Chapter 1: Propositions and Predicates ~ 31
EXAMPLE 1.35
Test the validity of the following argument:
All integers are irrational numbers.
Some integers are powers of 2.
Therefore, some irrational number is a power of 2.
Solution
Let Z(x) denote 'x is an integer'.
Let I(x) denote 'x is an irrational number'.
Let P(x) denote 'x is a power of 2'.
The premises are:
(i) 'ix (Z(x) => I(x))
(ii) ::Ix (Z(x) /\ P(x))
The conclusion is ::Ix (I(x) /\ P(x)).
1. ::Ix (Z(x) /\ P(x))
2. Z(b) /\ PCb)
3. Z(b)
4. PCb)
5. 'ix (Z(x) => l(x))
6. Z(b) => l(b)
7. l(b)
8. l(b) /\ PCb)
9. ::Ix (I(x) /\ P(x))
Hence the argument is valid.
Premise (ii)
1, Existential instantiation
2, Simplification
2, Simplification
Premise (i)
5. Universal instantiation
3,6, Modus ponens
7,4 Conjunction
8, Existential instantiation.
SELF-TEST
Choose the correct answer to Questions 1-5:
1. The following sentence is not a proposition.
(a) George Bush is the President of India.
(b) H is a real number.
(c) Mathematics is a difficult subject.
(d) I wish you all the best.
2. The following is a well-formed formula.
(a) (P /\ Q) => (P v Q)
(b) (P /\ Q) => (P v Q) /\ R)
(c) (P /\ (Q /\ R)) => (P /\ Q))
(d) -, (Q /\ -, (P V -, Q)
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