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What is the difference between ⊢ and ⊨?

  1. Oct 25, 2011 #1
    What is the difference between ⊢ and ⊨?

    how to call them?
     
  2. jcsd
  3. Oct 25, 2011 #2

    Bacle2

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    Science Advisor

    Maybe someone else can add more, but:

    AFAIK A ⊢ B means that B can be deduced/derived from A, or there is a proof of B from A.

    And M ⊨ N means M is a model for N , i.e., M is an interpretation in which all wffs in N are mapped into truths. Look up the meaning of interpretation.

    The first '⊢' is syntactic, dealing with provability, so that B can be deduced/proved from A, i.e., there is a finite collection of sentences starting with A, where each is either a theorem, or follows from previous sentences by the application of some inference rule(s), the last sentence of which is B. 'A' on the left may be the empty set, in which case B is a tautology.
    This deals only with formal relations between sentences, and not with their actual inner content.

    The second , i.e., '⊨' deals with semantics, or the notion of truth. Informally, M is a possible world in which the wffs are all feasible/realizable. This deals with truth.
     
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