Why is truth paradoxical

Another kind of substructural approach works by attacking various structural rules associated with transitivity of consequence. The difference is in how consequence is defined on these models. In all cases, consequence amounts to the absence of a countermodel, but there are different understandings available of what a model has to be like to be a countermodel to an argument. The motivating idea is that valid arguments must preserve truth, and must also preserve falsity backwards in a certain sense: if a valid argument has all its premises but one true and its conclusion false, then the remaining premise must be false.

This allows for counterexamples to cut, but not to simple transitivity, and allows for consistency to be maintained. The resulting logic is weaker than classical logic. A different kind of approach without cut is developed and explored in Barrio et al.

On this approach, a countermodel to an argument cannot assign the third value to any sentence that occurs in the argument. It also has the curious feature that every argument valid in classical logic remains valid.

That is, all the counterexamples to cut and simple transitivity involve appeal to capture, release, or some other special behaviour of the truth predicate. Despite this classical flavour, these approaches are also dialetheist; the claim that the liar sentence is both true and not true turns out to be a theorem. Such a claim is forced to take the third value, and so there can be no countermodel to any argument involving it.

Perhaps because of the importance of the rule of cut in proof theory, nontransitive approaches are often studied via proof systems rather than via models. There are helpful philosophical remarks on cut in Schroeder-Heister , which also notes some relations between noncontractive and nontransitive approaches. A third possibility for a substructural approach to paradoxes comes from attacking reflexivity , the principle that every sentence entails itself.

There is a close analogy between reflexivity and transitivity, as explained in Frankowski ; Girard et al. Nonreflexive approaches to paradoxes have so far been less-explored, but seem to be a promising direction for further work; see French and Meadows for more.

See also Malinowski for general work on nonreflexive logics. We have now seen a range of options for responding to the Liar paradox by reconsidering basic logic. There are also a number of approaches that leave classical logic unchanged, and try to find other ways of defusing the paradox. One hallmark of most of these approaches is a willingness to somehow restrict the range of application of capture and release, to block the paradoxical reasoning. This other view takes the main feature of truth to be that it reports a non-trivial semantic property of sentences e.

Many approaches within classical logic embody the idea that a proper understanding of this feature allows for restricted forms of capture and release, and this in turn allows the paradox to be blocked, without any departure from classical logic. We will consider a number of important approaches to the paradox within classical logic, most of which embody this idea in some form or another.

Instead, Tarski proposed that the truth predicate for a language is to be found only in an expanded metalanguage. Of course, this process does not stop. And so on. The process goes on indefinitely.

At each stage, a new classical interpreted language is produced, which expresses truth for languages below it. For more on the mathematics of this sort of hierarchy of languages, see Halbach Why is there no Liar paradox in this sort of hierarchy of languages?

Because the restriction that no truth predicate can apply to sentences of its own language is enforced as a syntactic one. There is no Liar paradox because there is no Liar sentence. One is that in light of naturally occurring cases of self-reference, his ruling Liar sentences syntactically not well-formed seems overly drastic. Another important problem was highlighted by Kripke As Kripke notes, any syntactically fixed set of levels will make it extremely hard, if not impossible, to place various non-paradoxical claims within the hierarchy.

For instance, if Jc says that everything Michael says is true , the claim has to be made from a level of the hierarchy higher than everything Michael says. This is impossible. It is also difficult to explain what level of the hierarchy an utterance winds up at when it can be coherently assigned a level.

What makes it such that it involves truth at one level rather than another? If such predicates are allowed, we are back in paradox, so defenders of the Tarskian hierarchy must say they are not possible. Explaining why is a problem for all hierarchical views. See Glanzberg for further discussion. One goal has been to work out which ones are well-motivated, and how to implement them. One way to do this was suggested by Kripke himself.

But it does make a restricted form true. What happens to the Liar sentence on this approach? As in the three-valued case, the Liar is interpreted as falling within the gap. Thus, the construction can be done without any implicit appeal to many-valued logic. Related issue bear in the classical case. We will discuss a few in turn. It also, as we observed, applies to all the sentences which are well-behaved in the sense of obeying the T-schema or capture and release. Kripke labeled this being grounded.

Herzberger The idea is that the determinate sentences are the ones with well-defined semantic properties. Where we have no such well-defined semantic properties, we should not expect the truth predicate to report anything well-behaved, nor should we expect properties like capture and release to hold. The notion of grounding has spawned its own literature, with Leitgeb a key impetus.

See also Bonnay and van Vugt , Meadows , and Schindler Another view which makes use of a form of determinateness is advocated by McGee The theory has many components, including a mathematically sophisticated approaches to truth related to the Kripkean ideas we have been discussing, in a setting which holds to classical logic. McGee relies on two notions: truth and definite truth. Definite truth is a form of the idea we glossed as determinateness. But, McGee describes this idea using some very sophisticated logical techniques.

We will mention them briefly, for those familiar with the technical background. It is thus different from the grounding notion we just discussed. McGee treats definitely as a predicate , on par with the truth predicate, and not as an operator on sentences as some developments do.

With the right notion of definite truth, McGee shows that a partially interpreted language containing its own truth predicate can meet restricted forms of capture and release put in terms of definite truth.

Indeed, McGee shows that these conditions can be met within a theory of both truth and definite truth, where truth meets appropriate forms of capture and release, and also where a formal statement of bivalence for truth comes out definitely true. McGee thus provides a theory which has strongly self-applicative truth and definite truth, within a classical setting. Thus, definite truth meets weaker forms of capture and release than truth itself. Furthermore, McGee suggests that this behavior of truth and definite truth makes truth a vague predicate.

We have now surveyed some important representatives of approaches to resolving the Liar within classical logic. There are a number of others, many of them involving some complex mathematics. We will pause to mention a few of the more important of these, though given the mathematical complexity, we will only gesture towards them. There is an important strand of work in proof theory, which has sought to develop axiomatic theories of self-applicative truth in classical logic, including work of Cantini , Feferman , , Friedman and Sheard , Halbach , and Horsten The idea is to find ways of expressing rules like capture and release that retain consistency.

Options include more care about how proof-theoretic rules of inference are formulated, and more care about formulating restricted rules. The main ideas are discussed in the entry on axiomatic theories of truth , to which we will leave the details.

These connections are explored further in work of Burgess and McGee We also pause to mention work of Aczel combining ideas about inductive definitions and the lambda calculus. Another family of proposed solutions to the Liar are contextualist solutions. These also make use of classical logic, but base their solutions primarily on some ideas from the philosophy of language. They take the basic lesson of the Liar to be that truth predicates show some form of context dependence , even in otherwise non-context-dependent fragments of a language.

They seek to explain how this can be so, and rely on it to resolve the problems faced by the Liar. One way of thinking about why the truth predicate is not well-behaved on the Liar sentence is that there is not really a well-defined truth bearer provided by the Liar sentence.

To make this vivid as discussed by C. Parsons , suppose that truth bearers are propositions expressed by sentences in contexts, and that the Liar sentence fails to express a proposition. This is the beginnings of an account of how the Liar winds up ungrounded or in some sense indeterminate.

But, it is an unstable proposal. We can reason that if the Liar sentence fails to express a proposition, it fails to express a true proposition. And, we have shown that this sentence says something true, and so expresses a true proposition. Thus, from the assumption that the Liar sentence is indeterminate or lacks semantic status, we reason that it must have proper semantic status, and indeed say something true. We are hence back in paradox. First of all, in a setting where sentences are context dependent, the natural formulation of a truth claim is always in terms of expressing a true proposition, or some related semantically careful application of the truth predicate.

But more importantly, to the contextualist, the main issue behind the Liar is embodied in the reasoning on display here. It involves two key steps. First, assigning the Liar semantically defective status—failing to express a proposition or being somehow indeterminate.

Second, concluding from the first step that the Liar must be true—and so not indeterminate or failing to express a proposition—after all. Both steps appear to be the result of sound reasoning, and so the conclusions reached at both must be true. The main problem of the Liar, according to a contextualist, is to explain how this can be, and how the second step can be non-paradoxical.

Such reasoning is explored by Glanzberg c and C. For a critical discussion, see Gauker Thus, contextualists seek to explain how the Liar sentence can have unstable semantic status, switching from defective to non-defective in the course of this sort of inference.

They do so by appealing to the role of context in fixing the semantic status of sentences. Sentences can have different semantic status in different contexts. Thus, to contextualists, there must be some non-trivial effect of context involved in the Liar sentence, and more generally, in predication of truth.

One prominent contextualist approach, advocated by Burge and developed by Koons and Simmons , starts with the idea that the Tarskian hierarchy itself offers a way to see the truth predicate as context dependent. Context then sets the level of the truth predicate. This idea can be seen as an improvement on the original Tarskian approach in several respects. First, once we have a contextual parameter, the need to insist that Liar sentences are never well-formed disappears.

Burge and the postscript to C. Parsons consider briefly how Kripkean techniques could be applied in this setting. Though he works in a very different setting, ideas of Gaifman , can be construed as showing how even more subtle ways of interpreting a context-dependent truth predicate can be developed. With suitable care, other problems for the Tarskian hierarchy can be avoided as well.

This approach gives substance to the idea that the Liar sentence is context dependent. This amounts to being true at some higher level of the hierarchy. Depending how the Burge view is spelled out technically, it will either have full capture and release at each level, or capture and release with the same restrictions as the closed-off Kripke construction.

The view that posits contextual parameters on the truth predicate does face a number of questions. For instance, it is fair to ask why we think the truth predicate really has a contextual parameter, especially if we mean a truth predicate like the one we use in natural language.

Merely noting that such a parameter would avoid paradox does not show that it is present in natural language. Furthermore, whether it is acceptable to see truth as coming in levels at all, context-based or not, remains disputed. Not all those who advocate contextual parameters on the truth predicate agree about the role of hierarchy. Finally, the Burgean appeal to Gricean mechanisms to set levels of truth has been challenged. Contextualist approaches come in many varieties, each of which makes use of slightly different apparatus.

With contextualist theories the choice often turns on issues in philosophy of language as well as logic. We already noted a different way of developing contextualist ideas from Gaifman , We will now briefly review a few more alternatives. Another contextualist approach, stemming from work of C. Parsons , seeks to build up the context dependence of the Liar sentence, and ultimately the context dependence of the truth predicate, from more basic components.

The key is to see the context dependence of the Liar sentence as derived from the context dependence of quantifier domains. Quantification enters the picture when we think about how to account for predication of truth when sentences display context dependence. In such an environment, it does not make good sense to predicate truth of sentences directly. Not all sentences will have the right kind of determinate semantic properties to be truth bearers; or, as we have been putting it, not all sentences will express propositions.

The current contextualist proposal starts with the observation that quantifiers in natural language typically have context-dependent domains of quantification. In particular, this domain must expand in the course of the reasoning about the semantic status of the Liar.

Proposals for how this expansion happens, and how to model the truth predicate and the relation of expressing a proposition in the presence of the Liar, have been explored by Glanzberg , a , building on work of C.

Defenders of this approach argue that it does better in locating the locus of context dependence than the parameters on truth predicates view. Another variant on the contextualist strategy for resolving the Liar, developed by Barwise and Etchemendy and Groeneveld , relies on situation theory rather than quantifier domains to provide the locus of context dependence.

Situation theory is a highly developed part of philosophy of language, so we shall again give only the roughest sketch of how their view works. Situations are classified by what are called situation types. A proposition involves classifying a situation as being of a situation type. There is a sense in which this proposition cannot be expressed. But there is a core observation in common between these two points, and the details do not matter for our purposes here.

This idea clearly has a lot in common with the restriction on quantifier domains view. In particular, both approaches seek to show how the domain of contents expressible in contexts can expand, to account for the instability of the Liar sentence. For discussion of relations between the situation-theoretic and quantifier domain approaches, see Glanzberg a. Barwise and Etchemendy discuss relations between their situation-based and a more traditional approach in Ch.

For a detailed match-up between the Barwise and Etchemendy framework and a Burgean framework of indexed truth predicates, see Koons It is a key challenge to contextualists to provide a full and well-motivated account of the source and nature of the shift in context involved in the Liar, though of course, many contextualists believe they have met this challenge.

In favor of the contextualist approach is that it takes the revenge phenomenon to be the basic problem, and so is largely immune to the kinds of revenge issues that affect other approaches we have considered. But, it may be that there is another form of revenge which might be applied. To achieve this, it must presumably be denied that there are any absolutely unrestricted quantifiers. Glanzberg b, argues this is the correct conclusion, but it is highly controversial. For a survey of thinking about this, see the papers in Rayo and Uzquiano Another approach to the Liar, advocated by Gupta , Herzberger , Gupta and Belnap , and a number of others, is the revision theory of truth.

But it is a distinctive approach. We will sketch some of the fundamentals of this view. For a discussion of the foundations of the revision theory, and its relations to contextualism, see L. More details, and more references, may be found in the entry on the revision theory of truth. The revision theory of truth starts with the idea that we may take the T-schema at face value. Indeed, Gupta and Belnap take up a suggestion from Tarski , that the instances of the T-schema can be seen as partial definitions of truth; presumably with all the instances together, for the right language or family of languages constituting a complete definition.

At the same time, the revision theory holds fast to classical logic. Thus, we already know, we have the Liar paradox for any language with enough expressive resources to produce Liar sentences.

In response, the revision theory proposes a different way of approaching the semantic properties of the truth predicate. Better in many respects. The Liar sentence never stabilizes under this process. We reach an alternation of truth values which will go on for ever. This shows, according the revision theory, that truth is a circular concept. As such, it does not have an extension in the ordinary sense.

Rather, it has a rule for revising extensions, which never stabilizes. Sequences of values we generate by such revision rules, starting with a given initial hypothesis, are revision sequences.

We leave to a more full presentation the important issue of the right way to define transfinite revision sequences. See the entry on revision theories of truth. The characteristic property of paradoxical sentences like the Liar sentence is that they are unstable in revision sequences: there is no point in the sequence at which they reach a stable truth value.

This classifies sentences as stably true, stably false, and unstable. The revision theory develops notions of consequence based on these, and related notions. See the entry on revision theories of truth for further exposition of this rich theory.

So long as we have EFQ as classical logic does , this results in triviality. Most of the proposed solutions we have considered with the exception of the revision theory try to avoid this result somehow, either by restricting capture and release or departing from classical logic.

But there is another idea that has occasionally been argued, that the Liar paradox simply shows that the kinds of languages we speak, which contain their own truth predicates, are inconsistent. This is not an easy view to formulate.

Though Tarski himself seemed to suggest something along these lines for natural languages, specifically , it was argued by Herzberger that it is impossible to have an inconsistent language. In contrast, Eklund takes seriously the idea that our semantic intuitions, expressed, for instance, by unrestricted capture and release, really are inconsistent. Eklund grants that this does not make sense if these intuitions have their source simply in our grasp of the truth conditions of sentences.

But he suggests an alternative picture of semantic competence which does make sense of it closely related to conceptual role views of meaning.

He suggests that we think of semantic competence in terms of a range of principles speakers are disposed to accept in virtue of knowing a language. Those principles may be inconsistent. But even so, they determine semantic values. Semantic values will be whatever comes closest to satisfying the principles—whatever makes them maximally correct—even if nothing can satisfy all of them due to an underlying inconsistency. Eklund thus supports an idea suggested by Chihara But along the way, he suggests that the source of the paradox is our acceptance of the T-schema by convention, he suggests , in spite of its inconsistency.

A related, though distinct, view is defended by Patterson , Patterson argues that competence with a language puts one in a cognitive state relating to an inconsistent theory—one including the unrestricted T-schema and governed by classical logic. He goes on to explore how such a cognitive state could allow us to successfully communicate, in spite of relating us to a false theory. A different sort of inconsistency theory is advocated by Scharp Scharp argues that truth is an inconsistent concept, like the pre-relativistic concept of mass.

As such, it is unsuitable for careful theorizing. What we need to do, according to Scharp, is replace the inconsistent concept of truth with a family of consistent concepts that work better.

Scharp develops just such a family of concepts, and offers a theory of them. There is much more to say about the Liar paradox than we have covered here: there are more approaches to the Liar variants we have mentioned, and more related paradoxes like those of denotation, properties, etc.

There are also more important technical results, and more important philosophical implications and applications. Our goal here has been to be more suggestive than exhaustive, and we hope to have given the reader an indication of what the Liar paradox is, and what its consequences might be. The Paradox and the Broader Phenomenon 1. Basic Ingredients 2. Significance 4. Some Families of Solutions 4. FLiar : FLiar is false. ULiar : ULiar is not true.

Barwise, H. Keisler, and K. Kunen eds , Amsterdam: North-Holland, 31— Asenjo, F. Beall, Jc, Ross T. Brady, Allen P. French and H. Wettstein eds , Boston: Wiley-Blackwell. Priest, R. Routley, and J. Norman eds , Munich: Philosophia Verlag, — Burgess, John P. Dowden, Bradley H. Vardi ed. Koons, Robert C. Martin, Robert L. Reprinted in Martin 47— Thesis, University of Queensland. Pearce and H.

Wansing eds , Berlin: Springer, — Bluhm and C. Nimtz eds , Paderborn: Mentis, 27— Sorensen, Roy A. References are to the translation by J. Well, notice that the conclusion of the argument is a bald contradiction: the claim on the blackboard is both true and false. Now, the principle of noncontradiction says that you can never accept a contradiction. So the paradox must be of the second kind: there must be something wrong with the argument.

Or must there? According to this theory, some contradictions are actually true, and the conclusion of the Liar Paradox is a paradigm example of one such contradiction. One thing that drives the view is that cogent diagnoses of what is wrong with the Liar argument are seemingly impossible to find. Suppose you say, for example, that paradoxical sentences of this kind are simply meaningless or neither true nor false, or some such. Then what if Professor Greene had written on the board:.

Everything written on the board in Room 33 is either false or meaningless. If this were true or false, we would be in the same bind as before. We are back with a contradiction. This sort of situation often called a strengthened paradox affects virtually all suggested attempts to explain what has gone wrong with the reasoning in the Liar Paradox.

At any rate, even after two and a half thousand years of trying to come up with an explanation of what is wrong with the argument in the Liar Paradox, there is still no consensus on the matter.

Maybe, then, we have just been trying to find a fault where there is none. Of course, this means junking the principle of noncontradiction. But why should we accept that anyway? Were they? Interestingly, virtually everything else that Aristotle ever defended has been overthrown — or at least seriously challenged.

The principle of noncontradiction is, it would seem, the last bastion! Naturally, there is more to be said about the matter — as there always is in philosophy. If you ask most modern logicians why there can be no true contradictions, they will probably tell you that everything follows logically from a contradiction, so if even one contradiction were true, everything would be true.

Clearly, everything is too much! This principle of inference that everything follows from a contradiction sometimes goes by its medieval name, ex falso quodlibet , but it is often now called by a more colorful name: explosion.

There is, in fact, a connection between explosion and the principle of noncontradiction. That is, everything follows from a contradiction. Evidently, if this argument is invoked against dialetheism, it is entirely question-begging, since it takes for granted the principle of noncontradiction, which is the very point at issue.

Moreover, for all its current orthodoxy, explosion seems a pretty implausible principle of inference. It tells us, after all, that if, for example, Melbourne were and were not the capital of Australia, Caesar would have invaded England in Explosion would itself seem to be a pretty paradoxical consequence of whatever it is supposed to follow from. Unsurprisingly, then, the last 40 years or so have seen fairly intensive investigations of logics according to which explosion is not correct.

These are called paraconsistent logics , and there is now a very robust theory of such logics. In fact, the mathematical details of these logics are absolutely essential in articulating dialetheism in any but a relatively superficial way.

But the details are, perhaps, best left for consenting logicians behind closed doors. You might think that there is another problem for dialetheism: if we could accept some contradictions, then we could never criticize someone whose views were inconsistent, since they might just be true. Suppose that I am charged with a crime. In court, I produce a cast-iron alibi, showing that I was somewhere else. The prosecutor accepts that I was not at the crime scene, but claims that I was there anyway.

We certainly want to be able to say that this is not very sensible! But the fact that it is rational to accept some contradictions does not mean that it is rational to accept any contradiction. If the principle of noncontradiction fails, then contradictions cannot be ruled out by logic alone. But many things cannot be ruled out by logic alone, though it would be quite irrational to believe them. The claim that the earth is flat is entirely consistent with the laws of logic.

And no one has yet mastered the trick of being in two places at the same time, as both we and the prosecutor know. Indeed, if you consider all the statements you have met in the last 24 hours including the ones in this article , the number that might plausibly be thought to be dialetheias is pretty small.