Expressive Limitations and the Liar’s Revenge: A Strict-Tolerant Solution and A Pragmatic Solution For Dialetheism
Ho-Yin Lui

Abstract:
In this thesis, I examine three different approaches to the Liar paradox and its revenge that respect the naive principles of truth. These approaches are the paracomplete gap approaches, the paraconsistent dialetheic approaches, and the strict-tolerant dialetheic approaches.
It is often argued that these approaches are either expressively incomplete or suffer from revenge paradoxes. Firstly, on these approaches, certain important semantical notions, if formalized in an obvious way, cannot get the desired interpretation. In paracomplete logics, the claim that the Liar sentence is neither true nor false, if formalized as ¬(T<λ> ∨ T<¬λ>), does not come out true. In paraconsistent logics and strict-tolerant logics, the claim that a sentence A is just true, if formalized as T<A> ∧ ¬T<¬A>, can still be a contradiction. Secondly, to fix these problems, one may suggest that we introduce some extra connectives to increase the expressive power of the theory in question. However, it is often argued that adding extra connectives gives rise to revenge paradoxes: given that semantical notions such as truth-value gaps and just true are expressible, they breed some liar-like paradoxes with which the theory cannot deal.
The first task of the thesis is that I argue that while the paracomplete gap approaches and the paraconsistent dialetheic approaches are plagued with revenge paradoxes, the strict-tolerant dialetheic approaches can resist the revenge arguments which makes use of the material biconditional or the semantic equivalence to represent self-reference.
The issue of expressive limitations is sometimes formulated as a problem of expressing disagreement. It has been argued that dialetheists have trouble in expressing what they disagree about to their opponents. Suppose that a dialetheist disagrees with A. Asserting ¬A (or ¬T<A>) cannot do the job, because ¬A (or ¬T<A>) is compatible with A. This is known as the exclusion problem.
It has been suggested that pragmatic implicatures accounts for how dialetheists communicate what they disagree about to their opponents: when a dialetheist asserts that it is not the case that A (or A is not true), his assertion will implicate that he does not accept A. It is often claimed that this suggestion does not work, because pragmatic implicatures cannot act on embedded sentences. In this thesis, we present some linguistic evidence that implicatures can arise at the subsentential level. The second task of the thesis is to develop an account of embedded implicatures. Our account explains how dialetheists communicate disagreement to their opponents through implicatures.