Metadata of the book and chapters that will be visualized online
Book series name
Studies in Universal Logic
Book title
The Life and Work of Leon Henkin
Book subtitle
Essays on His Contributions
Book copyright year
2014
Book copyright holder
Springer International Publishing Switzerland
Chapter title
Leon Henkin
Corresponding Author
Family name
Manzano
Particle
Given Name
María
Suffix
Division
Department of Philosophy, Logic and Aesthetics, Faculty of
Philosophy, Edificio FES
Author
Organization
University of Salamanca
Address
Campus Unamuno, 37007, Salamanca, Spain
E-mail
[email protected]
Family name
Alonso
Particle
Given Name
Enrique
Suffix
Division
Department of Logic and Philosophy of Science, Faculty of
Philosophy
Organization
Autonomous University of Madrid
Address
Avda. Tomás y Valiente s/n, Campus Canto Blanco, 28049,
Madrid, Spain
E-mail
Abstract
[email protected]
Leon Henkin was born in 1921 in Brooklyn, New York, in the heart of a Jewish family
that originally came from Russia. He died at the beginning of November in 2006. He
was an extraordinary logician, an excellent teacher, a dedicated professor, and an
exceptional person overall. He had a huge heart and he was passionately devoted to
his ideas of pacifism and socialism (in the sense of belonging to the left). He not only
believed in equality, but also worked actively to see that it was brought about.
Keywords
History of logic – Biographical studies of mathematicians – Foundational studies in
logic – Henkin
Chapter title
Lessons from Leon
Corresponding Author
Family name
Resek
Particle
Given Name
Diane
Suffix
Division
Department of Mathematics
Organization
San Francisco State University
Address
Thornton Hall 937, 1600 Holloway Ave, San Francisco, CA, 94132,
USA
E-mail
Abstract
[email protected]
This paper describes some lesson learned from Leon Henkin while the author wrote a
dissertation on Cylindric Algebras and collaborated with Henkin on several projects in
math education. The lessons were about writing, lecturing, relating to administrators,
and making decisions.
Keywords
Algebraic logic
Chapter title
Tracing Back “Logic in Wonderland” to My Work with Leon Henkin
Corresponding Author
Family name
Movshovitz-Hadar
Particle
Given Name
Nitsa
Suffix
Division
Abstract
Organization
Technion, Israel Institute of Technology
Address
Technion City, Haifa, 3200003, Israel
E-mail
[email protected]
The book Logic in Wonderland , I co-authored with Atara Shriki, was published in
early 2013, about 40 years after I started my Ph.D. thesis with Leon Henkin. The time
gap did not diminish his influence. A few anecdotes from my early days with Leon and
two sample tasks from the book recently published illustrate it.
Keywords
Leon Henkin – Thesis advisor – Mathematical humor – Logic in Wonderland – The
teaching and learning of logic
Chapter title
Henkin and the Suit
Corresponding Author
Family name
Visser
Particle
Given Name
Albert
Suffix
Division
Philosophy, Faculty of Humanities
Organization
Utrecht University
Address
Janskerkhof 13, 3512BL, Utrecht, The Netherlands
E-mail
[email protected]
Chapter title
A Fortuitous Year with Leon Henkin
Corresponding Author
Family name
Feferman
Particle
Given Name
Solomon
Suffix
Abstract
Division
Department of Mathematics
Organization
Stanford University
Address
Stanford, CA, 94305-2125, USA
E-mail
[email protected]
This is a personal reminiscence about the work I did under the direction of Leon
Henkin during the last year of my graduate studies, work that proved to be fortuitous
in the absence of Alfred Tarski, my thesis advisor.
Keywords
Completeness of predicate calculus – Henkin’s proof of completeness –
Incompleteness theorems – Formal consistency statements – Arithmetization of
metamathematics – Interpretability of theories
Chapter title
Leon Henkin and a Life of Service
Corresponding Author
Family name
Wells
Particle
Given Name
Benjamin
Suffix
Division
Departments of Mathematics and of Computer Science, College of
Arts and Sciences
Abstract
Organization
University of San Francisco
Address
2130 Fulton St., San Francisco, CA, 94117, USA
E-mail
[email protected]
For 45 years, Leon Henkin provided dedicated, unstinting service to people learning
mathematics. During most of that time, the author had personal contact with him.
Henkin’s seminars, projects, letters, and advice influenced the author’s career path
on many control points.
Keywords
Henkin – Logic – Mathematics education – Service – Personal recollection
Chapter title
Leon Henkin and Cylindric Algebras
Corresponding Author
Family name
Donald Monk
Particle
Given Name
J.
Suffix
Abstract
Division
Mathematics Department
Organization
University of Colorado
Address
Boulder, CO, 80309-0395, USA
E-mail
[email protected]
This is a description of the contributions of Leon Henkin to the theory of cylindric
algebras.
Keywords
Cylindric algebras – Permutation models – Dilation – Twisting
Chapter title
A Bit of History Related to Logic Based on Equality
Corresponding Author
Family name
Andrews
Particle
Given Name
Peter
Given Name
B.
Suffix
Abstract
Division
Department of Mathematical Sciences
Organization
Carnegie Mellon University
Address
Pittsburgh, PA, 15213-3890, USA
E-mail
[email protected]
This historical note illuminates how Leon Henkin’s work influenced that of the author.
It focuses on Henkin’s development of a formulation of type theory based on equality,
and the significance of this contribution.
Keywords
Type theory – Equality – Henkin – Axiom – Extensionality
Chapter title
Pairing Logical and Pedagogical Foundations for the Theory of Positive Rational
Numbers—Henkin’s Unfinished Work
Corresponding Author
Family name
Movshovitz-Hadar
Particle
Given Name
Suffix
Division
Nitsa
Abstract
Organization
Technion, Israel Institute of Technology
Address
Technion City, Haifa, 3200003, Israel
E-mail
[email protected]
Five different ways of “founding” the mathematical theory of positive rational
numbers for further logical development are presented as Leon Henkin outlined in
1979 in the form of notes for a future paper, suggesting that pairing them up with five
representation models could possibly yield further pedagogical development. In turn,
he thought that this would indicate a large variety of ways in which this number
system, as any other, can be introduced in a classroom setting—many of them quite
different from traditional ways. His wish was to explore how varying modes of
deductive development can be mirrored in varying classroom treatments rooted in
children’s experience and activities. This dream never came to full fruition.
Preliminary ideas about corresponding pedagogical embodiments for each logical
development are briefly presented here for the next generation of researchers in
mathematics-education and curriculum developers to contemplate with, bearing in
mind that Henkin’s work itself may need additional polishing.
Keywords
Logical foundation – Pedagogical foundation – Rational numbers – The teaching and
learning of rational numbers – Leon Henkin
Chapter title
Leon Henkin the Reviewer
Corresponding Author
Family name
Martínez Vidal
Particle
Given Name
Concha
Suffix
Author
Division
Department of Logic and Moral Philosophy, Faculty of Philosophy
Organization
University of Santiago de Compostela
Address
Praza de Mazarelos, 15782, Santiago de Compostela, Spain
E-mail
[email protected]
Family name
Úbeda Rives
Particle
Given Name
José
Given Name
Pedro
Suffix
Division
Department of Logic and Philosophy of Science, Faculty of
Philosophy and Education Sciences
Organization
University of Valencia
Address
Av. Blasco Ibáñez, 30, 46010, Valencia, Spain
E-mail
[email protected]
Abstract
In this chapter, we intend to look at Henkin’s reviews, a total of forty-six. The books
and papers reviewed deal with a large variety of subjects that range from the
algebraic treatment of logical systems to issues concerning the philosophy of
mathematics and, not surprisingly—given his active work in mathematical education
—one on the teaching of this subject. Most of them were published in The J ournal
of Symbolic Logic and only one in the Bulletin of the American
Mathemat ic al S ociety . We will start by sorting these works into subjects and
continue by providing a brief summary of each of them in order to point out those
aspects that are originally from Henkin, and what we take to be mistakes. This
analysis should disclose Henkin’s personal views on some of the most important
results and influential books of his time; for instance, Gödel’s discovery of the
consistency of the Continuum Hypothesis with the axioms of set theory or
Church’s Introduction to Logic . It should also provide insight into how various
outstanding results in logic and the foundations of mathematics were seen at the time.
Finally, we will relate Henkin’s reviews to Henkin’s major contributions.
Keywords
Leon Henkin – Reviews – Logic systems – Type theory – Metalogic – Algebraic logic
– Philosophy of logic and mathematics – Mathematical education
Chapter title
Henkin’s Theorem in Textbooks
Corresponding Author
Family name
Alonso
Particle
Given Name
Enrique
Suffix
Division
Department of Logic and Philosophy of Science, Faculty of
Philosophy, Avda. Tomás y Valiente s/n
Abstract
Organization
Autonomous University of Madrid
Address
Campus Canto Blanco, 28049, Madrid, Spain
E-mail
[email protected]
Our aim in this paper is to examine the incorporation and acceptance of Henkin’s
completeness proof in some textbooks on classical logic. The first conclusion of this
paper is that the inclusion of Henkin’s completeness proof into the standards of Logic
was neither quick nor easy. Surprising as it may seem today, most of the textbooks
published in the 1950s did not include a section for this proof, nor presented it in any
way. A question we should try to answer is at what moment does Henkin’s proof of
completeness for first order logic begin to be considered as a part of the standards of
elementary logic. This point brings us to a discussion on the way in which the specific
gains of Henkin’s proof have been assessed in literature. The possibility of using
Henkin’s methods in a wide variety of formal systems made completeness a general
property belonging to foundations of logic, leaving the realm of model theory for
quantification languages where it was previously located.
Keywords
History of logic – Case studies in history of mathematics – Completeness in
elementary logic
Chapter title
Henkin on Completeness
Corresponding Author
Family name
Manzano
Particle
Given Name
María
Suffix
Division
Abstract
Organization
Universidad de Salamanca
Address
Salamanca, Spain
E-mail
[email protected]
The Completeness of Formal Systems is the title of the thesis that Henkin
presented at Princeton in 1947 under the supervision of Alonzo Church. A few years
after the defense of his thesis, Henkin published two papers in the Journal of
Symbolic Logic : the first, on completeness for first-order logic (Henkin in J. Symb.
Log. 14(3):159–166, 1949), and the second one, devoted to completeness in type
theory (Henkin in J. Symb. Log. 15(2):81–91, 1950). In 1963, Henkin published a
completeness proof for propositional type theory (Henkin in J. Symb. Log.
28(3):201–216, 1963), where he devised yet another method not directly based on his
completeness proof for the whole theory of types.
In this paper, these tree proofs are analyzed, trying to understand not just the result
itself but also the process of discovery, using the information provided by Henkin in
Bull. Symb. Log. 2(2):127–158, 1996.
In the third section, we present two completeness proofs that Henkin used to teach us
in class. It is surprising that the first-order proof of completeness that Henkin
explained in class was not his own but was developed by using Herbrand’s theorem
and the completeness of propositional logic. In 1963, Henkin published An
extension of the Craig – Lyndon interpolation theorem , where one can find a
different completeness proof for first-order logic; this is the other completeness proof
Henkin told us about.
We conclude this paper, by introducing two expository papers on this subject. Henkin
was an extraordinary insightful professor, and in 1967, he published two works that
are very relevant for the subject addressed here: Truth and provability (Henkin in
Philosophy of Science Today, pp. 14–22, 1967) and Completeness (Henkin in
Philosophy of Science Today, pp. 23–35, 1967).
Keywords
Henkin – Truth – Provability – Completeness – Type theory – First-order logic –
Propositional type theory – Equality – Interpolation – Craig – Herbrand
Chapter title
The Countable Henkin Principle
Corresponding Author
Family name
Goldblatt
Particle
Given Name
Robert
Suffix
Abstract
Division
School of Mathematics, Statistics and Operations Research
Organization
Victoria University of Wellington
Address
PO Box 600, Wellington, 6140, New Zealand
E-mail
[email protected]
This is a revised and extended version of an article that encapsulates a key aspect of
the “Henkin method” in a general result about the existence of finitely consistent
theories satisfying prescribed closure conditions. This principle can be used to give
streamlined proofs of completeness for logical systems, in which inductive
Henkin-style constructions are replaced by a demonstration that a certain theory
“respects” some class of inference rules. The countable version of the principle has a
special role and is applied here to omitting-types theorems, and to strong
completeness proofs for first-order logic, omega-logic, countable fragments of
languages with infinite conjunctions, and a propositional logic with probabilistic
modalities. The paper concludes with a topological approach to the countable
principle, using the Baire Category Theorem.
Keywords
Deducibility – Inference – Finitely consistent – Maximally consistent – Countable
Henkin Principle – Lindenbaum’s Lemma – Completeness – Omitting types –
Probabilistic modality – Archimedean inference – Baire Category Theorem
Chapter title
Reflections on a Theorem of Henkin
Corresponding Author
Family name
Gunther
Particle
Given Name
William
Suffix
Author
Division
Department of Mathematical Sciences
Organization
Carnegie Mellon University
Address
Pittsburgh, PA, 15213, USA
E-mail
[email protected]
Family name
Statman
Particle
Given Name
Suffix
Richard
Abstract
Division
Department of Mathematical Sciences
Organization
Carnegie Mellon University
Address
Pittsburgh, PA, 15213, USA
E-mail
[email protected]
The λδ -calculus is the λ -calculus augmented with a discriminator which
distinguishes terms. We consider the simply typed λδ -calculus over one atomic type
variable augmented additionally with an existential quantifier and a description
operator, all of lowest type. First we provide a proof of a folklore result which states
that a function in the full type structure of [n ] is λδ -definable from the description
operator and existential quantifier if and only if it is symmetric, that is, fixed under the
group action of the symmetric group of n elements. This proof uses only elementary
facts from algebra and a way to reduce arbitrary functions to functions of lowest type
via a theorem of Henkin. Then we prove a necessary and sufficient condition for a
function on [n ] to be λδ -definable without the description operator or existential
quantifier, which requires a stronger notion of symmetry.
Keywords
Lambda calculus – Lambda delta calculus – Types – Typed lambda calculus – Simply
typed lambda calculus – Type theory – Classical type theory – First-order logic –
Henkin semantics – Typed lambda calculus semantics – Delta discriminator –
Description operator
Chapter title
Henkin’s Completeness Proof and Glivenko’s Theorem
Corresponding Author
Family name
Parlamento
Particle
Given Name
Franco
Suffix
Abstract
Division
Department of Mathematics and Computer Science
Organization
University of Udine
Address
via Delle Scienze 206, 33100, Udine, Italy
E-mail
[email protected]
We observe that Henkin’s argument for the completeness theorem yields also a
classical semantic proof of Glivenko’s theorem and leads in a straightforward way to
the weakest intermediate logic for which that theorem still holds. Some refinements of
the completeness theorem can also be obtained.
Keywords
Consistency – Completeness – Intermediate logics
Chapter title
From Classical to Fuzzy Type Theory
Corresponding Author
Family name
Particle
Novák
Given Name
Vilém
Suffix
Division
Institute for Research and Applications of Fuzzy Modelling,
National Supercomputing Centre IT4 Innovations
Abstract
Organization
University of Ostrava
Address
30. dubna 22, 701 03, Ostrava 1, Czech Republic
E-mail
[email protected]
Higher-order logic—the type theory (TT)—is a powerful formal theory that has various
kinds of applications, for example, in linguistic semantics, computer science,
foundations of mathematics and elsewhere. It was proved to be incomplete with
respect to standard models. In fifties and sixties of the last century, L. Henkin proved
that there is an axiomatic system of TT that is complete if we relax the concept of
model to the, so called, generalized one. The difference is that domains of functions in
generalized models need not contain all possible functions but only subsets of them.
Henkin then proved that a formula of type o (truth value) of a special theory T of the
theory of types is provable iff it is true in all general models of T . Mathematical fuzzy
logic is a special many-valued logic whose goal is to provide tools for capturing the
vagueness phenomenon via degrees. It went through intensive development and
many formal systems of both propositional as well as first-order fuzzy logic were
proved to be complete. This endeavor was crowned in 2005 when also higher-order
fuzzy logic (called the Fuzzy Type Theory, FTT) was developed and its completeness
with respect to general models was proved. The proof is based on the ideas of the
Henkin’s completeness proof for TT. This paper addresses several complete formal
systems of the fuzzy type theory. The systems differ from each other by a chosen
algebra of truth values. Namely, we focus on three systems: the Core FTT based on a
special algebra of truth values for fuzzy type theory—the EQ-algebra, then IMTL-FTT
based on IMTLΔ -algebra of truth values and finally the Ł-FTT based on MVΔ -algebra
of truth values.
Keywords
Fuzzy type theory – EQ-algebra – Residuated lattice – IMTL-algebra – MV-algebra –
Higher-order fuzzy logic – Mathematical fuzzy logic – Δ -operation
Chapter title
The Henkin Sentence
Author
Family name
Halbach
Particle
Given Name
Volker
Suffix
Division
Organization
New College
Address
OX1 3BN, Oxford, England
Corresponding Author
E-mail
[email protected]
Family name
Visser
Particle
Given Name
Albert
Suffix
Abstract
Division
Philosophy, Faculty of Humanities
Organization
Utrecht University
Address
Janskerkhof 13, 3512BL, Utrecht, The Netherlands
E-mail
[email protected]
In this paper we discuss Henkin’s question concerning a formula that has been
described as expressing its own provability. We analyze Henkin’s formulation of the
question and the early responses by Kreisel and Löb and sketch how this discussion
led to the development of provability logic. We argue that, in addition to that, the
question has philosophical aspects that are still interesting.
Keywords
Self-reference – Fixed Points – Second Incompleteness Theorem – Provability Logic
Chapter title
April the 19th
Corresponding Author
Family name
Manzano
Particle
Given Name
María
Suffix
Abstract
Division
Department of Philosophy
Organization
University of Salamanca
Address
Salamanca, Spain
E-mail
[email protected]
This paper is about my book (Manzano, Extensions of first-order logic, 1996),
published by Cambridge University Press in 1996. The main purpose of it being to
pinpoint Henkin’s influence concerning the translation technique proposed in the
book.
Several extensions of first order logic are introduced in Extensions, while trying to
pursue the thesis that most reasonably logical systems can be naturally translated
into many-sorted first order logic. I did credit most of the ideas involved in my
translation to Henkin’s papers (Completeness in the theory of types, 1950, and
Banishing the rule of substitution for functional variables, 1953).
Keywords
Henkin – Translations – Correspondence theory – Many-sorted logic – Extensions of
first-order logic – Type theory – General models – Comprehension schema – Rule of
substitution
Chapter title
Henkin and Hybrid Logic
Corresponding Author
Family name
Blackburn
Particle
Given Name
Patrick
Suffix
Division
Author
Organization
Roskilde Universitet
Address
Roskilde, Denmark
E-mail
[email protected]
Family name
Huertas
Particle
Given Name
Antonia
Suffix
Division
Author
Organization
Universitat Oberta de Catalunya
Address
Barcelona, Spain
Family name
Manzano
Particle
Given Name
María
Suffix
Division
Author
Organization
Universidad de Salamanca
Address
Salamanca, Spain
Family name
Jørgensen
Particle
Given Name
Klaus
Given Name
Frovin
Suffix
Division
Abstract
Organization
Roskilde Universitet
Address
Roskilde, Denmark
Leon Henkin was not a modal logician, but there is a branch of modal logic that has
been deeply influenced by his work. That branch is hybrid logic, a family of logics that
extend orthodox modal logic with special propositional symbols (called nominals) that
name worlds. This paper explains why Henkin’s techniques are so important in
hybrid logic. We do so by proving a completeness result for a hybrid type theory called
HTT, probably the strongest hybrid logic that has yet been explored. Our
completeness result builds on earlier work with a system called BHTT, or basic hybrid
type theory, and draws heavily on Henkin’s work. We prove our Lindenbaum Lemma
using a Henkin-inspired strategy, witnessing ◊-prefixed expressions with nominals.
Our use of general interpretations and the construction of the type hierarchy is
(almost) pure Henkin. Finally, the generality of our completeness result is due to the
first-order perspective, which lies at the heart of both Henkin’s best known work and
hybrid logic.
Keywords
Hybrid logic – Modal logic – Higher-order logic – Rigidity – Henkin constants – Henkin
models – Bounded fragment
Chapter title
Changing a Semantics: Opportunism or Courage?
Author
Family name
Andréka
Particle
Given Name
Hajnal
Suffix
Author
Division
Alfréd Rényi Institute of Mathematics
Organization
Hungarian Academy of Sciences
Address
Budapest, PF 127, 1364, Hungary
E-mail
[email protected]
Family name
Benthem
Particle
van
Given Name
Johan
Suffix
Corresponding Author
Division
Institute for Logic, Language and Computation (ILLC)
Organization
University of Amsterdam
Address
P.O. Box 94242, 1090 GE, Amsterdam, The Netherlands
E-mail
[email protected]
Family name
Bezhanishvili
Particle
Given Name
Nick
Suffix
Author
Division
Institute for Logic, Language and Computation (ILLC)
Organization
University of Amsterdam
Address
P.O. Box 94242, 1090 GE, Amsterdam, The Netherlands
E-mail
[email protected]
Family name
Németi
Particle
Given Name
István
Suffix
Abstract
Division
Alfréd Rényi Institute of Mathematics
Organization
Hungarian Academy of Sciences
Address
Budapest, PF 127, 1364, Hungary
E-mail
[email protected]
The generalized models for higher-order logics introduced by Leon Henkin and their
multiple offspring over the years have become a standard tool in many areas of logic.
Even so, discussion has persisted about their technical status, and perhaps even their
conceptual legitimacy. This paper gives a systematic view of generalized model
techniques, discusses what they mean in mathematical and philosophical terms, and
presents a few technical themes and results about their role in algebraic
representation, calibrating provability, lowering complexity, understanding fixed-point
logics, and achieving set-theoretic absoluteness. We also show how thinking about
Henkin’s approach to semantics of logical systems in this generality can yield new
results, dispelling the impression of adhocness.
Keywords
Henkin models – Definable predicates – General frames – Absoluteness – General
models for recursion and computation