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In Baghdad in the mid twelfth century Abū al-Barakāt proposes a radical new procedure for finding the conclusions of premise-pairs in syllogistic logic, and for identifying those premise-pairs that have no conclusions. The procedure makes no use of features of the standard Aristotelian apparatus, such as conversions or syllogistic figures. In place of these al-Barakāt writes out pages of diagrams consisting of labelled horizontal lines. He gives no instructions and no proof that the procedure will yield correct results. So the reader has to work out what his procedure is and whether it is correct. The procedure turns out to be insightful and entirely correct, but this paper may be the first study to give a full description of the procedure and a rigorous proof of its correctness.
Ibn Sīnā (11th century, greater Persia) proposed an analysis of arguments by reductio ad absurdum. His analysis contains, perhaps for the first time, a workable method for handling the making and discharging of assumptions in a formal proof. We translate the relevant text of Ibn Sīnā and put his analysis into the context of his general approach to logic.
In this chapter we discuss, first, how the Arabic logicians up to the end of the tenth century took over Greek material and added to it material of their own and how they reshaped the subject of logic in the process. We have included references to the young Averroes, although he wrote in the twelfth century, inasmuch as he belongs in the tradition of al-Fārābī (d. 950). After this we turn to the formal innovations of Avicenna's in the early eleventh century. Many of the questions that we discuss are treated also in Street (2004).
THE GREEK LOGICAL HERITAGE
Arabic logic as a branch of philosophy was heir to ancient Greek logic, and it belonged essentially to the Peripatetic tradition. Arabic grammar, Islamic jurisprudence and Islamic disputative theology (kalām) developed independent methods of reasoning and inevitably there was some interaction between these methods and those of logic as a philosophical discipline. This interaction ranged from conflict to absorption. The Greek Peripatetic logic was embodied in Aristotle's logical texts, which later became known as the Organon, together with the commentaries on them by Roman Empire scholars of various philosophical persuasions. These commentaries were the product of an activity which had run for eight centuries when the Arab philosophers became aware of it.
The Arabic Organon was in fact the extended Organon first contemplated in Late Antiquity, which began with Porphyry's Isagoge as an introduction and went on to include Aristotle's Rhetoric and Poetics. But what was only programmatic in Late Antiquity became a reality for the Arabic logicians. They conceived the Organon as embodying a system of logic. The formal heart of the system lay in its third book, the Prior Analytics, which aims to give the general theory of reasoning or of the syllogism (qiyās). The first two treatises, i.e. Categories (although its place here was challenged, in particular by Avicenna) and On Interpretation, are preparatory to the formal part. The remaining volumes adapt the theory of reasoning to different fields of human activity: to scientific activity, but also to social fields of communication. Logic as providing a method for science was the object of Aristotle's Posterior Analytics, while logic as providing a tool in order to systematise various fields of social communication was the object of the rest of the books of the Organon.
Abstract. Starting from any language provided with sentence meanings and a grammar, and using the principle that the meaning of a phrase is what it contributes to the meanings of sentences containing it, we derive a semantics for the whole language. The semantics is necessarily compositional and carries a structure of semantic categories. With a further assumption on the grammatical heads of phrases, we can assume that the meanings of head words are functions from meanings to meanings, in the Frege style. The paper sketches these results and considers how far the semantics of Husserl, Frege and Tarski can be seen as examples of this pattern.
When Alfred Tarski wrote his famous definition of truth  (1933) for a formal language, he had several stated aims. His chief aim was to define truth of sentences. Giving correct meanings of other expressions of the language was nowhere in his list of aims at all; it was a happy accident that a general semantics fell out of his truth definition.
So the following facts, all very easily proved, came to me as a surprise. Given any notion of meaning for sentences (for example, a specification of when they are true and when not), and assuming some simple book-keeping conditions, there is a canonical way of extending this notion to a semantics for the whole language. I call it the fregean extension; it is determined up to the question which pairs of expressions have the same meaning. Tarski's semantics for first-order logic is the fregean extension of the truth conditions for sentences. Afewmore book-keeping conditions guarantee that the fregean extension can be chosen to have good functional properties of the kind often associated with Frege and with type-theoretic semantics.
Tarski himselfwas certainly interested in the question howfar his solution of his problem was canonical, and we can learn useful things from his discussion of the issue. But the main results below on fregean extensions come closer to the linguistic and logical concerns of Frege and Husserl, a generation earlier than Tarski. Husserl has been unjustly neglected by logicians, and Frege's innovations in linguistics deserve to be better known.
We can use the compositional semantics of Hodges  to show that any compositional semantics for logics of imperfect information must obey certain constraints on the number of semantically inequivalent formulas. As a corollary, there is no compositional semantics for the ‘independence-friendly’ logic of Hintikka and Sandu (henceforth IF) in which the interpretation in a structure A of each 1 -ary formula is a subset of the domain of A (Corollary 6.2 below proves this and more). After a fashion, this rescues a claim of Hintikka and provides the proof which he lacked:
… there is no realistic hope of formulating compositional truth-conditions for [sentences of IF], even though I have not given a strict impossibility proof to that effect.
(Hintikka  page 110ff.) One curious spinoff is that there is a structure of cardinality 6 on which the logic of Hintikka and Sandu gives nearly eight million inequivalent formulas in one free variable (which is more than the population of Finland).
We thank the referee for a sensible change of notation, and Joel Berman and Stan Burris for bringing us up to date with the computation of Dedekind's function (see section 4). Our own calculations, utterly trivial by comparison, were done with Maple V.
The paper Hodges  (cf. ) gave a compositional semantics for a language with some devices of imperfect information. The language was complicated, because it allowed imperfect information both at quantifiers and at conjunctions and disjunctions.
We consider structures A consisting of an abelian group with a subgroup AP distinguished by a 1-ary relation symbol P, and complete theories T of such structures. Such a theory T is (κ, λ)-categorical if whenever A, B are models of T of cardinality λ with AP=BP and |AP|=κ, there is an isomorphism from A to B which is the identity on AP. We state all true theorems of the form: If T is (κ, λ)-categorical then T is (κ′, λ′)-categorical.
Relative categoricity has been for a long time one of the least popular branches of stability theory. In 1986 Saharon Shelah published a formidable paper giving a deep classification under a weak form of the generalised continuum hypothesis, and concluding ‘There are no particular problems (especially if you have read §4)’. I am not sure whether his paper contains any results that one could explain to a non-specialist. Since Shelah's paper there has been nothing like the Paris breakthrough that suddenly made classical stability theory all the rage in 1979. To the best of my knowledge nobody has made a serious attempt at a geometric theory outside the context of covers. Let me add a morsel of temptation: relative categoricity was probably the first area where first-order stability methods were applied to unstable theories.
§1. Introduction. I dedicate this essay to the two-dozen-odd people whose refutations of Cantor's diagonal argument (I mean the one proving that the set of real numbers and the set of natural numbers have different cardinalities) have come to me either as referee or as editor in the last twenty years or so. Sadly these submissions were all quite unpublishable; I sent them back with what I hope were helpful comments. A few years ago it occurred to me to wonder why so many people devote so much energy to refuting this harmless little argument—what had it done to make them angry with it? So I started to keep notes of these papers, in the hope that some pattern would emerge.
These pages report the results. They might be useful for editors faced with similar problem papers, or even for the authors of the papers themselves. But the main message to reach me is that there are several points of basic elementary logic that we usually teach and explain very badly, or not at all.
In 1995 an engineer named William Dilworth, who had published a refutation of Cantor's argument in the Transactions of the Wisconsin Academy of Sciences, Arts and Letters, sued for libel a mathematician named Underwood Dudley who had called him a crank ( pp. 44f, 354).
These notes should be read with those of Zoé Chatzidakis. We report some results from Hrushovski and Pillay. The main items in this paper are
an analogue for pseudofinite fields of ZiPber's Irreducibility Theorem (Theorem 23);
lemmas relating simplicity properties of an algebraic group G to its restriction G(F) to a pseudofinite field F (§6);
a fast though non-effective model-theoretic proof of a result of Matthews, Vaserstein and Weispfeiler on reduction at primes (Theorem 33; see also the similar argument in the last section of).
The main difference from is that I avoid the local stability arguments of Hrushovski and Pillay. In fact the proof of the Irreducibility Theorem removes all the stability arguments beyond the ‘(S1) property’, without adding anything in their place. (Later I quote the Theorem of §3, whose proof—at least in its present guise—uses forking, canonical bases and the definability of types.) It took several shoves to remove the parts of the argument that rely on stability; Frank Wagner and John Wilson delivered the final push during the Blaubeuren meeting. I think a fair comment would be that stability theory has powerful methods for showing that things are first-order definable, and this was the role that it played in the original argument. But sometimes, after the event, one sees that other devices may do the job faster.
And every hair a sheave shall be, And every sheave a golden tree.
George Peele, The Old Wife's Tale (1595).
Direct products are everywhere in algebra. They are the first general form of construction that the student meets – thanks to groups and vector spaces. From rings to automata they are part of the everyday scenery.
Why do so many types of mathematical object allow this construction? G. Birkhoff gave part of the answer when he showed that every class defined by identities is closed under direct products. J. C. C. McKinsey showed the same for classes defined by universal Horn sentences. A. I. Mal′tsev pointed out the intimate link between direct products and another common construction with a quite different history: presenting a structure by generators and relations.
Once these basic facts about direct products are established, we can set out in at least three directions. First, there is plenty more to say about generators and relations. They can be used to describe not just single structures, but constructions taking one class of structures to another. Many well-known constructions (such as tensor products and polynomial rings) have this form, and we can study the logical features which they share. This topic has recently had a shot of adrenalin through the development of software specification languages based on ‘initial semantics’.
The second useful direction is the study of other constructions based on direct products.
Now that we have structures in front of us, the most pressing need is to start classifying them and their features. Classifying is a kind of defining. Most mathematical classification is by axioms or defining equations – in short, by formulas. This chapter could have been entitled ‘The elementary theory of mathematical classification by formulas’.
Notice three ways in which mathematicians use formulas. First, a mathematician writes the equation ‘y = 4x2’. By writing this equation one names a set of points in the plane, i.e. a set of ordered pairs of real numbers. As a model theorist would put it, the equation defines a 2-ary relation on the reals. We study this kind of definition in section 2.1.
Or second, a mathematician writes down the laws
(*) For all x, y and z, x ≤y and y ≤ z imply x ≤ z;
for all x and y, exactly one of x≤y,y≤x,x = y holds.
By doing this one names a class of relations, namely those relations ≤ for which (*) is true. Section 2.2 lists some more examples of this kind of naming. They cover most branches of algebra.
This is an up-to-date and integrated introduction to model theory, designed to be used for graduate courses (for students who are familiar with first-order logic), and as a reference for more experienced logicians and mathematicians. Model theory is concerned with the notions of definition, interpretation and structure in a very general setting, and is applied to a wide variety of other areas such as set theory, geometry, algebra (in particular group theory), and computer science (e.g. logic programming and specification). Professor Hodges emphasises definability and methods of construction, and introduces the reader to advanced topics such as stability. He also provides the reader with much historical information and a full bibliography, enhancing the book's use as a reference.
Every person had in the beginning one only proper name, except the savages of Mount Atlas in Barbary, which were reported to be both nameless and dreamless.
In this first chapter we meet the main subject-matter of model theory: structures.
Every mathematician handles structures of some kind – be they modules, groups, rings, fields, lattices, partial orderings, Banach algebras or whatever. This chapter will define basic notions like ‘element’, ‘homomorphism’, ‘substructure’, and the definitions are not meant to contain any surprises. The notion of a (Robinson) ‘diagram’ of a structure may look a little strange at first, but really it is nothing more than a generalisation of the multiplication table of a group.
Nevertheless there is something that the reader may find unsettling. Model theorists are forever talking about symbols, names and labels. A group theorist will happily write the same abelian group multiplicatively or additively, whichever is more convenient for the matter in hand. Not so the model theorist: for him or her the group with ‘·’ is one structure and the group with ‘+’ is a different structure. Change the name and you change the structure.
This must look like pedantry. Model theory is an offshoot of mathematical logic, and I can't deny that some distinguished logicians have been pedantic about symbols. Nevertheless there are several good reasons why model theorists take the view that they do. For the moment let me mention two.