Bulletin of Symbolic Logic 16 (1):1-36 (2010)

Authors
Peter Koellner
Harvard University
Abstract
In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing.
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DOI 10.2178/bsl/1264433796
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References found in this work BETA

Ontological Relativity.W. V. Quine - 1968 - Journal of Philosophy 65 (7):185-212.
Nominalist Platonism.George Boolos - 1985 - Philosophical Review 94 (3):327-344.
Completeness in the Theory of Types.Leon Henkin - 1950 - Journal of Symbolic Logic 15 (2):81-91.
Philosophy of Logic.Willard V. O. Quine - 1986 - Philosophy 17 (3):392-393.
Plural Quantification Exposed.Øystein Linnebo - 2003 - Noûs 37 (1):71–92.

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Citations of this work BETA

Infinity and a Critical View of Logic.Charles Parsons - 2015 - Inquiry: An Interdisciplinary Journal of Philosophy 58 (1):1-19.
Structural-Abstraction Principles.Graham Leach-Krouse - 2015 - Philosophia Mathematica:nkv033.

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