Question

Give natural deduction proofs with the following assumptions and conclusion:

from assumption ∀x∀yR(x, y) to conclusion ∀y∀xR(x, y);

please help! It's logical philosophy.

Answer 1

See explanation below

Explanation:

R(x,y) means x and y are related according to previous established relation.

For example R(x,y) could be “x<y” if x, y are integer numbers, or R(x,y) could be “x and y are friends” if x and y are people.

The set which x and y belongs to, need not be the same.

∀x∀y R(x, y) means: for every x in a set S and every y in a set T, x and y are related.

The logical conector “and” is commutative, that is to say, the sentence “for every x in a set S and every y in a set T x and y are related” is the same as “for every y in a set T and every x in a set S x and y are related”.

This last sentence is ∀y∀x R(x, y).

So, they are the same thing.