Question

If two formulas are logically equivalent, then they must entail each other.

A. True
B. False

Answer 1

False, If two formulas are logically equivalent, it means that they have the same truth value for every possible interpretation. However, logical equivalence does not necessarily imply entailment. Two formulas can be logically equivalent without one entailing the other. Entailment refers to one formula logically implying another formula, but not necessarily being equivalent to it.

Answer 2

Option (A), The statement that two formulas which are logically equivalent must entail each other is True. In logic and arguments, logically equivalent formulas always share the same truth conditions.

Step-by-step explanation:

If two formulas are logically equivalent, they indeed entail each other, which means that they have exactly the same truth conditions. The truth of one formula guarantees the truth of the other and vice versa. This is the fundamental property of logically equivalent statements. Therefore, the statement 'If two formulas are logically equivalent, then they must entail each other.' is True.

Now, in the context of the provided options, the word that is closest in meaning to hypothesis is suggestion (d). A hypothesis is a proposed explanation for a phenomenon, a starting point for further investigation. While fact (a) refers to something proven to be true, law (b) refers to a statement that describes an observable occurrence in nature that appears to always be true, formula (c) generally refers to a mathematical relationship or rule expressed in symbols, and conclusion (e) is a judgment or decision reached after consideration.

When assessing logical statements and arguments, one must critically analyze premises to ensure they logically support the conclusion. This involves examining whether the conditions of necessity and sufficiency are met, and if necessary, providing a counterexample to disprove a conditional or universal affirmative statement.

In terms of probability, events A and B are mutually exclusive if the probability of them occurring at the same time (P(A AND B)) is zero. In this case, when calculating the probability of either event occurring (P(A OR B)), we can simply add their individual probabilities without considering their intersection since P(A AND B) = 0.

The commutative property, illustrated as A+B = B+A, shows the order of operation does not affect the outcome of the addition, just as with the addition of ordinary numbers (2 + 3 = 3 + 2), and this property holds true in many mathematical operations.