David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Review of Symbolic Logic 3 (3):374-414 (2010)
Kripke’s theory of truth succeeded in providing a trivalent semantics for a language that contains its own truth predicate and means of self-reference; but it did so by radically restricting the expressive power of the logic. In Kripke’s analysis, the Liar (e.g. This very sentence is not true) receives the indeterminate truth value; but the logic cannot express the fact that the Liar is something other than true: in order to do so, a weak negation not* would be needed, but it would also make the logic inconsistent (because the ‘Super Liar’ This very sentence is not* true could not be assigned any truth value). Taking a hint from the quantificational form of the problematic sentences (… is something other than true), we define a hierarchy of negations which each quantifies over a domain of truth values, assimilated to ordinals. The resulting logic has as many negations and truth values as there are ordinals. Unlike Kripke’s logic, it enjoys a form of expressive completeness. And although the logic is not monotonic, we show that under broad conditions we can construct a variety of fixed points; one of them emulates Kripke’s ‘least fixed point’, while another one assigns a different truth value to each Super Liar.
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References found in this work BETA
Saul A. Kripke (1975). Outline of a Theory of Truth. Journal of Philosophy 72 (19):690-716.
Laurence Horn (1989). A Natural History of Negation. University of Chicago Press.
Emmanuel Chemla (2009). Presuppositions of Quantified Sentences: Experimental Data. [REVIEW] Natural Language Semantics 17 (4):299-340.
Michael Glanzberg (2004). A Contextual-Hierarchical Approach to Truth and the Liar Paradox. Journal of Philosophical Logic 33 (1):27-88.
Hartry Field (2003). A Revenge-Immune Solution to the Semantic Paradoxes. Journal of Philosophical Logic 32 (2):139-177.
Citations of this work BETA
Toby Meadows (2016). Sets and Supersets. Synthese 193 (6):1875-1907.
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