Categories and Types in Logic, Language, and Physics: Essays by Claudia Casadio, Bob Coecke, Michael Moortgat, Philip Scott

By Claudia Casadio, Bob Coecke, Michael Moortgat, Philip Scott (eds.)

For greater than 60 years, Jim Lambek has been a profoundly inspirational mathematician, with groundbreaking contributions to algebra, type thought, linguistics, theoretical physics, good judgment and facts conception. This Festschrift was once prepare at the party of his ninetieth birthday. The papers in it provide a great photograph of the a number of examine components the place the effect of Jim Lambek's paintings may be felt. the quantity contains contributions via widespread researchers and via their scholars, exhibiting how Jim Lambek's rules maintain inspiring upcoming generations of scholars.

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Extra info for Categories and Types in Logic, Language, and Physics: Essays Dedicated to Jim Lambek on the Occasion of His 90th Birthday

Example text

When π is a CyM-PN and we focus on the conclusion C of π as an output so that all the other conclusions are expressed by the linear negations A⊥ of formulas A belonging to a finite sequence of formulas of CyMLL, the derivable sequent corresponding to π is of the form Γ ⊥ , C. e. by putting the inputs before and the output after . e. e. reading B as B ⊥⊥ , therefore writing B ⊥ A⊥ . The above considerations are summarized in the following statement: each sequent of the form Γ C may be considered as the reading of a possible CyMPN modulo the focalization on one of the conclusions (the conclusion C on the right side of the sequent).

1 . A2 ) ... ... ... ... ... . f (A1 ○ (1/A 2 )) ∥A1 ∥ . ... ... .... .... ..... 1) f (A. 1 ) .. .. ... . ... ... ... A2 ) (0 .. .... . . ... .... . (0 ∥A2 ∥● ) g(A2 )) Type Similarity for the Lambek-Grishin Calculus 43 A1 A2 1 (A1 A2 ) 1 (A1 A2 ) ((A1 ((1 A2 ) (1 A2 )) ((1 A2 ) 1) A2 ) ((1 A2 ) 1) A2 ) (((A1 A2 ) ((A1 1) A2 ) A2 ) (0 A1 ((1 A2 ) A2 ) ((1/A2 ) a 1∗ 1 1)) A1 (0 ((1/A2 ) ((1 A2 ) A1 (0 (((1/A2 ) (1 A2 )) ((1 A2 ) 1))) A1 (0 ((1 ((1/A2 ) A2 )) ((1 A2 ) 1))) ((1 A2 ) ((1 A2 ) 1))) (0 ((1 1) A1 ((0 (1 1)) ((1 A2 ) 1)) A1 (((0 1) ((1 A2 ) 1)) 1) ((0 ((1 1) A2 ) 1)) A1 (1 ((1 A2 ) 1)) (A 1) ((1 A2 ) 1) ((1 A1 ((A1 A2 ) A2 ) A2 ) ((1 1) A2 ) e a c e/ a a 1∗ 0 a 1∗ 1) Fig.

1 A)/0) B C) M. A (((1 B)/0) C) Completeness Results We consider models built upon algebraic structures featuring two binary operations × and +, related by linear distributivity. Their definition derives from the linear distributive categories of [2] by turning their arrows into equivalences. Definition 7. A linearly distributive algebra is a 6-tuple A = ⟨A, ×, +, , ⊺, ⟩ of type ⟨2, 2, 1, 0, 0⟩ satisfying 1. 2. 3. 4. 5. Associativity. (A × B) × C = A × (B × C); (A + B) + C = A + (B + C) Commutativity.

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