Thread: Interpretation and satisfaction

I need help with the following question on satisfaction in FOL:

The sentence α : ∀x(odd(x) ⊃ even(s(x))) is not logically implied by the set S of 4 sentences:
even(0) odd(s(0)) ∀x(even(x) ⊃ even(s(s(x)))) ∀x(even(s(x)) ⊃ odd(x))
Prove that this is so by identifying an interpretation which is a model for S but not a model for α.
What should be changed to S such that the sentence α becomes logically implied?

Thanks in advance.

Originally Posted by agaton

I need help with the following question on satisfaction in FOL:

The sentence α : ∀x(odd(x) ⊃ even(s(x))) is not logically implied by the set S of 4 sentences:
even(0) odd(s(0)) ∀x(even(x) ⊃ even(s(s(x)))) ∀x(even(s(x)) ⊃ odd(x))
Prove that this is so by identifying an interpretation which is a model for S but not a model for α.
What should be changed to S such that the sentence α becomes logically implied?

Thanks in advance.

The problem with S is that it does not exclude the possibility that odd is true for all x without the same being true for even.