wffs |
Definition. A WELL-FORMED FORMULA of predicate logic is
any expression in accordance with the following six rules: |
(1) | Sentence letters are wffs. |
(2) | An n-place predicate letter followed by n names is a wff. |
(3) | Expressions of the form α=β where α and β are names are wffs. |
Comment. Although the placement of the identity symbol superficially resembles that of a connective, it is in fact a special two-place predicate. For historical reasons alone it is placed between a and b rather than in front of them. | |
atomic sentence | [Definition. Wffs of the form specified in rules 1-3 are the ATOMIC SENTENCES of predicate logic.] |
Comment. We adopt the practice of omitting super-scripts from predicates. | |
(4) | Negations, conjunctions, disjunctions, conditionals, and biconditionals of wffs are wffs. |
Comment. The formation rules of chapter 1 are subsumed by this clause. | |
(5) | If Φ is a wff, then the result of replacing a t least one occurrence of a name in Φ by a new variable α (i.e.,α not in Φ) and prefixing ∀α is a wff. |
universal wff | [Definition. Such wffs are called UNIVERSALLY QUAN TIFIED wffs, or UNIVERSAL wffs.] |
(6) | If Φ is a wff, then the result of replacing at least one occurrence of a name in Φ by a new variable α (i.e., α not in Φ) and prefixing ∃α is a wff. |
existential wff | [Definition. Such wffs are called EXISTENTIALLY QUANTIFIED wffs, or EXISTENTIAL wffs.] |
(7) | Nothing else is a wff. |