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Download PDF by Dr. Wojciech Buszkowski, Dr. Witold Marciszewski, Prof. Dr.: Categorial Grammar

By Dr. Wojciech Buszkowski, Dr. Witold Marciszewski, Prof. Dr. Johan van Benthem

This e-book is dedicated to the mathematical foundations of categorial grammar together with type-theoretic foundations of arithmetic, grammatical different types and different themes with regards to categorial grammar and to philosophical and linguistic purposes of this framework. the amount comprises 3 elements. the 1st, introductory half, comprises the editor's addresses and survey chapters about the background (W. Marciszewski) and present tendencies of the self-discipline (J.van Benthem). the second one half includes 10 chapters dedicated to categorial grammar right, and the 3rd half 7 chapters dedicated to components as regards to categorial grammar. lots of the contributions are unique papers, yet 5 of them are reprints of classics (M.J. Cresswell, P.T. Geach, H. Hiz, J. Lambek, T. Potts).

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G . , in 'Julia weeps bitterly' , the adverb may be regarded as denoting a function from verbs to (complex) verbs . Thus , the interpretation of natural language expressions involves a hierarchy of functions , and the task of a categorial grammar is to assign suitable types of function ("categories") to linguistic expressions , so as to make the puzzle "fit" . This paper is concerned with some logical (rather than empirical) aspects of the categorial mode of description . More precisely , there are two basic types t ("truth value") and e ("en­ tity") , with complex types generated recursively by the rule If a, b are types, then so is (a, b) .

Then the uppermost type on the path contains (t,t) as a subtype , and it has not been withdrawn : a contradiction , as (e,e) does not meet the first requirement. Case 2: the path started with an introduction : ( t , t) -t . Then its upper­ most type contains t as a subtype , whence it must have been withdrawn (as the only non-withdrawn type (e,e) does not contain t) . But this is which impossible , as the proof tree could then only be of the form ( ' t) ' does not use (e,e) . : Withdrawal of types in uppermost position on these "major premise paths" is quite possible ; witness the first example of a tree displayed above .

B for any permutation A ' Proof: As every permutation is a composition of interchanges between neighbours it suffices to show that , if Ab1b 2 C =? d is derivable, then so is Ab 2b 1 C =? d . The following example shows the general principle at work : a 1 a2b 1 b 2c 1 c2 =? d, a1a2b1b 2c1 =? (c2 ,d) , a1a2b1b2 =? (c l ' (c2 ,d)), a2b1b2 =? (a1 ,(c l ' (c2 , d))) , b1b2 =? (a2 , (a l ' (cl ' (c2 , d)))) , b 1 =? (b2 ,(a2 , (a l ' (c p (c2 , d))))) , b 2b 1 =? b 2 { b 2 ,{a2 ,{a l ' (c l ' (c2 ,d))))) =?

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