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Math Help - logic

  1. #1
    Junior Member
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    logic

    Hi guys just a quick question

    Is the following true or false. give argument or counterexample, as appropriate.


    { A,B,\neg C} is an inconsistent set of sentences iff the argument from A,B as premises to C as conclusion is valid.

    Thanks in advance
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  2. #2
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    From left to right:

    Assume {A, B ~C} is an inconsistent set of premises. Therefore (A&B&~C) is a contradiction. Therefore, ~(A&B&~C) is true, which by de Morgan's law is equivalent to ~Av~BvC. If ~A, then A, B |-- C is valid vacuously because one of the premises, A, is false. If ~B, the same applies, mutatis mutandis. If C, then the sequent is valid trivially.

    From right to left:

    Assume A, B |-- C is valid. Assume, for reductio, that A, B, and ~C are all true. Then, since A is true and B is true, by the validity of the sequent, C is true, which means that ~C is false, contra hyp.
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  3. #3
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    Recall that consistency pertains to provability and validity pertains to entailment.

    And you are familiar with the notation '|-' for "proves" and '|=' for "entails"?

    Have you proven that the argument from {A B} to C is valid if and only if {A B} proves C (i.e., {A B} |= C if and only if {A B} |- C)?

    (I.e., more generally, have you proven the soundness and completeness theorems for whatever system - I surmise first order logic - you're talking about?)

    Suppose we have proven the soundness and completeness theorems:

    Prove left to right in the problem:
    If {A B ~C} is inconsistent, then {A B ~C} |- C , so, by soundness, {A B ~C} |= C.
    Now, suppose, A and B are both satisfied in a model M with variable assignment v, while assuming, toward a contradiction, that C is not satisfied in M with v. Then ~C is satisfied in M with v, so {A B ~C} are satisfied in M with v, so, since {A B ~C} |= C, we have {A B ~C C} satisfied in M with v, which is impossible since no formula C and ~C are both satisfied in a model with a variable assignment.

    Prove right to left in the problem:
    If {A B} |= C then, by completeness, {A B} |- C, so {A B ~C} is inconsistent, since, by monotonicity, {A B ~C) |- C and {A B ~C} |- ~C.
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  4. #4
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    Thanks so much for the help guys=)
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