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 for (following the de nullo ‘animal’ ‘mortal’ 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 is in a environment we can substitute for de nullo ‘flies’ ‘bird’ ‘flies’. If we extend to include A then B’, then modus ponens is simply a specific ‘A<B’ ‘If instance of the rule (one where contains only A):  dictum de omni (18) If Smith is tall then Jones is short Smith is tall Jones is short And is simply an instance of the rule (again, the instance modus tollens dictum de nullo 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)
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)
Dynamic_Lexicon.docx