602 Appendix A • Logic and Proofs
having modern logical symbolism available, logicians expressed these proposi-
tions in words:
All A's are B's
Some A's are B's- Universal positive
-Existential positive
(UP)
(EP)
No A's are B's -Universal negative (UN)
Some A's are not B's -Existential negative (EN)Statements of this type are very common in mathematics as well as in
ordinary conversation. Our knowledge of the principles of quantification and
negation will enable us to understand these statements better and to avoid
much confusion.
Examples A.2.14 Some categorical propositions:
(a) All math courses are interesting. (UP)
(b) Some cars are too expensive. (EP)
(c) No wars are justifiable. (UN)
( d) Some functions are not differentiable. (EN) DThe four categorical propositions are special cases of quantified statements.
Let A(x) = "x is an A,'' and B(x) = "x is a B." Then it is easy to translate
the categorical propositions into the symbolism of symbolic logic, as shown in
Table A.11:
Table A.11
Categorical Proposition Name Translation
All A's are B's UP \Ix, [A(x) ~ B(x)]
Some A's are B's EP 3x 3 [A(x) /\ B(x)]
No A's are B's UN \Ix, [A(x) ~ "'B(x)]
Some A's are not B's EN 3x 3 [A(x) /\ "'B(x)]Note: The translation of UN used in Table A.11 needs a word of explanation.
An exact literal translation of UN would be ",..., 3x 3 [A(x)/\ B(x)]" However,
by the principle of quantifier negation, this is equivalent to the form used in
Table A. 11. (See Theorem A.2.15.)
The importance of Table A.11 is to show that the classical categorical
propositions can be handled within the context of quantified propositional
functions. In this context, it is easy to derive the negations of the categori-
cal propositions.