A sentence S divides all its possible interpretations into two classes, those that are models of it and those that are not. In this way it defines a class, namely the class of all its models, written Mod(S). To take a legal example, the sentence
The first person has transferred the property to the second person, who thereby holds the property for the benefit of the third person.
defines a class of structures which take the form of labelled 4-tuples, as for example (writing the label on the left):
- the first person = Alfonso Arblaster;
- the property = the derelict land behind Alfonso's house;
- the second person = John Doe;
- the third person = Richard Roe.
This is a typical model-theoretic definition, defining a class of structures (in this case, the class known to the lawyers as trusts).
We can extend the idea of model-theoretic definition from a single sentence S to a set T of sentences; Mod(T) is the class of all interpretations that are simultaneously models of all the sentences in T. When a set T of sentences is used to define a class in this way, mathematicians say that T is a theory or a set of axioms, and that T axiomatises the class Mod(T).
Take for example the following set of first-order sentences:
∀x∀y∀z (x + (y + z) = (x + y) + z).
∀x (x + 0 = x).
∀x (x + (−x) = 0).
∀x∀y (x + y = y + x).
Here the labels are the addition symbol ‘+’, the minus symbol ‘−’ and the constant symbol ‘0’. An interpretation also needs to specify a domain for the quantifiers. With one proviso, the models of this set of sentences are precisely the structures that mathematicians know as abelian groups. The proviso is that in an abelian group A, the domain should contain the interpretation of the symbol 0, and it should be closed under the interpretations of the symbols + and −. In mathematical model theory one builds this condition (or the corresponding conditions for other function and constant symbols) into the definition of a structure.
Each mathematical structure is tied to a particular first-order language. A structure contains interpretations of certain predicate, function and constant symbols; each predicate or function symbol has a fixed arity. The collection K of these symbols is called the signature of the structure. Symbols in the signature are often called nonlogical constants, and an older name for them is primitives. The first-order language of signature K is the first-order language built up using the symbols in K, together with the equality sign =, to build up its atomic formulas. (See the entry on classical logic.) If K is a signature, S is a sentence of the language of signature K and A is a structure whose signature is K, then because the symbols match up, we know that Amakes S either true or false. So one defines the class of abelian groups to be the class of all those structures of signature +, −, 0 which are models of the sentences above. Apart from the fact that it uses a formal first-order language, this is exactly the algebraists' usual definition of the class of abelian groups; model theory formalises a kind of definition that is extremely common in mathematics.
Now the defining axioms for abelian groups have three kinds of symbol (apart from punctuation). First there is the logical symbol = with a fixed meaning. Second there are the nonlogical constants, which get their interpretation by being applied to a particular structure; one should group the quantifier symbols with them, because the structure also determines the domain over which the quantifiers range. And third there are the variables x, y etc. This three-level pattern of symbols allows us to define classes in a second way. Instead of looking for the interpretations of the nonlogical constants that will make a sentence true, we fix the interpretations of the nonlogical constants by choosing a particular structure A, and we look for assignments of elements of A to variables which will make a given formula true in A.
For example let Z be the additive group of integers. Its elements are the integers (positive, negative and 0), and the symbols +, −, 0 have their usual meanings. Consider the formula
v1 + v1 = v2.
If we assign the number −3 to v1 and the number −6 to v2, the formula works out as true in Z. We express this by saying that the pair (−3,−6) satisfies this formula in Z. Likewise (15,30) and (0,0) satisfy it, but (2,−4) and (3,3) don't. Thus the formula definesa binary relation on the integers, namely the set of pairs of integers that satisfy it. A relation defined in this way in a structure A is called a first-order definable relation in A. A useful generalisation is to allow the defining formula to use added names for some specific elements of A; these elements are called parameters and the relation is then definable with parameters.
This second type of definition, defining relations inside a structure rather than classes of structure, also formalises a common mathematical practice. But this time the practice belongs to geometry rather than to algebra. You may recognise the relation in the field of real numbers defined by the formula
v12 + v22 = 1.
It's the circle of radius 1 around the origin in the real plane. Algebraic geometry is full of definitions of this kind.
During the 1940s it occurred to several people (chiefly Anatolii Mal'tsev in Russia, Alfred Tarski in the USA and Abraham Robinson in Britain) that the metatheorems of classical logic could be used to prove mathematical theorems about classes defined in the two ways we have just described. In 1950 both Robinson and Tarski were invited to address the International Congress of Mathematicians at Cambridge Mass. on this new discipline (which as yet had no name — Tarski proposed the name ‘model theory’ in 1954). The conclusion of Robinson's address to that Congress is worth quoting:
[The] concrete examples produced in the present paper will have shown that contemporary symbolic logic can produce useful tools — though by no means omnipotent ones — for the development of actual mathematics, more particularly for the development of algebra and, it would appear, of algebraic geometry. This is the realisation of an ambition which was expressed by Leibniz in a letter to Huyghens as long ago as 1679.
In fact Mal'tsev had already made quite deep applications of model theory in group theory several years earlier, but under the political conditions of the time his work in Russia was not yet known in the West. By the end of the twentieth century, Robinson's hopes had been amply fulfilled; see the entry on first-order model theory.
There are at least two other kinds of definition in model theory besides these two above. The third is known as interpretation (a special case of the interpretations that we began with). Here we start with a structure A, and we build another structure B whose signature need not be related to that of A, by defining the domain X of B and all the labelled relations and functions of B to be the relations definable in A by certain formulas with parameters. A further refinement is to find a definable equivalence relation on X and take the domain of B to be not X itself but the set of equivalence classes of this relation. The structure B built in this way is said to be interpreted in the structure A.
A simple example, again from standard mathematics, is the interpretation of the group Z of integers in the structure N consisting of the natural numbers 0, 1, 2 etc. with labels for 0, 1 and +. To construct the domain of Zwe first take the set X of all ordered pairs of natural numbers (clearly a definable relation in N), and on this set X we define the equivalence relation ∼ by
(a,b) ∼ (c,d) if and only if a + d = b + c
(again definable). The domain of Z consists of the equivalence classes of this relation. We define addition on Z by
(a,b) + (c,d) = (e,f) if and only if a + c + f = b + d + e.
The equivalence class of (a,b) becomes the integer a − b.
When a structure B is interpreted in a structure A, every first-order statement about Bcan be translated back into a first-order statement about A, and in this way we can read off the complete theory of B from that of A. In fact if we carry out this construction not just for a single structure A but for a family of models of a theory T, always using the same defining formulas, then the resulting structures will all be models of a theory T′ that can be read off from T and the defining formulas. This gives a precise sense to the statement that the theory T′ is interpreted in the theory T. Philosophers of science have sometimes experimented with this notion of interpretation as a way of making precise what it means for one theory to be reducible to another. But realistic examples of reductions between scientific theories seem generally to be much subtler than this simple-minded model-theoretic idea will allow. See the entry on intertheory relations in physics.
The fourth kind of definability is a pair of notions, implicit definability and explicit definability of a particular relation in a theory. See section 3.3 of the entry on first-order model theory.
Unfortunately there used to be a very confused theory about model-theoretic axioms, that also went under the name of implicit definition. By the end of the nineteenth century, mathematical geometry had generally ceased to be a study of space, and it had become the study of classes of structures which satisfy certain ‘geometric’ axioms. Geometric terms like ‘point’, ‘line’ and ‘between’ survived, but only as the primitive symbols in axioms; they no longer had any meaning associated with them. So the old question, whether Euclid's parallel postulate (as a statement about space) was deducible from Euclid's other assumptions about space, was no longer interesting to geometers. Instead, geometers showed that if one wrote down an up-to-date version of Euclid's other assumptions, in the form of a theory T, then it was possible to find models of T which fail to satisfy the parallel postulate. (See the entry on geometry in the 19th centuryfor the contributions of Lobachevski and Klein to this achievement.) In 1899 David Hilbert published a book in which he constructed such models, using exactly the method of interpretation that we have just described.
Problems arose because of the way that Hilbert and others described what they were doing. The history is complicated, but roughly the following happened. Around the middle of the nineteenth century people noticed, for example, that in an abelian group the minus function is definable in terms of 0 and + (namely: −a is the element b such that a + b = 0). Since this description of minus is in fact one of the axioms defining abelian groups, we can say (using a term taken from J. D. Gergonne, who should not be held responsible for the later use made of it) that the axioms for abelian groups implicitly define minus. In the jargon of the time, one said not that the axioms define the function minus, but that they define the concept minus. Now suppose we switch around and try to define plus in terms of minus and 0. This way round it can't be done, since one can have two abelian groups with the same 0 and minus but different plus functions. Rather than say this, the nineteenth century mathematicians concluded that the axioms only partially define plus in terms of minus and 0. Having swallowed that much, they went on to say that the axioms together form an implicit definition of the concepts plus, minus and 0 together, and that this implicit definition is only partial but it says about these concepts precisely as much as we need to know.
One wonders how it could happen that for fifty years nobody challenged this nonsense. In fact some people did challenge it, notably the geometer Moritz Pasch who in section 12 of his Vorlesungen über Neuere Geometrie (1882) insisted that geometric axioms tell us nothing whatever about the meanings of ‘point’, ‘line’ etc. Instead, he said, the axioms give us relations between the concepts. If one thinks of a structure as a kind of ordered n-tuple of sets etc., then a class Mod(T) becomes an n-ary relation, and Pasch's account agrees with ours. But he was unable to spell out the details, and there is some evidence that his contemporaries (and some more recent commentators) thought he was saying that the axioms may not determine the meanings of ‘point’ and ‘line’, but they do determine those of relational terms such as ‘between’ and ‘incident with’! Frege's demolition of the implicit definition doctrine was masterly, but it came too late to save Hilbert from saying, at the beginning of his Grundlagen der Geometrie, that his axioms give ‘the exact and mathematically adequate description’ of the relations ‘lie’, ‘between’ and ‘congruent’. Fortunately Hilbert's mathematics speaks for itself, and one can simply bypass these philosophical faux pas. The model-theoretic account that we now take as a correct description of this line of work seems to have surfaced first in the group around Giuseppe Peano in the 1890s, and it reached the English-speaking world through Bertrand Russell's Principles of Mathematics in 1903.