In a Consistent System, Can a True Sentence Imply a False One?
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From: herbzet on 19 Jul 2010 01:48 Aatu Koskensilta wrote: > herbzet writes: > > > If the system is consistent, then there is a structure in which > > B -> A is true. > > Sure, but what of it? Whenever we speak of truth or falsity we must have > some specific interpretation in mind. The interpretation which most naturally suggests itself is one in which the axioms are true.
From: herbzet on 19 Jul 2010 01:54 Aatu Koskensilta wrote: > herbzet writes: > > > If this consistent system, so interpreted such that it has false > > axioms, proves that A -> B, does that mean that A implies B? > > No. That's what I'm saying. > What of it? That we need to clarify what it means to assert that a true statement can imply a false statement in a consistent system. -- hz
From: Aatu Koskensilta on 19 Jul 2010 01:55 herbzet <herbzet(a)gmail.com> writes: > The interpretation which most naturally suggests itself is one in which > the axioms are true. Suggests how? If we consider for instance the theory PA + "PA is inconsistent" I doubt any interpretation on which the axioms are true suggests itself to anyone. In any case, my point was simply that whenever we have some interpretation of a formal language in mind there are many consistent theories that prove falsehoods, and in particular prove false implications; and if we don't have some specific interpretation in mind it makes no sense to speak of a theory proving truths or falsehoods. The completeness theorem is wholly irrelevant. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 19 Jul 2010 01:57 herbzet <herbzet(a)gmail.com> writes: > That we need to clarify what it means to assert that a true > statement can imply a false statement in a consistent system. What's there to clarify? The observation that a consistent system can prove false implications is on the face of it completely transparent. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: herbzet on 19 Jul 2010 02:27
Aatu Koskensilta wrote: > In any case, my point was simply that > whenever we have some interpretation of a formal language in mind there > are many consistent theories that prove falsehoods, Sure. > and in particular prove false implications; By "false implications" here you mean sentences of the form 'A -> B' with A true and B false under interpretation. However, this is a misleading use of the term "implication", as you have agreed that a formal proof in T of 'A -> B' does not mean that A implies B. > and if we don't have some specific > interpretation in mind it makes no sense to speak of a theory proving > truths or falsehoods. Of course -- tell it to Charlie-Boo. > The completeness theorem is wholly irrelevant. Oh, I don't know about that. -- hz |