Page 20 - Chapter 2

 EXAMPLE 1 • Equivalence (p  q)

((p ⇒ q) ∧ (q ⇒ r)) ⇒ (p ⇒ r) is They are usually stated “p if and only
tautology, then the argument if q”. We need to prove both p=>q and

q=>p by the tautology mentioned in

example 3.
Algorithm :

Step one: Assuming p is true, show q
must be true.

is universally valid, and so is a rule of Step two: Assuming q is true, show p

inference. must be true.

 EXAMPLE 2 • Modus Ponens
Is the following argument valid? p is true, and p=>q is true, so q is true

If you invest in the stock market, (Theorem 4(g) in Section 2.2.)

then you will get rich.
If you get rich, then you will be

happy. p

p=>q
∴ q

∴ If you invest in the stock market,
then you will be happy.

let p: you invest in the stock market,

q: you will get rich, r: you will be happy  EXAMPLE 4
The above argument is of the form Is the following argument valid?

given in Example 1, hence the argument Smoking is healthy.
is valid! If smoking is healthy, then cigarettes are

prescribed by physicians.

 EXAMPLE 3 ∴ Cigarettes are prescribed by physicians
p: Smoking is healthy.
The tautology
q: cigarettes are prescribed by physicians
(p  q) ((p ⇒ q) ∧ (q ⇒ p)) The argument is valid since it is of form

means that the following two arguments modus ponens.
are valid

p ⇒ q
p  q q ⇒ p

∴ (p ⇒ q) ∧ (q ⇒ p) ∴ p  q