller and Lambek, 2007]. Call lexical category any ree compact bicategory C(D) with a single 0-cell 2, generated by some category D. Think o the objects o D as basic types and o the morphisms o D as basic morphisms. For simplicity, the canonical associativity and unit isomorphisms o the tensor product (1-cell composition) are replaced by identities, or example A (B C) = A B C = (A B) C, A = A = A. The iterated tensor products are assimilated to strings o objects. That explains why the symbol or the tensor product may be omitted in lexical categories Saying that C(D) is compact means that every 1-cell (object) Γ has a let adjoint Γ l and a right adjoint Γ r. Then Γ is a right adjoint to its let adjoint Γ l, thus Γ lr Γ. Hence the objects (1-cells) o C(D) are the unit, the objects o D, their iterated right o let adjoints and the strings built rom these. An iterated adjoint A (z) is even i z = (2n)l or z = (2n)r. t is odd i z = (2n+1)l or z = (2n + 1)r. By convention, A (0) = A. A similar convention applies to the morphisms o D. Capital latin letters designate objects o D, capital greek letters objects o C(D). The morphisms, i.e. the 2-cells, o C(D), are represented by graphs where the vertices are labelled by iterated adjoints o objects o D and the oriented links are labelled by morphisms o D. The irst our rules constitute a cut-ree axiomatisation o Compact Bilinear Logic. They imply the ith, the Cut rule. n the presentation below, each rule is ollowed by the corresponding morphism (on the let) and its proo-graph (on the right). Graphs display the domain o the morphism above, the codomain below. The links are directed, because they are morphisms o D. The basic type at the tail o the link is the domain o the link and the basic type at the head its codomain in D. n the case where the label is an identity, it is generally omitted. A double line stands or the collection o links o some previously constructed graph. Axioms 1 = 2 deinitially equivalent to compact closed category

23 5.1 The lexical category 5 COMPOSTONAL SEMANTCS Units z is even :A B D A (z) B (z) A (z) (z) = B (z) g : Γ, : A B D i z is odd :A B D B (z) A (z) B (z) (z) = A (z) z is even Γ A B A (z)r Γ B (z) z is odd (1 A (z)r g 1 B (z)) η (z) = A (z)r Γ B (z) Γ A B B (z)r Γ A (z) (1 B (z)r g 1 A (z)) η (z) = B (z)r Γ A (z) Counits g : Γ, : A B D z is even Γ A B A (z) Γ B (z)r ǫ (z) (1 A (z) g 1 B (z)r) = A (z) Γ B (z)r z is odd Γ A B B (z) Γ A (z)r ǫ (z) (1 B (z) g 1 A (z)r) = B (z) Γ A (z)r

24 5.1 The lexical category 5 COMPOSTONAL SEMANTCS 1-Cell Composition g : Γ, h : Θ Λ Γ Θ Λ Γ Θ Λ g h = Γ Θ Λ Cut g : Γ, h : Θ Γ Θ Γ Θ h g = Any morphism o C(D) can be obtained rom the morphisms o D without Cut by the irst our rules only, see [Preller and Lambek, 2007]. All paths in graphs obtained without Cut have length one. We compute the cut-ree graph o g by connecting the graph o g : Γ to the graph o h : Θ at their joint interace Θ and replace any path with endpoints in Γ or by a link with the same endpoints and direction, see the examples below. Choosing z = 0, g = 1, but : A B D arbitrary in the Unit and Counit rules we obtain η = A r A ǫ = A A r η l = Γ Θ B A l ǫ l = B l A n the particular case where = 1 A (z), the results are the unit η A : A r A and the counit ǫ A : A A r or the right adjoint and, as A = A lr, the unit η A l and counit ǫ A l or the let adjoint. The ollowing our examples o composition by walking paths illustrate how cut-elimination can be proved. The last o two o them are the axioms concerning units and counits. ( 1 A l) η A l = A A l B A l = B A l = η l. ǫ B l (1 B l ) = B l A B l B = B l A = ǫ l

25 5.2 Pregroup grammars 5 COMPOSTONAL SEMANTCS B l (ǫ B l 1 A l) (1 B l η l) = B l B A l = B l = l A l A l (ǫ A 1 B ) (1 A η ) = A A r B = A B Units o adjunction give rise to nested graphs. The same holds or counits. For example, let : A B, g : C D be morphisms in D A B = (1 A r η g l 1 B ) η = η (gl ) A r B g A r D C l B = g A r D C l B etc. Assume : A B, g : B C. Then (ǫ 1 C ) (1 A η g ) = (1 C ǫ l) (η g l 1 A ) = g A A A g g A B r C = C B l A = g B C C. The beneit o orienting and labelling links becomes evident when computing the meaning o strings o words where the graphs are given by the grammar in Section Pregroup grammars According to [Lambek, 1999], a pregroup grammar is determined by a partially ordered set B and a dictionary. The elements o B stand or grammatical notions and are called basic types, e.g. c 2, n 2 and s or plural count noun, plural noun

26 5.2 Pregroup grammars 5 COMPOSTONAL SEMANTCS phrase and sentence. The dictionary is a inite list o pairs word : T where the type T is an object o C(B). Every unctor rom B into a semantic category C extends to a unctor F : C(B) C that preserves right and let adjoints F(T l ) = F(T) = F(T r ) F( l ) = F() = F( r ) and every derivation o compact bilinear logic to a morphism o C. A lexicon is a inite list o triples word : T :: word, where T a type and word a meaning expression in the language o compact closed categories with two distinguished objects E and S. Thus, a pregroup lexicon is a pregroup dictionary, enriched by ormal meaning expressions in the language or compact closed categories. Dictionaries are suicient or recognising languages, but do not assign meanings. The added meaning expressions are the compact bilinear version o the lambda-terms o higher order logic, see [Moot and Retoré, 2011]. all : n 2 c l 2 :: all E E birds: c 2 :: some : n 2 c l 2 :: some E E bird E ly : n r 2 s:: ly E S who : c r 2 c 2 s l n 2 :: who E E S E do : n r si l d :: do E S S E not : d r ii l d :: not E S S E The basic types corresponding to the mini-lexicon above are c 2,n 2,d,s,i, partially ordered by the equalities and c 2 < n 2. The basic types d and i stand or dummy noun phrase and ininitive. A unctor interpreting the basic types c 2,n 2,d by the distinguished space E and the basic types i and s a by the distinguished space S also maps the type T in the entry word : T :: word to the codomain o word. Thus, the lexicon deines an obvious unctor rom B to the semantic category C, which maps the inequality c 2 < n 2 to 1 E. The meaning morphisms in the lexicon above a ormal expressions in the language o compact closed categories, represented by their graphs. For example, let E = V B all = η all = some = η some = all E E some E E all : E E some : E E

27 5.2 Pregroup grammars 5 COMPOSTONAL SEMANTCS do = η (1E do ) = not = η (1E not ) = do E S S E not E S S E do : S S not : S S who = who E E S E who : E S E The algebraic expression corresponding to the graph above is who = c (1 E who) (d E 1 S ) : E E S E where the morphism c : E S E E E E S E is the permutation that irst switches the last two actors o E S E E and then the third and the second actor. The meaning o grammatical strings involves besides the meanings o the words a syntactical analysis o the string. A string o words word 1...word n is grammatical i there are entries word 1 : T 1 :: word 1,..., word n : T n :: word n, and a basic type b such that T 1...T n b is provable in compact bilinear logic. Due to a theorem in [Lambek, 1999] the graph o the proo involves only underlinks. Such a graph is called a reduction. t designates a uniquely determined morphism o C(B). AnyunctorF : C(B) C mapsthereductiontoamorphismothesemantic category C and thus assigns the ollowing meaning to the grammatical string m(word 1...word n ) = F(r) (word 1... word n ). The string all birds ly has the ollowing reduction to the sentence type r 0 = all l n 2 c 2 birds c 2 ly n 2 r s s

28 5.2 Pregroup grammars 5 COMPOSTONAL SEMANTCS and thereore its meaning in C is the morphism F(r 0 ) (all bird ly) = all bird ly (E E ) (E) (E S) S = ly all bird S. Note that that any unctor on B to a compact closed category assigns a meaning-morphism to grammatical strings via these deinitions. An arbitrary unctor, however, does not guarantee that the resulting meaning relects the logical content o natural language. Thereore we add postulates that make the morphisms relect the logical content o the words. We require that S is the two-dimensional space o truthvalues with basis vectors, and that the basis vectors o E = V B stand or individuals, pairs o individuals and so on. The vectors noun : E are Boolean, morphisms word : E S are predicates on E, the morphisms word : E E are intrinsic projectors. Some words are completely determined by their logical content, namely do = 1 S, not = not S. Moreover, who : E S E is the explicitly deined morphism satisying who(b ) = b who(b ) = 0, or b B. Taking the postulates into account, the meaning o the sentence all birds ly is m(all birds ly) = ly bird : S The string birds who ly has a reduction to the plural noun phrase type, namely birds who ly ( c 2 ) (c r 2 c 2 s l n 2 ) (n r 2 s) r 1 = n 2