MODR 2640 Lecture : R March 1, 2012.pdf
19 Nov 2012
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Remember-- (x)fx requires that every substitution instance of fx is true. Some holds for more complex sentences: (x)(fx gx) requires every substitution instance of (fx gx) is true. Works the same way for overlapping quantifiers. (x)(y)lxy. Here we want to remove the (x) quantifier. It applies to everything that follows so we need every substitution instance of (y)lxy to be true. When taking a substitution instance, only free occurrences of the variable get replaced. So with 2 individuals we have: (y)lay (y)lby. So we have: (laa lab) (lba lbb) Now suppose that we have two existential quantifiers. ( x)( y)lxy. We start from the leftmost quantifier (always). ( y)lay ( y)lby (laa lbb) (lba lbb) Things get harder when the quantifiers are not of the same type. (x)( y)lxy everyone loves someone or other ( y)(x)lxy there is some person who is loved by everyone. We can often understand better by looking at expansions. (x)( y)lxy.
