According to the wiki https://en.wikipedia.org/wiki/Soundness,
"In mathematical logic, a logical system has the soundness property if and only if its inference rules prove only formulas that are valid with respect to its semantics. In symbols, if $\displaystyle A1,A2,...An\vdash C$, then $\displaystyle A1,A2,...\models C$, where $\vdash$ means derivation, $\models$ means tautological entailment."
But, it also says "An argument is sound if and only if
- The argument is valid, and 2. All of its premises are true."
All men are mortal. Socrates is a man. Therefore, Socrates is mortal. The argument is valid (because the conclusion is true based on the premises, that is, that the conclusion follows the premises) and since the premises are in fact true, the argument is sound.
My question is that according to the first definition of "soundness" in the wiki, I think that one of premises can be false in a sound argument as long as its conclusion is false. But the second definition of "soundness" says that all of its premises should be true to be a sound argument.
What am I missing?