What are the two equivalent translations of an "unless" statement?¬p ∨ qp → q¬p ∨ ¬qp → qp ∨ qp → ¬qp ∨ q¬p → q

To analyze the translations of an "unless" statement, let's first clarify what "unless" means logically. The phrase "p unless q" can be interpreted as "if not q, then p," which translates to the logical expression $\neg q \to p$. This can also be rearranged and expressed using disjunction.

Step 1: Understanding the "Unless" Statement

The statement "p unless q" means:

• If q is false (not true), then p must be true.

Step 2: Equivalent Translations

We can derive two equivalent forms of an "unless" statement:

1. The implication form: $\neg q \to p$

2. The disjunctive form: $p \lor q$ This can be explained as either p is true, or q is true, which encompasses the idea that if q is false, then p must hold.

Step 3: Confirming Logical Equivalence

To confirm that these two forms are equivalent, we can analyze them:

• If $\neg q$ is true, then $p$ must be true.
• Conversely, if $p$ is true, it does not matter if $q$ is true or false.

Thus, both representations cover the meaning of the "unless" relationship.

Final Answer

The two equivalent translations of the statement "p unless q" are:

1. $\neg q \to p$
2. $p \lor q$

Both formulations convey that if q is not true, then p must be true, thereby representing "unless" in logical terms.