In a Consistent System, Can a True Sentence Imply a False One?
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From: herbzet on 18 Jul 2010 21:33 Nam Nguyen wrote: > Daryl McCullough wrote: > > Jim Burns says... > > > >> Charlie-Boo wrote: > >>> In a consistent system, can a true sentence imply a false one? > >> The most straightforward answer to your question is "No". > >> > >> The reason is hard to express in a way that actually > >> makes the situation clearer, in much the same way that > >> it is hard to explain /why/ a bachelor cannot be married. > >> > >> 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 > >> > >> Another way of looking at it is that having > >> true sentences imply only true sentences is what > >> implications are /for/. If they didn't do that, then we > >> would be spending our time looking at some other > >> logical function, or something, anything else which > >> served a similar purpose: pushing out the envelope of > >> the known. > > > > 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. > > > > So, for instance, if A is a false statement, and A is an *axiom*, > > and B is a true statement, then of course > > > > B -> A > > > > is derivable. > > > > What you can say is this: If a system is consistent, then a provably > > true statement can never imply a provably false statement. Not every > > true statement is provably true, and not every false statement is provably > > false. > > > > On the other hand, a *sound* system has the property that only > > true statements are provable. So for a sound system, a true statement > > can never imply a false statement. > > It goes without saying that by "true statements" we assume to mean > "arithmetically true statements". Hm -- probably you're right that Daryl meant that. [...] -- hz
From: Daryl McCullough on 18 Jul 2010 22:29 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. 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. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 18 Jul 2010 22:31 In article <4C43AA70.2E4BE5CA(a)gmail.com>, herbzet says... > > > >Aatu Koskensilta wrote: >> Charlie-Boo writes: >> >> > In a consistent system, can a true sentence imply a false one? >> >> Sure. > >Nah -- implication is truth-preserving, by definition. Let's consider the theory T with the following axiom: 0 = S(0) That's a false sentence. 0 = 0 That's a true sentence. But 0 = 0 -> 0 = S(0) is derivable in this theory. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 18 Jul 2010 22:36 herbzet says... > > > >Daryl McCullough wrote: >> Jim Burns says... >> >Charlie-Boo wrote: > >> >> In a consistent system, can a true sentence imply a false one? > >[...] > >> What you can say is this: If a system is consistent, then a provably >> true statement can never imply a provably false statement. Not every >> true statement is provably true, and not every false statement is provably >> false. > >What is a "provably true statement"? Is this a redundant way of saying >"provable statement"? No, a provably true statement is one that is both provable and true. >Is "provably false statement" a way of saying "negation of a provable >statement"? No, a provably false statement is one that is both disprovable (the negation of a provable statement) and false. -- Daryl McCullough Ithaca, NY
From: herbzet on 18 Jul 2010 22:59
Daryl McCullough wrote: > herbzet says... > >Aatu Koskensilta wrote: > >> Charlie-Boo writes: > >> > >> > In a consistent system, can a true sentence imply a false one? > >> > >> Sure. > > > >Nah -- implication is truth-preserving, by definition. > > Let's consider the theory T with the following axiom: > > 0 = S(0) > > That's a false sentence. Yes, as Nam pointed out, it appears that by "false" you mean "arithmetically false". I didn't assume Charlie-Boo meant this, which may be my mistake. > 0 = 0 > > That's a true sentence. But > > 0 = 0 -> 0 = S(0) > > is derivable in this theory. Sure, by positive paradox A -> (B -> A). Do you see our (yours and mine) statements as contrary assertions? -- hz P.S. -- The fun part here is trying to rattle Aatu's cage. |