In a Consistent System, Can a True Sentence Imply a False One?
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From: herbzet on 18 Jul 2010 23:22 Daryl McCullough wrote: > herbzet says... > >Jim Burns wrote: > > > >> If A is a false sentence, then ~A is a true one. > >> If B is a true sentence and B implies A and A is false, > >> then we can assert > >> ~A > >> B > >> B -> A > >> from which it follows > >> A & ~A > > > >Specifically, we have (classically): > > > >1) ~A Given > >2) B Given > >3) B -> A Given > > The fact that A is false means that ~A is true, but it doesn't > mean that ~A is derivable in the system. Indeed, if the system is consistent and proves A, then ~A is not provable, as Jim Burns was pointing out. > I guess it was a little ambiguous what it means to say that > B implies A. I assumed that he meant that the implication > B -> A is derivable in the system, not that it is true. If the system is consistent, then there is a structure in which B -> A is true. -- hz
From: George Greene on 19 Jul 2010 00:16 On Jul 18, 1:17 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Charlie said *consistent* system, not *sound* system. A consistent > system only guarantees that you can't derive a contradiction. There > is no requirement that you can't derive false conclusions. You are basically saying that it is possible for a "system" to have A FALSE AXIOM. "System" is a COMPLETELY ILlegitimate term here. People who MEAN "theory" SHOULD SAY "theory".
From: George Greene on 19 Jul 2010 00:20 On Jul 18, 10:59 pm, herbzet <herb...(a)gmail.com> wrote: > > >Nah -- implication is truth-preserving, by definition. > DMC: > > Let's consider the theory T with the following axiom: > > > 0 = S(0) > > > That's a false sentence. That's a false AXIOM, more to the point. And WHO CARES that YOU think it's false??? This allegedly false axiom IS NOT false IN ANY model OF THESE AXIOMS! The fact that this sentence WAS MADE AND DESIGNATED an AXIOM *means* that WE ARE INTENTIONALLY RESTRICTING OUR ATTENTION to models OF THESE axioms! The fact that this sentence is false some OTHER model than you NORMALLY, typically, STANDARDLY, consider STANDARD *IS*JUST*IRRELEVANT*. THERE IS NO SUCH THING AS A FALSE AXIOM. That's an axiom.
From: herbzet on 19 Jul 2010 00:42 George Greene wrote: > > herbzet wrote: > > > >Nah -- implication is truth-preserving, by definition. > > DMC: > > > Let's consider the theory T with the following axiom: > > > > > 0 = S(0) > > > > > That's a false sentence. > > That's a false AXIOM, more to the point. > And WHO CARES that YOU think it's false??? > This allegedly false axiom IS NOT false IN ANY model OF THESE AXIOMS! > The fact that this sentence WAS MADE AND DESIGNATED an AXIOM *means* > that WE ARE INTENTIONALLY RESTRICTING OUR ATTENTION to models OF THESE > axioms! The fact that this sentence is false some OTHER model than > you NORMALLY, > typically, STANDARDLY, consider STANDARD > *IS*JUST*IRRELEVANT*. > > THERE IS NO SUCH THING AS A FALSE AXIOM. > > That's an axiom. Lol. -- hz
From: Jesse F. Hughes on 19 Jul 2010 01:01
George Greene <greeneg(a)email.unc.edu> writes: > "System" is a COMPLETELY ILlegitimate term here. May I be the first to note the fascinating progression of your capitalization skills here? Wow. Prefixes now. Whole new worlds are being born. -- Jesse F. Hughes .... one of the main causes of the fall of the Roman Empire was that, lacking zero, they had no way to indicate successful termination of their C programs. -- Robert Firth |