| SENTENCE LETTERS | A, B, C, ... |
| CONNECTIVES | ~, &, v, →, ↔ |
| NAMES | a, b, c, d, ... |
| VARIABLES | u, v, w, x, y, z, ... |
| PREDICATE LETTERS | A1, B1, C1, ... A2, B2, C2, ... A3, B3, C3 ... |
| IDENTITY SYMBOL | = |
| QUANTIFIERS | ∀, ∃ |
| PARENTHESES | (, ) |
The definition of a well-formed formula includes these additions:
An n-place predicate letter followed by n names is a wff
If α and β are names, then α=β is a wff
These add new atomic wffs to the system. The definitions for conjunctions, disjunctions, conditionals, biconditionals, and negations are taken over from sentential logic unchanged.
If Φ is a wff, then the result of replacing at least one occurrence of a name in Φ by a new variable α and prefixing ∀α is a wff.
If Φ is a wff, then the result of replacing at least one occurrence of a name in Φ by a new variable α and prefixing ∃α is a wff.
| i. | Fz |
| ii. | ∀xGac |
| iii. | ∀xGcax |
| iv. | ∃x∀y(Gxy & Gyx) |
| v. | ∀x(Gxy ↔ ∃yHy) |
| vi. | ∃x(Ax → ∀xFxx) |
| vii. | ∀x∀y(Fxy → ∀z(Hxyz & Jz)) |
| viii. | ∀xFxx ↔ ∀x∀yFxy |
| ix. | ~∀x~∃z(Hz v Jx) |
| x. | Ga → ∀x~(Ha v Fxx) |
| xi. | P → Gab |
| xii. | ~(P & ~∃xFx |
| xiii. | ∀x(Fx) & P |
| xiv. | ∃y(Fyy & P) |
| xv. | ∀xyz(Fzx ↔ Hxyz) |
| xvi. | b=b |
| xvii. | (a=a) |
| xviii. | P=c |
| xix. | Fa=Fa |
| xx. | ∀z(Fz → a=b) |
| xxi. | ∀x(x=x) |
| xxii. | ∃x(Fx = Gx) |
| xxiii. | ~∀x(Fx & ∃y x=y) |
| xxiv. | (~a=b ↔ ~∀x(Fxa & Fbx)) |
| xxv. | ∀x∃y(~x=y → y=~x) |