OJPP  Vol.4 No.1 , February 2014
Model Theories of Set Theories and Type Theory
Abstract: This paper is divided into three parts. In the first part, we review the historical background of a system of logic devised by Henry S. Leonard to allow for reasoning using existence as a predicate. In the second part, we consider various directions in which his logic could be further developed, syntactically, semantically, and as an adjunct to quantifier elimination and set theory. In the third and final part, we develop proofs of some underlying results of his logic, using modern notation but retaining his axioms and rules of inference.
Cite this paper: Jones, R. (2014). Model Theories of Set Theories and Type Theory. Open Journal of Philosophy, 4, 54-58. doi: 10.4236/ojpp.2014.41008.

[1]   Baldwin, J. (1985). Definable second-order quantifiers. In J. Barwise, & S. Feferman (Eds.), Model-theoretic logics (pp. 445-477). New York: Springer-Verlag.

[2]   Barcan, R. C. (1946). A functional calculus of first order based on strict implication. Journal of Symbolic Logic, 11, 1-16.

[3]   Barcan, R. C. (1946). The deduction theorem in a functional calculus of first order based on strict implication. Journal of Symbolic Logic, 11, 115-118.

[4]   Barcan, R. C. (1947). The identity of individuals in a strict functional calculus of second order. Journal of Symbolic Logic, 12, 12-15.

[5]   Büchi, J. R. (1962). On a decision method in restricted second order arithmetic. In E. Nagel, P. Suppes, & A. Tarski (Eds.), Logic, methodology and philosophy of science (pp. 1-11). Stanford: Stanford University Press.

[6]   Chang, C. C., & Keisler, H. J. (1990). Model theory (3rd ed.). Amsterdam: Elsevier.

[7]   Fischer, M. J., & Michael, O. R. (1974). Super-exponential complexity of Presburger arithmetic. In R. M. Karp (Ed.), Complexity of computation (pp. 27-41). Providence, RI: American Mathematical Society.

[8]   Goldblatt, R. (2011). Quantifiers, propositions and identity: Admissible semantics for quantified modal and substructural logics. Cambridge: Cambridge University Press.

[9]   Godel, K. (1931). über formal unentscheidbare Satze der principia mathematica und verwandter Systeme, I. Monatshefte für Mathematik und Physik, 38, 173-198.

[10]   Gurevich, Y. (1985). Monadic second-order theories. In J. Barwise, & S. Feferman (Eds.), Model-theoretic logics (pp. 479-506). New York: Springer-Verlag.

[11]   Jech, T. (2006). Set theory: The third millenium edition (3rd ed.). Berlin: Springer-Verlag.

[12]   Jones, R. M. (1962). A note on obversion. Mind, 284, 541-542.

[13]   Jones, R. M. (1964). Formal results in the logic of existence. Philosophical Studies, 15, 7-10.

[14]   Jones, R. M. (2013) Review of Robert Goldblatt. Philosophia mathematica, 21, 115-123.

[15]   Kleene, S. C. (1950). Introduction to metamathematics. New York: D. van Nostrand.

[16]   Lambert, K. (1967). Free logic and the concept of existence. Notre Dame Journal of Formal Logic, 8, 133-144.

[17]   Leonard, H. S. (1969). The logic of existence. Philosophical Studies, 7, 49-64.

[18]   Lindstrom, P. (1969). On extensions of elementary logic. Theoria, 35, 1-11.

[19]   Monk, J. D. (1976). Mathematical logic. New York: Springer-Verlag.

[20]   Presburger, M. (1929). über die Vollstandigkeit eines gewissen systems der Arithmetik ganzer Zahlen, in welchem die addition als einzige operation hervortritt in Comptes Rendus du I congrès de Math? maticiens des Pays Slaves, Warszawa, 92-101, Addendum, 395.

[21]   Quine, W. Van O. (1948). On what there is. Review of Metaphysics, 2, 21-38.

[22]   Smoryński, C. (1991). Logical number theory I: An introduction. Berlin: Springer-Verlag.

[23]   Whitehead, A. N., & Bertrand, R. (1910). Principia mathematica. Cambridge: Cambridge University Press.