4 The Syntax of Predicate Logic

Predicate letters:

Where k and n are either missing or a numeral ‘1’ ‘2’

The arity of a predicate letter is

The value of its upper index. If a predicate letter does not have an upper index its arity is 0

Atomic formulae of L2

If Z is a predicate letter of arity n and each of t1, . . . , tn is a variable or constant, then Zt1 . . . tn is an atomic formula of L2 (sentences letters are also atomic formulae)

Formulae of L2

(i) All atomic formulae of L2 are formulae of L2.
(ii) If φ and ψ are formulae of L2 then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are formulae of L2
(iii) If v is a variable and φ is a formula then ∀vφ and ∃vφ are formulae of L2.
(iv) Nothing else is a formula of L2

Free occurrence of a variable:

(i) All occurrences of variables in atomic formulae are free.
(ii) The occurrences of a variable that are free in φ and ψ are also free in ¬φ, φ ∧ ψ, φ ∨ ψ,
φ → ψ, and φ ↔ ψ.
(iii) In a formula ∀vφ or ∃vφ no occurrence of the variable v is free; all occurrences of variables other than v that are free in φ are also free in ∀vφ and ∃vφ.

An occurrence of a variable is bound in a formula iff

It is not free

A variable occurs freely in a formula iff

There is at least one free occurence of the variable in the formula

Sentence of L2

A formula of L2 is a sentence of L2 iff no variable occurs freely in the formula.