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Math Help - Universal Quantifier Introduction Rule

  1. #1
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    Universal Quantifier Introduction Rule

    Anyone can advise if line (6) is correct?
    Restriction for UI is that the x variable must not occur free in any of the assumptions. From line (2), it is clear that -ϕ is bounded, and from line (1), θ is bounded, but how about -θ?

    1 (1) ∀x (θ ↔ Ψ) Ass
    2 (2) ∀x (-Ψ ∨ -ϕ) Ass
    1 (3) (θ ↔ Ψ) UE, (1)
    2 (4) (-Ψ ∨ -ϕ) UE, (2)
    1,2 (5) (-ϕ ∨ -θ) Taut, (3), (4)
    1,2 (6) ∀x (-ϕ ∨ -θ) UI, (5)

    These formal proof questions are drowning me...
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  2. #2
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    Quote Originally Posted by tottijohn View Post
    Anyone can advise if line (6) is correct?
    Restriction for UI is that the x variable must not occur free in any of the assumptions. From line (2), it is clear that -ϕ is bounded, and from line (1), θ is bounded, but how about -θ?

    1 (1) ∀x (θ ↔ Ψ) Ass
    2 (2) ∀x (-Ψ ∨ -ϕ) Ass
    1 (3) (θ ↔ Ψ) UE, (1)
    2 (4) (-Ψ ∨ -ϕ) UE, (2)
    1,2 (5) (-ϕ ∨ -θ) Taut, (3), (4)
    1,2 (6) ∀x (-ϕ ∨ -θ) UI, (5)

    These formal proof questions are drowning me...
    You seem to be using the word,"bounded", in a rather strange way. At least it looks strange to me.
    After dropping the quantifiers, what has 'x' been replaced by?
    Last edited by PiperAlpha167; October 26th 2010 at 02:38 AM.
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  3. #3
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    From line (2), it is clear that -ϕ is bounded, and from line (1), θ is bounded, but how about -θ?
    You seem to be using the word,"bounded", in a rather strange way.
    Yes, "bounded" is not used in this sense, especially when talking about subformulas. The standard terminology is, "(An occurrence of) a variable x is free/bound in a formula θ" and "A (sub)formula θ is closed" (when all variables are bound in θ).

    In this derivation, the application of UI is correct because in the only two assumptions (1) and (2), x is not free. It does not matter that x may be free in the intermediate formulas (3)--(5) because they are not assumptions.

    After dropping the quantifiers, what has 'x' been replaced by?
    It isn't replaced, or, rather, it is replaced by itself. Presumably, x is free in θ, Ψ, and ϕ, but it does not have to be. E.g., using Universal Elimination, one gets x\ge 0 from \forall x.\,x\ge 0, as well as y\ge 0 from \forall x.\,y\ge 0 (in the latter two formulas, y is free).

    So, it is not true that θ is closed; most likely, it has a free occurrence of x.
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