1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Hardest logic problem!

  1. Nov 13, 2012 #1
    1. The problem statement, all variables and given/known data

    A universal (or ∀_1) formula is one of the form ∀x_1∀x_2...∀x_nθ, where θ is quantifi er-free.
    An existential (or ∃_1) formula is one of the form ∃_1∃_2...∃x_n θ.
    Let A be a substructure of B, and s valuation on A.

    a) Show that if ⊨_B ψ and ψ is universal, then ⊨_A ψ . Also show that if ⊨_A ψ and ψ is existential, then ⊨_B ψ .

    b) Use part (a) to show that the sentence (∃x Px) is not logically equivalent to any universal formula, and that (∀x Px) is not logically equivalent to any existential formula.

    2. Relevant equations


    3. The attempt at a solution

    My professor goes too quickly with satisfiability, and he expects me to solve such problem.

    Here are some thoughts I have..

    a) Since ψ is universal, we can say that ⊨_B ∀x_1∀x_2...∀x_n θ . I would say that since θ is quantifier free, no matter what type of symbols you have for θ, then, it must be just ∀x_1∀x_2...∀x_n θ = θ. I am thinking that I need to use induction principle for this part.

    Seems like I lack some understanding of how the proof should work. -___-

    b) I am not sure how to work out part b.
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted