What is the relation between consistency and soundness in mathematical logic?

Question

Are consistency and soundness the same or some how related in mathematical logic?

score 3 · Accepted Answer · answered Oct 13 '17 at 19:29

3

On notation, if $\Gamma$ is a collection of formulas and $P$ is a formula, then:

$\Gamma \vdash P$ means that, using the formulas in $\Gamma$, you can use the rules of inference to deduce $P$
$\Gamma \models P$ means that, in every way to interpret logic so that the formulas in $\Gamma$ are true, $P$ is also true
$\bot$ is a symbol denoting a contradictory proposition

Soundness is a property that first-order logic has:

$ \Gamma \vdash P $ implies $\Gamma \models P $

Inconsistency is a property that a set of formulas might have. "$\Gamma$ is inconsistent" means

$\Gamma \vdash \bot$

A consistent set of formulas is one that is not inconsistent.

answered Oct 13 '17 at 19:29

Does soundness only apply to first-order logic or any logical system? – K. Smith Oct 13 '17 at 19:34
@K.Smith: AFAIK, you could ask it of any logical system. Generally we're only interested in the ones that do have the property. – Oct 13 '17 at 19:51
Would you please also tell me the difference between validity and soundness? Because according to the following definition they look the same: Validity: one cannot arrive at false statements from true statements by applying theorems we know to be true; Soundness: every statement that’s proven is true. – K. Smith Oct 13 '17 at 20:38

score 1 · Answer 2 · answered Oct 13 '17 at 19:17

They are two separate concepts. Soundness is used to describe a logical argument, while consistency is used to describe a set of sentences.

A logical argument is sound if and only if the logical argument is valid and the premises are true.

A set of sentences is consistent if and only if it is possible for all the sentences to be true.

2 Answers2