By Ronald L. Trail
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Additional info for The grammar of Lamani
Func tion-elimination is like Modus Ponens , while function-introduction corre sponds to Conditionalization . Example 2 (a logical law) The logical law (a � (b � c)) => (b � (a � c)) is derivable as follows: a (a , (b ,c)) => (b ,c) a (a,(b,c)) b => (b ,c) b (b,c) b => c a (a ,(b ,c)) b => c (a,(b ,c)) b => (a ,c) ' (a,(b ,c)) => (b ,(a ,c)) . Nevertheless , other logical laws may fail . For instance , a => (b ,a) is unde- 42 JOHAN VAN BENTHEM rivable , as it requires a "vacuous" conditionalization .
Thus , type change rules are invoked to exhibit a semantic unity among apparently diverse multiple types that would have to be assigned on the traditional approach . Moreover, linguists have explored (or even re-invented) flexible types , or alternatively : multiple type combi nation modes, to create additional syntactic structures (though still operat ing on concatenated strings) , admitting strings as. interpretative units that would be non-unifiable on the earlier approach . An example is the constitu tent structure for complex noun phrases : (Determiner) (Common Noun(Relative Clause)) on the traditional categorial view , (Deter miner(Common Noun)) (Relative Clause) now becoming admissible too .
B 1 ,b 2)) is L-derivable . Example 4 In L one has : (a p (a 2 ,(b 1 ,b2))) a 1 a2 b 1 => b 2 ; (a l ' (a 2 ,(b 1 ,b2))) b 1 => (a p (a2 ,b 2)) ; ((a l ' (a 2 ,b 2)) , b2) (a 1 , (a2 , (b 1 , b 2))) b1 => b2 , etcetera . Now , if a1 , . . ,a 11 => (b 1 ,b 2 ) is derivable in L, then we are done, since the following sequent is then L-derivable : (a l ' ( . . (a11 ,b 2) . . )) ,b 2) => (b 1 ,b 2) . (The argument is this : a 1 , . . ,a11 => (b 1 ,b2) ; a 1 , . . ,a, , b 1 => b 2 ; b 1 => (a l ' ( .
The grammar of Lamani by Ronald L. Trail