# Since in general we want to be able to say that a statement of the form.docx

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Since in general we want to be able to say that a statement of the form xP ( x ) is false if and only if the statement P ( x ) is false for some x , this leads us into requiring, for example, that 1 . 5 > 1 1 . 5 > 2 be false. This is an example where p is true and q is false, and p q is true. In conclusion, if the truth or falsity of the statement p q is to depend only on the truth or falsity of p and q , then we cannot avoid the previous criterion in italics. See also the truth tables in Section 2.5. Finally, in this respect, note that the statements If I am not a pink elephant then 1 = 1 If I am a pink elephant then 1 = 1 and If pigs have wings then cabbages can be kings 1 are true statements. The statement p q is equivalent to ¬( p ¬ q ), i.e. not( p and not q ). This may seem confusing, and is perhaps best understood by considering the four different cases corresponding to the truth and/or falsity of p and q . It follows that the negation of x ( P ( x ) Q ( x )) is equivalent to the statement x ¬( P ( x ) Q ( x )) which in turn is equivalent to
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