Galois Connections in Categorial Type Logic Raffaella Bernardi joint work with Carlos Areces and Michael Moortgat Contents First Last Prev Next ◭
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Residuated operators in categorial type logic . . . . . . . . . . . . . . . . . . 5 3 Residuated and Galois connected functions . . . . . . . . . . . . . . . . . . . 6 4 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 Interpretation of the constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The base logic NL ( ✸ , · 0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9 7 Some useful derived properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 8 Linguistic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 9 Polarity Items (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 10 Polarity Items (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 11 Polarity Items (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 12 Typology of PIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 13 Options for cross-linguistic variation . . . . . . . . . . . . . . . . . . . . . . . . . 16 14 Greek (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 15 Greek (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 16 Italian (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 17 Italian (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Contents First Last Prev Next ◭
18 The point up till now . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 19 Connection with DMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 20 Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 21 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 22 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Contents First Last Prev Next ◭
1. Introduction ◮ Categorial type logic provides a modular architecture to study constants and variation of grammatical composition: ⊲ base logic grammatical invariants, universals of form/meaning assembly; ⊲ structural module non-logical axioms (postulates), lexically anchored options for structural reasoning. ◮ Up till now, research on the constants of the base logic has focussed on (unary, binary, . . . ) residuated pairs of operators. E.g. ⊲ Value Raising: A/C ⊢ B/C if A ⊢ B ; ⊲ Lifting theorem: A ⊢ ( B/A ) \ B . ◮ We extend the type-logical vocabulary with Galois connected operators and show how natural languages exploit the extra derivability patterns created by these connectives. Contents First Last Prev Next ◭
2. Residuated operators in categorial type logic The connectives • B, /B and A • , A \ of NL in [Lambek 58, 61] form residuated pairs of operators, i.e. ∀ A, B, C ∈ TYPE , [ RES 2 ] A ⊢ C/B iff A • B ⊢ C iff B ⊢ A \ C Similarly, the ✸ , ✷ ↓ connectives introduced in [Moortgat 95] form a residuated pair, i.e. ∀ A, B ∈ TYPE , A ⊢ ✷ ↓ B [ RES 1 ] ✸ A ⊢ B iff Contents First Last Prev Next ◭
3. Residuated and Galois connected functions Consider two posets A = ( A, ⊑ A ) and B = ( B, ⊑ B ), and functions f : A → B, g : B → A . The pair ( f, g ) is said to be residuated iff ∀ a ∈ A, b ∈ B [ RES 1 ] f ( a ) ⊑ B b iff a ⊑ A g ( b ) The pair ( f, g ) is said to be Galois connected iff ∀ a ∈ A, b ∈ B [ GC 1 ] b ⊑ B f ( a ) iff a ⊑ A g ( b ) Remark Let B ′ be a poset s.t. B ′ = ( B, ⊑ ′ def B ) where x ⊑ ′ B y = y ⊑ B x , and h : B → A . If ( f, h ) is a residuated pair with respect to ⊑ A and ⊑ ′ B , then it’s Galois connected with respect to ⊑ A and ⊑ B . b ⊑ B f ( a ) iff f ( a ) ⊑ ′ iff a ⊑ A h ( b ) B Contents First Last Prev Next ◭
4. Models Frames F = � W, R 2 0 , R 2 ✸ , R 3 • � W : ‘signs’, resources, expressions R 3 • : ‘Merge’, grammatical composition R 2 ✸ : ‘feature checking’, structural control R 2 0 : accessibility relation for the Galois connected operators Models M = � F, V � Valuation V : TYPE �→ P ( W ): types as sets of expressions Contents First Last Prev Next ◭
5. Interpretation of the constants { x | ∃ y ( R 2 V ( ✸ A ) = ✸ xy & y ∈ V ( A ) } V ( ✷ ↓ A ) { x | ∀ y ( R 2 = ✸ yx ⇒ y ∈ V ( A ) } { x | ∀ y ( y ∈ V ( A ) ⇒ ¬ R 2 V ( 0 A ) = 0 yx } V ( A 0 ) { x | ∀ y ( y ∈ V ( A ) ⇒ ¬ R 2 = 0 xy } { z |∃ x ∃ y [ R 3 zxy & x ∈ V ( A ) & y ∈ V ( B )] } V ( A • B ) = { x |∀ y ∀ z [( R 3 zxy & y ∈ V ( B )) ⇒ z ∈ V ( C )] } V ( C/B ) = { y |∀ x ∀ z [( R 3 zxy & x ∈ V ( A )) ⇒ z ∈ V ( C )] } V ( A \ C ) = Contents First Last Prev Next ◭
6. The base logic NL ( ✸ , · 0 ) Transitivity/Reflexivity of the derivability relation, plus ( res-l ) A • B ⊢ C iff A ⊢ C/B ( res-r ) A • B ⊢ C iff B ⊢ A \ C ( res-1 ) ✸ A ⊢ B iff A ⊢ ✷ ↓ B ( gal ) A ⊢ 0 B iff B ⊢ A 0 Soundness/Completeness A ⊢ B is provable iff ∀ F, V, V ( A ) ⊆ V ( B ) See [Areces, Bernardi & Moortgat 2001], also for Gentzen presentation, cut elimi- nation and decidability. Contents First Last Prev Next ◭
7. Some useful derived properties ✷ ↓ A ⊢ ✷ ↓ B (Iso/Anti)tonicity A ⊢ B implies ✸ A ⊢ ✸ B and B 0 ⊢ A 0 0 B ⊢ 0 A and A/C ⊢ B/C and C \ A ⊢ C \ B C/B ⊢ C/A and B \ C ⊢ A \ C A • C ⊢ B • C and C • A ⊢ C • B ✸✷ ↓ A ⊢ A A ⊢ ✷ ↓ ✸ A Compositions A ⊢ 0 ( A 0 ) A ⊢ ( 0 A ) 0 ( A/B ) • B ⊢ A A ⊢ ( A • B ) /B B • ( B \ A ) ⊢ A A ⊢ B \ ( B • A ) Closure Let ( · ) ∗ be 0 ( · 0 ), ( 0 · ) 0 , ✷ ↓ ✸ ( · ), X/ ( ·\ X ), ( X/ · ) \ X . ∀ A ∈ TYPE we have A ∗ ⊢ B ∗ if A ⊢ B, A ∗∗ ⊢ A ∗ A ⊢ A ∗ , Contents First Last Prev Next ◭
8. Linguistic Applications When looking at linguistic applications NL ( ✸ , · 0 ) offers: ◮ new (syntactic) derivability relations; ◮ new expressiveness on the semantic-syntactic interface; ◮ downward entailment relations. We will show how ◮ the new patterns can be used to account for polarity items; ◮ the new relation on the syntactic-semantic interface sheds light on possible connections between dynamic Montague grammar and categorial type logic. Contents First Last Prev Next ◭
9. Polarity Items (I) 1. * Any student left. 2. Some student left. 3. John didn’t see any student . 4. John didn’t see some student . Lexicon : ( np \ s ) / ( np \ ( 0 s ) 0 ) didn’t : q ( np, ( 0 s ) 0 , ( 0 s ) 0 ) any N : q ( np, ✷ ↓ ✸ s, ✷ ↓ ✸ s ) some N : s ⊢ ( 0 s ) 0 np ⊢ np ( \ L ) ( 0 s ) 0 �⊢ ✷ ↓ ✸ s np ◦ np \ s ⊢ ( 0 s ) 0 ( qL ) q ( np, ( 0 s ) 0 , ( 0 s ) 0 ) ◦ np \ s ⊢ ✷ ↓ ✸ s � �� � ���� Any student left q ( np, s 1 , s 2 ) def = ( np → s 1 ) → s 2 Contents First Last Prev Next ◭
10. Polarity Items (II) ❀ Wide Scope Negation ( ¬ GQ). np ⊢ np s ⊢ s 1 ( \ L ) np ⊢ np np ◦ np \ s ⊢ s 1 ( /L ) s 2 ⊢ ( 0 s ) 0 np ◦ (( np \ s ) /np ◦ np ) ⊢ s 1 ( qL ) np ◦ (( np \ s ) /np ◦ q ( np, s 1 , s 2 )) ⊢ ( 0 s ) 0 s ⊢ ✷ ↓ ✸ s np ⊢ np np ◦ np \ s ⊢ ✷ ↓ ✸ s ( \ L ) ( \ R ) ( np \ s ) /np ◦ q ( np, s 1 , s 2 ) ⊢ np \ ( 0 s ) 0 ( /L ) ◦ (( np \ s ) / ( np \ ( 0 s ) 0 ) )) ⊢ ✷ ↓ ✸ s np ◦ (( np \ s ) /np ◦ q ( np, s 1 , s 2 ) ���� � �� � � �� � � �� � John see didn’t GQ ◮ GQ: some student s 2 = ✷ ↓ ✸ s ✷ ↓ ✸ s �⊢ ( 0 s ) 0 ; ( 0 s ) 0 ⊢ ( 0 s ) 0 ◮ GQ: any student s 2 = ( 0 s ) 0 Contents First Last Prev Next ◭
11. Polarity Items (III) ❀ Narrow Scope Negation (GQ ¬ ). s ⊢ ( 0 s ) 0 np ⊢ np np ⊢ np s ⊢ s 1 ( \ R − L ) ( \ L ) np \ s ⊢ np \ ( 0 s ) 0 np ◦ np \ s ⊢ s 1 ( /L ) np ◦ (( np \ s ) / ( np \ ( 0 s ) 0 ) ◦ np \ s ) ⊢ s 1 np ⊢ np ( /L ) np ◦ (( np \ s ) / ( np \ ( 0 s ) 0 ) ◦ (( np \ s ) /np ◦ np )) ⊢ s 1 s 2 ⊢ ✷ ↓ ✸ s ( qL ) ◦ (( np \ s ) / ( np \ ( 0 s ) 0 ) )) ⊢ ✷ ↓ ✸ s np ◦ (( np \ s ) /np ◦ q ( np, s 1 , s 2 ) ���� � �� � � �� � � �� � see John didn’t GQ ◮ GQ: some student s 2 = ✷ ↓ ✸ s ✷ ↓ ✸ s ⊢ ✷ ↓ ✸ s ; ( 0 s ) 0 �⊢ ✷ ↓ ✸ s . ◮ GQ: any student s 2 = ( 0 s ) 0 Contents First Last Prev Next ◭
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