 Also standard are the definitions of the syntactic  concepts;[7]  namely those of consistency, maximal consistency and the property of having witnesses. In the present circumstance, a set of formulas contains witnesses if each existential formula comes with a witness (\(\exists x^{i}\varphi \in \Delta\) implies \(\varphi (t/x^{i})\in \Delta\) for a term \(t\) of sort \(i\)). All three are properties of formulas and/or sets of formulas and in their definition we use derivability; i.e., the deductive calculus.