Adrian Rezus - Academia.edu (original) (raw)
Papers by Adrian Rezus
CWI Syllabi, 1983
KNAW Narcis. Back to search results. Publication Abstract AUTOMATH (1983) Open access. Pagina-nav... more KNAW Narcis. Back to search results. Publication Abstract AUTOMATH (1983) Open access. Pagina-navigatie: Main. Title, Abstract AUTOMATH. Published in, CWI Syllabi, Vol. 160, p.1-188. Author, Rezus, A. Date, 1983-01-01. Language, English. Type, book. ...
"This thesis is intended to re-consider, in a modern setting, the foundational aspec... more "This thesis is intended to re-consider, in a modern setting, the foundational aspects of the theories of lambda-conversion and the corresponding combinatory variants, as originally proposed and worked out within the Church school (A. Church, S. C. Kleene, J. B. Rosser and, incidentally, A. M. Turing) in the thirties. The material is divided into seven Chapters and an Appendix. Chapter I: Lambda-terms [...], Chapter II: Calculi of lambda-conversion [...]. Chapter III: Rosser combinatory systems [...], Chapter IV: Lambda-definabiliy and numeral systems [...], Chapter V: Delta-conversion [...], Chapter VI: Logic and arithmetic: quanttifier-free systems [...], Chapter VII: Logic and arithmetic: arrows [...]. The type-free systems proposed and discussed in Chapters VI and VII are relevance-preserving, in the sense they do not allow formalizing, in the proposed context, classically and intuitionistically valid principles of the form "ex falso quodlibet". Finally [...], only theories possessing r.e. models admit of a type-free formalization in ordinal progression, on the pattern suggested here, thereby establishing the limits and stregth of the Church program. [...] The Appendix (Adequate numeral systems in \lambda\beta K) collects the examples of adequate numeral systems in [...] lambda-calculus, known so far to the author, in view of a possible classification." (From: "Adrian Rezus: Lambda-conversion and Logic, Ph.D. Dissertation, Rijksuniversiteit Utrecht, June 1981", Abstract, in: "Libertas Mathematica" [Arlington TX], vol. 2, 1982, pp. 182-185.)
Disponible depuis: http://www. dante. de/tex-archive/ …
Libertas Math, 1982
[MR 84b: 03019] The theorem of the title was asserted as Theorem 8 in a paper by J. Lukasiewicz a... more [MR 84b: 03019] The theorem of the title was asserted as Theorem 8 in a paper by J. Lukasiewicz and A. Tarski [see Tarski, "Logic, semantics and metamathematics", English translation, Clarendon Press, Oxford, 1956; MR 17, 1171]. It states: “Every finitely axiomatizable system of propositional calculus with the rules of substitution and detachment in which the two formulas CpCqp and CpCqCCpCqrr are provable can be axiomatized by a single formula.” According to the author, no proof of Tarski's theorem had been published prior to the present paper. The proof uses the formulas-as-types idea of H. B. Curry [Curry and R. Feys, "Combinatory logic", Vol I, Theorem 9E, North-Holland, Amsterdam, 1958; MR 20 #817] and the condensed detachment operator of C. A. Meredith [see, e.g., Meredith and A. N. Prior, Notre Dame J. Formal Logic 4 (1963), 171-187; MR 30 #1033]. Specifically, if \alpha and \beta are formulas of propositional calculus, and there is a substitution yielding \alpha' and \beta' where \alpha' is C\beta'\gamma, then the value of the metaformula (\alpha\beta) is the most general form of any such, whence we write (\alpha\beta) = \gamma. The metaformulas constructed from propositional calculus formulas as atoms by the operation of application thus en code proofs of the formulas which are their values. For example, let C and K abbreviate CCpCqrCqCpr and CpCqp, respectively. Then for any \alpha, (CK)\alpha = Cpp, and, whenever \alpha, \beta and \gamma have the requisite forms to give values to the metaformulas, ((C\alpha)\beta)\gamma = (\alpha\gamma)\beta. These examples suggest an analogy with the theory of combinators, and in fact the equality of metaformulas is largely analogous to combinatory equality, which the author exploits to obtain a proof of Tarski's theorem. Noting that CpCqp is not valid in systems of relevant logic, the author modifies and refnes his results to obtain single axioms for the purely implicational fragments of various relevant logics. He states as an open problem the existence of single axioms for relevant logics which include conjunction or disjunction. In an appendix he shows that every closed \lambda-calculus term is equal to an applicative combination of the single term \lambda xyz.y(\lambda u.z)(xz). (Review by B. Lercher.)
[MR 88a: 03034] The author presents an interpretation of P. Martin-Löf's type theory CST... more [MR 88a: 03034] The author presents an interpretation of P. Martin-Löf's type theory CST [Logic, methodology, and philosophy of science, VI (Hannover, 1979), 153-175, North-Holland, Amsterdam, 1982; MR 85d:03112] with one universe and without identity types. In the author's approach, following D. S. Scott [SIAM J. Comput. 5 (1976), no. 3, 522-587; MR 55 # 10262], types are translated into objects in a model of the type-free lambda calculus. The goal is to show that at least a subsystem of constructive type theory is intelligible from a classical non-constructivistic viewpoint. A second goal is to provide an abstract setting for the design of a large class of typed programming languages. The author's presentation includes an extensive discussion of related work on models of constructive type theory. (Review by Henry Africk.)
Tarski's Claim (TC = Theorem 8 in [13]) follows from simple conside-rations in (type-free) l... more Tarski's Claim (TC = Theorem 8 in [13]) follows from simple conside-rations in (type-free) lambda-calculus. The present note records essen-tially a proof of Lemma 1.1 in [16], ie TCL = the type-free lambda-calculus variant of TC, as well as a few historical ...
Irregular F is an open access online journal that publishes work from all areas and traditions of... more Irregular F is an open access online journal that publishes work from all areas and traditions of philosophy. High-quality theoretical research in social sciences and humanities may also be accepted. The journal focuses on research addressing novel or scarcely treated subject matter, remaining nevertheless faithful to the norms and standards of academic research. Irregular F promotes dialogues between various disciplines and the work of young researchers, but any valuable contribution from the aforementioned fields of study is welcome.
Information and Computation/information and Control - IANDC, 1983
![Research paper thumbnail of Tarski's Claim Thirty Years Later", Preprint [Nijmegen: October 1, 2010]](https://attachments.academia-assets.com/82050221/thumbnails/1.jpg)
](![Research paper thumbnail of Abstract Automath", Mathematisch Centrum [CWI], Amsterdam 1983 [Mathematical Centre Tracts 160]](https://attachments.academia-assets.com/82050216/thumbnails/1.jpg)
]("Abstract Automath", Mathematisch Centrum [CWI], Amsterdam 1983 [Mathematical Centre Tracts 160] [ISBN 90 6196 256 0], 1983
"Lambda-conversion and Logic", University of Utrecht, Utrecht 1981 [iv + 197 pp.], Jun 4, 1981
"This thesis is intended to re-consider, in a modern setting, the foundational aspects of th... more "This thesis is intended to re-consider, in a modern setting, the foundational aspects of the theories of lambda-conversion and the corresponding combinatory variants, as originally proposed and worked out within the Church school (A. Church, S. C. Kleene, J. B. Rosser and, incidentally, A. M. Turing) in the thirties. The material is divided into seven Chapters and an Appendix. Chapter I: Lambda-terms [...], Chapter II: Calculi of lambda-conversion [...]. Chapter III: Rosser combinatory systems [...], Chapter IV: Lambda-definabiliy and numeral systems [...], Chapter V: Delta-conversion [...], Chapter VI: Logic and arithmetic: quanttifier-free systems [...], Chapter VII: Logic and arithmetic: arrows [...]. The type-free systems proposed and discussed in Chapters VI and VII are relevance-preserving, in the sense they do not allow formalizing, in the proposed context, classically and intuitionistically valid principles of the form "ex falso quodlibet". Finally [...], only theories possessing r.e. models admit of a type-free formalization in ordinal progression, on the pattern suggested here, thereby establishing the limits and stregth of the Church program. [...] The Appendix (Adequate numeral systems in \lambda\beta K) collects the examples of adequate numeral systems in [...] lambda-calculus, known so far to the author, in view of a possible classification." (From: "Adrian Rezus: Lambda-conversion and Logic, Ph.D. Dissertation, Rijksuniversiteit Utrecht, June 1981", Abstract, in: "Libertas Mathematica" [Arlington TX], vol. 2, 1982, pp. 182-185.)
Irregular F is an open access online journal that publishes work from all areas and traditions of... more Irregular F is an open access online journal that publishes work from all areas and traditions of philosophy. High-quality theoretical research in social sciences and humanities may also be accepted. The journal focuses on research addressing novel or scarcely treated subject matter, remaining nevertheless faithful to the norms and standards of academic research. Irregular F promotes dialogues between various disciplines and the work of young researchers, but any valuable contribution from the aforementioned fields of study is welcome.
Information and Computation/information and Control - IANDC, 1983
![Research paper thumbnail of Tarski's Claim Thirty Years Later", Preprint [Nijmegen: October 1, 2010]](https://attachments.academia-assets.com/82050202/thumbnails/1.jpg)
](![Research paper thumbnail of Abstract Automath", Mathematisch Centrum [CWI], Amsterdam 1983 [Mathematical Centre Tracts 160]](https://attachments.academia-assets.com/82050204/thumbnails/1.jpg)
]("Abstract Automath", Mathematisch Centrum [CWI], Amsterdam 1983 [Mathematical Centre Tracts 160] [ISBN 90 6196 256 0], 1983
"Lambda-conversion and Logic", University of Utrecht, Utrecht 1981 [iv + 197 pp.], Jun 4, 1981
"This thesis is intended to re-consider, in a modern setting, the foundational aspects of th... more "This thesis is intended to re-consider, in a modern setting, the foundational aspects of the theories of lambda-conversion and the corresponding combinatory variants, as originally proposed and worked out within the Church school (A. Church, S. C. Kleene, J. B. Rosser and, incidentally, A. M. Turing) in the thirties. The material is divided into seven Chapters and an Appendix. Chapter I: Lambda-terms [...], Chapter II: Calculi of lambda-conversion [...]. Chapter III: Rosser combinatory systems [...], Chapter IV: Lambda-definabiliy and numeral systems [...], Chapter V: Delta-conversion [...], Chapter VI: Logic and arithmetic: quanttifier-free systems [...], Chapter VII: Logic and arithmetic: arrows [...]. The type-free systems proposed and discussed in Chapters VI and VII are relevance-preserving, in the sense they do not allow formalizing, in the proposed context, classically and intuitionistically valid principles of the form "ex falso quodlibet". Finally [...], only theories possessing r.e. models admit of a type-free formalization in ordinal progression, on the pattern suggested here, thereby establishing the limits and stregth of the Church program. [...] The Appendix (Adequate numeral systems in \lambda\beta K) collects the examples of adequate numeral systems in [...] lambda-calculus, known so far to the author, in view of a possible classification." (From: "Adrian Rezus: Lambda-conversion and Logic, Ph.D. Dissertation, Rijksuniversiteit Utrecht, June 1981", Abstract, in: "Libertas Mathematica" [Arlington TX], vol. 2, 1982, pp. 182-185.)
RomanianTeX, A multi-lingual LaTeX2e package for Romanian, 1986 (Online: www.ctan.org/tex-archive/language/romanian/RomanianTeX.)
A multilanguage LaTeX2e package for Romanian.
Libertas Mathematica [Arlington TX], 1982
[MR 84b: 03019] The theorem of the title was asserted as Theorem 8 in a paper by J. Lukasiewicz a... more [MR 84b: 03019] The theorem of the title was asserted as Theorem 8 in a paper by J. Lukasiewicz and A. Tarski [see Tarski, "Logic, semantics and metamathematics", English translation, Clarendon Press, Oxford, 1956; MR 17, 1171]. It states: “Every finitely axiomatizable system of propositional calculus with the rules of substitution and detachment in which the two formulas CpCqp and CpCqCCpCqrr are provable can be axiomatized by a single formula.” According to the author, no proof of Tarski's theorem had been published prior to the present paper. The proof uses the formulas-as-types idea of H. B. Curry [Curry and R. Feys, "Combinatory logic", Vol I, Theorem 9E, North-Holland, Amsterdam, 1958; MR 20 #817] and the condensed detachment operator of C. A. Meredith [see, e.g., Meredith and A. N. Prior, Notre Dame J. Formal Logic 4 (1963), 171-187; MR 30 #1033]. Specifically, if \alpha and \beta are formulas of propositional calculus, and there is a substitution yielding \alpha' and \beta' where \alpha' is C\beta'\gamma, then the value of the metaformula (\alpha\beta) is the most general form of any such, whence we write (\alpha\beta) = \gamma. The metaformulas constructed from propositional calculus formulas as atoms by the operation of application thus en code proofs of the formulas which are their values. For example, let C and K abbreviate CCpCqrCqCpr and CpCqp, respectively. Then for any \alpha, (CK)\alpha = Cpp, and, whenever \alpha, \beta and \gamma have the requisite forms to give values to the metaformulas, ((C\alpha)\beta)\gamma = (\alpha\gamma)\beta. These examples suggest an analogy with the theory of combinators, and in fact the equality of metaformulas is largely analogous to combinatory equality, which the author exploits to obtain a proof of Tarski's theorem. Noting that CpCqp is not valid in systems of relevant logic, the author modifies and refnes his results to obtain single axioms for the purely implicational fragments of various relevant logics. He states as an open problem the existence of single axioms for relevant logics which include conjunction or disjunction. In an appendix he shows that every closed \lambda-calculus term is equal to an applicative combination of the single term \lambda xyz.y(\lambda u.z)(xz). (Review by B. Lercher.)
CWI Syllabi, 1983
KNAW Narcis. Back to search results. Publication Abstract AUTOMATH (1983) Open access. Pagina-nav... more KNAW Narcis. Back to search results. Publication Abstract AUTOMATH (1983) Open access. Pagina-navigatie: Main. Title, Abstract AUTOMATH. Published in, CWI Syllabi, Vol. 160, p.1-188. Author, Rezus, A. Date, 1983-01-01. Language, English. Type, book. ...
"This thesis is intended to re-consider, in a modern setting, the foundational aspec... more "This thesis is intended to re-consider, in a modern setting, the foundational aspects of the theories of lambda-conversion and the corresponding combinatory variants, as originally proposed and worked out within the Church school (A. Church, S. C. Kleene, J. B. Rosser and, incidentally, A. M. Turing) in the thirties. The material is divided into seven Chapters and an Appendix. Chapter I: Lambda-terms [...], Chapter II: Calculi of lambda-conversion [...]. Chapter III: Rosser combinatory systems [...], Chapter IV: Lambda-definabiliy and numeral systems [...], Chapter V: Delta-conversion [...], Chapter VI: Logic and arithmetic: quanttifier-free systems [...], Chapter VII: Logic and arithmetic: arrows [...]. The type-free systems proposed and discussed in Chapters VI and VII are relevance-preserving, in the sense they do not allow formalizing, in the proposed context, classically and intuitionistically valid principles of the form "ex falso quodlibet". Finally [...], only theories possessing r.e. models admit of a type-free formalization in ordinal progression, on the pattern suggested here, thereby establishing the limits and stregth of the Church program. [...] The Appendix (Adequate numeral systems in \lambda\beta K) collects the examples of adequate numeral systems in [...] lambda-calculus, known so far to the author, in view of a possible classification." (From: "Adrian Rezus: Lambda-conversion and Logic, Ph.D. Dissertation, Rijksuniversiteit Utrecht, June 1981", Abstract, in: "Libertas Mathematica" [Arlington TX], vol. 2, 1982, pp. 182-185.)
Disponible depuis: http://www. dante. de/tex-archive/ …
Libertas Math, 1982
[MR 84b: 03019] The theorem of the title was asserted as Theorem 8 in a paper by J. Lukasiewicz a... more [MR 84b: 03019] The theorem of the title was asserted as Theorem 8 in a paper by J. Lukasiewicz and A. Tarski [see Tarski, "Logic, semantics and metamathematics", English translation, Clarendon Press, Oxford, 1956; MR 17, 1171]. It states: “Every finitely axiomatizable system of propositional calculus with the rules of substitution and detachment in which the two formulas CpCqp and CpCqCCpCqrr are provable can be axiomatized by a single formula.” According to the author, no proof of Tarski's theorem had been published prior to the present paper. The proof uses the formulas-as-types idea of H. B. Curry [Curry and R. Feys, "Combinatory logic", Vol I, Theorem 9E, North-Holland, Amsterdam, 1958; MR 20 #817] and the condensed detachment operator of C. A. Meredith [see, e.g., Meredith and A. N. Prior, Notre Dame J. Formal Logic 4 (1963), 171-187; MR 30 #1033]. Specifically, if \alpha and \beta are formulas of propositional calculus, and there is a substitution yielding \alpha' and \beta' where \alpha' is C\beta'\gamma, then the value of the metaformula (\alpha\beta) is the most general form of any such, whence we write (\alpha\beta) = \gamma. The metaformulas constructed from propositional calculus formulas as atoms by the operation of application thus en code proofs of the formulas which are their values. For example, let C and K abbreviate CCpCqrCqCpr and CpCqp, respectively. Then for any \alpha, (CK)\alpha = Cpp, and, whenever \alpha, \beta and \gamma have the requisite forms to give values to the metaformulas, ((C\alpha)\beta)\gamma = (\alpha\gamma)\beta. These examples suggest an analogy with the theory of combinators, and in fact the equality of metaformulas is largely analogous to combinatory equality, which the author exploits to obtain a proof of Tarski's theorem. Noting that CpCqp is not valid in systems of relevant logic, the author modifies and refnes his results to obtain single axioms for the purely implicational fragments of various relevant logics. He states as an open problem the existence of single axioms for relevant logics which include conjunction or disjunction. In an appendix he shows that every closed \lambda-calculus term is equal to an applicative combination of the single term \lambda xyz.y(\lambda u.z)(xz). (Review by B. Lercher.)
[MR 88a: 03034] The author presents an interpretation of P. Martin-Löf's type theory CST... more [MR 88a: 03034] The author presents an interpretation of P. Martin-Löf's type theory CST [Logic, methodology, and philosophy of science, VI (Hannover, 1979), 153-175, North-Holland, Amsterdam, 1982; MR 85d:03112] with one universe and without identity types. In the author's approach, following D. S. Scott [SIAM J. Comput. 5 (1976), no. 3, 522-587; MR 55 # 10262], types are translated into objects in a model of the type-free lambda calculus. The goal is to show that at least a subsystem of constructive type theory is intelligible from a classical non-constructivistic viewpoint. A second goal is to provide an abstract setting for the design of a large class of typed programming languages. The author's presentation includes an extensive discussion of related work on models of constructive type theory. (Review by Henry Africk.)
Tarski's Claim (TC = Theorem 8 in [13]) follows from simple conside-rations in (type-free) l... more Tarski's Claim (TC = Theorem 8 in [13]) follows from simple conside-rations in (type-free) lambda-calculus. The present note records essen-tially a proof of Lemma 1.1 in [16], ie TCL = the type-free lambda-calculus variant of TC, as well as a few historical ...
Irregular F is an open access online journal that publishes work from all areas and traditions of... more Irregular F is an open access online journal that publishes work from all areas and traditions of philosophy. High-quality theoretical research in social sciences and humanities may also be accepted. The journal focuses on research addressing novel or scarcely treated subject matter, remaining nevertheless faithful to the norms and standards of academic research. Irregular F promotes dialogues between various disciplines and the work of young researchers, but any valuable contribution from the aforementioned fields of study is welcome.
Information and Computation/information and Control - IANDC, 1983
"Abstract Automath", Mathematisch Centrum [CWI], Amsterdam 1983 [Mathematical Centre Tracts 160] [ISBN 90 6196 256 0], 1983
"Lambda-conversion and Logic", University of Utrecht, Utrecht 1981 [iv + 197 pp.], Jun 4, 1981
"This thesis is intended to re-consider, in a modern setting, the foundational aspects of th... more "This thesis is intended to re-consider, in a modern setting, the foundational aspects of the theories of lambda-conversion and the corresponding combinatory variants, as originally proposed and worked out within the Church school (A. Church, S. C. Kleene, J. B. Rosser and, incidentally, A. M. Turing) in the thirties. The material is divided into seven Chapters and an Appendix. Chapter I: Lambda-terms [...], Chapter II: Calculi of lambda-conversion [...]. Chapter III: Rosser combinatory systems [...], Chapter IV: Lambda-definabiliy and numeral systems [...], Chapter V: Delta-conversion [...], Chapter VI: Logic and arithmetic: quanttifier-free systems [...], Chapter VII: Logic and arithmetic: arrows [...]. The type-free systems proposed and discussed in Chapters VI and VII are relevance-preserving, in the sense they do not allow formalizing, in the proposed context, classically and intuitionistically valid principles of the form "ex falso quodlibet". Finally [...], only theories possessing r.e. models admit of a type-free formalization in ordinal progression, on the pattern suggested here, thereby establishing the limits and stregth of the Church program. [...] The Appendix (Adequate numeral systems in \lambda\beta K) collects the examples of adequate numeral systems in [...] lambda-calculus, known so far to the author, in view of a possible classification." (From: "Adrian Rezus: Lambda-conversion and Logic, Ph.D. Dissertation, Rijksuniversiteit Utrecht, June 1981", Abstract, in: "Libertas Mathematica" [Arlington TX], vol. 2, 1982, pp. 182-185.)
Irregular F is an open access online journal that publishes work from all areas and traditions of... more Irregular F is an open access online journal that publishes work from all areas and traditions of philosophy. High-quality theoretical research in social sciences and humanities may also be accepted. The journal focuses on research addressing novel or scarcely treated subject matter, remaining nevertheless faithful to the norms and standards of academic research. Irregular F promotes dialogues between various disciplines and the work of young researchers, but any valuable contribution from the aforementioned fields of study is welcome.
Information and Computation/information and Control - IANDC, 1983
"Abstract Automath", Mathematisch Centrum [CWI], Amsterdam 1983 [Mathematical Centre Tracts 160] [ISBN 90 6196 256 0], 1983
"Lambda-conversion and Logic", University of Utrecht, Utrecht 1981 [iv + 197 pp.], Jun 4, 1981
"This thesis is intended to re-consider, in a modern setting, the foundational aspects of th... more "This thesis is intended to re-consider, in a modern setting, the foundational aspects of the theories of lambda-conversion and the corresponding combinatory variants, as originally proposed and worked out within the Church school (A. Church, S. C. Kleene, J. B. Rosser and, incidentally, A. M. Turing) in the thirties. The material is divided into seven Chapters and an Appendix. Chapter I: Lambda-terms [...], Chapter II: Calculi of lambda-conversion [...]. Chapter III: Rosser combinatory systems [...], Chapter IV: Lambda-definabiliy and numeral systems [...], Chapter V: Delta-conversion [...], Chapter VI: Logic and arithmetic: quanttifier-free systems [...], Chapter VII: Logic and arithmetic: arrows [...]. The type-free systems proposed and discussed in Chapters VI and VII are relevance-preserving, in the sense they do not allow formalizing, in the proposed context, classically and intuitionistically valid principles of the form "ex falso quodlibet". Finally [...], only theories possessing r.e. models admit of a type-free formalization in ordinal progression, on the pattern suggested here, thereby establishing the limits and stregth of the Church program. [...] The Appendix (Adequate numeral systems in \lambda\beta K) collects the examples of adequate numeral systems in [...] lambda-calculus, known so far to the author, in view of a possible classification." (From: "Adrian Rezus: Lambda-conversion and Logic, Ph.D. Dissertation, Rijksuniversiteit Utrecht, June 1981", Abstract, in: "Libertas Mathematica" [Arlington TX], vol. 2, 1982, pp. 182-185.)
RomanianTeX, A multi-lingual LaTeX2e package for Romanian, 1986 (Online: www.ctan.org/tex-archive/language/romanian/RomanianTeX.)
A multilanguage LaTeX2e package for Romanian.
Libertas Mathematica [Arlington TX], 1982
[MR 84b: 03019] The theorem of the title was asserted as Theorem 8 in a paper by J. Lukasiewicz a... more [MR 84b: 03019] The theorem of the title was asserted as Theorem 8 in a paper by J. Lukasiewicz and A. Tarski [see Tarski, "Logic, semantics and metamathematics", English translation, Clarendon Press, Oxford, 1956; MR 17, 1171]. It states: “Every finitely axiomatizable system of propositional calculus with the rules of substitution and detachment in which the two formulas CpCqp and CpCqCCpCqrr are provable can be axiomatized by a single formula.” According to the author, no proof of Tarski's theorem had been published prior to the present paper. The proof uses the formulas-as-types idea of H. B. Curry [Curry and R. Feys, "Combinatory logic", Vol I, Theorem 9E, North-Holland, Amsterdam, 1958; MR 20 #817] and the condensed detachment operator of C. A. Meredith [see, e.g., Meredith and A. N. Prior, Notre Dame J. Formal Logic 4 (1963), 171-187; MR 30 #1033]. Specifically, if \alpha and \beta are formulas of propositional calculus, and there is a substitution yielding \alpha' and \beta' where \alpha' is C\beta'\gamma, then the value of the metaformula (\alpha\beta) is the most general form of any such, whence we write (\alpha\beta) = \gamma. The metaformulas constructed from propositional calculus formulas as atoms by the operation of application thus en code proofs of the formulas which are their values. For example, let C and K abbreviate CCpCqrCqCpr and CpCqp, respectively. Then for any \alpha, (CK)\alpha = Cpp, and, whenever \alpha, \beta and \gamma have the requisite forms to give values to the metaformulas, ((C\alpha)\beta)\gamma = (\alpha\gamma)\beta. These examples suggest an analogy with the theory of combinators, and in fact the equality of metaformulas is largely analogous to combinatory equality, which the author exploits to obtain a proof of Tarski's theorem. Noting that CpCqp is not valid in systems of relevant logic, the author modifies and refnes his results to obtain single axioms for the purely implicational fragments of various relevant logics. He states as an open problem the existence of single axioms for relevant logics which include conjunction or disjunction. In an appendix he shows that every closed \lambda-calculus term is equal to an applicative combination of the single term \lambda xyz.y(\lambda u.z)(xz). (Review by B. Lercher.)