Logic - The Syntax of Predicate Logic

sentence of L₂

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

atomic formula of L₂

If Z is a predicate letter of arity n and each of t₁,...,tₙ is a variable or a constant, then Zt₁...tₙ is an atomic formula of L₂ (eg P, Q, R, Q¹x, P³x₁c₄y etc)

formula of L₂

Something is a formula of L₂ iff it is i) an atomic formula (a predicate letter of arity n followed by n variables or constants) (ii) of the form ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ), or (φ ↔ ψ) where φ and ψ are formulas, or (iii) of the form ∃vφ or ∀vφ where v is a variable and φ is a formula.

arity

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

free variable

Variables are free iff they are in: i) an atomic formula ii) formulas of the form ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ), or (φ ↔ ψ) where φ and ψ are formulas, if they are free in at least one of φ and ψ iii) formulas of the form ∃vφ or ∀vφ where v is a variable and φ is a formula, if they are free in φ and distinct from v.

predicate letter

a capital letter, preferably one that reminds us of the meaning of the predicate and that is used in conjunction with variables to symbolize propositional functions (eg P¹, Q², R)

constant

a letter (a, b, c, a₁, b₁...) that translates proper names and similar expressions

variable

a letter such as x, y, z, that has not been specifically assigned any particular meaning but derives meaning from the predicate letters.

bound variable

all variables that aren't free are bound (these are variables referenced by a quantifier and contained in the following formula)

designator

an expression that intends to denote a single object

quantifier

an expression ∀v or ∃v where v is a variable

predicate

connects designators and expresses a certain relation between the designators