Semantic Models and Reality
Speaking on semantic models as mediators I extend the standard terminology and, what is perhaps more important, embed it into philosophy. The reason is that I consider the points (a)-(c) provide hints for a philosophical enterprise consisting formulating and defending SR.
The main issue concerns the thesis (a) reformulated in the following way[106]:(a*) Semantic models as representations of the world are not directly accessible from theories.
Using another terminology (see above), one might say that given T, its intended model M, that is, assumed to be a representation of the real world, is not directly derivable from T. In other words, the formal object of T does not entirely determine its material object. Hence, something must be add to STT in order to show in which circumstances semantic models of theories represent the real world.
(GM) suggests that M(T) and T are equivalent. However, if the issue concerns SR, it is more convenient to begin with T and its semantics. The valuation function V correlates expressions with their references in M(T). This machinery is too weak in order to capture the relation between M(T) and the world (W for brevity).[107] In fact, provided that V is already given, the pair n = organizes the T- semantics. If M(T) is to be characterized as a mediator, n has no second element to be mediated by the model in question. Thus, n must be extend to the triple H' =. Since V acts from the T (more precisely, its language) to M(T), it does not reach W. In order to fill this gap, we need a link between W and M(T) assuring that the latter is a mediator connecting T and W-items. Consider the function V': W M(T) and its composition with V, that is, O = V' • V.[108] This
construction displays the required link between T, M(T) and empirical data derived from W and the role of M(T) as a mediator.
Informally speaking, V reaches the real world via t V'. The latter is not a valuation function in the strict model-theoretic sense. Consequently, although W is not a model of T, we can still say that M(T) semantically represents W. It is not a problem of semantics to demonstrate how V acts. For instance, mathematical modeling or simulation can be helpful for deciding whether and how a given linguistic item applies to a piece of W.In the ordinary practice of science and the daily life, V and V are mixed, because, due to ways of learning and using language, learning V automatically qualifies V. Hence, the material object of knowledge is usually identified with the formal one. Both are distinguished in special cases, for instance, searching new theories, considering psychic abnormality, comparing reality with fiction, etc. On the other hand, semantics provides tools for a general philosophical picture. Although V is not a valuation function in the strict sense O can play this role. Consequently, we can introduce the concept of empirical satisfaction and empirical truth. Consider the following theoretical sentence:
(ii) If p is a particle completing the Standard Model, it should have such and such properties.
This assertion is true in M(ii) in the semantic sense. Also we can say that the open formula
(iii) If x is a particle completing the Standard Model, it has such and such properties, is satisfied by a hypothetical particle p possessing the properties in question, provided that (iii) is not empty-satisfied, that is, because its antecedent is false. Before discovering the Higgs boson, it was not known whether p really existed or not. although V valued the variable x in (iii) by p. In other words, p existed in M(ii), but its reality in W was an open question. The Higgs boson was just identified as the particle p. Clearly, V does not justify of substituting x by the term ‘the Higgs boson'. On the other hand, O, if restricted to the considered case, allows the substitution operation in question.
Call O the empirical valuation function. We say:
(SDTe) A sentence A of a theory T is empirically true in a model M(T) under O if and only if A is satisfied by the empty sequence of objects.
The simplest example concerning satisfaction is as follows
(iv) An object o satisfies the formula Px under O if and only if O(x) = o and o e O(P). Although the ordinary parlance admits to say that if A is true, under O, it is true about W, more proper would be say that A is true in M(T) with respect to O. However, the former way of speaking hast its justification in fact that empirical valuations map W on M(T) and help in setting which semantic model is standard or intended with respect to data.[109]
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