# Jan Kalicki

### Born: 28 January 1922 in Warsaw, Poland

Died: 25 November 1953 in Contra Costa County, USA

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**Jan Kalicki**attended secondary school in his home town of Warsaw, graduating from the school in May 1939. During the summer of 1939 as Kalicki waited to begin his university education, Europe moved closer to war. On 1 September Germany invaded Poland and Warsaw was bombed on the following day. Warsaw was quickly encircled by the invading German armies and on 17 September the Russian armies invaded Poland from the east occupying Lvov five days later. On 27 September Warsaw surrendered to the attacking German forces and the University of Warsaw was closed down.

There was an impressive collection of mathematicians at the University of Warsaw at this time including Borsuk, Łukasiewicz, Mazurkiewicz, Sierpiński, Mostowski, Kuratowski and others. They, together with colleagues in other disciplines, organised an underground version of the University of Warsaw which was strongly opposed by the Nazi authorities. There was a strategy by the Nazis to put an end to the intellectual life of Poland and to achieve this they sent many academics to concentration camps and murdered others. Borsuk, for example, was imprisoned after the authorities found that he was helping to run the underground university. It was in these extraordinarily difficult conditions that Kalicki received his university education, studying mathematics and philosophy. Kuratowski wrote of the underground university in which Kalicki was educated:-

When the Soviet forces came close to Warsaw in 1944, the Warsaw Resistance rose up against the weakened German garrison. However German reinforcements arrived and put down resistance. Around 160,000 people died in the Warsaw Uprising of 1944 and the city was left in a state of almost total devastation. Kalicki survived and after the war ended in 1945 he was awarded an M.A. in both philosophy and mathematics. After spending 1945-46 as a teaching assistant at the Universities of Lòdz and Warsaw he went to London in England. Financed by a British Council Scholarship for two years, Kalicki studied at the University of London, receiving his doctorate in mathematical logic in July 1948. While studying in London he married Mireya Jaimes-Freyre in 1947. They had one son who was given the name of Jan Kalicki, the same name as his father.Due to that underground organisation, and in spite of extremely difficult conditions, scientific work and teaching continued, though on a considerably smaller scale of course. The importance of clandestine education consisted among others in keeping up the spirit of resistance, as well as optimism and confidence in the future, which was so necessary in the conditions of occupation. The conditions of a scientist's life at that time were truly tragic. Most painful were the human losses.

By the time Kalicki was awarded his doctorate Poland was a communist controlled country which Kalicki felt would not encourage free thinking academic studies. He decided to remain in England and was appointed to a mathematics lectureship in Woolwich Polytechnic, London. After a year in this post he went to the University of Leeds where he was appointed as a Lecturer in Mathematics. He remained in Leeds until 1951 when he accepted an appointment in the United States as Visiting Assistant Professor of Mathematics in the University of California, Berkeley.

Kalicki worked on logical matrices and equational logic and published 13 papers on these topics from 1948 until his death five years later. Logical matrices were defined by Łukasiewicz and Tarski in 1930. In

*On Tarski's matrix method*(1948), Kalicki defines the sum , the product and also a partial ordering of logical matrices. His operations of addition "+" and multiplication "." are associative and commutative, and there is an identity for each operation. If

*A*is a logical matrix then let

*E*(

*A*) be the set of well-formed expressions which satisfy

*A*. Kalicki's operations of addition and multiplication now satisfy

*E*(

*A*+

*B*) =

*E*(

*A*) +

*E*(

*B*) and

*E*(

*A*.

*B*) =

*E*(

*A*).

*E*(

*B*).

In 1950 he defined an algebra binary connective

*D*on truth tables in

*Note on truth-tables*published in the

*Journal of Symbolic Logic*. The truth tables involve a finite number of truth values, some designated and some undesignated. Corresponding to each truth table

*M*, Kalicki considers the set of all formulae formed from variables and the connective

*D*, which are tautologies for

*M*. The product

*A*.

*B*of two truth tables

*A*and

*B*is defined to be a truth table formed from

*A*and

*B*in such a way that the set of tautologies corresponding to it is the intersection of the tautology sets for

*A*and

*B*. In a similar way the sum

*A*+

*B*of two truth tables is defined having its tautology set as the union of the tautology sets of

*A*and

*B*. Kalicki defines relations of isomorphism, equivalence, and inclusion of truth tables. Particular truth tables

*O*and

*U*are defined which play the role of the zero element and the identity element in the algebra. The resulting system resembles a lattice. The next paper in the same part of the

*Journal of Symbolic Logic*is also by Kalicki. It is

*A test for the existence of tautologies according to many-valued truth-tables*in which Kalicki gives an effective procedure to decide whether the set of tautologies determined by a given truth table with a finite number of truth-values is empty. Two years later another paper by Kalicki in the

*Journal of Symbolic Logic*is

*A test for the equality of truth-tables*which gives a necessary and sufficient condition for the equivalence of two truth-tables. However for the problem of deciding whether two infinite-valued truth-tables are equal is unsolvable as Kalicki showed in a paper which was not published until 1954 after his death.

Kalicki's first appointment at the University of California was as Visiting Assistant Professor of Mathematics at Berkeley where Tarski was working, but after one year he obtained a permanent appointment as Assistant Professor of Mathematics at the Davis campus. After one further year he returned to Berkeley in 1953, this time as Assistant Professor of Philosophy. He had only been in post for three months when he died in a car accident on a quiet Contra Costa County road.

B Mates, L Henkin and A Tarski, give details of Kalicki's character in an obituary:-

It is primarily as a person, even more than as a scholar, that he is missed by those who knew him. In this regard, we recall that there was a certain sweetness and depth about him that is difficult to describe. He was always straightforward and absolutely sincere, modest nearly to the point of shyness. He had unbounded confidence that almost anybody could be brought to an understanding of the fundamental results of modern logic if only he, Kalicki, could find the right words. His very optimism in this regard endeared him to the students. Indeed, one can say that it even contributed toward making itself well-founded. Kalicki had remarkable success with the type of students who consider themselves hopelessly incapable of mastering things mathematical. In him they found a person who had no inclination to deride or blame them for being slow and who saw in their perplexities only an indication that he must contrive to illuminate the subject from still another point of view. At the same time, he was always able to produce something of special interest for the students of greater aptitude. His great affection for the students was warmly returned by them, and their mourning upon his death will never be forgotten by those who witnessed it.

**Article by:** *J J O'Connor* and *E F Robertson*

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