In a Consistent System, Can a True Sentence Imply a False One?
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From: Charlie-Boo on 18 Jul 2010 09:45 In a consistent system, can a true sentence imply a false one? C-B
From: Jim Burns on 18 Jul 2010 10:50 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. Any more explanation than this starts to tread on "What is a consistent system?", "What is a true or false sentence?", or "What is an implication?" territory. These questions have answers and reasons behind the answers, too. I don't know if addressing them helps clarify the original question or just extends the conversation. Jim Burns
From: Aatu Koskensilta on 18 Jul 2010 11:07 Charlie-Boo <shymathguy(a)gmail.com> writes: > In a consistent system, can a true sentence imply a false one? Sure. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Daryl McCullough on 18 Jul 2010 13:17 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. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 18 Jul 2010 20:51
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". So "soundness" is a relative concept: relative what we mean by "arithmetic" and there's no absolutely "sound" system. Of course. -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan --------------------------------------------------- |