From the ALWS archives: A selection of papers from the International Wittgenstein Symposia in Kirchberg am Wechsel

Observations, operational definitions and observables

The object language applies to a domain consisting of physical systems. Physical systems possess properties and the attribution of a property to an individual system constitutes an atomic fact about the system. It is expressed by an atomic proposition, i.e. true atomic propositions are statements about observed atomic facts. The observations of atomic facts all involve the use of a standard of measure. The result of an observation follows from a comparison between a representation of the standard and the system. It determines a value from the standard, a predicate of the first kind.

Observations/measurements are based on operational definitions, i.e. definitions that specify the applied standard of measure, the laws/rules on which the measurements are based and the instruction of the actions to be performed to make a measurement. The operational definitions are formulated in a separate language. They provide intensional interpretations of the predicates expressing results of observations. The measurement of the colour of a system is an example. The measuring device is then a colour chart where each of the colours is named and the rule of application is to compare the colour of the system with the colours on the colour chart and pick out the one identical to the colour of the system. The name of the colour picked denotes the result of the measurement.

Each operational definition is symbolised by an observable that simulates the act of observation; the observable is an injective map between the domain and the set of predicates (of the first kind) that maps a system to the predicate representing a property possessed by the system (Piron 1975). The set of possible values of an observable represent mutually exclusive properties of the systems of the domain; no two properties corresponding to different values of the same observable can be possessed by any system. A system cannot at the same time weight 1 kg and 2 kg. Weight is therefore an observable. Other examples of observables are position in space, number of systems in a given container and colour of systems.

Names, predicates of the first kind and truth

Let L(D;N∪V,P₁∪P₂ stand for the object language for a domain D. N denotes the set of names, V the set of variables, P₁ the set of predicates of the first kind and P₂ the set of predicates of the second kind¹. The distinction between predicates of the first and second kind is semantic and made possible by the intensional interpretation. The names of the systems are given by a map ν,

     ν:D→N; d↦ν(d)

that to a system d in the domain D associates the name n by ν(d) = n . ν is an isomorphism; by convention, there is a unique name for every system and there is exactly the same number of names and systems.

Each observable δ determines an atomic fact about a system d∊D by

     δ:D→P₁; d↦δ(d)

For each δ there exists a unique (injective) map π defined by the condition of commutativity of the diagram

The diagram relates the simulation of measurements determining atomic facts assigning a property to a system and the formulation of an atomic sentence expressing such a fact by the juxtaposition pn of the name n referring to the system d and the predicate p referring to the property, i.e. if π(ν(d))=δ(d), for ν(d)=n and π(n) =p. The commutativity condition π(ν(d))=δ(d) is thus the truth condition of the object language. It equates a proposition about the system d with a statement of the result of a measurement on d with respect to the observable δ.

The Tarski interpretation can be derived from the above interpretation. By taking the inverse images of the predicates of the first kind with respect to the observables we get the extensions of the predicates. The opposite is however, not the case. The reason is that intensional definitions contain much more information than extensional enumerations. In particular, operational definitions give meaning to predicates of the first kind at the outset while those of the second kind are introduced by terminological definitions.

Predicates of the second kind

One distinguishes between two kinds of observables referring to two kinds of properties, properties that do not change in time and thus serves to identify the system, and properties that change. The corresponding observables are identification and state observables respectively.

The systems can be classified with respect to the identification observables. One starts with one of the observables and uses its values to distinguish between the systems to construct classes. Thus, one gets a class for each value of the observable, the class of systems that possess the particular property, e.g. the class of all red systems, the class of all green systems etc. The procedure can be continued recursively until the set of identification observables is exhausted. The result is a hierarchy of classes with respect to the set inclusion relation. The basic entities of the classification are the leaf classes.

The classes are referred to by predicates of the second kind which thus are ordered naturally in a taxonomy that constitutes a linguistic representation of the classification. The satisfaction conditions defining the classes are intensional definitions of the predicates of the second kind. They are of the form

• pn₁ = p₁n₁ ∧ p₂n₁ ∧ ... ∧ p_mn₁
• pn₂ = p₁n₂ ∧ p₂n₂ ∧ ... ∧ p_mn₂
• ∙
• ∙
• pn_k = p₁n_k ∧ p₂n_k ∧ ... ∧ p_mn_k

where p is the predicate of the second kind referring to the class defined by the satisfaction condition, p₁, p₂, p₃, … p_m are the values of the different observables and n₁, n₂, … n_k, stand for the names of the systems for which the atomic propositions are true. The set of intensional definitions constitute terminological definitions belonging to the ontology of the language.

In the framework of the property language the subject-predicate form of the atomic propositions whose function is to attribute a property to a system can be considered fundamental. Seemingly atomic propositions like “pn” where p is a predicate of the second kind are only abbreviations of sentences being conjunctions of atomic sentences. For example the sentence “S-2003 is a Car” hides the description of what falls under the concept referred to by the predicate of the second kind “Car”.

Properties

Predicates of the first kind refer to properties of systems. A property is something in terms of which a system manifests itself and is observed, and by means of which it is characterised and identified. To an observer a system appears as a collection of properties. The properties of a system are thus in a natural way mentally separated from the system. The separation is made possible by the fact that the ‘same’ property is possessed by more than one system. It is expressed by the commutativity of the following diagrams

where E is the abstract representation of the set of properties of the systems in D; the ε are injective maps that simulates the ‘mental’ separation of properties from the systems. In the case of coloured systems for example, the condition of commutativity means that if a system appears as red then it possesses the property redness. It is assumed that each predicate of the first kind refers to a unique potential property.

The property space E is a construction characterised by the diagram. The E chosen is a natural extension of the set of properties that can be associated to the systems of the domain as reflected in the set of predicates available in the standards of the operational definitions..

The maps ρ:E→P₁; e ↦ ρ(e) can be considered as naming maps for the properties, e.g. a point in abstract space is named by a set of coordinates. To describe the properties we need a formal language, the property language L(E,P₁∪W,Q), were P₁ denotes the set of names, W the set of variables and Q the set of predicates. The property language is associated with the diagrams

The symbolic entities of E also belong to the property language. The maps ρ thus symbolises the kinds of measurements inside the language.

Ontologies

The ontology of the property language is a set of definitions relating the terms of the vocabulary. It consists of an axiom system giving implicit definitions of what is chosen to be the primary terms and a set of terminological definitions introducing secondary terms which serves to simplify the discourse. The axioms are formalised accounts of the operational definitions which constitute a basis for the interpretation of the primary terms. The axioms define a semantic structure that pictures structural properties of the domain as seen through the operational definitions (Blanché 1999).

The Theory of Special Relativity offers an example for how operational definitions impose an axiomatic structure on the property language. It is derived from an operational definition of time and distance measurements based on a definition of simultaneity of distant events with respect to a given observer, the physical law claiming the velocity of light to be constant and independent of the velocity of the emitting source and the Principle of Relativity which postulates that the laws of physics are the same in all inertial frames of reference. Presently, the standard unit of time defines the second to be 9 192 631 770 periods of the radiation emitted from the transition between two hyperfine levels in the ground state of Caesium 133. Moreover, 1 meter is equal to the distance covered by a light ray in empty space in 1/299 792 458 seconds. Length measurements are thus based on time measurements.

The axioms of the property language express the content of the operational definitions which constitute the basis for empirical investigations. Their truth can therefore not be ascertained through empirical investigations. They are “hinge” propositions expressing statements about structural properties of the property language that reflects structural properties of the domain as seen through the “lenses” of the operational definitions (Wittgenstein 1975). The object language can only be indirectly tested by means of the models of systems of the domain formulated in the language.