If a is any individual variable and α is any wff, every occurrence of a in α is said to be bound (by the quantifiers) when occurring in the wffs (∀a)α and (∃a)α.
The following are also valid, again where α is any wff:(∀x)(∀y)α ≡ (∀y)(∀x)α (∃x)(∃y)α ≡ (∃y)(∃x)α The extensions of these lead to the following rule: 2. If a wff contains an unbroken sequence either of universal or of existential quantifiers, these quantifiers may be rearranged in any order and the resulting wff will be equivalent to the original wff.
Thus ∀ may be taken as primitive, and ∃ introduced by the definition(∃a)α =Df ∼(∀a)∼α,in which a is any variable and α is any wff; alternatively, ∃ may be taken as primitive, and ∀ introduced by the definition(∀a)α =Df ∼(∃a)∼α.
A common notation for this purpose is (ιx)ϕx, which may be read as "the thing that is ϕ" or more briefly as "the ϕ." In general, where a is any individual variable and α is any wff, (ιa)α then stands for the single value of a that makes α true.
If "=Df" is used to mean "is defined as," then the relevant definitions can be set down as follows: = Df ∼ = Df = Df in which α and β are any wffs of PC.
And, since (p ≡ q) ≡ [(p ⊃ q) · (q ⊃ p)] is valid, any wff containing ≡ can be transformed into an equivalent containing ⊃ and · but not ≡, and thus in turn by the previous steps it can be further transformed into one containing ∼ and ∨ but neither ≡ nor ⊃ nor ·.
One important feature of S5 but not of the other systems mentioned is that any wff that contains an unbroken sequence of monadic modal operators (Ls or Ms or both) is probably equivalent to the same wff with all these operators deleted except the last.
It should be noted that the rules, though designed to ensure unambiguous sense for the wffs of PC under the intended interpretation, are themselves stated without any reference to interpretation and in such a way that there is an effective procedure for determining, again without any reference to interpretation, whether any arbitrary string of symbols is a wff or not.
In general, a predicate variable followed by any number of individual variables is a wff of the predicate calculus.
Similarly, since ∼p ∨ q has the same truth table as p ⊃ q, (p ⊃ q) ≡ (∼p ∨ q) is valid, and any wff containing ⊃ can therefore be transformed into an equivalent wff containing ∼ and ∨ but not ⊃.
