A Tale of Excluding the Middle

he paper discusses a number of interconnected points concerning negation, truth, validity and the liar paradox. In particular, it discusses an argument for the dialetheic nature of the liar sentence which draws on Dummett’s teleological account of truth. Though one way of formulating this fails, a different way succeeds. The paper then discusses the role of the Principle of Excluded Middle in the argument, and of the thought that truth in a model should be a model of truth.

The liar paradox in the predictive mind

Most discussions frame the Liar Paradox as a formal logical-linguistic puzzle. Attempts to resolve the paradox have focused very little so far on aspects of cognitive psychology and processing, because semantic and cognitive-psychological issues are generally assumed to be disjunct. I provide a motivation and carry out a cognitive-computational treatment of the liar paradox based on a cognitive-computational model of language and conceptual knowledge within the Predictive Processing (PP) framework. I suggest that the paradox arises as a failure of synchronization between two ways of generating the liar situation in two different (idealized) PP sub-models, one corresponding to language processing and the other to the processing of meaning and world-knowledge. In this way, I put forward the claim that the liar sentence is meaningless but has an air of meaningfulness. I address the possible objection that the proposal violates the Principle of Unrestricted Compositionality, which purportedly regulates the conceptual competence of thinkers.

Not Wanted: On Scharp’s Solution to the Liar

Kevin Scharp argues that the concept of truth is defective, and is therefore unable to play its intended role in natural language truth-conditional semantics. As such, for this theoretical purpose, Scharp constructs two replacements: ascending truth and descending truth. Scharp applies the resultant theory, AD semantics, to the liar sentence, thereby obtaining a novel solution to the liar paradox. The aim of the present paper is fourfold. First, I show that, contrary to Scharp’s claims, AD semantics in fact yields an inconsistency when applied to standard liar sentences. Second, I diagnose the problem: AD semantics mishandles negation. I propose an alternative treatment, resulting in what I call AD* semantics. Third, I show that AD* semantics gives Scharp the resources required to respond to an alleged revenge paradox that has been raised against his view. Finally, I argue that, these consequences notwithstanding, it remains unclear whether AD* semantics provides an adequate account of alethic paradoxes more generally.

Jalāl ad-Dīn ad-Dawānī’s Solution to the Liar Paradox and Its Reception in Qāḍī Mubārak and Mullā Mubīn

I examine the views of Jalāl ad-Dīn ad-Dawānī (d. 1502) on the Liar paradox and their reception in the work of Qāḍī Mubārak (d. 1748) and Mullā Mubīn (d. 1810). Dawānī argues that the Liar sentence is neither true nor false since it is not the kind of utterance that is capable of bearing a truth-value (i.e., it is not truth-apt). In the course of justifying this view, he proposes a criterion for a sentence’s being truth-apt and attempts to counter a number of objections. I address two of these: one involves certain intuitively true or false self-referential sentences and the other is the ‘strengthened Liar.’ I then argue that both Qāḍī Mubārak and Mullā Mubīn present a version of the solution Dawānī gives in his Sharḥ at-Tahdhīb and, moreover, that Dawānī does not endorse this solution in all his other works. Furthermore, the solution they attribute to Dawānī differs slightly from the one he gives in his Sharḥ at-Tahdhīb in terms of how the major premise is justified. I present evidence which shows that this modification was inspired by Mīr Zāhid al-Harawī’s (d. 1689) gloss on Dawānī’s Sharḥ at-Tahdhīb.

Semantic paradoxes and theories of truth

The Cretan philosopher Epimenides said that Cretans always lie. Assuming, for the sake of argument, the mendacity of all other statements by Cretans, we get a paradox: if what Epimenides said was true, it must have been a lie, whereas if what he said was a lie, it would have made his statement true. The citizens of Crete have long since forgiven the insult, but semantics has never recovered. Alfred Tarski perceived the consequences of Epimenides’ paradox with particular clarity. Our common-sense intuitions about truth follow the paradigm: ‘Snow is white’ is true if and only if snow is white. As Tarski rigorously shows, if the language we are describing (the object language) is the same as the language in which we are formulating our theory (the metalanguage), this paradigm will be inconsistent with the rudimentary laws of syntax. The conclusion Tarski drew was that, if we are to develop a satisfactory theory of truth, our metalanguage must be essentially richer in expressive power than the object language. Since there is no human language essentially richer than English (or any other natural language), there can be no satisfactory theory of truth for English. One earnestly hopes that this is not the end of the matter. Tarski’s analysis leaves open the prospect that we can develop a fully satisfactory theory of truth for a substantial fragment of English; also the prospect that we can develop a theory of truth for English as a whole which, while not fully satisfying our intuitions, is none the less useful and illuminating. Both prospects have been substantially advanced by Saul Kripke’s ‘Outline of a Theory of Truth’, which exploits the idea that there are truth-value gaps, statements which are neither true nor false, and that Epimenides’ insult was one of them. Invocation of truth-value gaps does not resolve the paradox in any straightforward way. If we let the phrase ‘the simple liar sentence’ refer to the sentence ‘The simple liar sentence is false’, we see that we can readily account for the paradoxical features of the sentence by declaring the sentence neither true nor false; but if we let the strengthened liar sentence be ‘The strengthened liar sentence is not true’, we get a sentence we cannot dispose of so tidily. If the strengthened liar is neither true nor false, then it is not true; but that it is not true is precisely what the sentence says. Truth-value gaps have not vanquished the liar paradox. Nor have any of the alternatives, the most prominent of which are a contextualist account, which sees the English word ‘true’ as radically ambiguous, and so-called ‘revision theory’, which investigates the cyclic reasoning that occurs when we try to evaluate the simple liar sentence: if the sentence is true, then it must be false; but if, then, it is false, it must be true; and so on. While these approaches have not eliminated the paradox, they have opened new approaches that have greatly improved our prospects for finding a comfortable way to live with it.

Dealing with Liars

In this final chapter, I want to investigate what the theory of truth developed so far tells us about the Liar paradox (and other paradoxes involving truth). I’ll begin by reviewing the paradox (§9.1) and why it is such a difficult issue to solve (§9.2). I’ll then argue that approaches which weaken the underlying logic (§§9.3–9.5), or restrict or alter the T-scheme (§9.6), are not good options. Instead, I’ll argue that our account of propositions from chapter 8 already provides an adequate solution: there is no proposition for the Liar sentence to express (§9.8). I discuss the consequences of this view for meaning and logic in §9.9.

Thinking about the Liar, Fast and Slow

In the past, experimental philosophers have explored the psychological underpinning of a number of notions in philosophy, including free will, moral responsibility, and more. But prior to this chapter, although a number of philosophers have speculated on how ordinary folks might, or should, think about the liar paradox, no one had systematically explored the psychological underpinnings of the Liar itself. The authors take on this task. In particular, the chapter investigates the status of a liar sentence, L = ‘Sentence L is false’. The thesis, arrived at by interpreting the data the authors have accrued, is that reflective thinkers (some of whom possess a modicum of philosophical expertise) judge L to be neither true nor false (as opposed to false or true), and the authors see this as some evidence for the claim that L is neither true nor false.

The Liar without Truth

Liar sentences say nothing, according to this chapter—which, it claims, we can, in effect, prove. But extending the proof as the chapter does appears to result in revenge. The solution to this problem is to restrict the laws of logic by distinguishing expressing a falsehood from failing to express a truth. But the question that presses is how we can signify that a given sentence—a liar sentence, for example—fails to express a truth without being mired in paradox. To this end, the chapter revisits the sort of bilateral system that Rumfitt (2000) has discussed. The chapter shows that there is a way of developing Aristotle’s conception of truth into a definition of truth that does not yield a contradiction, even when applied to a semantically closed language. If successful, the proposal will enable us to reject a Strengthened Liar as untrue without asserting its negation.