Syntax of L2

Question 1

What is the arity of a predicate letter?

Answer 1

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.

Question 2

What are the constants of L2?

Answer 2

a, b, c, a1, b1, c1, a2, b2, c2, . . . are constants.

Question 3

What are the variables of L2?

Answer 3

Variables: x, y, z, x1, y1, z1, x2, y2, z2, . . . are variables.

Question 4

What are the atomic formulae of L2?

Answer 4

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.

Question 5

What is a quantifier?

Answer 5

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

Question 6

What is the definition of formulae of L2?

Answer 6

(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.

Question 7

What are the definitions for free occurence of a variable?

Answer 7

(i) All occurrences of variables in atomic formulae are free.
(ii) 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.
(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 occurrence of the variable in
the formula.

Question 8

What is a sentence of L2?

Answer 8

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