30
Formal Logic
The inference rules are also truth-preserving. For example, suppose that P
and P S Q are both true wffs in a proof sequence. Then Q is deducible from these
two wffs by modus ponens. If P and P S Q are both true, then—by the truth table
for implication—Q is also true.
The derivation rules, like the tautological equivalencies of Section 1.1, represent recipes or patterns for transforming wffs. A rule can be applied only when the
wffs exactly match the pattern.
pRaCtiCe 10 Give a next step and a justification for a proof sequence that begins
1. (A ` B′) S C hyp
2. C′
hyp
ReMIndeR
To use a derivation rule,
wffs must exactly match
the rule pattern.
eXAMPLe 13
Suppose that (A S B) ~ C and A are two hypotheses of an argument. A proof
sequence for the argument could begin with the following steps:
1. (A S B) ~ C hyp
2. A
hyp
Unlike Example 12, however, nothing further can be done. Modus ponens requires
the presence of wffs matching the pattern P and P S Q. In P S Q, the main connective is an implication. The wff (A S B) ~ C has disjunction, not implication, as
its main connective. Modus ponens does not apply, nor does anything else.
Now we are ready to work our way through a complete proof of an argument.
eXAMPLe 14
Using propositional logic, prove that the argument
A ` (B S C ) ` [(A ` B) S (D ~ C′)] ` B S D
is valid.
We must produce a proof sequence that begins with the hypotheses and ends
with the conclusion. There are four hypotheses, so this gives us lots of “ammunition” to use in the proof. The beginning of the proof is easy enough because it just
involves listing the hypotheses:
1. A
hyp
2. B S C
hyp
3. (A ` B) S (D ` C′) hyp
4. B
hyp
Our final goal is to arrive at D, the conclusion. But without even looking ahead,
there are a couple of fairly obvious steps we can take that may or may not be helpful.
5. C
2, 4, mp
6. A ` B
1, 4, con
7. D ~ C ′ 3, 6, mp
At least at this point we have introduced D, but it’s not by itself. Note that from
step 5 we have C, which we haven’t made use of. If only we had C S D, we’d be
Formal Logic
The inference rules are also truth-preserving. For example, suppose that P
and P S Q are both true wffs in a proof sequence. Then Q is deducible from these
two wffs by modus ponens. If P and P S Q are both true, then—by the truth table
for implication—Q is also true.
The derivation rules, like the tautological equivalencies of Section 1.1, represent recipes or patterns for transforming wffs. A rule can be applied only when the
wffs exactly match the pattern.
pRaCtiCe 10 Give a next step and a justification for a proof sequence that begins
1. (A ` B′) S C hyp
2. C′
hyp
ReMIndeR
To use a derivation rule,
wffs must exactly match
the rule pattern.
eXAMPLe 13
Suppose that (A S B) ~ C and A are two hypotheses of an argument. A proof
sequence for the argument could begin with the following steps:
1. (A S B) ~ C hyp
2. A
hyp
Unlike Example 12, however, nothing further can be done. Modus ponens requires
the presence of wffs matching the pattern P and P S Q. In P S Q, the main connective is an implication. The wff (A S B) ~ C has disjunction, not implication, as
its main connective. Modus ponens does not apply, nor does anything else.
Now we are ready to work our way through a complete proof of an argument.
eXAMPLe 14
Using propositional logic, prove that the argument
A ` (B S C ) ` [(A ` B) S (D ~ C′)] ` B S D
is valid.
We must produce a proof sequence that begins with the hypotheses and ends
with the conclusion. There are four hypotheses, so this gives us lots of “ammunition” to use in the proof. The beginning of the proof is easy enough because it just
involves listing the hypotheses:
1. A
hyp
2. B S C
hyp
3. (A ` B) S (D ` C′) hyp
4. B
hyp
Our final goal is to arrive at D, the conclusion. But without even looking ahead,
there are a couple of fairly obvious steps we can take that may or may not be helpful.
5. C
2, 4, mp
6. A ` B
1, 4, con
7. D ~ C ′ 3, 6, mp
At least at this point we have introduced D, but it’s not by itself. Note that from
step 5 we have C, which we haven’t made use of. If only we had C S D, we’d be