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
But if we add a negation, as in the following argument, then we have to use the de nullo rule. Notice that here we can substitute ‘animal’ for ‘mortal’ (following the de nullo paradigm where we go from broader to narrower.) (15) Some man is not mortal Every animal is mortal (animal<mortal) Some man is not (an) animal These two paradigms cover the Aristotelian syllogisms, but they cover a number of other apparently valid inferences that fall outside of the Aristotelian paradigm as well. It works just as well for the following arguments. (16) Every man is mortal (man<mortal) Socrates is a man Socrates is mortal (17) No man flies Every bird flies (bird<flies) No man (is a) bird Here, since ‘flies’ is in a de nullo environment we can substitute ‘bird’ for ‘flies’. If we extend ‘A<B’ to include ‘If A then B’, then modus ponens is simply a specific instance of the dictum de omni rule (one where  contains only A): (18) If Smith is tall then Jones is short Smith is tall Jones is short And modus tollens is simply an instance of the dictum de nullo rule (again, the instance where contains only A). (19) If is tall then Jones is short Jones is not short Smith is not tall The solution to De Morgan’s example be as in (20): (20) man is an animal (man<animal)
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