  A straightforward solution to these problems is to abandon classical rules for the quantifiers and to adopt rules for free logic \((\mathbf{FL})\) instead. The rules of \(\mathbf{FL}\) are the same as the classical rules, except that inferences from \(\forall xRx\) (everything is real) to \(Rp\) (Pegasus is real) are blocked. This is done by introducing a predicate ‘\(E\)’ (for ‘actually exists’) and modifying the rule of universal instantiation. From \(\forall xRx\) one is allowed to obtain \(Rp\) only if one also has obtained \(Ep\). Assuming that the universal quantifier \(\forall x\) is primitive, and the existential quantifier \(\exists x\) is defined by \(\exists xA =_{df} {\sim}\forall x{\sim}A\), then \(\mathbf{FL}\) may be constructed by adding the following two principles to the rules of propositional logic