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I'm having some trouble understanding the meaning of "unqualified notion of truth" in the following passage from pages 252-253 of Logic: The Laws of Truth by Nicholas J. J. Smith:

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.

Some of the terms used in the excerpt don't seem to have a standard definition, so I'd just like to briefly summarise my understanding of the sense in which these terms are used in the book:

  • Without content, a closed wff cannot be said to be true or false, i.e. cannot be said to represent a proposition. Translation, valuation and axiomatisation are different ways of giving a closed wff content.
  • Translation views the contents as intensions (functions from wws to values). The nonlogical symbols that make up the wff are assigned intensions by a glossary. Assuming compositionality of content, these intensions are combined by the logical operators in the wff to form the intension of the wff as a whole. This intension is a function from wws to truth values. When the intensions for the nonlogical symbols are applied to a ww, we obtain an assignment of values to these symbols. These values, together with a domain, give us a model, $\mathfrak{M}$. The closed wff can now be regarded as representing a proposition about the world represented by $\mathfrak{M}$. This proposition is true if the closed wff is true in $\mathfrak{M}$ and false otherwise. The model obtained by applying the intensions to the actual ww (the ww that represents the world as it actually is) is called the actual model.
  • Valuation gives content to a closed wff by directly assigning values to its nonlogical symbols, yielding a model called the intended model. As in translation, the closed wff can now be regarded as representing a proposition about the world represented by the intended model, which is true if the closed wff is true in that model and false otherwise.
  • Axiomatisation gives content to a closed wff by restricting the set of possible values that its nonlogical components can take. This is achieved by a system of axioms (or postulates), which is a set of wffs on the same language fragment as the original wff. The truth or falsity of the original wff can only be evaluated in models in which all the axioms are true. A closed wff which is true in every model in which all the axioms are true is a theorem. The set of all such wffs is a theory.

NB: ww is my abbreviation for "way of the world". Partial previews of the sections of the book that cover translation and valuation can be found here and here, respectively.

My questions:

  • In what way are the notions of truth in the actual and intended models "unqualified"?

    In translation and valuation, the truth/falsity of a closed wff is always determined relative to a model. So, isn't the notion of truth in a model a "qualified" notion of truth, in that it is qualified by/conditinal on that particular model.

  • Why does the author single out the actual and intended models in particular? Is that to say that only the notion of truth in these models is unqualified?

My main points of confusion are the first and second sentences of the quoted excerpt, so I'd be grateful if someone could clarify them. I had originally asked this question on Philosophy Stack Exchange. While the original post does have a few answers, I am still very hazy as to what the excerpt is saying.

user51462
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  • The terminology is quite idiosyncratic: the concept of "ww" (the way the world is, page 245) is simply that of possible world: "a proposition is a claim about the world—it represents things as being thus and so. It is true if the world is the way the proposition represents the world to be—if things are thus and so—and otherwise it is false. Implicit in this picture are the ideas that there are (in principle) different ways the world could be, and that a proposition gets a truth value when confronted with such a way." – Mauro ALLEGRANZA Nov 10 '23 at 06:49
  • Applied to "standard" first order logic, there is only one "world": the actual one. We use different domains (natural numbers, point in space) and functions and relations on them to define an interpretation (what the author calls "translation") to give meaning to non-logical symbol. – Mauro ALLEGRANZA Nov 10 '23 at 06:51
  • Using an interpretation and valuation for free variables, we get (page 204) the usual definition of: "true in $\mathfrak M$". "There are eight types of closed wff [...]. So we have eight rules telling us how the truth values of closed wffs are determined relative to a given model." – Mauro ALLEGRANZA Nov 10 '23 at 06:52
  • Also valuation is used differenttly from usual (see Interpretation, variable assignment and valuation). But basically the idea are the usual one for predicate logic semantics (page 248): "wffs have many models, in some of which they are true and in others false, but once we fix on a particular model, it is natural to think of a wff as saying something about the things in the domain of this model and as being simply true if it is true relative to the chosen model." – Mauro ALLEGRANZA Nov 10 '23 at 07:25
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    But IMO, the key-concept is that of "unqualified (common sense) notion of truth" vs "formal notion of truth in a model".See page 249: "Once one of them [models] has been singled out as the intended model, it is then natural to speak of a wff as simply “true” (as opposed to “true in such and such a model”) if it is true in the intended model." – Mauro ALLEGRANZA Nov 10 '23 at 07:29
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    Thank you @MauroALLEGRANZA, I think your last point is correct. I actually emailed Smith to ask for clarification, and he very kindly replied as follows: "You are right that the notion of truth on a model is a relative or qualified notion, but the thought is that when we pick out one special model (the actual one, or the intended one), we get an absolute or unqualified notion: truth relative to that model." I think this is what you are referring to in your last point as well. – user51462 Nov 10 '23 at 11:59
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    Yes thanks; consider that "intended model" is not a formal definition. The intended model of natural numbers is exactly the "common sense" structure of numbers we have learned at school (see also Intended Interpretation). In this sense, we consider e.g. the usual laws of arithmetic, like $n+m=m+n$, as true, without "qualification". – Mauro ALLEGRANZA Nov 10 '23 at 13:02

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