Section 1.2 Propositional Logic
33
eXAMPLe 16
The rule of hypothetical syllogism (hs) is
From P S Q and Q S R, one can derive P S R.
This rule is making the claim that
(P S Q) ` (Q S R) S (P S R)
is a valid argument. The proof sequence for this argument looks just like that for
Practice 12. Because it is a legitimate derivation rule, hypothetical syllogism can
be used to justify a step in a proof sequence.
eXAMPLe 17
Use propositional logic to prove
(A′ ~ B) ` (B S C ) S (A S C )
The following proof sequence will do.
1. A′ ~ B hyp
2. B S C hyp
3. A S B 1, imp
4. A S C 2, 3, hs
Without use of the new rule, we could still have produced a proof sequence by
essentially proving the new rule as part of this proof:
1. A′ ~ B hyp
2. B S C hyp
3. A S B 1, imp
4. A
hyp
5. B
3, 4, mp
6. C
2, 5, mp
Additional rules thus can shorten proof sequences but at the expense of having to
remember additional rules!
pRaCtiCe 13 Prove
(A S B) ` (C′ ~ A) ` C S B
Verbal arguments
An argument in English (an attorney’s trial summary, an advertisement, or a
political speech) that consists of simple statements can be tested for validity by a
two-step process:
1. Symbolize the argument using propositional wffs.
2. Prove that the argument is valid by constructing a proof sequence for it
using the derivation rules for propositional logic.