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VIII. CHAPTER EIGHT: PREDICATE LOGIC SEMANTICS
A. General Theory
1. If there are no unicorns, what is the truth value of the sentence (x)(Ux Mx),where
Ux =x is a unicorn, and Mx = x is mortal?
2. What about the truth value of the sentence ( x)(Ux Mx)?
A. Answers
B. Proving invalidity. Prove that the following arguments are invalid by either the
interpretation method or by the expansion method:
(1) 1. ( x)(Ax Bx) (3) 1. (x)(~ Ax Bx)
2. ( x) ~ Ax 2. ~ ( x) ~ Ax
/ ( x) ~ Bx / ~ ( x)Bx
(2) 1. (x)[(Ax Bx) Cx] (4) 1. ~ ( x)(~ Ax Bx)
2. ~ (Ba Cb) 2. ( x)(Ax Cx)
3. ~ (Ca Ab) / ~ (x)(Cx ~ Bx)
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/ ~ (~ Aa ~Ba)
(5) 1. (x)[(Ax Bx)Cx]
/ (x)[(Ax Bx)Cx]
B. Suggested Answers
(2) In a two individual universe of discourse, the argument amounts to
(3) Let the domain of discourse be restricted to the positive integers, and let Ax = x is
(4) Restrict the domain of discourse to two individuals, a and b. Then the expansion
(5) Let the domain of discourse be restricted to the positive integers, and let Ax = x >
C. Proving consistency. Show that the premises of the arguments below are consistent.
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(1) 1. (x)[(Ax Bx) Cx]
2. ( y)(Ay ~ Cy)
3. ( z)(Bz ~ Cz)
/ ( x)(~ Ax ~ Bx)
(2) 1. (x)[(Ax Bx) Cx]
2. Aa Ba
3. ~ Cb / ~ Bb
C. Suggested Answers
(1) Let the domain be the positive integers, Ax = x > 1, Bx = x is odd, Cx = x is > 2: