# Soundness in propositional and first order logic

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

1. The argument is valid, and 2. All of its premises are true."

For instance,

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?

• Which book do you learn logic now? Having read many introduction books on logic, I highly recommend Logic, Sets and Recursion by Robert Causey for beginners. It's much rigorous and systematic than others, say Kahane's. – Eric Jul 29 '17 at 14:56

## 2 Answers

These just two different meanings of the word "sound". The first defines what it means for a logical system to be sound, while the second defines what it means for a particular argument to be sound.

If a logical system is sound, you can trust the proofs generated by that system. So if $A_1,\dots,A_n\vdash B$, then you can be confident that whenever the premises hold, the conclusion also holds. Soundness of the system has nothing to do with truth of the premises in any particular argument.

On the other hand, if a particular argument is sound, then you can trust the conclusion. You need to know that the premises are all true, and that the steps in the argument came from a sound logical system, so the conclusion must be true.

• Thank you for your reply. If a logical system is sound, is it possible that one of the premises, let's say A_k, is false and the conclusion B is false as well? What happens if a logical system is sound and one of its statements in the logical system is false? – Tim Lee Jul 29 '17 at 13:55
• First question: yes, that's certainly possible. Second question: what do you mean by "one of its statements in the logical system"? – Alex Kruckman Jul 29 '17 at 13:57
• My apologies to asking with an incorrect usage of terms in logic. I would like to make sure the following. (1) If a logical system is sound, it is possible that one of the premises, let's say A_k, is false and the conclusion B is false. (2) If a logical system is sound, it is not possible that one of the premises, let's say A_k, is false and the conclusion B is true. Thanks. – Tim Lee Jul 29 '17 at 14:06
• (1) is correct. – Alex Kruckman Jul 29 '17 at 14:20
• Assuming that a logical system is a sound first-order logical system, is it OK to say that (2) is not possible? – Tim Lee Jul 29 '17 at 14:23

I think where you are having a difficulty is with the question 'can a VALID argument FORM have a false conclusion and false premises?' The answer is yes! All men are from Mars. Socrates is a man. Therefore, Socrates is from Mars. But notice that although this argument is valid (syntactically i.e. in virtue of its form) it is unsound since one of its premises is false and the conclusion is also false.

Keep in mind the distinction between validity as a syntactic notion and validity (and soundness) as semantic notions. The rules of logic operate at the syntactic level not only to preserve syntactic validity but also the semantic notions (which we usually care more about in actual practice).

A false conclusion in a sound argument is impossible. For if the conclusion is false, then the argument is unsound (no matter what the truth values of the premises are). If the premises are true but the conclusion is false, then we have an invalid argument (semantically speaking, since it is possible for an argument to be invalid in virtue of its syntactic form). If the premises are false, then it doesn't matter what the truth value of conclusion is: a false antecedent implies anything (in classical logic at least): the argument is invalid (semantically speaking) and unsound.