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I am trying to prove that an argument is valid iff it is derivable. Apparently it is some difficult proof that my book left out.

W. G.
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  • Need to specify what arguments are "valid" and what are "derivable" for this. For example the derivation rules for a proof, and does "valid" mean existence of a model in some particular universe. – coffeemath Apr 10 '17 at 00:52
  • Derivable refers to syntactic consequence and valid refers to a semantic consequence. – W. G. Apr 10 '17 at 00:55
  • One still needs to know the system your question is assuming is used. For example in Peano arithmetic I think that theory is undecidable. – coffeemath Apr 10 '17 at 00:57
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    Here is a link https://books.google.com/books?id=vJLqBwAAQBAJ&pg=PA34&lpg=PA34&dq=abstract+mathematics+derivability&source=bl&ots=4F_mkaLVf1&sig=nOVvp_qYpCrUOkOTKm6ocujuthQ&hl=en&sa=X&ved=0ahUKEwiB7L3O15jTAhUD2IMKHTEfAtcQ6AEINTAD#v=onepage&q=derivable&f=false – W. G. Apr 10 '17 at 01:02
  • On page 35 it says "that is, an argument is valid if and only if it is derivable" – W. G. Apr 10 '17 at 01:03
  • In that link it seems the system is propositional logic, rather than something like Peano arithmetic. Propositional logic is decidable by truth tables, so I agree with "valid iff derivable" in this case. – coffeemath Apr 10 '17 at 01:09
  • Let us continue this discussion in chat. – W. G. Apr 10 '17 at 01:35
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    See any math log textbook, under "Propositional calculus: Completeness". – Mauro ALLEGRANZA Apr 10 '17 at 06:12
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    See page 32: "An argument is valid if the conclusion necessarily follows from the premises. [...] an argument is valid if we cannot assign truth values to the component statements used in the argument in such a way that the premises are all true but the conclusion is false." – Mauro ALLEGRANZA Apr 10 '17 at 09:21
  • @coffeemath: Whether the theory is undecidable is irrelevant here. "Valid" here means "true in every model". – user21820 Apr 10 '17 at 11:36
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    But this question is a bad question because it does not specify what "valid" and "derivable" mean. @W.G.: Don't expect everyone to know what you mean by them; write out the definitions you are using instead of just linking to it. – user21820 Apr 10 '17 at 11:39
  • And by the way, the book is also a bad book! It says "It is not that there is anything logically wrong with inconsistent premises, they are simply of no use to mathematicians, since we can derive anything from [Pages 38 to 39 are not shown in this preview.]". This is nonsense! Inconsistent premises are useless to everyone because they cannot be given meaningful interpretation in the real world! And later they say that forall x in U ( P(x) ) implies exists x in U ( P(x) ), without stating that U is non-empty, and so it's nonsense! – user21820 Apr 10 '17 at 12:09
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    Derivable means a chain of statements connected by metastatements and valid refers to exactly what Mauro Allegranza said. I apologize if the question is worded poorly and hoped to provide additional information concerning the universe regarding the link. The rules of inference that the book is using for these metastatements are both valid and derivable by checking them. – W. G. Apr 10 '17 at 12:10
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    @W.G.: That is a nonsensical definition. What on earth does "connected by metastatements" mean? I told you already that your book is terrible; that's the problem! If you are really interested in logic proper, I suggest the free online references here. There may be better ones, but I haven't found them. – user21820 Apr 10 '17 at 13:04

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