Leon Henkin

In 1991 he was granted the title of Professor Emeritus at the University of Berkeley and retired.

After he retired, Henkin continued to work on math teaching projects. From 1991, he took part on a summer courses program at Mills College intended to give talented women from across the nation education in mathematics in order to prepare them for college. Finally, Ginette and Henkin moved to Oakland, where Henkin died a few years later, in November 2006.

Henkin was always grateful to Tarski, as it was thanks to him that he was able to settle in Berkeley. After Tarski's death in 1983, he wrote in a personal letter: “I write to tell you that Alfred Tarski, who came to Berkeley in 1942 and founded our great Center for the Study of Logic and Foundations, died Wednesday night, at age 82 [...]. It was he who brought me to Berkeley in 1953, so I owe much to him personally as well as scientifically.”

Among the research trips that Henkin did throughout the years are his visits to universities in Hanover, Princeton, Colorado, as well as to several European Universities, such as Oxford (in the United Kingdom), and others in Yugoslavia, Spain, Portugal and France. In 1979, with his second Fulbright Grant, Henkin spent a year in Israel, in Haifa, at the Department of Science Education of the Technion University. On this occasion he also visited two universities in Egypt. In 1982 he first visited Spain. He gave conferences at several universities, including those in Barcelona, Madrid and Seville.

Although Henkin's first encounter with teaching mathematics was as a professor, later in life he began to do research in mathematics' teaching as well. Some of his writings in this field are: "Retracing Elementary Mathematics", "New directions in secondary school mathematics" or "The roles of action and of thought in mathematics education". From 1979 onwards he put special emphasis on this facet of his research and the last doctoral theses he directed are related to the teaching of mathematics or the integration of minority groups in research.

Henkin had an active role in research and teaching, but his activities at the university went far beyond that. In addition to the dedication he put in his teaching as well as and in guiding the Group in Logic and the Methodology of Science, he held some administrative positions; he was director of the Department of Mathematics from 1966 to 1968, and subsequently from 1983 to 1985. One of the activities to which he devoted most energy was the teaching of mathematics, on which he also did some research.

Henkin liked to write expository articles, for some of which he received awards such as the Chauvenet Prize (1964), for the article "Are Logic and Mathematics Identical? " or the Lester R. Ford Award, for the article "Mathematical Foundations of Mathematics".

On some occasions Henkin attended to his children's schools to talk to elementary school children about maths, talking to them about "the negative numbers", or "how to subtract by addition". Around that time (about 1960), Henkin began to alternate his research work in mathematics with research work in teaching mathematics; the latter became increasingly frequent.

Some of the social projects he formed or participated in are the following. Between 1957 and 1959 he was part of the Summer Institutes, aimed at mathematics teachers and dedicated to improving high school and college education. In 1958 the National Science Foundation authorized the committee of the American Mathematical Society –which had been interested for some years in the use of films and visual material for mathematics education– to produce experimental films for this purpose, accompanied by printed manuals with appendices that would go deeper into the content and problems to be solved. Henkin participated in this project with a film on mathematical induction, whose supplementary manual was printed by the American Mathematical Society. The film was broadcast in the series "Mathematics Today". Between 1961 and 1964, he participated in a series of courses for elementary school teachers, organized by the Committee on the Undergraduate Program in Mathematics. Also around that time, he promoted the Activities to Broaden Opportunity initiative, which sought to provide opportunities for promising students from ethnic minority groups by offering them summer courses and scholarships. He took part in the SEED (Special Elementary Education for the Disadvantaged) program, which encouraged college students to participate in elementary education, as well as in SESAME (Special Excellence in Science and Mathematics Education), the interdisciplinary doctoral program created by members of various science departments, whose purpose was to research teaching and learning of science, engineering, and mathematics. Between 1960 and 1968 he participated in a series of conferences in mathematics schools, and was involved in the development of several films produced by the National Council of Teachers of Mathematics (NCTM). These films dealt with topics such as the integer system and the rational number system. He also participated in support courses for female calculus students and convinced the mathematics department to allow graduate students to receive the same financial support for working as elementary school teachers as they did for working as assistant teachers in college. "He not only believed in equality, but also worked actively to see that it was brought about."

From 1953, most of Henkin's academic activity revolved around Berkeley, where he collaborated with a solid research group in Logic. He remained there for almost all his academic life, except for some periods in which he traveled abroad with scholarships and grants of diverse institutes, like the one-year stay he had in Amsterdam or the one in Israel with the Fulbright Research Grants he was awarded (in 1954 and 1979 respectively).

In 1952 Tarski had managed to obtain a permanent position at Berkeley for Henkin. However, Henkin did not want to accept it, as he was sympathetic to the protests recently raised by the controversial oath of allegiance that had been required of university professors since 1950.  Once the oath requirement disappeared, Henkin accepted Tarski's offer and settled in Berkeley in 1953.

Tarski not only offered Henkin a job opportunity, but also provided him with a very fertile interdisciplinary collaborative environment for the development of Logic. Tarski had founded the Center for the Study of Logic and Foundations in Berkeley, but with Henkin's help he was able to bring together a group of logicians, mathematicians and philosophers who formed the Group in Logic and the Methodology of Science, which is still active today. As part of this project they created an interdisciplinary postgraduate program culminating in a Ph.D. Tarski and Henkin boosted the project by organizing important congresses and conferences on Logic, following Tarski's conception of "logic as a common basis for the whole of human knowledge". The intense activity that took place in Berkeley in the 1950s and 1960s on metalogic was largely due to the activity of Tarski and Henkin, both in teaching and research. Many results of what are today crucial to Model Theory came as a result of the academic activity in Berkeley that took place in those years.

In addition to his courses and supervision of graduate students, Henkin's role in the scholars education was significant. Tarski had invited him to Berkeley with a clear purpose. As a mathematician, Henkin had a key role in Tarski's project to make Berkeley a center of development of logic, bringing together mathematicians, logicians and philosophers. Henkin aided him to carry out the project, helping him in the creation of the interdisciplinary Group in Logic and the Methodology of Science, whose successful performance was largely due to Henkin's drive. Part of this project was the creation of an interdisciplinary university program that culminated in a Ph.D. in "Logic, Methodology and Philosophy of Science". He also collaborated in organising important meetings and conferences that promoted interdisciplinary collaboration united by logic. The outcome was that in the 1950s and 1960s there was a vibrant development of logic in Berkeley, from which many advances in Model Theory emerged.

In 1949 "The completeness of the first order functional calculus" was published, as well as "Completeness in the theory of types" in 1950. Both presented part of the results exposed in the dissertation "The completeness of formal systems" with which Henkin received his Ph.D. degree at Princeton in 1947. One of Henkin's best known results is that of the completeness of First-Order Logic, published in the above-mentioned 1949 article, which appears as the first theorem of the 1947 dissertation. It states the following:

Among the other theorems of completeness given by Henkin, the most relevant is perhaps that of the completeness of Church's Theory of Types, which is the first of the completeness theorems Henkin proved. Then, he adapted the method developed in that proof to prove the completeness of other deductive systems. This method has continued to be used to give proofs of completeness in both classical and non-classical logics, and it has become the usual proof of completeness for First-Order Logic in Logic textbooks. When Henkin published this result in 1949, completeness was not even part of the canonical subjects covered by the textbooks; some twenty years later, this theorem, along with its proof and corollaries, was part of virtually every Logic textbook. As for non-classical logics, Henkin's method can be used, among other things, to extend the completeness of Fuzzy Logic from first order to higher order, producing a complete Fuzzy Type Theory; it also offers a way to obtain results that link classical logic with intuitionist logic; and it allows one to test results of completeness in other non-classical logics, as in the cases of Hybrid Type Theory and Equational Hybrid Propositional Type Theory.

Having obtained his Ph.D. degree, Henkin spent two more years at Princeton working on post-doctoral studies. During this time, in 1948, he met Ginette Potvin, during a trip to Montreal with his sister Estelle and Princeton mathematics graduate student Harold Kuhn. Ginette would become his wife in 1950, a half year after Estelle married Harold. After completing his second year of postdoctoral studies at Princeton in 1949, Leon returned to California, where he entered the mathematics department at the University of Southern California. There he held the position of assistant professor until 1953.

Despite being one of his best known results, Henkin got to the proof of the completeness of first-order logic "accidentally", trying to prove a completely different result. The order of publication of his articles and even the order of presentation of the theorems in his 1947 dissertation does not reflect the evolution that followed the ideas that led him to his completeness results. However, Henkin simplifies the difficult task of tracing the development and shaping of his ideas by his article "The discovery of my completeness proofs", published in 1996. In it, he describes the process of the development of his dissertation. He doesn't only explain the content of his work, but he also explains the ideas that led to it, from his first logic courses in College until the end of the writing of his thesis.

Once the war was over, Henkin returned to Princeton in 1946, where he was still required to write a dissertation to complete his Ph.D. studies. Upon his return he joined the logic course that Church had begun a month earlier on Frege's theory of "sense and reference". In this course he discovered Church's theory of types, which he found extremely interesting. The questions he asked about it eventually led him to give his proof of the completeness of the theory of types, which he was able to adapt to also give a new proof of the completeness of First-Order Logic. These results, as well as others that other that emerged from the same ideas, came to take part in Henkin's doctoral dissertation, which was titled "The completeness of formal systems", with which he graduated in June 1947. The dissertation itself was not published, although parts of it were rewritten and published in articles, and. Many years later, Henkin wrote the article "The discovery of my completeness proofs", which contains a detailed review of the contents of his dissertation. The procedures used in it have become frequent methods of proofs in various branches of logic.

In his last year at Columbia, in 1941, Professor F. J. Murray, knowing that Henkin was a mathematics student interested in Logic, suggested that they review together the monograph by Gödel recently published at Princeton on the consistency of the axiom of choice with the generalized continuum hypothesis. Although the meetings they had to discuss it were scarce and Leon ended up revising this monograph practically alone, the experience was considered by him as the most enriching one in his formation at Columbia. According to Henkin, then began to take form some of the ideas that became the starting-point of his doctoral dissertation.

Henkin began his graduate studies at Princeton in 1941, studying under the direction of Church. The Ph.D. program he attended consisted of two years of mathematics courses, after which he was to take a "qualifying" oral examination to show he was well educated in at least three branches of mathematics; with this he would receive a M.A. degree. He would then have another two years to write a doctoral dissertation containing an original research, after which he would get the degree of Ph.D.

In 1941 the United States entered the Second World War, altering Henkin's plans. He had to rush his oral qualification exam, with which he obtained the degree of M. A. and left Princeton to take part in the Manhattan Project. This interruption would last four years, during which he contributed his mathematical knowledge working on radar problems and in the design of a plant to separate uranium isotopes. Most of his work required numerical analysis to solve partial differential equations. During this period, all of his work and readings on logic were completely suspended.

In 1940, Henkin decided to apply for admission to a doctoral program, without having fully defined what path to follow in his research. He was accepted to three universities, from which he chose Princeton, since the renowned logician Alonzo Church was there, although at the time Henkin was unaware of his work.

The following year, in the fall semester of 1939, Henkin took a second course of Logic with Nagel, in which formal systems of propositional logic and First-Order Logic were addressed. These constituted his first experience with the mathematical treatment of deductive systems. The course did not go into metalogical results that established a relationship between the semantics and syntactics, and the issue of completeness was not addressed at all. However, Nagel proposed to Henkin as an independent project the reading of the proof of the completeness of propositional logic given by Quine, which had appeared a few months before in the Journal of Symbolic Logic. This reading was highly significant for Henkin, not so much because of the content itself, but because with it he discovered that he could understand the research on logic and mathematics that was taking place at the time. According to Henkin, although he managed to follow Quine's demonstration, he did not manage to capture the idea of the proof: "I simply noted that the aim of the paper was to show that every tautology had a formal proof in the system of axioms presented, and I expended my utmost effort to check Quine's reasoning that this was so, without ever reflecting on why author and reader were making this effort. This strictly limited objective also kept me from wondering how the author thought of putting the steps of the proof together; the result was that I failed to get 'the idea of the proof', the essential ingredient needed for discovery."

In 1937 Leon entered Columbia University as a mathematics student. It was during his time at this institution that he developed an interest in logic, which would determine the course of his academic career. His first contact with logic was through B. Russell's book, "Mysticism and Mathematics", which drew his interest during a visit to the library. This interest was increased and cultivated by some courses. Although the mathematics department of the University did not offer courses in Logic (these were offered by the Philosophy department), Leon was one of the few mathematics students interested in that discipline and he decided to attend them. In the fall of 1938, in his second year as a Columbia University student, he participated in a first course in Logic taught by Ernest Nagel, who had contributed to the creation of the Association of Symbolic Logic two years earlier. This course brought him closer to Russell's book "Principles of Mathematics", where he first encountered the axiom of choice; Russell's presentation made a strong impression on him and led him to explore the Principia Mathematica that Russell wrote with Whitehead a few years later. He was struck by the general ideas of Type Theory and by the mysterious axiom of reducibility. Both the axiom of choice and Type Theory later played an important role in his doctoral dissertation.

Henkin is mainly known for his completeness proofs of diverse formal systems, such as type theory and first-order logic (the completeness of the latter, in its weak version, had been proven by Kurt Gödel in 1929). To prove the completeness of Type Theory, Henkin introduces new semantics, based on certain structures, called general models (also known as Henkin models). The change of semantics that he proposed permits to provide a complete deductive calculus for Type Theory and for Second-Order Logic, amongst other logics. Henkin methods have aided to prove various model theory results, both in classical and non-classical logics. Besides logic, the other branch on which his investigations were centered was algebra; he specialized in cylindric algebras, in which he worked together with A. Tarski and D. Monk. As for the philosophy of mathematics, although the works in which he explicitly approaches it are scarce, he can be considered to have a nominalist position.

This is the strong version of the completeness theorem, from which the weak version is obtained as a corollary. The latter states the result for the particular case in which S {\displaystyle S} is the empty set, this is to say, the deductive calculus of first order logic is capable of deriving all valid formulas. The weak version, known as Gödel's completeness theorem, had been proved by Gödel in 1929, in his own doctoral thesis. Henkin's proof is more general, more accessible than Gödel's and more easily generalizable to languages of any cardinality. It approaches completeness from a new and fruitful perspective and its greatest quality is perhaps that its proof can be easily adapted to prove the completeness of other deductive systems. Other results central to model theory are obtained as corollaries of the strong completeness of the first order logic proved by Henkin. From it follows, for example, the following result for a first order language  L {\displaystyle L} :

Leon Albert Henkin (April 19, 1921, Brooklyn, New York - November 1, 2006, Oakland, California) was an American logician, whose works played a strong role in the development of logic, particularly in the theory of types. He was an active scholar at the University of California, Berkeley, where he made great contributions as a researcher, teacher, as well as in administrative positions. At this university he directed, together with Alfred Tarski, the Group in Logic and the Methodology of Science, from which many important logicians and philosophers emerged. He had a strong sense of social commitment and was a passionate defensor of his pacifist and progressive ideas. He took part in many social projects aimed at teaching mathematics, as well as projects aimed at supporting women's and minority groups to pursue careers in mathematics and related fields. A lover of dance and literature, he appreciated life in all its facets: art, culture, science and, above all, the warmth of human relations. He is remembered by his students for his great kindness, as well as for his academic and teaching excellence.

Leon Albert Henkin was born on April 19, 1921, in Brooklyn, New York, to a Jewish family that had emigrated from Russia a generation earlier. The first of the family to emigrate was Abraham Henkin, the eldest of the brothers of Leon's father. According to Leon, his father had been extremely proud of him since he was just a boy. His high expectations were evident in the name he gave him: he chose to name his son Albert after a series of articles on Einstein's theory of relativity that the New York Times published shortly before Henkin's birth. His family was sympathetic with pacifist and progressive ideas, and although he was not religious, he had deeply rooted Jewish traditions. Leon grew up surrounded by tight family ties; he was very close to his cousins, with whom he lived during his childhood in Brooklyn.