Page 80 - HOW TO PROVE IT: A Structured Approach, Second Edition

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66 Quantificational Logic
Let’s translate this last formula back into colloquial English. Leaving
asidethefirstquantifierforthemoment,theformula∀y(R(x, y) → L(x, y))
means that for every person y,if x is related to y then x likes y. In other
words, x likes all his relatives. Adding ∃x to the beginning of this, we get the
statement “There is someone who likes all his relatives.” You should take a
minute to convince yourself that this really is equivalent to the negation of
the original statement “Everyone has a relative he doesn’t like.”

For another example of how the quantifier negation laws can help us un-
derstand statements, consider the statement “Everyone who Patricia likes, Sue
doesn’t like.” If we let L(x, y) stand for “x likes y,” and we let p stand for
Patricia and s for Sue, then this statement would be represented by the formula
∀x(L(p, x) →¬L(s, x)). Now we can work out a formula equivalent to this
one as follows:

∀x(L(p, x) →¬L(s, x))
is equivalent to ∀x(¬L(p, x) ∨¬L(s, x)) (conditional law),
which is equivalent to ∀x¬(L(p, x) ∧ L(s, x)) (DeMorgan’s law),
which is equivalent to ¬∃x(L(p, x) ∧ L(s, x)) (quantifier negation law).

Translating the last formula back into English, we get the statement “There’s
no one who both Patricia and Sue like,” and this does mean the same thing as
the statement we started with.
We saw in Section 2.1 that reversing the order of two quantifiers can some-
times change the meaning of a formula. However, if the quantifiers are the same
type (both ∀ or both ∃), it turns out the order can always be switched with-
out affecting the meaning of the formula. For example, consider the statement
“Someone has a teacher who is younger than he is.” To write this symbolically
we first write ∃x(x has a teacher who is younger than x). Now to say “x has a
teacher who is younger than x” we write ∃y(T (y, x) ∧ P(y, x)), where T (y, x)
means “y is a teacher of x” and P(y, x) means “y is younger than x.” Putting
this all together, the original statement would be represented by the formula
∃x∃y(T (y, x) ∧ P(y, x)).
Now what happens if we switch the quantifiers? In other words, what does
the formula ∃y∃x(T (y, x) ∧ P(y, x)) mean? You should be able to convince
yourself that this formula says that there is a person y such that y is a teacher of
someone who is older than y. In other words, someone is a teacher of a person
who is older than he is. But this would be true in exactly the same circumstances
as the original statement, “Someone has a teacher who is younger than he is”!
Both mean that there are people x and y such that y is a teacher of x and y is