Proof of validity-1 - Short(brief/indirect truth-table...

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Short (brief/indirect) truth-table method N.Apostol The Procedure of Short-Table Proof 1. Put argument form in propositional form. 2. Put an F to the left of the conclusion and a T to the left of each premiss. 3. Start with that proposition (whether conclusion or premise) which allows for only one assignment of truth-values to make it as required by the values to its left (false, if it is the conclusion you start with; true, if it is a permise you start with) 4. Fill in the truth values for the rest of the propositions, carrying the "locked-in" values and assigning new values, as required for the rest of the simple propositions 5. If all the premises can be made true and the conclusion false, the argument is INVALID and the procedure is complete. If not, then we enter into the "Columbo Syndrome." "Columbo Syndrome" means that just like the detective of that popular series starring Peter Falk, who knew who commited the crime but had to prove it, so we know that the argument is valid, but we have to prove it. Proving it means showing that under every possible truth substitution, one of the premises must be false. . We need show only the instance where the conclusion forces at least one premise to become false; or all true (relevant) premises force the conclusion to become true . NOTE: never stat by making the conclusion true never start by making the premises false Test the following arguments for validity. In other words, determine whether the following arguments are valid or invalid. ( Important: all answers require proof; that is, all answers require an adequate justification, whether arguments are valid or invalid). CASE ( I ): A B ~A A&B ~(A&B) AvB ~A->B A->B A v B T T F T F T T T :. ~A -> B T F F F T T T F F T T F T T T T F F T F T F F T CASE (I) P1 P2 C CASE (II) C P2 P1 In order to test for validity, only one combination of truth values between premises and conclusion will prove (reveal) the nature of the argument: all true premises and false conclusion. 1) If it is possible to assign certain truth values to the sentences A and B such that all premises are true and conclusion false, the argument is invalid. 2) If it is impossible to assign any truth values to the sentences A and B which will make all premises true and the conclusion false, then the argument is valid. In other words, every
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single combination of truth values between A and B will never produce all true premises and a false conclusion . This only happens due to the way the sentences are related by the logical connectives ( &, v, ->, <->, ~ ) within the argument. It is that relation which either allows for all true premises and a false conclusion, proof of invalidity, or it blocks the possibility of all true premises and a false conclusion, the proof of validity. Very important: validity and invalidity are a function (a property) of an argument given the way the atomic sentences (A, B, etc.) are related. We (or anyone) cannot make a given argument valid or invalid, they are valid or invalid as a result of the already existing relation
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Proof of validity-1 - Short(brief/indirect truth-table...

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