This paper has, as its point of departure, the foundational
axiomatic theory of E. De Giorgi (1996, Scuola Normale
Superiore di Pisa, Preprints di Matematica 26, 1), based on two
primitive notions of quality and relation. With the introduction of
a unary relation, we develop a system totally based on the sole
primitive notion of relation. Such a modification enables a definition
of the concept of dynamic unary relation. In this way we construct a
simple language capable to express other well known theories such
as Robinson-s arithmetic or a piece of a theory of concatenation. A
key role in this system plays an abstract relation designated by “( )",
which can be interpreted in different ways, but in this paper we will
focus on the case when we can perform computations and obtain
results.
[1] H. B., Curry, Foundations of Mathematical Logic, New York: McGraw-
Hill, 1963.
[2] E. De Giorgi, M. Forti, M. and G. Lenzi, Verso i sistemi assiomatici del
2000 in matematica, logica e informatica, Scuola Normale Superiore di
Pisa, Preprints di Matematica (26), 1-19, 1996.
[3] E. Engeler, Foundations of Mathematics: Questions of Analysis, Geometry
and Algorithmics, Berlin: Springer, 1993.
[4] M. Forti and G. Lenzi, A general axiomatic framework for the foundations
of mathematics, logic and computer science, Rend. Mat. Acc. Naz. Sci.,
(XL), 1-32, 1997.
[5] A. Grzegorczyk and K. Zdanowski, Undecidability and Concatenation,
in: Ehrenfeucht, A., Marek, V. W., Srebrny, M. (eds). Andrzej Mostowski
and foundational studies, Amsterdam: IOS Press, 2008.
[6] L. Obojska, "Primary relations" in a new foundational axiomatic framework,
Journal of Philosophical Logic, 36 (6), 641-657, 2007.
[7] G. Peano, Arithmetices principia nova methodo expositia, in: Opere
scelte, vol. 2, 20-55, Rome: Cremonese, 1958.
[8] W.V.O. Quine, Concatenation as a basis for arithmetic, Journal of
Symbolic Logic, 11 (4), 105-114, 1946.
[9] G. C. Rota, Husserl and the Reform of Logic, in: M. Kac, G. C. Rota, J.
Schwartz, Discrete Thoughts, Basel: Birkhauser, 1992.
[10] B. Smith, Logic and Formal Ontology, in: Husserl-s Phenomenology,
(ed.) J. N. Mohanty and W. McKenna, Lanham: University Press of
America, 29-67, 1989.
[11] V. Svejdar, Weak Theories and Essential Incompleteness, The Logica
Yearbook 2007: Proc. of the Logica07 Int. Conference, 213-224, 2008.
[12] V. Svejdar, Relatives of Robinson Arithmetic, The Logica Yearbook
2008: Proc. of the Logica08 Int. Conference, 253-263, 2009.
[13] A. Tarski, A. Mostowski and R. M. Robinson, Undecidable Theories,
Amsterdam: North-Holland, 1953.
[1] H. B., Curry, Foundations of Mathematical Logic, New York: McGraw-
Hill, 1963.
[2] E. De Giorgi, M. Forti, M. and G. Lenzi, Verso i sistemi assiomatici del
2000 in matematica, logica e informatica, Scuola Normale Superiore di
Pisa, Preprints di Matematica (26), 1-19, 1996.
[3] E. Engeler, Foundations of Mathematics: Questions of Analysis, Geometry
and Algorithmics, Berlin: Springer, 1993.
[4] M. Forti and G. Lenzi, A general axiomatic framework for the foundations
of mathematics, logic and computer science, Rend. Mat. Acc. Naz. Sci.,
(XL), 1-32, 1997.
[5] A. Grzegorczyk and K. Zdanowski, Undecidability and Concatenation,
in: Ehrenfeucht, A., Marek, V. W., Srebrny, M. (eds). Andrzej Mostowski
and foundational studies, Amsterdam: IOS Press, 2008.
[6] L. Obojska, "Primary relations" in a new foundational axiomatic framework,
Journal of Philosophical Logic, 36 (6), 641-657, 2007.
[7] G. Peano, Arithmetices principia nova methodo expositia, in: Opere
scelte, vol. 2, 20-55, Rome: Cremonese, 1958.
[8] W.V.O. Quine, Concatenation as a basis for arithmetic, Journal of
Symbolic Logic, 11 (4), 105-114, 1946.
[9] G. C. Rota, Husserl and the Reform of Logic, in: M. Kac, G. C. Rota, J.
Schwartz, Discrete Thoughts, Basel: Birkhauser, 1992.
[10] B. Smith, Logic and Formal Ontology, in: Husserl-s Phenomenology,
(ed.) J. N. Mohanty and W. McKenna, Lanham: University Press of
America, 29-67, 1989.
[11] V. Svejdar, Weak Theories and Essential Incompleteness, The Logica
Yearbook 2007: Proc. of the Logica07 Int. Conference, 213-224, 2008.
[12] V. Svejdar, Relatives of Robinson Arithmetic, The Logica Yearbook
2008: Proc. of the Logica08 Int. Conference, 253-263, 2009.
[13] A. Tarski, A. Mostowski and R. M. Robinson, Undecidable Theories,
Amsterdam: North-Holland, 1953.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:64789", author = "Lidia Obojska", title = "Relational Framework and its Applications", abstract = "This paper has, as its point of departure, the foundational
axiomatic theory of E. De Giorgi (1996, Scuola Normale
Superiore di Pisa, Preprints di Matematica 26, 1), based on two
primitive notions of quality and relation. With the introduction of
a unary relation, we develop a system totally based on the sole
primitive notion of relation. Such a modification enables a definition
of the concept of dynamic unary relation. In this way we construct a
simple language capable to express other well known theories such
as Robinson-s arithmetic or a piece of a theory of concatenation. A
key role in this system plays an abstract relation designated by “( )",
which can be interpreted in different ways, but in this paper we will
focus on the case when we can perform computations and obtain
results.", keywords = "language, unary relations, arithmetic, computability", volume = "5", number = "12", pages = "2149-6", }