Leon Henkin

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.

Henkin studied primarily in New York City public schools; he attended Lincoln High School, where he graduated at age 16 to enter Columbia University.

Both in college and high school he was a member of the chess teams; he always preferred games that involved rational thinking to games of chance.

In the years of his high school education, Henkin considered becoming a math teacher and also came to desire to become a writer (as he later expressed in a personal letter).

Although he dedicated himself to university academic life, he never abandoned his interest in teaching elementary mathematics, to which he later actively contributed.

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, not equivalent to standard semantics, based on 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 Tarski and Donald 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.

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.

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."

Just before Henkin began his second year at Columbia, World War II broke out.

This had several repercussions on his life.