Val(x, iff x = Kansas ‘Kansas’) b. Val(x, flat’) iff x is flat ‘is Val(x, iff x leaps ‘leaps’) So axiom (3a) tells us that x is the semantic value of just in case x is identical ‘Kansas’, to Kansas. (3b) tells us that x is a semantic value of just in case x is flat. ‘flat’ The nonterminal nodes now tell us how to compute the semantic value of higher level syntactic structure (like the verb phrase and the sentence as a whole) from the semantic values of the words. Since this is a very simple language we only need the following rules to cover all the cases. (4) a. Val(T, [S NP VP ]) iff for some x, Val(x, NP) and Val(x, VP) b. Val(x, [a ]) iff Val(x, ranges over categories, and ranges over  )   categories and lexical items) c. Val(T, [S S1 S2 ]) iff ‘and’ Val(T, S1) and Val(T, S2) d. Val(T, [S S1 S2 ]) iff ‘or’ either Val(T, S1) or Val(T, S2) e. Val(T, [S is not the case that’ S1]) iff ‘it it is not the case that Val(T, S1) Finally, we need some derivation rules that tell us how to mechanically apply the axioms. In this case we only need two rules. The first one tells us that if we derive something of the form iff we are entitled to swap in for at any point in the derivation. The  ,   second rule does a simple cleanup to get rid of the variables we have introduced. Derivation Rule (SoE) ........  iff  therefore ........ Derivation Rule (SoI) iff for some x, x = and ....x....   therefore  iff ........ these axioms and the two derivation rules we can, in a straightforward way, derive
Val(x, ‘Kansas’) iff x = Kansas b. Val(x, ‘is flat’) iff x is flat Val(x, ‘leaps’) iff x leaps So axiom (3a) tells us that x is the semantic value of ‘Kansas’, just in case x is identical to Kansas. (3b) tells us that x is a semantic value of ‘flat’ just in case x is flat. The nonterminal nodes now tell us how to compute the semantic value of higher level syntactic structure (like the verb phrase and the sentence as a whole) from the semantic values of the words. Since this is a very simple language we only need the following rules to cover all the cases. (4) a. Val(T, [S NP VP ]) iff for some x, Val(x, NP) and Val(x, VP) b. Val(x, [a  ]) iff Val(x, )  ranges over categories, and  ranges over categories and lexical items) c. Val(T, [S S1 ‘and’ S2 ]) iff Val(T, S1) and Val(T, S2) d. Val(T, [S S1 ‘or’ S2 ]) iff either Val(T, S1) or Val(T, S2) e. Val(T, [S ‘it is not the case that’ S1]) iff it is not the case that Val(T, S1) Finally, we need some derivation rules that tell us how to mechanically apply the axioms. In this case we only need two rules. The first one tells us that if we derive something of the form  iff , we are entitled to swap in  for  at any point in the derivation. The second rule does a simple cleanup to get rid of the variables we have introduced. Derivation Rule (SoE) ........  iff  therefore ........ Derivation Rule (SoI)  iff for some x, x =  and ....x.... therefore  iff ........ these axioms and the two derivation rules we can, in a straightforward way, derive
Dynamic_Lexicon.docx