Online research in philosophy

Both in dealing with the semantic paradoxes and in dealing with vagueness and indeterminacy, there is some temptation to weaken classical logic: in particular, to restrict the law of excluded middle. The reasons for doing this are somewhat different in the two cases. In the case of the semantic paradoxes, a weakening of classical logic (presumably involving a restriction of excluded middle) is required if we are to preserve the naive theory of truth without inconsistency. In the case of vagueness and indeterminacy, there is no worry about inconsistency; but a central intuition is that we must reject the factual status of certain sentences, and it hard to see how we can do that while claiming that the law of excluded middle applies to those sentences. So despite the different routes, we have a similar conclusion in the two cases.

Truth values are, properly understood, merely proxies for the various relations that can hold between language and the world. Once truth values are understood in this way, consideration of the Liar paradox and the revenge problem shows that our language is indefinitely extensible, as is the class of truth values that statements of our language can take – in short, there is a proper class of such truth values. As a result, important and unexpected connections emerge between the semantic paradoxes and the set-theoretic paradoxes.

The aim of this paper is to show that Graham Priest's dialetheic account of semantic paradoxes and the paraconsistent logics employed cannot achieve semantic universality. Dialetheism therefore fails as a solution to semantic paradoxes for the same reason that consistent approaches did. It will be demonstrated that if dialetheism can express its own semantic principles, a strengthened liar paradox will result, which renders dialetheism trivial. In particular, the argument is not invalidated by relational valuations, which were brought into paraconsistent logic in order to avoid strengthened liar paradoxes.

The paradoxes of self-reference are genuinely paradoxical. The liar paradox, Russell’s paradox and their cousins pose enormous difficulties to anyone who seeks to give a comprehensive theory of semantics, or of sets, or of any other domain which allows a modicum of self-reference and a modest number of logical principles. One approach to the paradoxes of self-reference takes these paradoxes as motivating a non-classical theory of logical consequence. Similar logical principles are used in each of the paradoxical inferences. If one or other of these problematic inferences are rejected, we may arrive at a consistent (or at least, a coherent) theory. In this paper I will show that such approaches come at a serious cost. The general approach of using the paradoxes to restrict the class of allowable inferences places severe constraints on the domain of possible propositional logics, and on the kind of metatheory that is appropriate in the study of logic itself. Proof-theoretic and model-theoretic analyses of logical consequence make provide different ways for non-classical responses to the paradoxes to be defeated by revenge problems: the redefinition of logical connectives thought to be ruled out on logical grounds. Non-classical solutions are not the “easy way out” of the paradoxes.

emantic pathologies of self-reference include the Liar (‘this sentence is false’), the Truth-Teller (‘this sentence is true’) and the Open Pair (‘the neighbouring sentence is false’ ‘the neighbouring sentence is false’). Although they seem like perfectly meaningful declarative sentences, truth value assignment to their uses seems either inconsistent (the Liar) or arbitrary (the Truth-Teller and the Open-Pair). These pathologies thus call for a resolution. I propose such a resolution in terms of relative-truth: the truth value of a pathological sentence use varies with the context of its assessment. It always has a determinate truth value, but this truth value is relative to the context of its assessment. I start by considering a fairly esoteric pathology: the Truth-Teller, that is, sentences which assert nothing but their own truth. I make the case that truth value of a given truth-teller use must in general depend on the context of its assessment, and that one can indeed change its truth value at will. I then show how the notion of assessment-sensitive truth can help us provide solutions to other semantic paradoxes such as the Liar and the Open Pair and that those solutions are immune to revenge problems. I conclude by situating my proposal among the main approaches to the semantic paradoxes, and by drawing a very broad moral about pathological self-reference and intentionality.

I propose a solution to the aletheic paradoxes on which truth predicates are assessment-sensitive. Truth is not an antecedently plausible topic for a semantic relativist treatment; nevertheless, the aletheic paradoxes give us good reason to think that truth is an inconsistent concept, and there are good reasons to think that semantic relativism is appropriate for inconsistent concepts, especially those that display what I call empirical inconsistency. Thus, I show that a promising version of the best approach to the paradoxes is an application of semantic relativism to truth itself—arguing from results about the paradoxes and general considerations about language use to aletheic assessment-sensitivity. The paper is divided into three parts, the first on the aletheic paradoxes, the second on semantic relativism, and the third on assessment-sensitivity with respect to truth predicates. The first contains an overview of my preferred approach to the paradoxes, which entails that truth is an inconsistent concept that that should be replaced (for certain purposes) by a team of consistent concepts that can do its work without causing troubling paradoxes. The second part provides an overview of semantic relativism and its rivals. The third considers which treatment is most appropriate for inconsistent concepts in general and truth in particular. In it, I propose an assessment-sensitivity view of truth, discuss some prominent objections to semantic relativism, and review some issues that arise for approaches to the aletheic paradoxes.

The vast majority of approaches to the liar paradox generate new paradoxes that are structurally similar to the liar (often called revenge paradoxes). There is a complex group of issues surrounding revenge paradoxes, the expressive powers of natural languages, and the adequacy of approaches to the liar. My goal is to provide a precise framework against which these issues can be formulated and discussed. The centerpiece of this framework is the notion of internalizability: a semantic theory is internalizable for a language if and only if there exists an extension of the language such that (i) the theory is expressible in that extended language, and (ii) the theory assigns meanings to all the relevant sentences of that extended language. The framework is applied to three examples from the literature: Reinhardt and McGee on theories that require expressively richer metalanguages, Field on revenge-immunity, and Gupta on semantic self-sufficiency.

It is “the received wisdom” that any intuitively natural and consistent resolution of a class of semantic paradoxes immediately leads to other paradoxes just as bad as the first. This is often called the “revenge problem”. Some proponents of the received wisdom draw the conclusion that there is no hope of any natural treatment that puts all the paradoxes to rest: we must either live with the existence of paradoxes that we are unable to treat, or adopt artificial and ad hoc means to avoid them. Others (“dialetheists”) argue that we can put the paradoxes to rest, but only by licensing the acceptance of some contradictions (presumably in a paraconsistent logic that prevents the contradictions from spreading everywhere).

I express my dissatisfaction with the common ways to treat the semantic paradoxes. Not only do they give rise to revenge paradoxes, they ignore the wisdom contained in the ordinary reaction to paradoxes. I instead propose an account that vindicates the ordinary reaction to paradox by putting the blame on us philosophers. It is the wrong conception of what a valid inference is, one that is central to “the ideal of deductive logic” that gives rise to the problem. The solution outlined gives us a new way to accept defeat in light of the paradoxes: the arguments that lead to them are based on valid forms of reasoning, but their conclusions are nonetheless rationally rejected.

Consideration of a paradox originally discovered by John Buridan provides a springboard for a general solution to paradoxes within the Liar family. The solution rests on a philosophical defence of truth-value-gaps and is consistent (non-dialetheist), avoids ‘revenge’ problems, imports no ad hoc assumptions, is not applicable to only a proper subset of the semantic paradoxes and implies no restriction of the expressive capacities of language.