and "there exists (at least one)" that extend over the domain of discourse. The calculus requires only the first notion "for all", but typically includes both: (1) the notion "for all x" or "for every x" is symbolized in the literature as variously as (x), ∀x, Πx etc., and the (2) notion of "there exists (at least one x)" variously symbolized as Ex, ∃x. The restriction is that the generalization "for all" applies only to the variables (objects x, y, z etc. drawn from the domain of discourse) and not to functions, in other words the calculus will permit ∀xf(x) ("for all creatures x, x is a bird") but not ∀f∀x(f(x)) [but if "equality" is added to the calculus it will permit ∀f:f(x); see below under Tarski]. Example: Let the predicate "function" f(x) be "x is a mammal", and the subject-domain (or universe of discourse) (cf Kleene 1967:84) be the category "bats": The formula ∀xf(x) yields the truth value "truth" (read: "For all instances x of objects 'bats', 'x is a mammal'" is a truth, i.e. "All bats are mammals"); But if the instances of x are drawn from a domain "winged creatures" then ∀xf(x) yields the truth value "false" (i.e. "For all instances x of 'winged creatures', 'x is a mammal'" has a truth value of "falsity"; "Flying insects are mammals" is false); However over the broad domain of discourse "all winged creatures" (e.g. "birds" + "flying insects" + "flying squirrels" + "bats") we can assert ∃xf(x) (read: "There exists at least one winged creature that is a mammal'"; it yields a truth value of "truth" because the objects x can come from the category "bats" and perhaps "flying squirrels" (depending on how we define "winged"). But the formula yields "falsity" when the domain of discourse is restricted to "flying insects" or "birds" or both "insects" and "birds". Kleene remarks that "the predicate calculus (without or with equality) fully accomplishes (for first order theories) what has been conceived to be the role of logic" (Kleene 1967:322). A new axiom: Aristotle's dictum – "the maxim of all and none" This first half of this axiom – "the maxim of all" will appear as the first of two additional axioms in Gödel's axiom set. The "dictum of Aristotle" (dictum de omni et nullo) is sometimes called "the maxim of all and none" but is really two "maxims" that assert: "What is