¡¡¡¡Let W stand for ¡°you work hard£¬¡± S stand for ¡°you will be successful in Amer
¡¡¡¡ica£¬¡° and L stand for ¡±you can lead a life of leisure.¡° Now the first senten
¡¡¡¡ce translates as W¡ª¡ª>S£¬ the second sentence as S¡ª¡ª>L£¬ and the conclusion as
¡¡¡¡W¡ª¡ª>L. Combining these symbol statements yields the following diagram£º
¡¡¡¡W¡ª¡ª>S
¡¡¡¡S¡ª¡ª>L
¡¡¡¡Therefore£¬ W¡ª¡ª>L
¡¡¡¡The diagram clearly displays the transitive property.
¡¡¡¡DeMorgan's Laws
¡¡¡¡~£¨A & B£© = ~A or ~B
¡¡¡¡~£¨A or B£© = ~A & ~B
¡¡¡¡If you have taken a course in logic£¬ you are probably familiar with these fo
¡¡¡¡rmulas. Their validity is intuitively clear£º The conjunction A&B is false wh
¡¡¡¡en either£¬ or both£¬ of its parts are false. This is precisely what ~A or ~B
¡¡¡¡says. And the disjunction A or B is false only when both A and B are false£¬
¡¡¡¡which is precisely what ~A and ~B says.
¡¡¡¡You will rarely get an argument whose main structure is based on these rules
¡¡¡¡¡ª¡ªthey are too mechanical. Nevertheless£¬ DeMorgan's laws often help simplify
¡¡¡¡£¬ clarify£¬ or transform parts of an argument. They are also useful with game
¡¡¡¡s.
¡¡¡¡Example£º £¨DeMorgan's Law£©
¡¡¡¡It is not the case that either Bill or Jane is going to the party.
¡¡¡¡This argument can be diagrammed as ~£¨B or J£©£¬ which by the second of DeMorga
¡¡¡¡n's laws simplifies to £¨~B and ~J£©¡£ This diagram tells us that neither of th
¡¡¡¡em is going to the party.
