﻿ 1.7 Quick test of logical inference

### 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:

 ¬p ∧ q ∧ r [p→(q∨s)] ∧ (¬r↔¬q)

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:

 [p→(q∨s)] ∧ (¬r↔¬q) p: F, q: T, r: T [(F→(T∨s)] ∧ (F↔F) T ∧ T T

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:

 ¬(p→q) ∧ r (¬q∨r) → (¬p∨s)

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 “¬(pq)∧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:

 (¬q∨r) → (¬p∨s) p: T, q: F, r: T (T∨T) → (F∨s) T → s s 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:

 p ↔ (¬r∨q) p ∨ ¬q

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:

 p ↔ (¬r∨q) p: F, q: T F ↔ (¬r∨T) F ↔ T F

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.