Difference between consistency and satisfiability
If a set of formula is consistent, there exist a model in which every formula is true. This is only if the set is satisfiable. But satisfiability is the fact that it can be true so what is the difference between the 2 notions ?
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$\begingroup$Consistency is a syntactic property. It means that there is no proof of contradiction from your axioms.
Satisfiability is a semantic property. It means that there is a model of the axioms.
In first-order logic (as well as propositional logic) the two notions are equivalent because the logic is sound and complete. Meaning a satisfiable theory is consistent, and a consistent theory is satisfiable.
Other logics, however, are not so lucky to have both of these properties and the two notions separate. In fact, if we do not assume the axiom of choice, then it is consistent that there is a theory which is consistent but not satisfiable.
$\endgroup$ 6 $\begingroup$Consistency is defined syntactically :
a set $\Gamma$ of formulas is inconsistent iff we can derive (in the proof system) a contradiction from it : $\Gamma \vdash \bot$.
In this way, we prove that :
a set $\Gamma$ is consistent iff it is satisfiable.
See also the post : Relationship between consistency, strong completeness and soundness.
$\endgroup$ $\begingroup$"satisfiable": A 1.-order-logic-statement is satisfiable if it can not be simplified to False without knowing anything about the predicates, that it contains. Also a 1.-order-logic-statement is satisfiable if the 2.-order-logic-formula, that is obtained by replacing all predicates in it with fictive variables of existential quantifiers, that are put around the 1.-order-logic-statement, is not false. Note, that since this 2.-order-logic-formula does not contain any predicates it can be only True or False.
"consistent": A 1.-order-logic-statement is consistent if it can not be simplified to False even with using "outside information" about the predicates, that it contains.
Here is a more exhaustive explanation about it with examples.
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