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I have been trying to solve some propositional proofs,

E.g.

(A ⊃ B)
(~A ⊃ B)
Therefore, B

And I know that this is valid argument. Can we ever know the real values of A and B from the truth table or is it just that we assume some values which we can use to prove the argument to be valid or invalid?

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See also the Entscheidungsproblem –  Xodarap Oct 31 '12 at 0:15

1 Answer 1

up vote 6 down vote accepted

We may consider all possible values of each atomic proposition, and compute the corresponding truth-values of the formulae that arise as a result. Then we may restrict to those rows of the truth-table in which the assumed propositions take the value 'true'. If any atomic proposition obtains a constant value across all of those rows, we may infer that it is correspondingly either 'true' or 'false'.

Taking the example from your question, we may compute:

  A  B  |  A ⊃ B  | ¬A ⊃ B
--------|---------|----------
  F  F  |    T    |    F
  F  T  |    T    |    T       ⇐ satisfies (A ⊃ B) and (¬A ⊃ B)
  T  F  |    F    |    T
  T  T  |    T    |    T       ⇐ satisfies (A ⊃ B) and (¬A ⊃ B)

If we delete all rows which do not satisfy the propositions which are asserted, we obtain

  A  B  |  A ⊃ B  | ¬A ⊃ B
--------|---------|----------
  F  T  |    T    |    T      
  T  T  |    T    |    T       

from which we infer that B is true, although A could still be either true or false.

Edited to add — in reference to your question, "is it just that we assume some values which we can use to prove the argument to be valid or invalid?" It isn't enough to assume some values to prove the argument valid — what we are doing here is showing that all possible values of A and B which satisfy the two imposed constraints (A ⊃ B and ¬A ⊃ B) which are used as hypotheses, validates the inference of B. By the same token, because we see that there are two different values of A are compatible with the imposed constraints, and so no valid inference can be made about A in either direction.

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Does this strategy work on any kind of propositional proof? We're looking for the cases where the propositions take the value 'true'. So, we are assuming all the premises and their conclusion to be true but for an argument to be valid we don't necessarily need to have all it's premises to be true. Is that just based on an assumption? –  user963241 Oct 30 '12 at 22:23
    
It does work on any propositional proof. The truth-table essentially describes all possible models for the atomic propositions, and the resulting values of various compound propositions. If restricting to those models in which some propositon Q is true (whether compound or atomic) also restricts some other proposition P (compound or atomic), this implies that the value of P is fixed by the truth of Q. –  Niel de Beaudrap Oct 30 '12 at 22:30
    
@user120321 I'm not sure what you mean by "assuming all the premises and their conclusion to be true". Just as in a deduction by rules of inference, we posit the propositions A ⊃ B and ¬A ⊃ B; these are taken as hypotheses in the normal way. Given that they are true, the truth-table allows us to infer that in all models where these hold, B is also true. By the completeness of propositional logic — the fact that all tautologies are theorems — it follows that any sort of inference of this sort, by inspection of all possible worlds satisfying a constraint, can also be done by deduction. –  Niel de Beaudrap Oct 30 '12 at 22:36
    
@user120321: after some reflection, I have added some commentary on assuming specific values of the premises which I hope addresses your comment above. –  Niel de Beaudrap Oct 30 '12 at 22:41

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