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Part of the book series: Studies in Universal Logic ((SUL))

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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 Journal of Symbolic Logic and only one in the Bulletin of the American Mathematical Society. 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.

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Notes

  1. 1.

    See J. Donald Monk [194].

  2. 2.

    See H.B. Enderton [30] for a detailed account of Church as a reviewer. All the information we include about the issue is taken from Enderton’s work.

  3. 3.

    L. Henkin: “The completeness of the first-order functional calculus” [52].

  4. 4.

    “Completeness in the theory of types” [59].

  5. 5.

    Note that in 1942 Henkin was Church’s student. This review has a strong technical character.

  6. 6.

    Pre-existent definitions (the usual ones in the foundations of mathematics) placed natural numbers/integers in one type while real numbers were defined as “special sets of integers,” hence they/their definition belonged in a higher functional type.

  7. 7.

    Works by Henkin that are contributions to the subject are: “Completeness in the theory of types” [59]; “On the primitive symbols of Quine’s ‘Mathematical Logic’ ” [68]; “A generalization of the concept of ω-consistency” [73]; “On the definitions of ‘formal deduction’ ” (with R. Montague) [195]; “A generalization of the concept of ω-completeness” [92]; “Some remarks on infinitely long formulas” [104]; “An extension of the Craig–Lyndon interpolation theorem” [107]; Logical Systems Containing Only a Finite Number of Symbols [114], and “Relativization with respect to formulas and its use in proofs of independence” [116].

  8. 8.

    Note that Church wrote a second version of this paper, in which he revised this proposal. Moreover, Church explains that along the years he doubted the viability of his initial proposal in his 1973 paper “Outline of a revised formulation of the logic of sense and denotation (Part I)” [25] what he takes to be an adequate proposal. The next year, Church published the second part [26], and in 1993, he published an alternative formulation [27].

  9. 9.

    Homogeneous theorems are homogeneous formulas that are derivable from axioms using the inference rules in the system, and a formula is homogeneous if only predicates with the same number of variables and in the same order occur in it.

  10. 10.

    This review could have also been included in Sect. 3.1 “Algebraic treatment of logic systems.”

  11. 11.

    “Algebraic logic can be divided into two main parts. Part I studies algebras which are relevant to logic(s), e.g. algebras which were obtained from logics (one way or another). Since Part I studies algebras, its methods are, basically, algebraic. One could say that Part I belongs to ‘Algebra Country’. Continuing this metaphor, Part II deals with studying and building the bridge between Algebra Country and Logic Country. Part II deals with the methodology of solving logic problems by (i) translating them to algebra (the process of algebraization), (ii) solving the algebraic problem (this really belongs to Part I), and (iii) translating the result back to logic. There is an emphasis here on step (iii), because without such a methodological emphasis one could be tempted to play the ‘enjoyable games’ (i) and (ii), and then forget about the ‘boring duty’ of (iii). Of course, this bridge can also be used backwards, to solve algebraic problems with logical methods.” (Hajnal Andréka, Istvan Németi, and Ildikó Sain [1, p. 133].)

  12. 12.

    See [171177], and [178].

  13. 13.

    In his book Algebraic logic [47], Halmos publishes his 10 main papers on polyadic algebra.

  14. 14.

    Cylindric algebras generalize Boolean algebras for each ordinal α by adding “distinguished elements” (the so called “diagonal elements” d κ,λ where κ and λ are less than α) to the elements of the Boolean algebra 0 and 1, and unary operations called “cylindrifications” (c κ where κ<α).

  15. 15.

    A hopefully complete list of Henkin’s works on the subject is: “The representation theorem for cylindric algebras” [78]; “La structure algébrique des théories mathématiques” [87]; “Cylindrical algebras” (with A. Tarski) [140]; the abstract “Cylindrical algebras of dimension 2. Preliminary report” [91]; Cylindric Algebras. Lectures presented at the 1961 Seminar of the Canadian Mathematical Congress [103]; “Cylindric algebras” (with A. Tarski) [141]; Cylindric Algebras, Part I (1971 with J.D. Monk and A. Tarski) [132]; “Cylindric algebras and related structures” (with J.D. Monk ) [131]; “Relativization of cylindric algebras” (with D. Resek) [137]; “Cylindric set algebras and related structures” (with J.D. Monk and A. Tarski) [133]; Cylindric Algebras, Part II (with J.D. Monk and A. Tarski) [134], and “Representable cylindric algebras” (with J.D. Monk and A. Tarski) [135].

  16. 16.

    Krasner in [156] dates that work in 1958, not in 1938. Henkin does not provide the explicit reference for [155].

  17. 17.

    Each tautology adopts only designated values, while non-tautologies don’t.

  18. 18.

    There are designated elements x and xy such that y is not a designated element.

  19. 19.

    Well, in fact, he quotes his other review of another work by Menger, when he discusses Menger’s book Calculus. A Modern Approach.

  20. 20.

    “On mathematical induction” [100]; with W.N. Smith, V.J. Varineau, and M.J. Walsh Retracing Elementary Mathematics [138]; “New directions in secondary school mathematics” [109]; “The axiomatic method in mathematics courses at the secondary level” [113]; “Linguistic aspects of mathematical education” [119]; “The logic of equality” [122]; with Nitsa Hadar “Children’s conditional reasoning, Part II: Towards a Reliable Test of Conditional Reasoning Ability” [129]; with Robert B. Davis “Aspects of mathematics learning that should be the subject of testing” [127]; with Robert B. Davis “Inadequately tested aspects of mathematics learning” [128]; with Shmuel Avital “On equations that hold identically in the system of real numbers” [126]; “The roles of action and of thought in mathematics education—One mathematician’s passage” [123].

  21. 21.

    This is a really short review. Its text goes: “This is a brief discussion of the following questions. Should semantics be considered as a part of, or as complementary to, symbolic logic? Does the formalization of theoretical physics require the introduction of new logical systems?”

  22. 22.

    The Manhattan project was a US government project that produced the first atomic bombs. According to the obituary published on the web page of the University of Berkeley (http://www.berkeley.edu/news/media/releases/2006/11/09_henkin.shtml), “During World War II, he worked in industry for the Manhattan project, first as a mathematician for the Signal Corps Radar Laboratory in Belmar, New Jersey; then in New York City on the design of an isotope diffusion plant; and finally as head of the separation performance group at Union Carbide and Carbon Corp. in Oak Ridge, Tenn.” In fact, Henkin himself explains the circumstances in [124, pp. 133–134, note 11]:

    “During the period May 1942–March 1946 I worked as a mathematician, first on radar problems and then, beginning January 1943, on the design of a plant to separate uranium isotopes. Most of my work involved numerical analysis to obtain solutions of certain partial difference-differential equations. During this period I neither read, nor thought about, logic.”

  23. 23.

    Mostowski himself uses the word “false” applied to proofs, but “incorrect” seems to be the right word to use; it is sentences, statements, or propositions that are true or false, and hence the conclusion of an intended proof can be false but not the proof itself.

  24. 24.

    Henkin quotes none of his works in these reviews of the two manuals mentioned. In his review of Church’s manual, he quotes no other work, whereas in his review of Beth’s, he quotes Beth’s work “The Foundations of Mathematics. A Study of the Philosophy of Science” [10].

  25. 25.

    The period from 1943 to 1947 in which he worked in his Ph.D. and in the Manhattan project (see note 22 above), years 1969, 1982, 1984, 1987, 1988, and the period from 1990 to 1994.

  26. 26.

    The Axiomatic Method [139] with P. Suppes and A. Tarski; The Theory of Models [125] with J.W. Addison and A. Tarski; Logic, Methodology and Philosophy of Science IV [224] with P. Suppes, A. Joja, and Gr.C. Moisil; and Proceedings of the Tarski Symposium [120].

  27. 27.

    La structure algébrique des theories mathématiques [87]; Cylindric Algebras. Lectures presented at the 1961 Seminar of the Canadian Mathematical Congress [103]; Retracing Elementary Mathematics [138] with W.N. Smith, V.J. Varineau, and M.J. Walsh; Logical Systems Containing Only a Finite Number of Symbols [114]; Cylindric Algebras, Part I [132] with J.D. Monk and A. Tarski; Cylindric Algebras, Part II [134] with J.D. Monk and A. Tarski; and Mathematics-Report of the Project 2061 Phase I Mathematics Panel [14] with D. Blackwell.

  28. 28.

    12 of those papers were written in cooperation with someone else: “On the definition of ‘formal deduction’ ” [195] with R. Montague; “Cylindrical Algebras” [140] and “Cylindric Algebras” [141] with A. Tarski; “Cylindric algebras and related structures” [131] with J.D. Monk; “Relativization of cylindric algebras” [137] with D. Resek; “A Euclidean construction?” [130] with W. Leonard; “Children’s conditional reasoning, Part II: Towards a reliable test of conditional reasoning ability” [129] with Nitsa Hadar; “Aspects of mathematics learning that should be the subject of testing” [127] and “Inadequately tested aspects of mathematics learning” [128] with Robert B. Davis; “Cylindric set algebras and related structures” [133] and “Representable cylindric algebras” [135] with J.D. Monk and A. Tarski; and “On equations that hold identically in the system of real numbers” [126] with Shmuel Avital.

  29. 29.

    See Appendix.

  30. 30.

    This classification coincides with that provided in 2000. Barwise’s [4, p. vii] “Mathematical logic is traditionally divided into four parts: model theory, set theory, recursion theory, and proof theory” is not that complete.

  31. 31.

    “The roles of action and of thought in mathematics education—One mathematician’s passage” [77]; with W.N. Smith, V.J. Varineau and M.J. Walsh Retracing Elementary Mathematics [138]; “New directions in secondary school mathematics” [109]; “The axiomatic method in mathematics courses at the secondary level” [113]; “Linguistic aspects of mathematical education” [119]; with Nitsa Hadar “Children’s conditional reasoning, Part II: Towards a Reliable Test of Conditional Reasoning Ability” [129]; with Robert B. Davis “Aspects of mathematics learning that should be the subject of testing” [127]; with Robert B. Davis “Inadequately tested aspects of mathematics learning” [128]; with D. Blackwell Mathematics-Report of the Project 2061 Phase I Mathematics Panel [14] and “The roles of action and of thought in mathematics education—One mathematician’s passage” [123].

  32. 32.

    See Enderton [30].

  33. 33.

    This is a really short review. Its text goes: “This is a brief discussion of the following questions. Should semantics be considered as a part of, or as complementary to, symbolic logic? Does the formalization of theoretical physics require the introduction of new logical systems?”

  34. 34.

    In fact, Henkin lists the three works, but he only reviews Robinson’s.

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Martínez Vidal, C., Úbeda Rives, J.P. (2014). Leon Henkin the Reviewer. In: Manzano, M., Sain, I., Alonso, E. (eds) The Life and Work of Leon Henkin. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-09719-0_10

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