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Formal Logic
The first step, translating the argument from verbal to symbolic form, is often
the most difficult. Look for keys in the verbal representation of the argument.
“If… then” and “either… or” indicate implication and disjunction, respectively. A
period (or sometimes a semicolon) signifies the end of a hypothesis. The separate
hypotheses are joined by conjunctions. “Therefore” is a big key word; it indicates
the end of the hypotheses and announces that the conclusion is about to be stated.
eXAMPLe 18
Consider the argument, “If interest rates drop, the housing market will improve.
Either the federal discount rate will drop or the housing market will not improve.
Interest rates will drop. Therefore the federal discount rate will drop.” Using
I Interest rates drop.
H The housing market will improve.
F The federal discount rate will drop.
the argument is
(I S H ) ` (F ~ H′) ` I S F
A proof sequence to establish validity is
1. I S H hyp
2. F ~ H′ hyp
3. I
hyp
4. H′ ~ F 2, comm
5. H S F 4, imp
6. I S F
1, 5, hs
7. F
3, 6, mp
eXAMPLe 19
Is the following argument valid? “My client is left-handed, but if the diary is not
missing, then my client is not left-handed; therefore, the diary is missing.” There
are only two simple statements involved here, so we symbolize them as follows:
L My client is left-handed.
D The diary is missing.
The argument is then
L ` (D′ S L′) S D
The validity of the argument is established by the following proof sequence.
1. L
hyp
2. D′ S L′
hyp
3. (D′)′ ~ L′ 2, imp
4. D ~ L′
3, dn
5. L′ ~ D
4, comm
6. L S D
5, imp
7. D
1, 6, mp