Answer
In order to justify that the rule of universal transitivity, which states that ∀x(P(x) → R(x)) is true if ∀x(P(x) → Q(x)) and ∀x(Q(x) → R(x)) are true keeping the domains of all quantifiers are the same we came across with these rules of inference listed
universal instantiation
hypothetical syllogism
universal generalization
Work Step by Step
A step by step answer to justify the rule of universal transitivity with the above-mentioned limitations is given below:
We assume that
∀x(P(x) →Q(x)) and ∀x(Q(x) →R(x))are true
hence the above given two statements are the premises.
Step Reason
1 ∀x(P(x) →Q(x)) premise
2 ∀x(Q(x) →R(x)) premise
3 P(c) →Q(c) universal instantiation from 1
4 Q(c) →R(c) universal instantiation from 2
5 P(c) →R(c) hypothetical syllogism from 3 and 4
6 ∀x(P(x) →R(x)) universal generalization from 5
thus have shown that if the premise
∀x(P(x) →Q(x)) and ∀x(Q(x) →R(x)) are true, then the conclusion ∀x(P(x) →R(x)) is also true.
