Operative Definitions and Three Level Semantics

Frege conceived logic as ranging over a universal domain, while the algebraic tra­dition, starting from De Morgan and Boole and developed in classical Tarskian semantics, abandons the idea of a universal domain, and assumes from the start the possibility of different domains of interpretation of the signs of the formal system.

An interpretation of Agazzi’s point has been given by Bottani (1997) with an argument on the different uses of ordinary language and scientific language. Among the objects of chemistry we don’t find sheets of paper, but chemical prop­erties, belonging to parts analysed with specific procedures; therefore sentences like “this sheet of paper has such a chemical composition” are not sentences belonging to chemistry, but “sentences of ordinary language that represent sen­tences of Chemistry as referred to objects that belongs to the domain of quantifica­tion of ordinary language, but not to the quantificational domain of Chemistry”.

A predicate is an expression that has a class of objects as extension and a set of possible worlds as intension (or a function that maps a class for every possible world). But here we have the problem posed by Agazzi and Putnam and discussed in the previous paragraph: how can we define the “intended” interpretation? How to distinguish among different interpretations of isomorphic models, among differ­ent classes of objects? Intensional logic, although it gives more than extensional logic, cannot give a specification of predicates, because it can give only a discrimi­nation concerning the relations among predicates; and different interpretations may keep the same structural relations (as clearly presented by Quine with his per­mutations). If we follow this line of though we are bound to accept the well known consequences on the incommensurability problem: we cannot determine which of two theories better fits the reality (the intended model) through a description of the meanings (as intensions) of their predicates.

Agazzi’s proposal is to insert in the definition of the meaning of a predicate also the operative procedures linked to the use of the predicate in a scientific and experimental context. Operative procedures are therefore intended as belonging to the meaning and not as a mere methodological aspect of the theory.

From a logical point of view Agazzi (1976, 1979) does not assume a priori a domain of individuals, against the standard logical assumptions according to which (i) the individuals of the domain are “given” and (ii), while the relations are not always decidable, the domain is decidable (given any individual we may know if it belongs to the domain or not). But the undecidability of relations is one of the main sources of undecidability, and the source of semantic ambiguity. Agazzi’s alternative is that a model M, instead of a pair of a domain and a set of relations (M = (D, R)) is a quadruplet M = (S, O, R, P) where S is a set of Scientific Instruments, O a set of Operations, R a set of Results and P a set of Operative Predicates, and every predicate belonging to P is an element of S x O x R. Let us take an operative predicate as “electrically charged”, constituted by the operation “put x on the disk of the electroscope”, by the Scientific Instrument Gold-leaf elec­troscope, and from the possible Result “the gold leaves spread apart”. The question that apparently comes to mind is what can we refer to with the variable x, given that, by assumption, we have not used a fixed domain of interpretation. We have however on the one hand the universal discourse domain, that precedes any defini­tion of the objects of the theory, and on the other hand we have a theory that creates its own object on the ground of its specific definitions applied to things of which we may informally speak in everyday language (electrons or quarks included).

It is not difficult to see the import of such theory from the point of view of a general theory of meaning. The main idea is the necessity of considering some­thing different from intension, but at the same time something which interact with intensions, without loosing the expressive power of intensional logics. Here the three level semantics which we have hinted at in the first paragraph becomes more complex, giving space to an element of procedural aspect which cannot eas­ily be reduced either to sense or to reference. Here however (Agazzi 2012a, 2014: Sect. 4.1) oscillates between a standard tripartition in kinds of expressions (proper names, predicates, sentences) and their sense and reference, and a different kind of analysis where predicates can be considered to have a reference in a “limiting sense”, while it is sensible to speak of the extension of a predicate. Agazzi (2012a: 12) remarks that “reference and extension are related notions, but are not identical, so that a Fregean and a Carnapian semantics are not really equivalent”. Making these further distinctions, he actually follows the suggestion given by Frege in a letter to Husserl, where Frege explicitly claims that for the predicates we need a tripartite division in sense, reference and extension.

Although new suggestions are presented in the general semiotic framework in Agazzi 2014 (especially the distinction between encoding and exemplifying at page 190-191), something seems still to be missing: the specific place of the oper­ational criteria of referentiality, that are implicitly connected with the old discus­sion on the operational meaning. Here a possible integration of the general structure provided in Agazzi 2014 might impinge upon the relation between the sense and the reference or denotation of a predicate.

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