hodges - model theory

7/28/2019 Hodges - Model Theory

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Introduction to model theory

Wilfrid Hodges

Queen Mary, University of London

This course is an introduction in two senses. First, it is for people whohavent studied model theory before, though I trust most people in the class

will have heard of it. I have tried to make most of the material accessi-ble to people coming from any of the main disciplines where model theoryis used: mathematics, philosophy, computer science, linguistics, cognitivepsychology.

And second, I have tried to start where model theory starts. This courseis an introduction to the basic notions rather than to the most impressiveachievements.

Since we are discussing the starting points, I dip back into history per-haps more than is usual in introductory courses.

Unfortunately I havent yet found time to put together complete notes.What you have here is a provisional schedule of topics, and some historicaland other material that I expect to refer to when we discuss these topics.

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DAY ONE: Schemas, formulas, models, structures

The texts quoted here illustrate important steps in the early developmentof the notion of a model.

From David Hilbert, Foundations of geometry (1899), [6] 9:

Consider a pair of numbers (x, y) from the field [the field ofalgebraic numbers] as a point and the ratios (u : v : w) of anythree numbers from as a line provided u, v are not both zero.Furthermore, let the existence of the equation

ux + vy + w = 0

mean that the point (x, y) lies on the line (u : v : w). Thereby,as is easy to see, Axioms I, 1-3 and IV are immediately satisfied.[Axiom I,1 said For every two points A, B there exists a line athat contains each of the points A, B.]

From Oswald and Young, Projective Geometry II (1918) [17]:

This is represented in fig. 19 and may be realized in a model bycutting out a rectangular strip of paper, giving it a half twist,

and pasting together the two ends.. . .

To complete the model it would be necessary to bring the twoedges labeled in fig. 18 into coincidence. This, however, is notpossible in a finite three-dimensional figure without letting thesurface cut itself. (Footnote: Plaster models showing this surfaceare manufactured by Martin Schilling of Leipzig.)

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DAY TWO: Truth definitions

From Tarski and Vaught, Arithmetical extensions of relational systems (1957)[16]:

A relational system is a sequence R = A, R0, . . . , R, . . .


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From Rudolf Carnap, Introduction to Symbolic Logic and its Applications

(1958), [2] p. 99:

Rules of designation

Primitive sign Intension Extension

a (the individual concept) ( the thing)moon moon

b (the individual concept) (the thing)sun sun

c (the individual concept) (the thing)Africa Africa

P the property the classof being spherical of spherical thingsQ the property the class

of being blue of blue thingsR the relation the class of pairs x, y such

greater than that x is greater than y

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DAY THREE: Model-theoretic entailment

From Alfred Tarski, On the concept of logical consequence (1936) [13] p.414ff:

Certain considerations of an intuitive nature will form our starting-point. Consider any class K of sentences and a sentence X whichfollows from the sentences of this class. From an intuitive stand-point it can never happen that both the class K consists onlyof true sentences and the sentences X is false. Moreover, sincewe are concerned here with the concept of logical, i.e. formal,consequence, and thus with a relation which is to be uniquely

determined by the form of the sentences between which it holds,this relation cannot be influenced in any way by empirical knowl-edge, and in particular by knowledge of the objects to which thesentence X or the sentences of the class K refer. The conse-quence relation cannot be affected by replacing the designationsof the objects referred to in these sentences by the designations ofany other objects. The two circumstances just indicated, whichseem to be very characteristic and essential for the proper con-cept of consequence, may be jointly expressed in the followingstatement:

(F) If, in the sentences of the class K and in thesentence X, the constantsapart from purely logicalconstantsare replaced by any other constants (likesigns being everywhere replaced by like signs), and ifwe denote the class of sentences thus obtained from Kby K, and the sentence obtained from X by X,then the sentence X must be true provided only thatall sentences of the class K are true.

. . .

The condition (F) could be regarded as sufficient for the sen-

tence X to follow from the class K only if the designations ofall possible objects occurred in the language in question. Thisassumption, however, is fictitious and can never be realised . . . .We must therefore look for some means of expressing the inten-tions of the condition (F) which will be completely independentof that fictitious assumption.

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Such a means is provided by semantics.

. . . One of the concepts which can be defined in terms of the con-cept of satisfaction is the concept of model. Let us assume thatin the language we are considering certain variables correspondto every extra-logical constant, and in such a way that everysentence becomes a sentential function if the constants in it arereplaced by the corresponding variables. Let K be any classof sentences. We replace all extra-logical constants which occurin the sentences belonging to L by corresponding variables, likeconstants being replaced by like variables, and unlike by unlike.In this way we obtain a class L of sentential function. An arbi-trary sequence of objects which satisfies every sentential functionof the class L will be called a model or realization of the classL of sentences (in just this sense one usually speaks of modelsof an axiom system of a deductive theory). If, in particular, theclass L consists of a single sentence X, we shall also refer to amodel of the class L as a model of the sentence X.

In terms of these concepts we can define the concept of logicalconsequence as follows:

The sentence X follows logically from the sentences ofthe class K if and only if every model of the class K

is also a model of the sentence X. . . .It seems to me that everyone who understands the content ofthe above definition must admit that it agrees quite well withcommon usage.

From Tarski, Mostowski and Robinson, Undecidable Theories (1953) [15] p.8:

A sentence is said to be a logical consequence of a set A ofsentences if it is satisfied in every realization R in which all sen-tences of A are satisfied; it is called logically true if it is satisfied

in every possible realization.

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The next two examples illustrate model-theoretic and other approaches

to defining and manipulating algebraic structures. Both authors are definingboolean algebras.

From E. V. Huntington: Postulates for the algebra of logic (1904). (Toremove a conflict with modern notations, I have used 0, 1 in place of Hunt-ingtons , . I have also used a symbol in place of a symbol which is notavailable in my TEX.)

In 1 we take as the fundamental concepts a class, K, with tworules of combination, and ; and as the fundamental propo-sitions, the following ten postulates:

Ia. a b is in the class whenever a and b are in the class.

Ib. a b is in the class whenever a and b are in the class.

IIa. There is an element 0 such that a 0 = a for every elementa.

IIb. There is an element 1 such that a 1 = a for every elementa.

IIIa. a b = b a whenever a,b,a b, and b a are in the class.

IIIb. a b = b a whenever a,b,a b, and b a are in the class.

IVa. a (b c) = (a b) (a c) whenever a,b,c,a b, a

c, b c, a (b c), and (a b) (a c) are in the class.IVb. a (b c) = (a b) (a c) whenever a,b,c,a b, a

c, b c, a (b c), and (a b) (a c) are in the class.

V. If the elements 0 and 1 in postulates IIa and IIb exist andare unique, then for every element a there is an element asuch that a a = 1 and a a = 0.

VI. There are at least two elements, x and y, in the class suchthat x = y.

. . .

DEFINITION. We shall write a b (or b a) when and onlywhen a b = b.

. . .

Any system (K, , , ) which obeys the laws of the algebra oflogic may be called a logical field . . .

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From Peter J. Cameron, Combinatorics (1994), [1] p193ff.

A partial order on X is a relation R on X which is

reflexive: (x, x) R for all x X;

antisymmetric: (x, y), (y, x) R imply x = y; and

transitive: (x, y), (y, z) R imply (x, z) R.

The pair (X, R) is called a partially ordered set, or poset forshort.

. . .

A lattice is a poset in which each pair of elements has a unique

greatest lower bound and a unique least upper bound. . . . Weuse the notation x y and x y for the g.l.b. and l.u.b. of x andy in a lattice.

. . .

(12.1.2) Proposition. Let X be a set, and two binaryoperations defined on X, and 0 and 1 two elements of X. Then(X, , , 0, 1) is a lattice if and only if the following axioms aresatisfied:

Associative laws: x (y z) = (x y) z and x (y z) =

(x y) z); Commutative laws: x y = y x and x y = y x;

Idempotent laws: x x = x x = x;

x (x y) = x = x (x y);

x 0 = 0, x 1 = 1.

. . .

A lattice L is distributive if it satisfies the two distributive laws

x (y z) = (x y) (x z),

x (y z) = (x y) (x z).

. . .

Among distributive lattices, a special class are the Boolean lat-tices. These are the distributive lattices L possessing a unaryoperation x x called complementation, satisfying

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(x y) = x y, (x y) = x y;

x x = 1, x x = 0.

(12.3.3) Theorem. A finite Boolean lattice is isomorphic tothe lattice of all subsets of a finite set X, with x interpreted asX\ x.

PROOF. Let L be a finite Boolean lattice. We have an embed-ding of L into P(X), where X is the set of join-indecomposableelements of L. To show that L = P(X), we show that any two

join-indecomposable elements are incomparablethen any set ofjoin-indecomposable elements is a down-set.

So suppose that a and b are distinct join-indecomposable ele-ments with a b. Then

a (b a) = (a b) (a a) = b 1 = b.

Since b is join-indecomposable and a = b, we must have b =b a a. . . .

The remaining two quotations illustrate the view that there is such athing as semantic reasoning.

From Johnson-Laird and Byrne, Deduction (1991) [9] p. 212:

[Against the view that there is little or no difference betweenmental models and formal rules of inference:]

[It is a] mistaken assumption that the semantic method of truthtables is not fundamentally different from the syntactic methodof proof in the propositional calculus . . . . In fact, there is a pro-found difference between the two sorts of psychological theories.They postulate different sorts of mental representation, and dif-ferent sorts of procedures: rule theories employ representationsthat are language-like and that contain variables, whereas men-tal models are remote from the structure of sentences and do

not contain variables. At the heart of deduction for rule theo-ries is the application of formal rules of inference, such as modusponens, to representations of the logical forms of sentences. Atthe heart of deduction for model theories is a search for alterna-tive models of the premises. The search makes no use of modusponens or any other formal rules of inference.

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From Hans Kamp and Uwe Reyle, From Discourse to Logic (1993) [10] p.

17ff.

. . . in the second part of the 19th century, the logician JohnVenn (18341923) developed a method, usually referred to asthe method of Venn-diagrams, for modelling the premises andconclusions of syllogistic inferences in a systematic way. To de-termine the validity of a given pattern it suffices to constructthe relevant Venn-diagram or diagrams for it. The answer canthen be read off these diagrams mechanically. In a Venn-diagramthe classes that are given in our syllogistic notation by the let-ters P , Q , R , . . . are represented by circles (or other closed curves)

whose relations of inclusion and intersection conform to the prem-ises of the pattern under consideration. If every such diagramalso verifies the conclusion of the pattern, then the pattern willbe valid, and otherwise not. . . . Venns method may be calleda semantic method inasmuch as it involves, through its dia-grams, the concept of an interpreting structure. The methodhas an analogue for the much richer notation of predicate logic. . . Both in the case of syllogistic and in that of predicate logicthe semantic method for analysing validity must be distinguishedfrom another, in which a small number of inference patterns areselected as basic and the validity of other patterns is established

by chaining two or more applications of the basic patterns to-gether. This second method . . . is known as the proof-theoreticor deductive method. It captures an aspect of deduction whichthe semantic definition of validity does not touch: in many caseswhere we reason from given premises to a certain conclusion itis only by moving in small steps that we succeed in arriving atthe final conclusion; in others, where we may see the inferencemore or less directly, it may nevertheless be necessary to breakit up into a chain of simple inferences to persuade others thatour conclusion really follows.

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DAY FOUR: Classes of structures

From Alfred Tarski, Contributions to the theory of models. I (1954), [14]:

Within the last years a new branch of metamathematics has beendeveloping. It is called the theory of models and can be regardedas a part of the semantics of formalized theories. The problemsstudied in the theory of models concern mutual relations betweensentences of formalized theories and mathematical systems inwhich these sentences hold. Every set of sentences determinesuniquely a class K of mathematical systems; in fact, the classof all those mathematical systems in which every sentence of

holds. is sometimes referred to as a postulate system for K;mathematical systems which belong to K are called models of. Among questions which naturally arise in the discussion ofthese notions, the following may be mentioned: Knowing somestructural (formal) properties of a set of sentences, what con-clusions can we draw concerning mathematical properties of thecorrelated class K of models? Conversely, knowing some math-ematical properties of a class K of mathematical systems, whatcan we say about structural properties of a set which con-stitutes a postulate system for K? Among publications in thisfield we may point out the articles and monographs [of Birkhoff,

Henkin, Abraham Robinson and Tarski].

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DAY FIVE: Telling one structure from another

From Anand Pillay, Geometric stability theory (1996), [12] Introduction:

Morley proved in the early 1960s that if T is a complete count-able first-order theory which has exactly one model (up to iso-morphism) of some given uncountable cardinality , then T hasexactly one model of cardinality for all uncountable cardinals. Such a theory is called uncountably categorical, or often just1-categorical. The classical examples are the theory of alge-braically closed fields (of a fixed characteristic) and the theory ofvector spaces (over a fixed field). Shelah, in the late 1960s, theninitiated a far-reaching programme of attempting, for an arbi-trary first-order theory T, either to classify the models of T (upto isomorphism) or to show such a classification to be impossible.This was classification theory, and at least for countable theo-ries, it reached a successful conclusion in the early 1980s. Clas-sifying the models of T amounted to, from Shelahs viewpoint,describing models by certain nice trees of cardinal invariants.If this could be done (for a given theory T), the class of mod-els of T was said to have a structure theorem. (Uncountablycategorical theories have the best structure theorem.) The im-possibility of such a classification was usually taken to amountto showing that T has 2 models of cardinality for all large .This was called a non-structure theorem. (There have been inthe meantime refinements of the structure/non-structure con-ceptual dichotomy, involving for example the question of whethermodels can be characterized by their theories in certain infinitaryor generalized logics.)

In order to implement his programme Shelah developed a seriesof fundamental dichotomies (on the class of first-order theories).The most important of these was stability. Roughly speaking,T is said to be stable if no model of T contains an infinite set oftuples on which some formula defines a linear ordering.

. . .

. . . the present period is a very exciting one for model theory, inwhich there are both grand unifying trends within model theory(in which geometric stability theory has an important role) andnew applications to and connections with core areas of mathe-matics. In fact, it is becoming increasingly clear that the scope

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of the ideas discussed above goes beyond stable theories, and

that, in a sense, the prior concentration on stability was the re-sult of the particular role of stability within Shelahs programme.But discussion and elaboration of these issues should be left foranother book.

From Phokion Kolaitis,Combinatorial Games in Database Theory (extracts)[11].

The study ofdatabase query languages has occupied a prominentplace in database theory during the past twenty years.

. . .Main Issues: Database Query Language L.

Determine whether or not a particular query Q is express-ible in L.

Characterize the class of queries that are expressible in L.

. . .

Question:

How are negative results established?

What are the main techniques used to obtain lower boundsfor expressibility?

Note:Similar issues have been studied extensively in mathematicallogic and, especially, in model theory.

. . .

Facts:

Combinatorial Games can be used to characterize the ex-pressive power of various logics.

These characterizations remain valid if only finite structuresare considered.

Combinatorial Games have been used extensively in FiniteModel Theory during the past twenty years.

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[15] Alfred Tarski, A. Mostowski and R. M. Robinson, Undecidable theories,

North-Holland, Amsterdam 1953.

[16] Alfred Tarski and Robert L. Vaught, Arithmetical extensions of rela-tional systems, Compositio Mathematica 13 (1957) 81102.

[17] Oswald Veblen and John Wesley Young, Projective GeometryII, Boston1918.

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