We devote this paper to the axiomatization and the computability of a variant of iteration-free PDL with fork. Concerning the axiomatization, our resuts are based on the following: although the program operation of fork is not modally definable in the ordinary language of PDL, it becomes definable in a modal language strengthened by the introduction of propositional quantifiers. Instead of using axioms to define the program operation of fork in the language of PDL enlarged with propositional quantifiers, we add an unorthodox rule of proof that makes the canonical model standard for the program operation of fork and we use large programs for the proof of the Truth Lemma. Concerning the computability, we prove by a selection procedure that our variant of PDL has a strong finite property, hence is decidable.