Investigation of Formulae in Predicate Calculus

These are the elements that I use for Predicate Calculus

Ø 

A ... Z (with or without scripts)

 a ... z (with or without scripts)

∈ , = 

∀ , ∃ 

¬ , ∧ , ∨ , ⇒ , ⇔ 

( , ) , [ , ] , { , }

Expressions in Predicate Calculus include any finite sequence of the elements listed above.

Examples:         Ø∃⇒⇒              ¬ [∈∀               A = B

An atomic formula in Predicate Calculus is either an atomic formula from Propositional Calculus or an expression of the form Q(q1, q2, ... ,qn) where Q is an n-place predicate with q1, q2, ... , qn as terms. However, in the special case of the two place predicates ∈ and =, terms are placed on the left and right side of the predicates.

Examples:         P         Ua         Yxz           X ∈ Y           A = B

The terms of a Predicate can be either a constant or a variable. I take u, v, w, x, y, z upper and lowercase with or without scripts to represent variables and the rest of the alphabet with or without scripts both upper and lowercase as constants. 

1.) Every atomic formula of Predicate Calculus is a wff of Predicate Calculus.

2.) If P is a wff, then ¬P is a wff.

3.) If P and Q are wff's, then P ∧ Q, P ∨ Q, P ⇒ Q, and P ⇔ Q are wff's.

4.) If P is a wff that contains at least one occurrence of x and no x-quantifier, then ∀x(P) and ∃x(P) are both wff's.

Nothing is a wff of Predicate Calculus unless it can be formed by repeated applications of 1 - 4.

Now it's time to make some wff's!