I just want to know the difference between this two symbols and what are the semantic meaning ?? And what is the difference betwwen this two formulas ????
A → B ⊢ ¬B → ¬A
A → B ⊨ ¬B → ¬A
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1Please check logical consequence. In short, the first means that assuming left side, you can prove the right side, the second means that for each interpretation in which left side is true (model), the right side is true as well. First is concerned with syntax, while the second with semantics of the formal system you are working with. – Ennar Oct 12 '16 at 11:17
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I know that, but i want to know what is the relation between the two , with common sens i assume that if the first formula is valid the the second is , but not the reverse ( A ⊢ B -> A⊨ B ) and not ( A⊨ B -> A ⊢ B ) so is that true ?? btw thank you for your answer :) – Lilo Oct 12 '16 at 11:24
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You can see also the post difference-between-some-terminologies-in-logic. – Mauro ALLEGRANZA Oct 12 '16 at 11:33
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5And also this one : meaning of symbols $\vdash$ and $\models$ with further links.. – Mauro ALLEGRANZA Oct 12 '16 at 11:35
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1Think of rephrasing the title and the first sentence then. It is strongly implied that you don't understand what the symbols $\vdash$ and $\vDash$ stand for. You probably want to see this. – Ennar Oct 12 '16 at 11:36
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2Propositional logic is Complete, i.e. $\Gamma \vdash \varphi$ iff $\Gamma \vDash \varphi$. – Mauro ALLEGRANZA Oct 12 '16 at 11:37
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So the propositional logic is complete and the predicate logic is not ??? – Lilo Oct 12 '16 at 11:42
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No; also predicate logic is. – Mauro ALLEGRANZA Oct 12 '16 at 11:47
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So what Incompleteness does prove the famous Gödel incompleteness theorem ??? – Lilo Oct 12 '16 at 11:49
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Another type on incompleteness. – Mauro ALLEGRANZA Oct 12 '16 at 11:54
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I see know more clearly, thank you so much @MauroALLEGRANZA :) – Lilo Oct 12 '16 at 11:55
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Related also to your previous post : in prop logic $\vDash \varphi$ means that $\varphi$ is a tautology and $\vdash \varphi$ means that it is a theorem of prop calculus. The same for FOL. – Mauro ALLEGRANZA Oct 12 '16 at 11:58
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In very general terms, without getting into the specifics of models, languages, the difference is one of syntax vs. semantics:
[Syntax] $P \vdash Q$ means that there is a proof (in whatever formal system you are using) of $Q$ assuming $P.$
[Semantics] $P \models Q$ means that in every situation where $P$ is true, $Q$ also is true; typically this means that, in terms of models, every model of $P$ is also a model of $Q.$
If your formal system is consistent and logically sound, then $P \vdash Q$ implies $P \models Q$ (since if you've proven something, it must be right).
On the other hand, it's conceivable for $P \models Q$ to be true but for there to be no proof of this fact in your formal system, so that $P \not\vdash Q.$ Intuitively, this can happen if your formal system isn't strong enough to capture the actual reason that any model of $P$ is also a model of $Q.$ (For first-order logic, this can't happen; that's the Completeness Theorem.)
I suppose it's also conceivable that $P\models Q$ "accidentally"; that is, it just so happens that any model of $P$ is also a model of $Q,$ but there's no reason for it. It's hard to formalize this precisely (the word "reason" is vague), and it certainly can't happen for first-order logic; I don't know of any examples at all, and I'm not sure how you would even demonstrate such a thing if it were ever true of some language. ("Demonstrating" it would seem to involve providing a reason it's true, after all.)
Mitchell Spector
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