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A Pure Logic of Residuation in Axiom & Transitivity Lambek Calculus with Modalities

Explore the pure logic of residuation in Axiom, Transitivity & Residuation Lambek Calculus with modalities. Learn how structures impact A and how absence of S affects A’s structure. Discover consequences, sequents, and structural postulates. Examples include non-peripheral extraction. Adapted proof-nets consider resource restructuring and represent modalities. Refer to Richard Moot’s thesis "Proof Nets for Linguistic Analysis" which uses a contraction criterion. Discover restructuring-1, restructuring-2, and contraction step.

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A Pure Logic of Residuation in Axiom & Transitivity Lambek Calculus with Modalities

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  1. now the extensions…

  2. NL: a pure logic of residuation • Axiom • Transitivity • Residuation

  3. Lambek calculus(sequents)

  4. + modalities If  brings you an A, then when provided with the structure S, it gives you an A with the structure S If  with S brings you an A, then without it it gives you an A which lacks S!

  5. A  A A  A []L ([]A)  A (A)  A L []R []A  A A  []A consequences R

  6. + structural postulates • Example :

  7. ¯ à ( n \ n ) /( s / [] np ), (( np , ( np \ s ) / np ), s \ s ) n \ n a Example: non peripheral extraction(that I met _ yesterday) .... (( np , (( np \ s ) / np , np )), s \ s ) s a à L ¯ à (( np , (( np \ s ) / np , [] np )), s \ s ) s a P 1 ¯ à ((( np , ( np \ s ) / np ), [] np ), s \ s ) s n n n n a a a P 2 ¯ à ((( np , ( np \ s ) / np ), s \ s ), [] np ) s n n \ n n a a ¯ à n \ n n \ n (( np , ( np \ s ) / np ), s \ s ) s / [] np a a

  8. ¯ à ( n \ n ) /( s / [] np ), (( np , ( np \ s ) / np ), s \ s ) n \ n a Example: non peripheral extraction(that I met _ yesterday) .... (( np , (( np \ s ) / np , np )), s \ s ) s a à L ¯ à (( np , (( np \ s ) / np , [] np )), s \ s ) s a P 1 ¯ à ((( np , ( np \ s ) / np ), [] np ), s \ s ) s n n n n a a a P 2 ¯ à ((( np , ( np \ s ) / np ), s \ s ), [] np ) s n n \ n n a a ¯ à n \ n n \ n (( np , ( np \ s ) / np ), s \ s ) s / [] np a a

  9. ¯ à (( np ( np \ s ) / np ), s \ s ) [] np ¯ à ( n \ n ) /( s / [] np ), (( np , ( np \ s ) / np ), s \ s ) n \ n a Example: non peripheral extraction(that I met _ yesterday) .... (( np , (( np \ s ) / np , np )), s \ s ) s a à L ¯ à (( np , (( np \ s ) / np , [] np )), s \ s ) s a P 1 ¯ à ((( np , ( np \ s ) / np ), [] np ), s \ s ) s n n n n a a a P 2 ¯ à ((( np , ( np \ s ) / np ), s \ s ), [] np ) s n n \ n n a a n \ n n \ n , s / a a

  10. ¯ à [] np ¯ à (( np ( np \ s ) / np ), s \ s ) [] np ¯ à ( n \ n ) /( s / [] np ), (( np , ( np \ s ) / np ), s \ s ) n \ n a Example: non peripheral extraction(that I met _ yesterday) .... (( np , (( np \ s ) / np , np )), s \ s ) s a à L ¯ à (( np , (( np \ s ) / np , [] np )), s \ s ) s a P 1 ¯ à ((( np , ( np \ s ) / np ), [] np ), s \ s ) s n n n n a a a P 2 ((( np , ( np \ s ) / np ), s \ s ), ) s n n \ n n a a n \ n n \ n , s / a a

  11. ¯ à [] np ¯ à [] np ¯ à (( np ( np \ s ) / np ), s \ s ) [] np ¯ à ( n \ n ) /( s / [] np ), (( np , ( np \ s ) / np ), s \ s ) n \ n a Example: non peripheral extraction(that I met _ yesterday) .... (( np , (( np \ s ) / np , np )), s \ s ) s a à L ¯ à (( np , (( np \ s ) / np , [] np )), s \ s ) s a P 1 ((( np , ( np \ s ) / np ), ), s \ s ) s n n n n a a a P 2 ((( np , ( np \ s ) / np ), s \ s ), ) s n n \ n n a a n \ n n \ n , s / a a

  12. ¯ à [] np ¯ à [] np ¯ à (( np ( np \ s ) / np ), s \ s ) [] np ¯ à ( n \ n ) /( s / [] np ), (( np , ( np \ s ) / np ), s \ s ) n \ n a Example: non peripheral extraction(that I met _ yesterday) .... (( np , (( np \ s ) / np , np )), s \ s ) s a à L ¯ à (( np , (( np \ s ) / np , [] np )), s \ s ) s a P 1 ) ((( np , ( np \ s ) / np , ), s \ s ) s n n n n a a a P 2 ((( np , ( np \ s ) / np ), s \ s ), ) s n n \ n n a a n \ n n \ n , s / a a

  13. ¯ à [] np ¯ à [] np ¯ à (( np ( np \ s ) / np ), s \ s ) [] np ¯ à ( n \ n ) /( s / [] np ), (( np , ( np \ s ) / np ), s \ s ) n \ n a Example: non peripheral extraction(that I met _ yesterday) .... (( np , (( np \ s ) / np , np )), s \ s ) s a à L ¯ à (( np , (( np \ s ) / np , [] np )), s \ s ) s a P 1 ) ((( np , ( np \ s ) / np , ), s \ s ) s n n n n a a a P 2 ((( np , ( np \ s ) / np ), s \ s ), ) s n n \ n n a a n \ n n \ n , s / a a

  14. ¯ à [] np ¯ à [] np ¯ à (( np ( np \ s ) / np ), s \ s ) [] np ¯ à ( n \ n ) /( s / [] np ), (( np , ( np \ s ) / np ), s \ s ) n \ n a Example: non peripheral extraction(that I met _ yesterday) .... (( np , (( np \ s ) / np , np )), s \ s ) s a à L ¯ à (( np , (( np \ s ) / np , [] np )), s \ s ) s a P 1 ) ((( np , ( np \ s ) / np , ), s \ s ) s n n n n a a a P 2 ((( np , ( np \ s ) / np ), s \ s ), ) s n n \ n n a a n \ n n \ n , s / a a

  15. adapted proof-nets • To take restructuring of resources into account • To represent modalities • see Richard Moot’s thesis • « Proof Nets for Linguistic Analysis »

  16. Uses a contraction criterion (graph-rewriting)

  17. a/a a/a a/a a/a

  18. a a/a a a/a a/a a/a a/a

  19. a a a/a a/a a a a/a a/a a/a a/a

  20. a a a a/a a/a a/a a a a a/a a/a a/a a/a

  21. a a a a/a a/a a/a a a a a/a a/a a/a a/a a/a

  22. a a a a/a a/a a/a a a a a/a a/a a/a a/a a/a

  23. a a/a a restructuring-1 a/a a/a a a a/a

  24. restructuring-2 a/a a/a a/a a a a/a

  25. contraction step a/a a/a a/a a

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