Methodological Foundations

Language Structures and the Language of Science

All kinds of languages can be characterized in syntactic (i.e., logical) terms or semantic terms. The determinant of the differentiation between commonsense language and scientific language is not so much the syntactic aspect; it is also, and above all, the semantic aspect, i.e., the modes of language interpretation (given that every language is interpreted).

In short, commonsense language has an experiential-type interpretation, as it considers phenomena according to a paradigm based on individual or group experiences (subjective opinions); instead, scientific language has an experimental-type interpretation, in principle.

Needless to say, controlled (laboratory) experiments imply a specific syntax, strictly comparable with the experimental analysis.

Giulio Bolacchi

Language as a Semantical and Syntactical System

Each language consists of a set of signs, ordered according to specific syntactic relations and interpreted in semantic terms. The concept of semantic interpretation of a particular syntax ultimately refers to the modes through which linguistic stimuli translate (or should translate) into sets of behaviors, which are corresponding responses to the signs by the interpreters towards whom signs act as stimuli.

It is important to keep the set of syntactic signs separate from the various sets of semantic signs: in fact, several semantic interpretations can correspond to each set of syntactic signs. Semantic interpretations can be many in the language domain referable to common sense. Let us take into consideration, for instance, the literary works of the same author or different authors; we find they ultimately use the same syntactical patterns, namely the same more or less basic syntactic relations (in short, the same propositional connectives, more or less diversified). Here, the semantic interpretation refers to the life experience of every individual author.

1. Internal Perspective and External Perspective

Semantics can be considered from two perspectives of analysis. The first perspective concerns the relations between signs within a given language and refers to the syntactic and semantic rules that characterize the conditions under which the linguistic terms are meaningful; in this sense, we can speak of semantic analysis internal to the language. The second perspective concerns the modes of using signs, i.e., the relations between signs and “interpreters” who use signs in a communication process; in this sense, we can speak of an analysis external to the language.

The internal perspective has been rigorously explained by R. Carnap, who delved deeply into the syntactic characterization and semantic interpretation of linguistic terms.

The external perspective has been analyzed by C. Morris, who first proposed a definition of sign in behavioristic terms. Although B. F. Skinner did not take into consideration, in his analysis of verbal behavior, Morris’s explanation of the sign-process, the latter could be reformulated within the current science of behavior.

(cf. Metodologia delle scienze sociali [Methodology of the social sciences], p. 6)

2. Rules of Formation, Designation, Truth

«An explanation of the concept of “system” rests, on the semantic level, on the rules of formation, rules of designation, and rules of truth, which determine the form of the sentences admitted within a system (sentence in S), the descriptive constants of the system (designation in S) and the truth-conditions for the sentences of the system (true in S), respectively.

The definition of the concept of “true in S” is the cornerstone of the whole system, not to determine the truth-values of the sentences of the system (which are determined through semantic rules of a different type), but to determine the truth-conditions of the statements of the system. In this sense, the rules of formation and the rules of designation constitute a preliminary characterization for the rules of truth; through these three types of rules, in particular the latter, new semantic concepts can be defined.

The rules of designation (which found the concept of “system” on the semantic level) establish mere conventions in the form of a definition of “designation in S,” enumerating the cases in which the relation of designation is to hold.[5] The term “designation” can also be used for compound expressions and sentences, in which case the rules of designation define by enumeration the preliminary term “direct designation”; through the latter, the more general term “designation” is defined recursively.[6]» IT

(Metodologia delle scienze sociali [Methodology of the social sciences], pp. 94)

3. Semantical System

«The rules of designation concern particularly those non-logical constants determining which attributes (properties and relations) are designated by predicates and which individuals are designated by individual expressions.[7] These rules of designation translate into (a) the interpretation of predicates and hence the determination of their level of abstraction; (b) the choice and specification of the degrees (number of arguments) of primitive predicates; (c) the interpretation of single expressions (intended as individual constants).

Alongside the rules of formation [8] and designation of the non-logical constants (predicates and individual expressions), an explanation of the concept of “system” must take into account further semantic rules that determine the meanings of the sentences in the system, with particular reference to the concepts of “truth “or” falsity ” […].»

«A rule of truth for atomic sentences is given by the consideration that an atomic sentence in S (consisting of a predicate followed by an individual constant) is true if and only if the individual designated by that individual constant has the property designated by that predicate.[17] […]

The rules of truth together constitute a recursive definition for “true in S” [20] because they determine, in combination with the rules of designation, for every sentence in S, a necessary and sufficient condition of its truth. Therefore, just as truth tables establish a truth condition for molecular sentences, rules of truth determine a truth condition for any given sentence i in S, although they do not determine the truth value of the sentence iIT

(Metodologia delle scienze sociali [Methodology of the social sciences], pp. 95, 96-97)

4. State-Descriptions and Ranges

«A state-description for a system S must determine, for every individual [29] designated in S and for each property designated by a primitive predicate in S, whether or not that individual has that property; and analogously for relations.[30] This means that a class of sentences in S, which for each atomic sentence contains either that sentence or its negation but not both, and no other sentences, is called a state description in S, because it contains a complete description of a possible state of the universe of individuals with respect to all properties [31] and relations designed by the predicates of the system. In this way, it is possible to formulate semantic rules that determine for every sentence in S whether or not it holds in a given state-description. […]

For state-descriptions to describe possible states, the interpretation of the individual constants and the primitive predicates must fulfill the requirement of logical independence. Where a system also contains inductive (probabilistic) logical structures, it must furthermore fulfill the requirement of completeness; namely, it must be sufficient for expressing all qualitative attributes occurring in the given universe.[33] The rules of ranges provide, together with the rules of designations for the predicates and the individual constants, an interpretation for all sentences of the system, since knowing the meaning of a sentence means knowing in which of the possible cases it holds and in which does not.[34]» IT

(Metodologia delle scienze sociali [Methodology of the social sciences], pp. 98-99)

5. Intension and Extension of Linguistic Terms

«The concepts of “intension” and “extension” explicate two fundamental semantic perspectives. The first one consists in determining the meaning of expressions, i.e., identifying the factors which univocally characterize the meaning of linguistic terms. The second consists in determining the semantic domain of validity of expressions, i.e., identifying the conditions of applicability of linguistic terms to the possible cases described within a given language.

Both determining the meaning of expressions and identifying their semantic domain of validity translate into a set of semantic rules; therefore, they pertain to a consideration of linguistic structures that may be called internal, in contradistinction to an external consideration of linguistic structures in behavioristic terms. Thus, using purely semantic rules and hence the internal language methodological perspective, the predicates, the phrases, and the individual expressions can be characterized not only in intentional terms as properties,[50] propositions,[51] and individual concepts;[52] they can also be characterized, respectively, as the classes of all entities [53] to which a given predicate can apply within a linguistic system, as values of truth or falsity (necessary and sufficient conditions for the truth of sentences in a linguistic system),[54] and as individuals (determined by the designation rules) to which the individual expressions refer.

Therefore, determining the intension and the extension of terms does not require referring to or investigating empirical situations;[55] intension and extension must be formulated using only the semantic rules of the language and, in the strict sense, belong to the metalanguage (in its standard or neutral formulation).[56]» IT

(Metodologia delle scienze sociali [Methodology of the social sciences], pp. 49-50)

6. The Semantics of Formal Systems

«Given that in formal logical languages (logical semantics) the signs characterized by the rules of designation are used to indicate every possible property or relation or any individual concept, the rules of designation do not determine, in this case, a particular interpretation for the individual predicates and expressions, because the choice of such an interpretation is irrelevant for both deductive and inductive logic (considered from a semantic point of view). A system like that – for Carnap – would not be, strictly speaking, a semantical system, but the skeleton, or rather the structure of the system.[65] It goes without saying that, with a view to concrete application, the system should be supplemented by choosing a finite number of predicates, specifying their degrees and giving an interpretation of these predicates by (semantical) rules of designation; the latter operation should be repeated for the individual expressions.[66] Such interpretations should satisfy the requirement of logical independence.[67] Thus, Carnap argues that there would be no need to lay down rules of designation with regard to the semantical systems of deductive and inductive logic.

However, it seems more accurate to state that such rules of designation should be laid down in formal semantic systems, too,[68] and hence in all systems of deductive and inductive logic commonly used by logicians as object languages [69] […]. Therefore, it cannot be said that the signs of formal languages do not require any rules of designation; the very fact that they are considered and used as signs postulates they are necessarily relevant within a semantic process […].» IT

(Metodologia delle scienze sociali [Metodology of the social sciences], pp. 51-52)

7. Language Structures

«The concept of “structure” must firstly be explicated in terms of predicates (properties and relations) and individual constants. The reference to predicates and individual constants results from the fact that the term “structure,” taken as an explicandum, is used to indicate the fundamental properties (i.e., the primary elements or essential components) of a given linguistic context.

The explication of “structure” through the concepts of “predicate” and “individual constant,” taken as basic elements, does not exhaust the domain of application of the structure. The latter can also be considered in terms of relations connecting the basic elements. These interconnections characterize the structure as a system.

The interconnections between the basic elements of the structure need to be further specified, because not all the interconnections between basic elements identify the system’s structure. The latter must be characterized with reference to the domain of application of the system’s predicates, which determines their level of abstraction. Only the fundamental postulates of a system with the highest level of abstraction characterize the system from a structural perspective. The structure is thus ultimately explicated in terms of the system’s predicates at the highest level of abstraction (primitive predicates).

From a semantic perspective, the concept of “structure” so defined is directly characterized by the rules of formation, designation, and truth of the system to which the structure refers.» IT

(Metodologia delle scienze sociali [Methodology of the social sciences], p. 103)

8. Relations between Language Structures

«The fundamental postulates influence the relations between the various structures, such that a widening or a narrowing of these postulates reflects directly on the domain of application of the structures and determines a widening or a narrowing of the system to which they belong.

Given that the introduction and construction of new language structures are obtained through the determination of higher-level predicates qualifying the individual expressions belonging to the new structures (constants and new types of variables of which the constants are values), the narrowing or widening of the domain of the structures identifies with the narrowing or widening of the domain of application of the fundamental predicates of the system and the primitive individual constants qualified by the predicates (which are values of the newly introduced variables).»

«Given that the relations between different structures can be configured with reference to the widening or narrowing of the domain of application of the system, depending on the introduction of predicates that have a broader or narrower domain of application than the domain of the pre-existing fundamental postulates, a comparison is possible between the domains of application of predicates (and consequently of the related individual constants or variables), which are thus characterized by different levels of abstraction.

The concept of “structure” of a particular system must therefore be explicated with reference to the domain of application of the predicates belonging to the system. However, further characterization is needed: indeed, given that predicates with different domains of application exist within a system, the structure of the system cannot be characterized by all the predicates of the system but only by the primitive predicates that having the broadest domain of application, that is, the highest level of abstraction within the system.» IT

(Metodologia delle scienze sociali [Metodology of the social sciences], pp. 92-93)

9. Abstraction and Semantic Homogeneity

«Abstraction is a methodological concept related to the introduction of different-level predicates in pre-existing languages, which can also be concretely identified languages, i.e., languages containing descriptive predicates concerning things at a particular space-time position.[7] Abstraction reflects the distinction between particularized phenomena, individualized in all their multiple facets within the complex experience in which they are included, and phenomena considered in terms of one or more specific aspects that are selected and isolated from the context of the total experience.»

«Both the level of abstraction and the identity in the level of abstraction are elements that must necessarily be established conventionally through the rules of designation of the linguistic system referred to, which interpret the individual constants and the primitive predicates of the system. It is precisely this interpretation that characterizes the level of abstraction at the very moment when the meanings of the terms are established.[10] Related to this interpretation is what we could name the semantic homogeneity of the system, which characterizes the identity in the level of abstraction.[11]

Therefore, the distinction between abstract language and concrete language, as well as the corresponding distinction between observational language and theoretical language, cannot be drawn so sharply. We can only speak of different linguistic levels related to different levels of abstraction of individual constants and predicates interpreted in the system. On the other hand, any linguistic level can be characterized as a theoretical level with respect to the lower levels of abstraction to which it refers, and in this sense, considered as a whole, constitutes a theoretical interpretation that can be characterized either in purely semantic terms or in syntactic terms (axiomatic); in the latter case, the formalization of the system is made explicit to the fullest extent.» IT

(Strutture teoriche e scienze sociali [Theoretical structures and social sciences], pp. 47; 48-49)

10. Truth-Conditions and Truth-Values

«The problem of the empirical or observational verifiability of structures works out into the problem of the distinction between a language with observational predicates and a language with non-observational predicates (theoretical or abstract). This distinction refers to a characterization of linguistic structures different from that based on abstraction levels. The problem of abstraction is a question internal to language; instead, the problem of observability, to which the problem of empirical verifiability of structures relates, is a question external to language that works out into an analysis aimed at identifying the denotatum of the linguistic expressions.

It means that, whereas the level of abstraction of language structures can be characterized through the semantic rules alone, the matter of observability and verifiability of linguistic structures postulate the recourse to extra-linguistic elements. The first problem can be formalized in terms of truth conditions, whereas the second concerns the existence or otherwise of truth valuesIT

(Metodologia delle scienze sociali [Methodology of the social sciences], pp. 83-84)

11. Theoretical System

«A theory stands as a language expressed in formal and abstract terms, i.e., a set of terms related to form a system (structure), which have a conventional domain of application so large that it can agree with any state-descriptions within a predetermined linguistic universe. Therefore, we can say that the theoretical terms are such as to a predetermined universe, when they hold in all state-descriptions belonging to the latter.[20]

This concept is made even more explicit where the theoretical language is expressed in axiomatic terms.» IT

(Strutture teoriche e scienze sociali [Theoretical structures and social sciences], p. 58)

12. Axiomatic System

«Any axiomatic system that is not formal or semiformal has two fundamental facets: the first is a logical one, and consists in the set of deductive relations considered from the perspective of the syntactical structure (descriptive syntax); the second is substantial and consists in the contents or meanings that found the semantic interpretation of formal relations (descriptive semantics).[6]

The deductive system’s postulates (or axioms) and theorems fall within this second facet and constitute its semantical interpretation. They possess a very high level of abstraction; namely, they represent the characterizing element of the theoretical structure of the system itself.» IT

(Strutture teoriche e scienze sociali [Theoretical structures and social sciences], pp. 45)

The principle of conventionality

«Two fundamental features characterize language structures:

(a) each sign has a constant and common signification to a given interpreter-family, being producible by the members of the latter (comsign) even under conditions different from those under which it was introduced originally (plurisituationality);[45]

(b) stable and systematic relations of combination between signs must exist, determined with reference to the different language structures.

From point (a), it necessarily follows the conventionality of complex sign processes (i.e., language structures or semantical systems). The principle of conventionality,[46] which establishes that the construction of a language and the choice of its particular features are a matter of convention, was first applied to axiomatic systems. However, every formalized axiomatic system, not to be meaningless, must rest directly on a corresponding semantical system from which an adequate logical interpretation of the terms (in particular, the rules of formation, the rules of designation, and the rules of deduction or transformation) can be derived.[47] It follows that the conventionality of a calculus or axiom system is strictly related to the semantical system interpreting its formal rules.

Therefore, the principle of conventionality properly applies to the semantical systems with regard to the rules of formation and deduction, and the syntactical (axiomatized) systems with regard to the primitive axiomatic terms. When syntactical systems have both uninterpreted terms and logical terms with their customary interpretation, they are called semiformal or semi-interpreted.[48]» IT

[45] C. Morris, Signs, language and behavior, pp. 32-42 (Prentice-Hall, 1947);
[46] R. Carnap, Introduction to semantics, p. 247;
[47] R. Carnap, Introduction to semantics, pp. 155 ff., Formalization of logic, pp. 69 ff.;
[48] R. Carnap, Logical foundations of probability, pp. 15-16.

(Metodologia delle scienze sociali [Methodology of the social sciences], p. 48)

The main semantical concepts in object language and metalanguage

«In general, it can be said that it is appropriate to distinguish (a) sounds and marks from (b) symbols (linguistic signs or verbal symbols), which are recognizable patterns of marks or sounds used for purposes of expression and communication. The latter must be distinguished from (c) signs (in the strict sense) or linguistic expressions, qualified by the association of a symbol (or linguistic sign) with a fixed and determined meaning.[15]. […]

We can, therefore, isolate the following terms or semantic figures, following Lewis’s schema [16]:

1) Terms of the object language.

2) Linguistic expressions, which belong to the not strictly semantical part of the metalanguage.

The nature of the nonsemantical part of the metalanguage is explained by Carnap,[17] who speaks of translation (version) of the expressions of the object language within the metalanguage.[18] It means these expressions have an extralinguistic, nonsemantical denotatum, as they are signs that do not denote other signs (as is the case in the metalanguage for the names of signs of an object language) but extralinguistic elements, even if they belong in a broad sense to the metalanguage (which is, by definition, a discourse about other signs). Although these terms are used within the metalanguage and belong to it, they do not relate to another object language; they belong to the nonsemantical part of the metalanguage, namely that part within which the sentences and terms of the object language, at large, can be translated. […]

3) Linguistic symbols, which also belong to the nonsemantical part of the metalanguage and are used in the (conventional) rules of designation pertaining to the object language.

It is, therefore, necessary to distinguish between (a) a sign-vehicle, considered in terms of the mere physical features that characterize it, (b) a symbol as an element or object physically identified and used for the purposes of expression and communication, and (c) a sign, as a symbol qualified by an interpretant and a denotatum, and hence by a certain range of meanings (field of meaningful applications) toward certain interpreters or within a particular language system.[21] […]

4) Names of the object language’s linguistic symbols within the metalanguage referring to that language.

5) Name of the linguistic expression (sign) within the metalanguage.

The intension of the name is given by the linguistic expression to which it refers, considered in its entirety as a symbol qualified by an interpretant (and denotatum) determining the field of meaning (the linguistic field of meanings, on an extra-linguistic basis), i.e., the intension (and extension). In short, the name’s intension is given by the set of conditions such that the object which satisfies them is a denotatum of the name (in the case in question, a sign). The extension is given, instead, by the sign existing within the object language. Both extension and intension have an extra-linguistic counterpart (denotatum and interpretant) and a metalinguistic counterpart, too, consisting of the terms of the metalanguage, which refer to them within the latter.[23]

The (conventional) rules of designation characterize, within a given language, the association of a symbol (not the name of a symbol) with a fixed meaning.[24] […] The symbol belongs only to the object language, whereas the name of the symbol belongs to the metalanguage whose object is terms and symbols of the language to which it refers.[27]» IT

[15] C.I. Lewis, An analysis of knowledge and valuation, p. 73 ff.

[16] C.I. Lewis, An analysis of knowledge and valuation, p. 100 ff.

[17] R. Carnap, Meaning and necessity, pp. 24, 25, 94, 155; Introduction to semantics, p. 88 ff.

[18] R. Carnap, Introduction to semantics, p. 53; Meaning and necessity, pp. 66, 161. On the extra-linguistic, nonsemantical denotatum of linguistic expressions belonging to the not strictly semantical part of the metalanguage, cf. Carnap, Meaning and necessity, pp. 93, 107.

[21] C.I. Lewis (An analysis of knowledge and valuation, pp. 73, 74, 101) distinguishes between marks and sounds (sign-vehicle), symbols (linguistic sign or verbal symbol), words or expressions, also postulating a further distinction between the expression and the meaning which is expressed, and defining “a word as an elementary expression of a meaning by a symbol.” (p. 73).

[23] The above concepts can be combined in the following ways:
(1) Symbol-Symbol: “A” is an abbreviation for “BC.”
(2) Symbol-Expression: “A” symbolizes the same meaning of ‘BC’ (in that case, “A” denotes A; ‘BC’ denotes the intension of A. There is no equivalence here but the fundamental semantical relation).
(3) Expression-Expression: ‘A’ has the same meaning as ‘BC.’
Based on these three combinations, other combinations can be obtained, one of which is relevant to the semantical concept of truth. It should be noted, however, that the translation of an expression in the metalanguage always refers to that expression’s intension (Lewis, An analysis of knowledge and valuation, pp. 98-99, 100-101; Carnap, Meaning and necessity, pp. 111-112).

[24] However, these rules must be expressed using the name of the symbol. It is, therefore, necessary to distinguish between conventional rules of designation and relation of designation, which qualifies the relation between sign and significatum, in the sense that each sign signifies its own significatum.
The rules of range (R. Carnap, Logical foundations of probability, pp. 70 ff., 78-80) that determine the extension of the terms within a given language presuppose necessarily the rules of designation and, hence, the intensional characterization of the signs (on the intension as primary characterization of a sign, cf. R. Carnap, Meaning and necessity, pp. 108, 112, 157, 203), though they do not have linguistic symbols as their object, rather signs or expressions in the strict sense, i.e., symbols already qualified in terms of intension.

[27] The names of the terms of the object language can be expressed in the metalanguage in two different manners: by a special notation consisting in enclosing the symbol of the object language in quotation marks (C.I. Lewis, An analysis of knowledge and valuation, pp. 100, 104); or by using new terms (linguistic expressions) of the metalanguage whose denotata are the terms of the object language. Either way, it is necessary to lay down appropriate rules; in the second case, they are rules of designation in all respects. The latter is Carnap’s method (Introduction to semantics, pp. 19-21, 32-33, 50 ff.; Logical foundations of probability, p. 55 ff.).
The distinction between the terms of the object language […] and the names of the terms of the object language is important because it allows, besides the definition of the concept of “truth” and the other semantical concepts, to distinguish between a purely semantical part and a non-semantical part within the metalanguage (Carnap, Meaning and necessity, pp. 24, 66, 94, 155, 161). The non-semantical part, which is usually indicated as the translation of the terms of the object language into metalinguistic terms […], expresses the intension of the terms of the object language. In fact, if these metalinguistic terms did not express an intension, but a translation of a sentence of the object language, then a sentence with the same intension and extension as the translated one would necessarily exist within the metalanguage. In that case, the translation could not belong to the metalanguage as such, nor its non-semantical part, but it would necessarily belong to another object language, and the two expressions would be synonymous, as they would have the same intension and extension (C.I. Lewis, An analysis of knowledge and valuation, p. 101 […]). IT

(Metodologia delle scienze sociali [Methodology of the social sciences], pp. 41-43, 59-61)

Intension and extension in the metalanguage

«Carnap explicates the two concepts of “intension” and “extension” of linguistic terms and states that the intension is to be considered as the primary semantical factor because, when it is given, it also uniquely determines the corresponding extension [39] […].

That said, there can be no question that, within a given metalanguage, it is possible to speak separately about the intentions and extensions of the terms of the corresponding object language (for example, about classes and properties in relation to predicates). The method of intension and extension is an alternative to the customary name-relation method, according to which the expression must be the name of only one of the semantic factors: the name-relation method leads indeed to unnecessary duplication of expressions in the object language and, in some cases, to an overt antinomy.[41]

The formulations in terms of intension and extension in the metalanguage do not entail any duplication of entities, because it is only a semantical distinction between two modes of speech, referring to the two semantical perspectives according to which it is possible to consider a sign within the language to which it belongs (duplication of expressions). However, although the method of extension and intension eliminates the distinction between two types of entities by transforming it into a distinction between two modes of speech, it does not eliminate the use of two expressions within the metalanguage, one for the intension, the other for the extension. Nevertheless, it is possible to construct a neutral metalanguage with only one kind of expression, within which even the apparent duplication of entities thus disappears. […]

The fact that it is possible to construct such a metalanguage containing contextual definitions of non-neutral terms (intensions and extensions)–namely, that it is possible to reintroduce the latter by contextual definitions–shows that the neutral method can still account for those semantical distinctions, making it evident that they do not presuppose a duplication of entities, but only a dual perspective that can be used to consider linguistic signs. Since the neutral metalanguage and the non-neutral metalanguage cover the same domain of application, the choice between them is only a matter of practical preference: the neutral formulation is simpler and more useful for a precise clarification of symbolic languages.[44]» IT

[39] R. Carnap, Meaning and necessity, p. 108.
[41] R. Carnap, Meaning and necessity, pp. 96 ff., 133-135, 203.
[44] R. Carnap, Meaning and necessity, pp. 157 ff., 168.

(Metodologia delle scienze sociali [Metodology of the social sciences], pp. 46-47)

The Language of Science

In the language of science, the multiplicity of syntactical structures tends (or should tend) to relate to the mathematical syntax. They tend to relate, in general terms, to that specific mathematical syntax characterized by the function (a particular correspondence such that each element of the domain of the function is associated with one and only one element of the codomain), which is strictly comparable with the controlled (laboratory) experiment. The semantical interpretation varies according to the type of experiments characterizing each specific branch of science. If it were not so, we could not distinguish between the various branches of science.

The correspondence between (mathematical) syntax and controlled (laboratory) experiments can entail more or less specific syntactical and semantical differentiations, depending on what sciences are considered. There are many distinctions in this regard: it is one thing to work in physics, another to work between physics and biology, or between physics, biology, and behavior science. On this matter, many methodological aspects of biology, as compared to physics, remain to be clarified; the same can be said of the relations between behavioral and neurobiological sciences. Here, the central point is that methodological developments have had as their object mainly physical phenomena and only secondarily biological phenomena.

1. Syntax and Semantics of Scientific Language

«Each science is a linguistic construction, quantitative in principle: (1) whose logical syntax is characterized, in the most abstract sense, by that specific subset of the Cartesian product (AxB), where to each first element of any ordered pair there corresponds one and only one second element (function); (2) whose semantic interpretation is susceptible of different levels of abstraction, which correspond to the order deriving from the (transitive) relation of inclusion among subsets defined by specific properties (two-place predicates).»

(A new paradigm for the integration of the social sciences, p. 317, 350)

2. Mathematical Function and Controlled Experiment

The scientific language «is characterized by the syntax of mathematical functions and by the strict correspondence between these functions and controlled (laboratory) experiments, which gives them a specific semantic interpretation.»

Galileo was the first to use this new perspective based on “sense-experiences and necessary demonstrations.”

«With Galileo, scientific knowledge gets rid of the Aristotelian concept of essence, with its lack of explanatory power, and replaces it with the mathematical concept of function. In this way, Galileo realized a methodological revolution that reveals a world of natural events considered not singly but as ordered pairs that satisfy the known properties of functions.»

«The function is the basic syntactic structure of scientific experimental language, and the experiment is the semantic interpretation of the function. Obviously, the function can be interpreted in a statistical way, too. The controlled experiment is the closed or isolated system by which to define semantically (by the rules of designation) the domain of application of ordered pairs in a functional relationship; or briefly, the closed or isolated system by which to assign an unambiguous semantic interpretation to a function.»

(On “social sciences” and science, pp. 470-471, 468)

3. Delimiting the Field of Knowledge

«The function denotes the system that delimits the field of knowledge (laboratory experiment) and/or the field of operation (technology), whereas parameters denote all those factors external to the system. This does not mean that relations (functions) between the system and the outside do not exist. It only means that these relations are not taken into consideration; i.e., they are assumed to be non-influential with reference to the given system. If the relations between system and outside (external system) are taken into consideration, they extend the boundaries of the system and define new partitions within the system (i.e., functions with a lower abstraction level compared to the basic functions).»

(A new paradigm for the integration of the social sciences, p. 318)

4. Intensional and Extensional Perspectives

«The syntax and the semantics of a specific (and formalized) scientific language express a paradigm of reference that identifies the domain of application of this language, i.e., the properties characterizing each set (namely, each element of any set) belonging to the given language. In this sense, each property designates a subset of this language, i.e., the subset of elements which have that property. Therefore both the intensional perspective of properties (expressing the operational fields to which science refers: the properties) and the extensional perspective of the quantitative concepts according to which the properties are explicated are equally important for the language of science.»

(A new paradigm for the integration of the social sciences, pp. 318-319)

5. Semantic Homogeneity and Rules of Designation

«The other requirement, i.e., the semantic homogeneity of all variables of the language of science, can be expressed in more abstract terms pointing out that all the subsets of variables of the language of science must belong to a defined superset.

Moreover, the semantic homogeneity of the variables belonging to a given scientific language implies that the meaning of all the predicates (properties) that define the sets of functions (which are ordered by the relation of set inclusion) can be (in principle) brought back to the experimental analysis. Thus, predicates and functions correspond to (extra-linguistic) experimental situations to which they are strictly related by the semantic rules of designation formulated within a specific metalanguage. Within the language of science, the rules of designation are made unambiguous by measurement systems. When an experiment cannot be directly carried out (owing to technical difficulties concerning the number or the type of variables), the correspondence between semantic predicates and experimental situations is replaced by statistical analysis, which can also work in the strictly experimental context.

Therefore, semantic homogeneity is chiefly due to the strict correspondence between the sets of functions of the language of science and the sets of experimental variables at any given level of abstraction defined by the relation of set inclusion, and this strict correspondence is guaranteed by the fact that both the elements belonging to the domain and the elements belonging to the range of the functions are defined using rules of designation founded on experimental operations or measurements of the same type. In this way, the rules of designation delimit the domain of application of science (and of specific sciences). This implies that every new partition hypothesis that is advanced within the language of science has to be brought back to specific experimental variables. No matter how, hypotheses on functions can be advanced, but they do not come to belong to the language of science until they correspond (by the rules of designation) to experimental (or statistical) variables.»

(On “social sciences” and science, pp. 473-474)

6. Equivalence Classes

«Obviously, there are other relations besides this one, among which is the equivalence relation that establishes the important logical concept of equivalence classes with respect to a specific partition of a given set.

The partition of S originates an equivalence relation in S of the type “from the same subset as” (in extensional terms) or “has the same property as” (in intensional terms concerning the property that defines the subset). The equivalence relation is between all pairs of elements in the same subset, so every subset of the partition is an equivalence class.»

The taxonomic language of science is based on partitions.

(On “social sciences” and science, p. 473)

7. Levels of Abstraction

«One requirement is that all the predicates (properties) belonging to the language of science should be ordered by the relation of set inclusion. The language of a given science, in its more abstract meaning, is a set of subsets of functions. This ordering is normally a partial ordering, because the sets are not each other comparable at all by the relation “is a proper subset of.” […]

The relation of inclusion is fundamental for the axiomatization of scientific language because it determines the level of abstraction of the statements belonging to this language.»

(On “social sciences” and science, p. 473)

«The order of the abstraction levels, which expresses the relation of inclusion from the most abstract (basic) to the less abstract (technological) sets of functions, is a partial order, as the (sub)sets of the same level are not comparable by the relation of inclusion. Each level of abstraction is defined by one or more proper subsets of functions, i.e., by one or more partitions of pairwise-disjoint subsets of functions corresponding to the equivalence classes that comprise functions of the same type (i.e., equivalent as to their experimental semantic interpretation).»

(A new paradigm for the integration of the social sciences, pp. 317-318; 350)

8. Theory and Technology

«The levels of abstraction can be made explicit or not in accordance with the state of progress of the research. They take their stand in a range from the highest levels of abstraction (which are expressed by the sets defined by the basic relations of a given scientific language) to lower levels (which are characterized by sets defined by a progressively larger number of two-place predicates). […]

Functions belonging to the highest levels of abstraction are poorer in semantic interpretations. Functions belonging to those sets characterized by lower levels of abstraction specify the theoretical predicates, i.e., they enrich them with more analytic meanings denoting situations (environments) where the multiplicity of variables and parameters tends toward individualization.

The case of technology points this out. Technological predicates define sets that express not only theoretical invariants (basic functions), but a much larger number of (specific) functions as well. These latter functions, which define the concrete environment where basic functions can work, specify the domains of application of the abstract basic functions. In this sense, the technological environment is richer (in semantic interpretations) compared to the poorer environment where the abstract basic functions take their place.»

(A new paradigm for the integration of the social sciences, p. 317)

9. Intersubjectiveness of Scientific Language

«There is a reason for scientific discourse in that it can be constructed as an intersubjective discourse, i.e., as a discourse whose meaningfulness cannot be challenged in principle, because founded on experimental predicates characterized by the methodological predicate of repeatability and related to each other by the function […].

Intersubjectiveness (intersubjective verifiability) is, thus, the only reference criterion for the widening of scientific discourse (which differs in this respect from ideological or evaluative discourses in the broad sense, which are subjective by definition). The widening of scientific discourse can be achieved: (1) by introducing new experimental predicates; (2) by introducing restrictive predicates that, from a methodological viewpoint, enlarge the domain of application of a given structure, and from a syntactical viewpoint, specify the more general predicates that qualify the structure (in this sense they can be called limiting predicates); (3) by introducing predicates of a more abstract level than a set of predicates that characterize a pre-existing structure, in such a way that the new structure can explicate the phenomena explicated within the less abstract structure, too, as a specification of the more general predicates of the new (more abstract) structure (the less abstract structure still holds within the postulates which qualify it); (4) by introducing new predicates through nominal definitions.» IT

(L’analisi scientifica del comportamento di scelta [The scientific analysis of choice behavior], II, pp. 11-12)

10. Axiomatization and the Cumulative Character of Science

«Scientific knowledge is by definition cumulative. This means that scientific explications that come in sequence, if they are scientific, cannot be contradicted within the language of science because they are guaranteed by the strict compatibility with the rules (constraints) that characterize this language. The explicative language of science, being in principle susceptible to axiomatization [3], continually deepens and widens in such a way that the new scientific knowledge be strictly consistent with previously acquired results, and every scientific explication can be considered to be a proper subset of a more abstract explication.

The “progress” of science lies in this very process of continuous reformulation of its language […], which still does not alter the fundamental predicates that define sets in the language of science. The reformulation of explications and theories to make them consistent with new (more abstract or more specific) ones, which arise from the continuous deepening and widening of the domains of application of the language of science, does not alter the scientific character of the pre-existent explications and theories, typical of previous phases of the process of scientific knowledge.»


[3] In abstract terms, the axiomatic method points out all the relations (especially the functions) occurring between all the sets of a given (scientific) language, so that all the statements of the language  can be logically deduced as theorems from the postulates or primitive statements, and if the postulate set is consistent no contradictory statement can be deduced from the set. A postulate set can be semantically interpreted by the rules of designation concerning a corresponding set of experimental results, not a set of experimental statements.

(On “social sciences” and science, pp. 468-469)

11. Structural Dynamics and Evolutionary Dynamics

The scientific method imposes a clear-cut distinction between repetition time and evolution time, respectively related to reversibility and irreversibility.

«Repetition time expresses a measure of intervals interpreted as length. In physics (and in everyday experience) it is realized by repeating identically a (cyclic) set of operations – such as the oscillation of a pendulum, of a balance wheel, of a quartz crystal – which is an index to which an invariant numerical value is related (Faggiani 1957, p. 7 ff.). Repetition time, as a rule, is used in scientific language as an implicit independent variable in the context of reversible processes. Evolution time expresses a different concept, which applies not to a relation concerning length, but rather to a concentration process or to a dissipation process […]; these cases refer to irreversible processes.»

«Every natural (physical), biological or behavioral phenomenon is characterized by evolution time, but the latter has to be considered as a parameter if one wants to explain the phenomenon in strict reversibility terms. Of course, also, the phenomena characterized by evolution time can be explicated from a scientific point of view, as is shown in physics; in this case, too, however, the two temporal perspectives cannot be confused. So, corresponding to reversibility and irreversibility, structural dynamics must be distinguished from evolutionary (cumulative) dynamics.»

(A new paradigm for the integration of the social sciences, pp. 320, 321)

Aristotelian intuition method and Galilean experimental method

«Intuition is the method through which Aristotle identifies and establishes the “first principles” or the universal and necessary properties of being (noncontradiction, identity, excluded third) and the axioms of mathematics and geometry, too, which as intuitions are immediately evident.

Intuition also pertains to Aristotle’s procedure to establish true premises of the “scientific syllogism”: the “abstractive induction,” by which it is possible to determine, even from a single particular case, the universal and necessary properties of sensible entities that change, i.e., to apprehend their “forms” or “essences” (universals which hold always). Nevertheless, how could the “intellect” realize a process of abstraction from a single, particular case to a “universal essence” without intuition? The “abstractive induction,” which starts even from a single particular and reaches a universal that holds always, would be meaningless if it were not based on intuition.

The other type of induction, which Aristotle defines as the procedure that goes from particulars to the universal and allows identifying a property common to a multiplicity of events, has limited validity within space and time and is therefore devoid of necessity and immutability. It cannot be inferred, from the occurrence of a given event, that the event will continue to occur always and with certainty. One cannot infer a universal that holds in every case and at all times (which requires intuition), but only a universal which holds “for the most part,” expressing compliance with a law referring to events that leave room for exceptions, which occur regularly.

Galileo contrasts intuition as a “principle of science” with the experimental method qualified by the functional relation, showing how “principles” can be justified within science. To found science on the “abstractive induction” (i.e., intuition) means to negate science; in fact, being characterized by subjectivity, intuition makes the various observers’ positions not comparable, in principle. Furthermore, the single event is of primary relevance for Galileo, as the controlled experiment refers to specific phenomena for which compliance with the law is assumed.

It follows that the concepts characterized as “necessary essences” (universals which hold always) are devoid of operational capacity: the Aristotelian concept of motion, for example, cannot be operated.

For Aristotle, intuition supported by induction expresses the universal premises on which, through the syllogism, the demonstrations of “science” are based; the syllogism makes evident the “cause,” understood as the reason for attributing a quality to an event or a subject. Galileo, on the contrary, states that it is necessary to know the functional relations between events rather than the events “in themselves” (as “essences”) to know the causes (understood as independent variables).

Aristotle answers the question “what is the event” and conceives science as knowledge by demonstration based on “causes” understood as “necessary essences.” Galileo answers the question “how does the event (dependent variable) work” with reference to other events and conceives science as knowledge by relation based on causes understood as independent variables.» IT

(Le scatole vuote della sociologia [The empty boxes of sociology], pp. 50-52)

Theoretical predicates and experimental predicates within the language of science

«The characterization of the basic schema of the scientific experiment as a relation between the independent variable (Vi), dependent variable (Vd), and constrained variable (Vc) requires the language of science to be constructed starting from identifying some experimental predicates (variables) and their relations. The theoretical construction must rest on these basic predicates, and new predicates can only be introduced as higher generalization-level predicates compared to experimental predicates. That means so-called primitive predicates, which are assumed to have the highest level of generalization within a given theoretical structure—the latter having a given level of abstraction – can only be introduced if they express attributes (properties and relations) that are also typical of experimental predicates. Furthermore, this means that primitive predicates presuppose a plurality of different-type predicates and that a theory having a level of generalization n+1, relative to experimental predicates assumed of level n, cannot be founded on a single type of experimental predicates. In this hypothesis, one could only define the various experimental (or operational) predicates in different terms, i.e., one could only introduce new nominal definitions.

Primitive predicates always work by characterizing all predicates belonging to a theoretical structure at a maximum level of generalization. The abstraction level may be higher or lower relative to those of other theoretical structures within the same scientific discourse (e.g. Newtonian mechanics and relativistic mechanics are theoretical structures with different levels of abstraction in the context of physical discourse). Introducing new predicates allows for enlarging the structure’s domain of application within the structure itself, i.e., with reference to the same abstraction level.

On these grounds, the distinction between observational and theoretical terms–a distinction corresponding to what Hempel (La formazione dei concetti e delle teorie nella scienza empirica [Fundamentals of concept formation in empirical science], p. 105) calls the level of empirical generalization and the level of theory construction–has no place.»

«Theoretical concepts are formulated by isolating some fundamental predicates, which identify the variables characterizing the experiment (independent, dependent, and constrained variable). It means that theoretical predicates predetermine the experiment, even when they have the minimum level of generalization. From these predicates, one can reach higher-level predicates, up to the maximum-level predicates, within a given structure in the system. Only predicates with the maximum generalization level can be qualified as theoretical; we name experimental predicates the lower-level ones.» IT

(L’analisi scientifica del comportamento di scelta [The scientific analysis of choice behavior], II, pp. 21; 25)

On the concept of functional relation in respect of time variable

«(a) The structural schema Rf (x, y; p1, p2,…, pn) represents the fundamental relation of scientific discourse. It expresses the relation between two variables when a third factor (a parameter) remains constant; by changing the experimental conditions, the latter can be considered either a dependent or independent variable, in which case, one of the two previously varying factors is kept constant.

(b) The structural schema Rf (x, y; p1, p2,…, pn) may or may not refer to time as a variable. Even so, time is not an effective independent variable within the scientific discourse […]. The methodological status of the time variable, thus, differs from the methodological status of the variables expressing typical predicates of the single sciences. Introducing time as an implicit independent variable in scientific laws means ordering the states according to an open series. At this level of analysis, states are characterized by the order of time.

(c) When the relation Rƒ (x, y; p1, p2,…, pn) does not contain time as an implicit independent variable, it cannot be considered as a relation generating a serial order. That is because such a relation possesses the property of asymmetry but not the properties of transitivity and connexity. In this hypothesis, no temporal ordering of states can be derived from the relation Rƒ (x, y; p1, p2,…, pn), even where the relation describes processes that take place over time.

When the relation Rf (x, y; p1, p2,…, pn) contains the time variable, we can speak of dynamic analysis; when it does not, we can speak of static analysis.

Scientific analyses, whether they are formulated from the static perspective or the dynamic perspective, are hence based on the fundamental relation Rf (x, y; p1, p2,…, pn). In one case, it is a mere asymmetrical relation expressing an interdependence between the factors taken into account, which cannot generate a (temporal) order of states. In the other case, it contains time as an implicit independent variable and, hence, is also an order relation from which a (temporal) succession of states is obtained.» IT

(Il problema del metodo nella sociologia [The problem of method in sociology], pp. 25-26)

On structural dynamics and cumulative dynamics: the concepts of time order and time direction

«An explanatory schema of history, i.e., of cumulative (or evolutionary) dynamics, cannot be based on a consideration of time as order. History consists of a succession of serially ordered states, where it is considered time as direction, not time as order. Here, the system’s states are ordered as to the change in the content of the states; namely, change is characterized in terms of evolution (cumulative process).

To get a better understanding of this point, let us consider the concept of dynamics. Every asymmetric, transitive, connected relation establishes a direction. It is, therefore, possible to distinguish between the ordering relation and its inverse; this distinction allows us to characterize the order and the counterdirectionality of the series. Hence, the serial order refers to the reversibility concept (meant as possibility of reversing the order of states). In this case, given an ordered set of states, there is a structural identity between the description in the direction of the original relation and the inverse description; namely, it is possible to describe the serial order in the opposite direction.

Let us now consider an open series and suppose that a reversal of the order of states cannot be assumed for such a series. Under this assumption, a structural diversity would exist between the description in the original-relation direction and the inverse description. What is the element that characterizes this structural diversity? It is a constraint that we place upon the ordering relation considered as an open series. The constraint expresses, in methodological terms, a particular type of serial-order specification obtained by characterizing the order of the states according to the diversity in the content of the states. The order of the states is, thus, determined as to the change in the content of the states; here, we can speak of irreversibility. On the contrary, in the case of reversibility, the order is indeterminate as to the content of the states.

To clarify this point, we observe that defining the concept of reversibility requires considering the characteristics that remain unchanged when the direction of time is reversed. These characteristics identify the meaning of the relation between any two states, which remains the same whether we describe the series in the direction of the original relation or we describe it in the opposite direction. Both reversibility and irreversibility refer to a series of states ordered according to a certain direction. When the direction is the original one (corresponding to the direction of the generating relation), we can speak of a description in +t. However, a second description can be given by considering the inverse of the generating relation; here, we can speak of a description in -t.

An inverse description of the process is conceivable both in the case of reversibility and in the case of irreversibility. However, in the reversibility case, the two descriptions are structurally identical, because the meaning of the relation remains unchanged in the two descriptions (it is possible to describe the process in the opposite direction). In the irreversibility case, instead, the inverse description (description in -t) is structurally different from the original one (description in +t); namely, the meaning of the relation is different depending on the two descriptions, because the process can only be described in the original direction.

There are, therefore, two fundamental characterizations. The first refers to the order considered as an asymmetrical, transitive, and connected relation, which is not structurally different from its inverse (time as order, reversibility). The second refers to the order considered as an asymmetrical, transitive, and connected relation, which is structurally different from its inverse (time as direction, irreversibility). We get the diversity of the structures by specifying the order as to the change in the states’ content. […]

This conclusion is of particular importance for the social sciences. It allows us to refute Weber’s methodological perspective (Il metodo delle scienze storico sociali [The method of historical-social sciences]), which fails to distinguish between the order of time and the direction of time and considers historicity as the fundamental methodological dimension for the explanation of social phenomena.

To clarify this point, let us consider, as a methodological example, a succession of physical states. These states can be ordered according to the change in their content by the “entropy” magnitude. An order of states is thereby obtained based on their increasing entropy quantities. It is important to point out that the order obtained is not a causal order but a historical order of states identified by an increasing succession of entropy quantities. Therefore, the second law of thermodynamics (entropy principle) merely states the irreversibility of physical processes but does not provide an explanation of way such irreversibility.» IT

(Il problema del metodo nella sociologia [The problem of method in sociology], pp. 27-29)

The logic of change within science

«The logic of change (deepening and widening) within science is completely different from the transition stages from pre-science to science. In this latter case, the problem has a sociological and psychological nature because it concerns the behavior of researchers (i.e., the trial-and-error sequences that researchers carry out). In fact, it may be that these ordered sets of operant behaviors take place in a pre-scientific context and/or are conditioned by hypotheses that do not conform to the constraints of science; under these circumstances, the erroneous operants of researchers do not belong to science.

It is, therefore, necessary to draw a distinction between two different historical perspectives: (a) the more traditional one examining the development of research behavior (what in the past was called natural philosophy) concerning every kind of trial-and-error behavior, without specific reference to the constraints of science, and mixing pre-scientific and scientific behavior in their historical development; and (b) a different perspective examining only the development of scientific thought, i.e., only the research behavior conforming to the constraints and to the cumulative results of science.

Two kinds of histories of science, therefore, exist: (a) socio-psychological history concerning all the research methods, in whatever manner carried out, and the socio-psychological conditionings to which the researchers have been subjected in the social context where they worked; and (b) history, purified from socio-psychological connotations, concerning the research methods conforming to the constraints and to the cumulative results of science, i.e., the history of the deepening and widening of the domains of application of scientific language only.

Kuhn (1962) did not pay much attention to this distinction, and his oversight is the origin of many misunderstandings. Indirectly, it led to a relativistic perspective, which tends to deny the validity (in terms of intersubjectivity) of scientific language itself by placing the scientific method on the same level as all the other possible methods of knowledge acquisition (first among them intuition).»

(On “social sciences” and science, p. 469)

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