In a Consistent System, Can a True Sentence Imply a False One?
in [Logic]
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From: herbzet on 19 Jul 2010 03:06 Aatu Koskensilta wrote: > > 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. Well, alright then. > 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. Right -- there you go. I don't think we need to fuzz things up with a double-hyphen '-->' -- hz
From: Aatu Koskensilta on 19 Jul 2010 03:07 herbzet <herbzet(a)gmail.com> writes: > I don't think we need to fuzz things up with a double-hyphen '-->' The double-hyphen is absolutely, positively essential. -- 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 03:11 Aatu Koskensilta wrote: > > herbzet <herbzet(a)gmail.com> writes: > > > I don't think we need to fuzz things up with a double-hyphen '-->' > > The double-hyphen is absolutely, positively essential. lol
From: George Greene on 19 Jul 2010 10:22 > George Greene <gree...(a)email.unc.edu> writes: > > THERE IS NO SUCH THING AS A FALSE AXIOM. > > > That's an axiom. On Jul 19, 1:21 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > That's a false axiom. Predictability is not always a virtue. That's a truism.
From: George Greene on 19 Jul 2010 10:26
On Jul 19, 1:13 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > A system may well be consistent even if some of its axioms are false. By definition, if the system is consistent, IT HAS A MODEL. By definition, IF WE HAVE DESIGNATED SOME PARTICULAR subset of the statements true in all models of the system AS AXIOMS, Thent it is THOSE models we CARE about! > We are here presupposing the formal language the system is formulated in > has some intended interpretation Who "We", Exactly?? > so that it makes sense to speak of truth or falsity. It is the theory and not the language that has the intended interpretations, and the interpretations intended ARE BY DEFINITION the ones THAT MAKE THE AXIOMS *TRUE*!! Choosing your axioms IS HOW you communicate which models you intend! And people REALLY need to QUIT saying "system" around here. This is frightfully ambiguous. |