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Rules of Inference problem

  1. Oct 3, 2016 #1
    1. The problem statement, all variables and given/known data

    Determine whether the following is valid:

    [tex] p \rightarrow \neg q , r \rightarrow q , r, \vdash \neg p [/tex]

    2. Relevant equations

    Modus Ponens, disjunctive syllogism, double negation.

    3. The attempt at a solution

    I've boiled it down to

    [tex] p \rightarrow \neg q , q, \vdash \neg p [/tex]

    However I do not understand how the book says to use disjunctive syllogism and double negation here. I've expanded the implication into its fundamental forum but I still cant see it.
  2. jcsd
  3. Oct 3, 2016 #2


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    These problems always depend on what rules of inference, axioms and replacement rules you've been given.
    ##p\to \neg q## is equivalent to ##\neg p\vee \neg q##. This may be specified as a replacement rule, or as the definition of one or the other of ##\to## or ##\vee##.

    Using that replacement, are you able to do the problem using DNE (##\neg\neg q\equiv q##) and then DS?
  4. Oct 4, 2016 #3
    I've got it now, thank you. I wasn't seeing how ##neg neg q \equiv q ## would fit to DS, but now it's clear.
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