Question

If we consider bare, uninterpreted closed wffs, we can say that they are true in some models and false in others, but we cannot say that they are simply true or false without qualification. Both translation and valuation yield (different) unqualified notions of truth: translation involves the notion of truth in the actual model; valuation involves the notion of truth in the intended model. Axiomatisation does not yield an unqualified notion of truth. Rather, the notion that emerges naturally in this context is that of a theorem: a wff that is true every model that makes all the axioms true. In other words, a theorem is a logical consequence of the axioms. A set of wffs that is closed under logical consequence - that is, one for which every wff that is a logical consequence of some wffs in the set is also in the set - is a theory. A set of axioms generates a theory: the set of all wffs that are true in every model in which all the axioms are true.

What does unqualified notion of truth mean in this passeage

Answer 1

The term 'unqualified notion of truth' in this context refers to the idea that bare, uninterpreted closed well-formed formulas cannot be simply classified as true or false without further qualification. Instead, we use the concept of theorems, which are logical consequences of axioms. A set of wffs that is closed under logical consequence is called a theory.

Step-by-step explanation:

The term 'unqualified notion of truth' in this context refers to the idea that bare, uninterpreted closed well-formed formulas (wffs) cannot be simply classified as true or false without further context or qualification. When considering these wffs in relation to models and axioms, different notions of truth emerge. Translation involves the notion of truth in the actual model, while valuation involves the notion of truth in the intended model. Axiomatization, on the other hand, yields the notion of a theorem, which is a wff that is true in every model that makes all the axioms true.

In other words, an unqualified notion of truth is not applicable to bare, uninterpreted closed wffs. Instead, we use the concept of theorems, which are logical consequences of the axioms. A set of wffs that is closed under logical consequence is called a theory. A set of axioms generates a theory that consists of all the wffs that are true in every model in which all the axioms are true.