1.7 Quick test of logical inference
There is a very short and effective way of checking the logical validity of inferences by
truth-value analysis, which, however, is only sometimes applicable.
In principle, testing whether an inference scheme is valid comes down to determining
whether it is possible for the conclusion to be false when the premise (or the conjunction of
the premises if there are more) is true. The testing can be simplified if it is clear that
the premise (the conjunction of the premises) is true in only one case or that
the conclusion is false in only one case. If the premise is true in only one case, we can
check whether in this case the conclusion can be false. The scheme will be valid if and only
if it cannot. Consider the following scheme as an example:
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¬p ∧ q ∧ r
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[p→(q∨s)] ∧ (¬r↔¬q)
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Obviously, the premise is true if and only if “p” is false and “q” and “r” are
true (a conjunction is true when all its members are true). In order to check whether in this case
the conclusion can be false, we replace the propositional letters in the conclusion with the corresponding
truth values and check by truth-value analysis whether it can obtain the value F:
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[p→(q∨s)] ∧ (¬r↔¬q)
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p: F, q: T, r: T
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[(F→(T∨s)] ∧ (F↔F)
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T ∧ T
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T
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The analysis shows that in the only case where the premise is true, the conclusion is also
true; therefore, this is a logically valid inference scheme.
Consider a similar example. We want to check whether the following inference scheme is valid:
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¬(p→q) ∧ r
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(¬q∨r) → (¬p∨s)
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Here again, the premise is true in a single case because the negation of a conditional is true (i.e.,
the conditional itself is false) only when its antecedent is true and its consequent is false, which means
that “¬(p→q)∧r” is true only when “p” is true, “q” is false, and “r”
is true. To test whether the scheme is valid, we replace in the conclusion those propositional
letters with those truth values and check whether it is possible for the conclusion to obtain the value F:
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(¬q∨r) → (¬p∨s)
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p: T, q: F, r: T
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(T∨T) → (F∨s)
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T → s
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s
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| T F |
The analysis shows that when the premise is true, the conclusion may be false, which
means that the inference scheme is invalid.
The quick test is also applicable when the conclusion is false in only one case.
Then we can check whether in this case the premise may be true. If it cannot, it will not
be possible for the conclusion to be false when the premise is true, and therefore the inference
scheme will be valid. Conversely, if it is possible for the premise to be true in the case in
question, the inference scheme will be invalid. As an example, let us check the validity of the
following scheme:
The conclusion is false only when “p” is false and “q” is true. By replacing in the premise
“p” and “q” with these truth values, we check whether the premise can be true in this case:
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p ↔ (¬r∨q)
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p: F, q: T
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F ↔ (¬r∨T)
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F ↔ T
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F
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It turns out that in this case the premise cannot be true; the scheme is therefore valid.
The quick test is a very effective method that is more often applicable than it might seem at first glance.
Often the conclusion of an argument is a simple sentence, a negation of a simple sentences, a disjunction or a
conditional of simple sentences. Then the conclusion is false in only one case, which makes the quick testing
applicable in the variant where we check whether the premise (the conjunction of the premises if they are more
than one) can be true in this case. The examples in exercise (2) below are an illustration.
Exercises
(Download the exercises as a PDF file.)
(1)
Determine by the quick test whether the following inference schemes are logically valid:
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4)
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(p∨q) → [(r∨s)→t]
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(p∧q) → (r→t)
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5)
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(r→q) ∧ (s→r) ∧ [p→(r∨s)]
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p → (q∨r)
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7)
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¬p ∧ ¬q ∧ s
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p ↔ [q→(r↔s)]
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8)
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(p∧¬q) ∨ [(r∧s)→p]
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¬p ∨ (q→r)
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10)
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¬(p∨q) ∧ r
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[(p∧r)→¬q] ∨ (¬p∧¬r)
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(2)
Prove by the quick test that the following arguments are logically valid:
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1)
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If I start a new job, I will have to buy a car, and if I go to the sea this summer, I will spend
half of my savings. But if I buy a car and spend half of my savings, I will have to live on 10 euros
a day. However, I cannot live on 10 euros a day. So either I won’t buy a car or I won’t go to the sea.
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2)
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If our representative runs for president, if he does a positive campaign, there will be a
runoff. If there is a runoff and he wins the election, he will not be reelected in four years.
However, if he supports the death penalty, he will win the election and be reelected in four years.
Therefore, if our representative runs for president, if he runs a positive campaign, he will not
support the death penalty.
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3)
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If the doctor injects the antibodies, the patient will have an allergic reaction, and if he
has an allergic reaction, his kidney will stop functioning. But if the doctor does not inject
the antibodies, the bacteria will spread to the bloodstream. The patient’s kidney will stop
functioning if the bacteria spreads to the bloodstream. If the kidney stops functioning, the
patient will not survive until the morning. Therefore, the patient will certainly not survive
until the morning.
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4)
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On New Year’s Eve, John drinks red wine. If he celebrates with friends, he drinks beer.
Therefore, if John celebrates the New Year with friends, he drinks red wine and beer.
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5)
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If Alice enrolls in Ancient Greek, she will also enroll in Latin. If she enrolls in Ancient
Greek and Latin, she will enroll in logic. But if she enrolls in Ancient Greek, then if she
enrolls in logic, she will enroll in mathematics. Therefore, if Alice enrolls in Ancient Greek,
she will also enroll in mathematics.
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6)
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If an economic crisis begins or international conflicts arise, if the government fails to act or
takes inadequate action, there will be neither economic growth nor political stability. If there
is no economic growth or taxes are raised, there will be protests. Therefore, if an economic
crisis begins, then if the government fails to act, there will be protests.
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7)
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If Peter has met Mary, he would have told her the news if he knew it. But Peter met
Mary and did not tell her the news. So he didn’t know it.
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8)
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If I go to the ball, I’ll have to buy а tailcoat. But if I buy a tailcoat, I will not be able
to pay my rent and repay my loan at the same time. If I don’t pay the rent, I'll have to hide
from the landlord, and I can’t do that. Besides, I'll have to repay the loan. So, I can’t go
to the ball.
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