Semantics and Completeness for Schematic Logic

Oliver William Tatton-Brown

DOI: http://dx.doi.org/10.12775/LLP.2020.021

Abstract


This paper gives a semantics for schematic logic, proving soundness and completeness. The argument for soundness is carried out in ontologically innocent fashion, relying only on the existence of formulae which are actually written down in the course of a derivation in the logic. This makes the logic available to a nominalist, even a nominalist who does not wish to rely on modal notions, and who accepts the possibility that the universe may in fact be finite.


Keywords


logic; nominalist; schematic logic; semantics; completeness

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References


Boolos, George, 1971, “The iterative conception of set”, The Journal of Philosophy 68 (8): 215–231. URL http://www.jstor.org/stable/2025204. DOI: http://dx.doi.org/10.2307/2025204

Burgess, John P., and Gideon Rosen, 1997, A Subject With No Object: Strategies for Nominalistic Interpretation of Mathematics, Oxford University Press: Oxford, New York.

Cori, René, and Daniel Lascar, 2001, Mathematical Logic: A Course with Exercises Part II, OUP Oxford: Oxford.

Feferman, Solomon, 1991, “Reflecting on incompleteness”, The Journal of Symbolic Logic 56 (1): 1–49. URL http://www.jstor.org/stable/2274902. DOI: http://dx.doi.org/10.2307/2274902

Field, Hartry, 1980, Science Without Numbers, Princeton University Press: Princeton, N.J.

Heck, Richard Kimberly, 2011, Frege’s Theorem, OUP Oxford: Oxford, New York. (Originally published under the name “Richard G. Heck, Jr”.)

Lavine, Shaughan, 1998, Understanding the Infinite, new edition, Harvard University Press: Cambridge, Mass.

Maddy, Penelope, 1997, Naturalism in Mathematics, Clarendon Press.

McGee, Vann, 1997, “How we learn mathematical language”, The Philosophical Review 106 (1): 35–68. URL http://www.jstor.org/stable/2998341. DOI: http://dx.doi.org/10.2307/2998341

Parsons, Charles, 2007, Mathematical Thought and its Objects, Cambridge University Press.

Quine, W.V., 1999, “Reply to Charles Parsons”, in pages 396–403 Lewis Edwin Hahn and Paul Arthur Schilpp (eds.), The Philosophy of W.V. Quine, Volume 18, Open Court Publishing Co: La Salle, Ill.

Shapiro, Stewart, 2000, Foundations without Foundationalism: A Case for Second-order Logic, new edition, Oxford University Press: Oxford, USA.

Shoenfield, J., 1982, “Axioms of set theory”, in Handbook of Mathematical Logic, North Holland: Amsterdam, New York.








ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

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