XIII: Interpretation of a theory
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Since 1915: Extension of the word „model“ in the modern logic
Further extension and „theory of models“
Wilfrid Hodges on terminology: models - pseudomodels – structures – systems - theories
Pseudomodels and abstract models
Around 1950: Alfred Tarski and Abraham Robinson: „Theory of models“
1960 Robinson: „standard models“ and „nonstandard models“
In science: The world as a model of the theory
A clash of terminology?
Since 1915: Extension of the word „model“ in the modern logic
The model concept was expanded once again by the Viennese philosopher Ludwig Wittgenstein (1921) – see Jörg Burkhardt (1965), Wolfgang Stegmüller (1966), Friedrich Waismann (1976, 446-470), Hans-Joachim Glock (1996), and for the Viennese philosophy: Kurt Rudolf Fischer (1991).
Some key phrases of Wittgenstein’s „Tractatus logico-philosophicus“ (1921; in English 1922) are:
„The world divides into facts.“
„The facts in logical space are the world.”
“The total reality is the world.”
„We make to ourselves pictures of facts.“
„The picture is a model of reality.”
“The proposition is apicture of reality.”
“The proposition is a model of the reality as we think it is.”
For bibliography and some citations of Wittgenstein’s „Tractatus logico-philosophicus“ (1921) see: Fig. 63.
Another extension of the meaning of „model“ was given by the German mathematician Hermann Weyl (1927). For him “model” belongs into the range of logic.
In the 1930’s logicians as Rudolf Carnap (1934ff – Fig. 64), Morris Raphael Cohen and his disciple Ernest Nagel (1934) as well as Alfred Tarski (1935; 1935/36 – Fig. 65) saw the model as "fulfilment" of axiomatic systems and formalized theories. Or in reverse: Theories, syntactically seen, were reconstructed as non-interpreted calculuses of axiomatic systems.
Two key sentences by Tarski in 1935 and 1936 are:
„By deductive theories I understand here the models (realizations) of the axiom system.“
and:
„An arbitrary sequence of objects which satisfies every sentential function of the class L' will be called a model or realization of the class L of sentences (in just this sense one usually speaks of models of an axiom system of a deductive theory).“
The key sentences of Rudolf Carnap in 1942 are:
„what is usually called the construction of a model for a set of postulates is the same as the semantical interpretation of a calculus“
and.
„What is usually called the construction of a model for a postulate set is the construction of an interpretation for this syntactical part.“
Further extension and „theory of models“
After World War II this kind of model theory expanded further. Two of the first essays in this area, in which the expression „theory of models“ occurs, are from the mathematician Chen Chung Chang (1954) and Alfred Tarski (1954/55).
Tarski’s definitions of 1953 (11, 12) ares legendary:
„A possible realization in which all valid sentences of a theory T are satisfied is called a model of T.“
and:
„Consistency and completeness can also be characterized in terms of models: a theory T is consistent if and only if it has at least one model; it is complete if and only if every sentence of T which is satified in one model is also satisfied in any other model of T. Two theories T1 and T2 are said to be compatible if they have a common consistent extension; this is equivalent to saying that the union of T1 and T2 is consistent.“
The first introduction to this theory of model was given by Abraham Robinson in 1963.
In an appendix to the papers of a symposium, edited by John W. Addison et al. (1965, 442-492) we find a collection of no less than 940 titles in the „Bibliography of the theory of models“.
At a symposium in honour of Alfred Tarski in summer 1971 in Berkeley Robert L. Vaught and Chen Chung Chang told the history of this kind of “theory of models” since 1915 till 1971 with vast bibliographies.
Shortly after Chen Chung Chang and Howard Jerome Keisler edited a comprehensive „Model Theory“ (1973). It was replaced 20 years later by Wilfrid Hodges’ „Model Theory“ (1993; reprints 1995 and 1997).
For definitions by Mary Hesse (1967), Stephen Cole Kleene (1964/73) and Elisabeth A. Lloyd (1998) see:
Fig. 66: Definitions of logical, mathematical and metamathematical models
further:
Fig. 67: Dictionary Definitions of „model“ in logic
Fig. 68: Encyclopaedia Britannica: „Model theory“
Wilfrid Hodges on terminology: models - pseudomodels – structures – systems – theories
Pseudomodels and abstract models
As Wilfrid Hodges points out, geometricians in the 19th century discovered some two-dimensional surfaces that could be described algebraically but not built in three-dimensional space. To model them they developed a pair of devices, tat could be used together or separately:
· „In pseudomodels we introduce a systematic and reversible distortion of some feature of what is being modelled.“
· „In abstract models we give up the attempt to make a physical model. Instead we take an abstract mathematical structure, for example four-dimensional euclidean space, and we give a mathematical definition of the set of points of this structure which form our model.“
Abstract models were at first known as „interpretations“ (viz. George Peacock, 1834; Eugenio Beltrami, 1868). Sometimes „interpretation“ was translated into German as „Bild“.
During the 1920s distorted models of geometrical surfaces and spaces came to be known as „pseudospaces“. Later in the 1920s mathematicians in the school of Hilbert describe distorted abstract models as „pseudomodels“ or simply „models“ (Fraenkel, Von Neumann, Weyl).
In 1936 Alfred Tarski took a further step in generalisation, and used „model“ to mean any systematically distorted interpretation. (His context was set-theoretical interpretations of formal languages – see Fig. 65.)
Around 1950: Alfred Tarski and Abraham Robinson: „Theory of models“
During the 1940s it became clear to a number of logicians that a notion was needed which would be a common generalisation of both pseudomodels and classes of algebraic structures (such as groups) that are defined by systems of axioms. Around 1950 two logicians, Alfred Tarski and Abraham Robinson, came to essentially the same generalisation. They seem to have worked independently; Tarski generalised his own „models“ (i. e. pseudomodels), while Robinson combined work of Carnap („Introduction to Semantics“, 1942) with Bourbaki’s notion of a „structure“. In 1954 Tarski proposed the name „theory of models“ for the discipline that dealt with this new notion. Carnap had earlier (1942, 240) tentatively used the name „theory of systems“ for the theory of models.
Tarski’s new notion of model was essentially a linguistic device. Probably the first appearance in print of „model“ in this new sense was the paper of Andrzej Mostowski (1952), which sets out the gory details of the linguistics. But it is true that moves were made to cut down the reference to language, e. g. by Patrick Suppes (1960, 2002) applying Tarski’s models to scientific modelling.
Tarski’s model-theoretic models are often known as semantic models. This name probably comes from the connection with his earlier mathematical truth definition; he had coined the name „semantic theory of truth“ for his own justification of it.
In 1965 John Addison published some recommendations for terminology and notation in model theory; the main lines of his recommendations are still followed today in mathematical model theory. He remarked that in model theory
a model is considered to be a structure related to a given theory, rather than a theory intended to explain a given realm of phenomena (1965, 438).
Thus for most model theorists, „Structure M is a model of theory T“ means that the sentences of the theory T, when interpreted as applying to the structure M, are all true.
But some mathematicians also use ‘model’ as a synonym for ‘structure’.
1960 Robinson: „standard models“ and „nonstandard models“
In 1960 Robinson found himself studying theories where one particular interpretation is „intended“, but other interpretations of the same formal language are used as a mathematical device. So these other interpretations were what had earlier been known as pseudomodels. But he introduced a new name: the intended interpretation was the „standard model“ of the relevant theory, and the pseudomodels were „nonstandard models“. His application of these notions to the foundations of mathematical analysis is known as „nonstandard analysis“.
In science: The world as a model of the theory<
As soon as the new notion of semantic model became available, philosophers of science were quick to see its relevance to their concerns. Tarski had explained the notion of a formal theory being „true in“ a structure. There was an obvious analogy with the way in which a formal scientific theory is or is not „true of“ some aspect of the real world. The analogy was particularly convincing for philosophers who regarded the ideal scientific theory as a formal theory in some mathematically defined language. To say that the theory is „true of“ some aspect of the world just means that if we assign certain features of the world to the non-logical constants of the theory, this makes the world into a model (in Tarski’s sense) of the theory.
But there was also a clash of terminology, because scientific theories had often been regarded as a kind of „model“.
A clash of terminology?
One popular solution to this terminological clash has been to think of a theory as standing not for a single model but for a class of models— the class of all its semantic models. One can finesse this idea by limiting to the class of „physically relevant“ models. For certain types of theory, particularly those consisting of systems of differential equations, one can pack down a whole family of semantic models into a single model together with a choice of real-number parameters. Some of the options are studied in Frederick Roy Suppe (1989), Theo A. F. Kuipers (2001) and Ilkka Niiniluoto (1999).
Patrick Suppes (1960) argues that the clash of terminology is only apparent.
Bibliography
Dr. phil. Roland Müller, Switzerland / Copyright © by Mueller Science 2001-2016 / All rights reserved
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