Results 1 to 2 of 2

Math Help - Modal sequents

  1. #1
    Newbie
    Joined
    Feb 2011
    Posts
    1

    Modal sequents

    1. A sequent with only atoms is valid iff some formula occurs on
    both sides.
    2. A, ϕ ⇒ B iff A ⇒ B, ϕ
    3. A ⇒ B, ϕ iff A, ϕ ⇒ B
    4. A, ϕ ∧ ψ ⇒ B iff A, ϕ, ψ ⇒ B
    5. A ⇒ B, ϕ ∧ ψ iff both A ⇒ B, ϕ and A ⇒ B, ψ

    These are facts about modal sequents...
    How do we prove them?

    For example 2. A, ϕ ⇒ B iff A ⇒ B, ϕ
    Where A and B = finite sets of wwfs
    We suppose A, ϕ ⇒ B first. Can we prove it without using contradiction method?
    Supposing A not ⇒ B, ϕ and showing contradiction.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,536
    Thanks
    778
    Welcome to the forum.

    A, ϕ ⇒ B iff A ⇒ B, ϕ
    The sequent A, ϕ ⇒ B is true, by definition, if \bigwedge A\land\neg\phi\to\bigvee B is true and A ⇒ B, ϕ is true iff \bigwedge A\to\bigvee B\lor\phi is true. Here \bigwedge A denotes the conjunction of all formulas in A, and \bigvee B denotes the disjunction of formulas in B. In the following, however, let's pretend that A is \bigwedge A and B is \bigvee B.

    So, assume that A\land\neg\phi\to B is true; we need to show A\to B\lor\phi. Assume A. If \phi is true, then the conclusion B\lor\phi is true and we are done. If \neg\phi is true, then the first assumption implies B, so again the conclusion B\lor\phi is true.

    For the converse, assume A\to B\lor\phi is true; we need to show that A\land\neg\phi\to B is true. Assume A and \neg\phi. Then the first assumption implies B\lor\phi. However, since \neg\phi is true, B is also true, as required.

    Ultimately, each of the equivalences in your question corresponds to a tautology, which can be verified using a truth table. For example, (5) corresponds to (A\to B\lor (\phi\land\psi))\leftrightarrow (A\to B\lor\phi)\land (A\to B\lor\psi).

    Why do you refer to these sequents as "modal"? They seem to come from regular, not modal logic.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Mean when larger than modal group
    Posted in the Statistics Forum
    Replies: 3
    Last Post: May 23rd 2011, 02:59 PM
  2. lower and upper bounds of modal class
    Posted in the Statistics Forum
    Replies: 7
    Last Post: May 7th 2010, 07:30 AM
  3. Modal logic: correct proof?
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: April 22nd 2010, 01:48 PM
  4. modal class
    Posted in the Statistics Forum
    Replies: 4
    Last Post: August 11th 2009, 07:39 PM
  5. second order differential in modal anlays
    Posted in the Calculus Forum
    Replies: 0
    Last Post: March 10th 2008, 02:58 AM

Search Tags


/mathhelpforum @mathhelpforum