I see that valid formula (proposition or statement) is the one that is valid under every interpretation. But this is a tautology. Is there any difference between tautology and valid formula? They also say that A,B ⊨ Φ is valid if it preserves the truth. Is it related with the first definition of validity and being tautology?

They also say that in addition to being valid, i.e. preserving the truth, your argument is sound when premises are also valid. But there is another definition of soundness, which tells that theory is sound if formula provability implies that formula is semantically valid, (if Г ⊢ P, then also Г ⊨ P). Are these two notions of soundness identical, related or simply homonyms?

I see that valid formula (proposition or statement) is the one that is valid under every interpretation. But this is a tautology. Is there any difference between tautology and valid formula?

Some people use "tautology" in logic in a wide sense, to mean any logically true wff. But others use "tautology" more narrowly to mean true in virtue of truth-functional structure (so, "valid by the truth-table test"). So, for example, $\forall xFx \to Fa$ would count as a tautology in the first, wide, sense but not in the second, narrow, sense. (How to use 'tautology', then, is a matter of terminological preference: I much prefer the second narrower usage as it gives us a label for a special class of logical truths which we need a label for.)

They also say that $A, B \vDash \Phi$ is valid if it preserves the truth

No they don't. Or at least, they shouldn't. It is the inference that preserve truth. Thus if $A, B \therefore \Phi$ necessarily preserves truth, then it is valid. But $A, B \vDash \Phi$ is not an inference: it is a meta-linguistic statement about the inference, and says the inference is valid.

They also say that in addition to being valid, i.e. preserving the truth, your argument is sound when premises are also valid.

Fine: an argument is sound if it has true premisses and a valid inference move.

But there is another definition of soundness, which tells that theory is sound if formula provability implies that formula is semantically valid.

As you almost say, this is a definition of what it is for a formal deductive system to be sound.

Arguments aren't formal deductive systems, and soundness for arguments is not soundness for formal deductive systems. The notions can't be identical. But there is some similarity between the two notions -- both have something to do with being trustworthy (an argument won't lead you astray about the truth if it starts with truths and proceeds by truth-preserving moves, i.e. is sound; likewise a formal system encoding inferences won't lead you astray about what's a valid inference if it is sound)

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    Hi Peter, you write: "I much prefer the second narrower usage as it gives us a label for a special class of logical truths which we need a label for." I'm interested in what you're saying here. What significance do the tautologies of first-order logic (as opposed to the validities) have? By which I mean: why do we need a name for them? Perhaps automated theorem proving? – goblin Jan 21 '14 at 21:41
  • @Peter I asked how two definition of valideness, being true in every interpretation and preserving truth are related. Do they mean the same? You try to draw the parallels between two definitions of soundness. Thanks for that. This is what I asked. But, I did not get your argument. I do not see how sound formal deductive system preserves the truth when valid premise is provided. – Val Apr 20 '14 at 10:36

I have seen the definition which said that valid means true in every interpretation but tautology is this subset of valid sentences which coincides with tautologies of sentential logic. So for example in sentential logic $a \iff a$ is tautology so in FOL $\forall x.P(x) \iff \forall x.P(x)$ is also tautology.

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