3 basic metalogic questions.

From: Yarden Katz (yarden_at_umd.edu)
Date: 06/03/04 

Date: Thu, 03 Jun 2004 18:32:32 -0500

Hi,

  I have three quick questions on metalogic. First, my textbook asks to:

  10.9 Show that if T U {~(B & C)} is satisfiable, then either T U
        {~B} is satisfiable or T U {~C} is satisfiable.

  It strikes me that the book's example proofs (which are usually 2-5
  sentences long) are really informal - maybe it's the text, or maybe
  that's just the way these type of things are proven. In any event, I
  did this exercise, but I can't help but feel as if I'm sort of
  "cheating" by kind of restating what I'm asked to prove. If the
  nature of these proofs as presented was more formal, I'd feel more
  confident. However, when I compare my proof to the other examples,
  I don't think that I'm any more informal. Here's my proof:

    For T U {~(B & C)} to be satisfiable, it cannot be that B(c) is
    true *and* C(c) is true, where c is the denotation of any element
    in our domain. In other words, it must be that:

      1. ~(B(c) & C(c)) (again for c being the denotation of any and
                           all elements in our domain)

    [note that when I say B(c) I really mean something like B^c or
    B^m, i.e., B in the interpretation c or in the interpretation m -
    I just thought the function notation is clearer (is it?)]

    From this it follows that,

      2. ~(~B(c) v ~C(c))

    For (2) to be satisfiable, it must be that either T U {~B} or T U
    {~C}. Q.E.D.

  Is this correct? Is it necessary to get more
  formal about proofs like this, that clearly don't take a lot ingenuity
  or insight?

  A second problem was:

  10.10(f) "Show that: If c does not occur in F(x) or G(x), and F(c) and G(c)
             are equivalent, then AxF(x) and AxG(x) are equivalent,
             and similarly for E."

  What exactly do they mean by "if c does not occur in F(x) or G(x)"?
  Does it mean that c is not one of the terms in the actual sentence
  F(x)? Can someone give a hint about approaching this problem?

  Finally, some of the exercises in my book, like this one:

    "Show that EyAxR(x, y) implies AxEy(x, y)"

  Are really so obvious/superficial that I am kind of at loss on how to approach
  them in a proof. I think I could probably do the above indirectly
  to somehow get a contradiction, but if I were not allowed to do it
  indirectly I'd really have no idea where to start. Can someone shed
  some light of this?

Thanks a lot!

P.S. All exercises from Boolos and Jeffrey, Logic and Computation, 4th edition. 

-- 
Yarden Katz <yarden@umd.edu>  |  Mind the gap

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