Question

2. Predicate logic 1 (1 pt.) In the domain of all people, consider the predicate disclosed(a,b) that is interpreted as "a has disclosed a secret to b. ,"

1. How are the following two expressions translated into plain English? Are the two expressions logically equivalent? - ∀a∃b disclosed (a,b). - ∃b∀a disclosed (a,b).
2. Can we claim that ∀a∃b disclosed (a,b)→∃b∀a disclosed (a,b) ? Discuss in detail.
3. Can we claim that ∃b∀a disclosed (a,b)→∀a∃b disclosed (a,b) ? Discuss in detail.

Answer 1

The two expressions are translated into plain English and shown to be not logically equivalent. The claims for conditional statements are discussed with counterexamples.

Step-by-step explanation:

The two expressions in plain English are:

1. For all people, there exists someone to whom they have disclosed a secret.
2. There exists someone to whom everyone with secrets has disclosed a secret.

These expressions are not logically equivalent.

In general, the assertion that ∀a∃b disclosed (a,b)→∃b∀a disclosed (a,b) is untrue. It is possible that someone has revealed a secret to one person but not to the others. Since at least one person has been told a secret, the left side is correct in this instance, but the right side is incorrect because not everyone has told everyone their secret.

Similarly, it is not generally true to say that ∃b∀a disclosed (a,b)→∀a∃b disclosed (a,b). While not everyone has revealed their secret to at least one person, there may be someone who has received a secret from everyone else. Since at least one person has learned everyone's secret, the left side of this argument is true in this instance, but the right side is untrue because not everyone has told anyone their secret.