Predicate Calculus

Question 1

What is special about ‘predicate calculus’ as opposed to other logic systems?

Answer 1

Have two new devices: Subject-predicate and quantifier-variable

Question 2

Define a PC model

Answer 2

A PC model M is an ordered pair ⟨D, I⟩ such that:
D is a non-empty set (the domain)
I is a function (interpretation) such that:
- I(𝛼)∈D, where 𝛼 is a constant
- if Π is a predicate then I(Π) is a n-place relation over D

Question 3

Define a PC-wff

Answer 3

(i) if Π is a n-place predicate and 𝛼1, 𝛼2,…, 𝛼n, are terms, Π𝛼1𝛼2…𝛼n is a PC-wff
(ii) if 𝜙, 𝜓 are wff and 𝛼 is a variable then (𝜙→𝜓) and ¬𝜙, etc and ∃𝛼𝜙 and ∀𝛼𝜙 are also PC-wffs
(iii) only what can be shown to be PC-wffs using (i) and (ii) are PC-wffs

Question 4

When is a variable bound?

Answer 4

when it is tied to a quantifier

Question 5

What is an open formula? what is a closed formula?

Answer 5

A formula is open if it contains a free variable, closed otherwise

Question 6

What is the scope of a quantifier?

Answer 6

For how much of the formula it applies.

Question 7

define consistency and inconsistency

Answer 7

A formula is inconsistent if you can prove a contradiction from it. It is consistent if it is not inconsistent.