Willem F Esterhuyse Posted February 11, 2023 Share Posted February 11, 2023 Say you got a formal proof: Line # Statement Reason 1 A Premise 2 ... ... n B m, Some reason n+1 A -> B n, Deduction Theorem. Now we have proven: A -> A -> B, so we proved: true -> B = true (by truth table of "therefore"). So we proved A -> true (by deduction theorem). Now A could be false, then we shouldn't be able to derive "true". Link to comment Share on other sites More sharing options...
Lorentz Jr Posted February 11, 2023 Share Posted February 11, 2023 (edited) 23 minutes ago, Willem F Esterhuyse said: Now we have proven: A -> A -> B, so we proved: true -> B = true You're mangling the associativity of the operator (it's right-associative). We've proved A -> (A -> B), not (A -> A) -> B. In English, it's "if A, then if A then B", not "if (if A then A) then B". Edited February 11, 2023 by Lorentz Jr Link to comment Share on other sites More sharing options...
Genady Posted February 11, 2023 Share Posted February 11, 2023 4 hours ago, Willem F Esterhuyse said: A could be false, then we shouldn't be able to derive "true". This is incorrect. Anything, true or false, can be derived from a false statement. See the truth table and the second dot here: Link to comment Share on other sites More sharing options...
swansont Posted February 11, 2023 Share Posted February 11, 2023 5 hours ago, Willem F Esterhuyse said: Now A could be false, then we shouldn't be able to derive "true". Your proof was contingent on A being true. Its conclusion is only valid in that case. It says nothing about the state of affairs if A is false. Link to comment Share on other sites More sharing options...
Willem F Esterhuyse Posted February 11, 2023 Author Share Posted February 11, 2023 (edited) I made an error: (true -> B) = B, (true -> B) not= true. Edited February 11, 2023 by Willem F Esterhuyse Link to comment Share on other sites More sharing options...
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