Chapter 4: The Syntax of Predicate Logic

Predicate letters are…

All expressions of the form Pᴷₙ, Qᴷₙ, or Rᴷₙ, where k is either missing (no symbol) or a numeral ‘1’, ‘2’, ‘3’,… and similarly, n is either missing (no symbol) or a numeral ‘1’, ‘2’, ‘3’,…

Arity Is…

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

Constants are…

a, b, c, a₁, b₁, c₁, a₂, b₂, c₂, a₃,…

Variables are…

x, y, z, x₁, y₁, z₁, x₂…

Atomic formula

If Z is a predicate letter of arity n and each of t₁,…,tₙ is a variable constant, then Zt₁…tₙ is an _____ ______ of 𝐿₂.

A quantifier is…

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

Formulae of 𝐿₂ include:

All atomic formulae of 𝐿₂,

If φ and ψ are ________ _ __, then ¬φ, (φ∧ψ), (φ∨ψ), (φ→ψ) and (φ↔ψ) are ________ _ __,

If v is a variable and φ is a formula then ∀vφ and ∃vφ are ________ _ __,

Nothing else is a ________ _ ___.

Free occurrences of variables 1

All occurrences of variables in atomic formulae are free.

Free occurrences of variables 2

The occurences of a varaiable that are free in φ and ψ are also free in ¬φ, φ∧ψ, φ∨ψ, φ→ψ, and φ↔ψ.

Free occurrences of variables 3

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.

Free occurrences of variables 4

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 occurrence of the variable in the formula.

A formula of 𝐿₂ is a sentence of 𝐿₂ iff…

no variable occurs freely in the formula.