- University of Gothenburg
- Faculty of Humanities
- Department of Philosophy, Linguistics and Theory of Science
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- Research Areas
- Logic
- The Lindström Lectures
- The Lindström Lectures 2013

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# The Lindström Lectures 2013

## Public lecture: Wilfrid Hodges on Ibn Sina on the foundations of logic

## Research lecture: Wilfrid Hodges on Ibn Sina on discharging assumptions in proofs

## Wilfrid Hodges

## Per (Pelle) Lindström

##
Wilfrid Hodges

The Department of Philosophy, Linguistics and Theory of Science at the University of Gothenburg is proud to announce a lecture series to celebrate the singular achievements of Pelle Lindström, former professor of logic at the department. Annually, a distinguished logician will be invited to deliver a general lecture to the public, and a specialized presentation at the logic seminar.

The Lindström Lectures series will be inaugurated this fall with lectures by Wilfrid Hodges, Emeritus Professor of Mathematics, Queen Mary, University of London.

*Thursday November 14, 2013. 16:00 - 17:30 in T302, Olof Wijksgatan 6.*

Ibn Sina (Avicenna, 11th century Iran) believed that the foundations of logic lie in metaphysics. He complained bitterly that this has led people to confuse logic itself with its foundations and dress up metaphysics as logic. His own description of the foundations of logic is in overtly ontological language. But from a modern perspective it becomes clear that in fact he is talking about methodological issues, like how to represent occurrences of a component within a compound, and whether the primitive notions of a theory should be stipulated from outside (as in Tarski) or incorporated into the objects (as in web ontology and object-based programming). This all has strong implications for any project to formalise Ibn Sina's logic. My own readings of some key passages are different from the traditional metaphysical ones, and seem to me more intelligible and highly comparable with some modern metalogical and metalinguistic views; but then I have a deaf ear for metaphysics.

*Friday November 15, 2013. 13:15 - 15:00 in T307, Olof Wijksgatan 6, Göteborg.*

Ibn Sina (Avicenna, 11th century Iran) wrote thousands of pages of commentary on Aristotle's logical works. Among them is a short section on how to understand proofs by reductio ad absurdum. For Ibn Sina the problem is how to read an RAA proof so that the conclusion self-evidently follows from the premises, making minimal changes to the written form of the proof. In remarks almost certainly based on the Arabic Euclid, he observes (like Frege) that an assumption is stated so as not to have to keep repeating it at every stage of the argument. He infers that to see what is really meant in the argument, one should restore the assumption at every step where it is used. (Again this parallels Frege, though the motivation is different.) A key question is how Ibn Sina knows that adding these assumptions preserves validity. Writing out the answer as a metaprinciple yields a powerful rule of inference, which needs not much else added to give a complete sequent calculus for first-order logic - though Ibn Sina would have strongly disapproved of treating it that way.

Professor Wilfrid Augustine Hodges is best known for his influential and wide-ranging work in mathematical logic, as expounded in his exquisitely crafted papers and books, including a definitive 780-page graduate text on model theory. His recent research work has focused on general semantics, cognitive aspects of logic, and history of logic, especially Arabic logic.

Professor Hodges attended New College, Oxford (1959–65), where he received degrees in both Literae Humaniores and (Christianic) Theology. In 1970 he was awarded a doctorate for a thesis in Logic. He lectured in both Philosophy and Mathematics at Bedford College, University of London. He was Professor of Mathematics at Queen Mary, University of London from 1987 to 2006. He has held visiting appointments in the department of philosophy at the University of California and in the department of mathematics at University of Colorado. He was President of the British Logic Colloquium, of the European Association for Logic, Language and Information and of the Division of Logic, Methodology, and Philosophy of Science. In 2009 he was elected a Fellow of the British Academy.

*This is a much condensed version of the obituary by Väänänen and Westerståhl in Theoria 2010 (76) pages 100-107.*

Per Lindström, or Pelle Lindström as he insisted on being called, was born on April 9, 1936, and spent most of his academic life at the Department of Philosophy, University of Gothenburg, in Sweden, where he was employed first as a lecturer (‘docent’) and, from 1991 until his retirement in 2001, as a Professor of Logic.

Lindström is most famous for his work in model theory. In 1964 he made his first major contribution, the so-called Lindström’s test for model completeness. In 1966 he proved the undefinability of well-order in L_{ω1ω} (obtained independently and in more generality by Lopez-Escobar). The same year he also introduced the concept of a Lindström quantifier, which has now become standard in model theory, theoretical computer science, and formal semantics.

It was his 1969 paper ‘On extensions of elementary logic’ (in Theoria), where he presented his famous characterizations of first-order logic—Lindström’s Theorem—in terms of properties such as compactness, completeness, and Löwenheim-Skolem properties, that was first recognized as a major contribution to logic. It laid the foundation of what has become known as abstract model theory. The proof was based on Ehrenfeucht-Fraïssé games, a concept he came up with independently, and on a new proof of interpolation. Several other characterizations of first-order logic followed in later papers.

Beginning at the end of the 1970’s, Lindström turned his attention to the study of formal arithmetic and interpretability. He started a truly systematic investigation of this topic, which had been somewhat dormant since Feferman’s pioneering contributions in the late 1950’s. In doing so he invented novel technically advanced tools, for example, the so-called Lindström fixed point construction, a far-reaching application of Gödel’s diagonalization lemma to define arithmetical formulas with specific properties.

Pelle Lindström had an exceptionally clear and concise style in writing mathematical logic. His 1997 book, Aspects of Incompleteness, remains a perfect example: it provides a systematic introduction to his work in arithmetic and interpretability. The book is short but rich in material.

Throughout his life, Pelle Lindström also took an active interest in philosophy. He participated in the debate following Roger Penrose’s new version of the argument that Gödel’s Incompleteness Theorems show that the human mind is not mechanical. He presented his own philosophy of mathematics, which he called ‘quasi-realism’, in a paper in The Monist in 2000. It is based on the idea that the ‘visualizable’ parts of mathematics are beyond doubt (and that classical logic holds for them). He counted as visualizable not only the ω-sequence

of natural numbers but also arbitrary sets of numbers, the latter visualizable as branches in the infinite binary tree, whereas nothing similar can be said for sets of sets of numbers, for example.

Pelle Lindström passed away in Gothenburg, Sweden, on August 21, 2009, after a short period of illness.

(Photo from Mathematisches Forschungsinstitut Oberwolfach)