- University of Gothenburg
- Faculty of Humanities
- Department of Philosophy, Linguistics and Theory of Science
- Research
- Research Areas
- Logic
- The Lindström Lectures
- The Lindström Lectures 2014

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# The Lindström Lectures 2014

### Public lecture of Joan Rand Moschovakis

Intuitionistic Analysis, Forward and Backward

### Public lecture of Yiannis Moschovakis

Frege's sense and denotation as algorithm and value

### Research lecture of Joan Rand Moschovakis

Now Under Construction: Intuitionistic Reverse Analysis

### Research lecture of Yiannis Moschovakis

The logical form and meaning of attitudinal sentences

### Per (Pelle) Lindström

The Department of Philosophy, Linguistics and Theory of Science at the University of Gothenburg launched 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.

We are proud to announce that the second series of Lindström Lectures will be delivered by Joan Rand Moschovakis and Yiannis Moschovakis during October 23 (Public Lectures) and October 24, 2014 (Research Lectures).

**Public Lectures, Thursday October 23:**

- Joan Rand Moschovakis: Intuitionistic Analysis, Forward and Backward. 15:15 - 16:15.
- Yiannis Moschovakis:Frege's sense and denotation as algorithm and value. 17:00 - 18:00.

**Research Lectures, Friday October 24:**

- Joan Rand Moschovakis: Now Under Construction: Intuitionistic Reverse Analysis. 13:15 - 15:00.
- Yiannis Moschovakis: The logical form and meaning of attitudinal sentences. 15:30 - 17:15.

Joan Rand Moschovakis is Emerita Professor of Mathematics at Occidental College, Los Angeles, California. She obtained her undergraduate degree in mathematics at the University of California Berkeley, and completed her doctoral degree in mathematics at the University of Wisconsin, Madison, under the direction of Stephen Kleene. She has also taught in the Graduate Program in Logic and Algorithms at the University of Athens, Greece. Her research interests include Foundations of Intuitionistic Analysis, Intuitionistic Interpretations of Classical Mathematics, Classical Interpretations of Intuitionistic Mathematics, Admissible Rules of Intuitionistic Logic, and History and Philosophy of Intuitionistic Logic.

Yiannis Nicholas Moschovakis is Emeritus Professor of Mathematics at University of Southern California, Los Angeles. During 1996-2005 he also served as professor of mathematics at the University of Athens. He obtained his undergraduate and master's degrees in mathematics at the Massachusetts Institute of Technology, and completed his doctoral degree at the University of Wisconsin, Madison, under the direction of Stephen Kleene. His research Interests include: Recursion Theory, Descriptive Set Theory, Foundations of Computer Science, Philosophy of Language and Mathematics.

Intuitionistic Analysis, Forward and Backward

*Thursday October 23, 2014. 15:15 - 16:15 in T302, Olof Wijksgatan 6.*

In the early 20th century the Dutch mathematician L. E. J. Brouwer questioned the universal applicability of the Aristotelian law of excluded middle and proposed basing mathematical analysis on informal intuitionistic logic, with natural numbers and *choice sequences* (infinitely proceeding sequences of freely chosen natural numbers) as objects. For Brouwer, numbers and choice sequences were mental constructions which by their nature satisfied mathematical induction, countable and dependent choice, bar induction, and a continuity principle contradicting classical logic. Half a century later, S. C. Kleene and R. E. Vesley developed a significant part of Brouwer's intuitionistic analysis in a formal system **FIM**. Kleene's function-realizability interpretation proved **FIM** consistent relative to its classically correct subsystem **B**, facilitating a detailed comparison of intuitionistic with classical analysis **C**. Continuing Brouwer's work into the 21st century, Wim Veldman and others are now developing an intuitionistic reverse mathematics parallel to, but diverging significantly from, both classical reverse mathematics as established by H. Friedman and S. Simpson, and constructive reverse mathematics in the style of E. Bishop. This lecture provides the basics of intuitionistic analysis and a sketch of its reverse development.

Frege's sense and denotation as algorithm and value

*Thursday October 23, 2014. 17:00 - 18:00 in T302, Olof Wijksgatan 6.*

In his classical 1892 article *On sense and denotation*, Frege associates with each declarative sentence its *denotation* (truth value) and also its *sense* (meaning) "wherein the mode of presentation [of the denotation] is contained". For example, *1 1=2* and *there are infinitely many prime numbers* are both true, but they mean different things - they are not *synonymous*. Frege [1892] has an extensive discussion of senses and their properties, including, for example, the claim that *the same sense has different expressions in different languages or even in the same language*; but he does not say *what senses are* or give an axiomatization of their theory which might make it possible to prove these claims. This has led to a large literature by philosophers of language and logicians on the subject, which is still today an active research topic. A plausible approach that has been discussed by many (including Michael Dummett) is suggested by the "wherein" quote above: in slogan form, * the sense of a sentence is an algorithm which computes its denotation*. Coupled with a rigorous definition of (abstract, possibly infnitary) *algorithms*, this leads to a rich theory of meaning and synonymy for suitably formalized fragments of natural language, including Richard Montague's *Language of Intensional Logic*. My aim in this talk is to discuss with as few technicalities as possible how this program can be carried out and what it contributes to our understanding of some classical puzzles in the philosophy of language.

Now Under Construction: Intuitionistic Reverse Analysis

*Friday October 24, 2014. 13:15 - 15:00 in T307, Olof Wijksgatan 6.*

Each variety of reverse mathematics attempts to determine a minimal axiomatic basis for proving a particular mathematical theorem. Classical reverse analysis asks which set existence axioms are needed to prove particular theorems of classical second-order number theory. Intuitionistic reverse analysis asks which intuitionistically accepted properties of numbers and functions suffice to prove particular theorems of intuitionistic analysis using intuitionistic logic; it may also consider the relative strength of classical principles which do not contradict intuitionistic analysis.

S. Simpson showed that many theorems of classical analysis are equivalent, over a weak system **PRA** of primitive recursive arithmetic, to one of the following set existence principles: recursive comprehension, arithmetical comprehension, weak Konig's Lemma, arithmetical transfinite recursion, Π^{1}_{1} comprehension. Intermediate principles are studied also. Intuitionistic analysis depends on function existence principles: countable and dependent choice, fan and bar theorems, continuous choice. The fan and bar theorems have important mathematical equivalents. W. Veldman, building on a proof by T. Coquand, recently showed that over intuitionistic two-sorted recursive arithmetic **BIM** the principle of open induction is strictly intermediate between the fan and bar theorems, and is equivalent to intuitionistic versions of a number of classical theorems. Intuitionistic logic separates classically equivalent versions of countable choice, and it matters how decidability is interpreted. R. Solovay recently showed that Markov's Principle is surprisingly strong in the presence of the bar theorem. The picture gradually becomes clear.

The logical form and meaning of attitudinal sentences

*Friday October 24, 2014. 15:30 - 17:15 in T307, Olof Wijksgatan 6.*

The language **L** over a fixed vocabulary K is an (applied) * typed λ-calculus* with additional constructs for *acyclic recursion* and * attitudinal application*, an extension of Montague's *Language of Intensional Logic* **LIL** as formulated by Daniel Gallin. It is denotationally interpretable in the classical typed λ-calculus over K, but intensionally richer: in particular, it can define the *referential intension* of each term A, an abstract algorithm which computes the denotation of A and provides a plausible explication of its meaning. The key mathematical construction of **L** is an effective *reduction calculus* which compiles each term A into an (essentially) unique *canonical form* cf(A), a denotational term which explicates the *logical form* of A and from which the referential intension of A can be read off. The central open problem about **L** (over a finite, interpreted vocabulary) is the * decidability of global synonymy* - and it is a problem about the classical, interpreted typed λ-calculus.

*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.