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Math Help - Interpretation and satisfaction

  1. #1
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    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.
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by agaton View Post
    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.
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