The Syntax of Predicate Logic

Question 1

Predicate letters

Answer 1

All expressions of the form Pkn, Qkn, Rkn are predicate letters where k and n are either missing or a numeral ‘1’, ‘2’ . . . .

Question 2

Arity

Answer 2

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 3

Constants

Answer 3

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

Question 4

Variables

Answer 4

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

Question 5

Atomic formulae of L2

Answer 5

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 6

Quantifier

Answer 6

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

Question 7

Formulae of L2

Answer 7

(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 8

Free occurrence of a variable

Answer 8

(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, andφ↔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 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 9

Sentence of L2

Answer 9

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