It's a theorem of K that $\diamond$ distributes to disjuncts and vice versa:
$$\diamond(p \lor q) ≡ \diamond p \lor \diamond q$$
Does it distribute to negated disjuncts? Is the following a licit proof in K?
- $\diamond(p → q)$
- $\diamond(\lnot p\lor q)$ ------------- Def. of material implication
- $\diamond \lnot p \lor \diamond q$ -------------- ◊ - distribution
Thanks in advance!
Yes, so far so good. The rule is better put schematically rather than by using propositional atoms, i.e. the rule is
$$\diamond(\alpha \lor \beta) \equiv (\diamond\alpha \lor \diamond\beta) \text{ for all wffs }\alpha, \beta$$
This makes it clear the rule applies generally, not just to propositional atoms, as perhaps using $p, q$ misleading suggests.