Determining truth values from one premise

Statement

Let $p$ be a true statement. For each expression, decide whether this alone determines its truth value; if so, state it.

(i) $p \lor q$ (ii) $q \to p$ (iii) $p \to q$ (iv) $\lnot p \to (q \lor r)$

Solution

(i) Determined: $p \lor q = T \lor q = T$ (a disjunction with one true member is true). (ii) Determined: $q \to p = q \to T = T$ (an implication with a true consequent is true). (iii) Not determined: $p \to q = T \to q$ equals $q$, which is unknown. (iv) Determined: $\lnot p = F$, and $F \to (q \lor r) = T$ (false antecedent).