4 The Syntax of Predicate Logic

Question 1

Define arity

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

Define atomic formulae of L2

Answer 2

If Z is a predicate letter of aroty n and each of t1,…,tn is a variable or constant then Zt1…tn is an atomic formula of L2.

Question 3

Define a quantifier in L2

Answer 3

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

Question 4

Define a formula of L2 (4 parts)

Answer 4

(i). All atomic formulae of L2 are formulae of L2
(ii). If ∅ and Ψ are formulae of L2 then ¬∅, (∅⋏Ψ), (∅⋎Ψ), (∅→Ψ) and (∅↔︎Ψ) are formulae of L2.
(iii). If v is a variable and ∅ is a formula then ∀v∅ and ∃v∅ are formulae of L2.
(iv). Nothing else is a formula of L2.

Question 5

All occurrences of variables in atomic formulae are —–

Answer 5

free

Question 6

An occurrence of a variable is bound in a formula iff ——–

Answer 6

it is not free.

Question 7

A variable occurs freely in a formula iff

Answer 7

there is at least one free occurrence of the variable in the formula.

Question 8

Define a sentence of L2

Answer 8

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

Question 9

Define a free occurrence of a variable (3 parts)

Answer 9

(i). All occurrences of variables in atomic formulae are free.
(ii). The occurrences of a variable that are free in ∅ and Ψ are also frfee in ¬∅, ∅⋏Ψ, ∅⋎Ψ, ∅→Ψ, and ∅↔︎Ψ.
(iii). In a formula ∀v∅ or ∃v∅ no occurrence of the variable v is free; all occurrences of variables other than v that are free in ∅ are also free in ∀v∅ and ∃v∅.

Question 10

Why mustn’t a dictionary formalizing a predicate logic sentence contain translations for variables?

Answer 10

Variables in sentences never refer to particular objects; they are only used for making universal or existence claims.