In a Consistent System, Can a True Sentence Imply a False One?
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From: Aatu Koskensilta on 19 Jul 2010 02:31 herbzet <herbzet(a)gmail.com> writes: > 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. By "implication" I meant just a statement of the form A --> B. This is, I believe, relatively standard usage in logic. -- 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 02:38 Aatu Koskensilta wrote: > > 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. But the proof-in-T is not itself a false implication under any interpretation, though its conclusion may be false under this or that interpretation. Look at the title of the thread: Is there an interpretation of a consistent theory T such that from its true axioms (if any exist) one can derive false theorems? I'd like to see it. -- hz
From: Aatu Koskensilta on 19 Jul 2010 02:42 herbzet <herbzet(a)gmail.com> writes: > Look at the title of the thread: Is there an interpretation of a > consistent theory T such that from its true axioms (if any exist) one > can derive false theorems? That's not the title of this thread. You're of course right that a true sentence never implies a false one. That notwithstanding, when in logic we talk about a sentence A implying another sentence B in a theory or system we usually have in mind the provability of the implication A --> B in the theory or system. -- 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 02:53 Aatu Koskensilta wrote: > herbzet <herbzet(a)gmail.com> writes: > > > 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. > > By "implication" I meant just a statement of the form A --> B. This is, > I believe, relatively standard usage in logic. The question of when A -> B means that A implies B is quite non-trivial. What is standard usage in logic is in this respect problematical, and the problem is in the equivocal use of the term "implies". Sure, in a consistent system, we can prove false theorems, but there will have to be false axioms used -- just using true axioms will not suffice. A counter-example would end this argument quickly. -- hz
From: Aatu Koskensilta on 19 Jul 2010 02:59
herbzet <herbzet(a)gmail.com> writes: > Sure, in a consistent system, we can prove false theorems, but there > will have to be false axioms used -- just using true axioms will not > suffice. No one's claimed otherwise. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |