In a Consistent System, Can a True Sentence Imply a False One?
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From: herbzet on 18 Jul 2010 21:29 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.
From: herbzet on 18 Jul 2010 21:29 Jim Burns wrote: > Charlie-Boo wrote: > > In a consistent system, can a true sentence imply a false one? > [...] 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 [...] herbzet likes this.
From: herbzet on 18 Jul 2010 21:30 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"? Is "provably false statement" a way of saying "negation of a provable statement"?
From: herbzet on 18 Jul 2010 21:31 Charlie-Boo wrote: > > In a consistent system, can a true sentence imply a false one? The default assumption around here for "system" is that you mean a classical, first-order system. With this proviso, a consistent system has a model, by the model existence theorem. That is, there is a structure in which all the axioms are true. Just something to think about. -- hz
From: herbzet on 18 Jul 2010 21:32
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 4) A (2),(3) modus ponens 5) A -> ( B -> (A & B)) a tautology ("strong adjunction") 6) A -> (~A -> (A & ~A)) (5) uniform substitution ~A/B 7) ~A -> (A & ~A) (4),(6) modus ponens 8) A & ~A (1),(7) modus ponens In a non-classiscal system that rejects strong adjunction as a theorem, the conclusion A & ~A may not be derivable. One reason to reject strong adjunction is that if you accept it, and also accept simplification (A & B) -> A and prefixing (B -> C) -> ((A -> B) -> (A -> C)) then you have: the derived inference rule (B -> C) |- (A -> B) -> (A -> C) 1) B -> C Given 2) (B -> C) -> ((A -> B) -> (A -> C)) Prefixing 3) (A -> B) -> (A -> C) (1),(2) modus ponens and thus 1) (A & B) -> A Simplification 2) (B -> (A & B)) -> (B -> A) (1) derived inference from prefixing 3) (A -> (B -> (A & B))) -> (A -> (B -> A)) (2) derived inference from prefixing 4) A -> (B -> (A & B)) strong adjunction 5) A -> (B -> A) (3),(4) modus ponens and this last formula is known as "positive paradox". Nobody likes it much. -- hz |