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On the Nature of Truth

Interview by Richard Marshall.

Truth is taken to be a primitive notion. The axiomatic approach differs from traditional definitional approaches such as the correspondence or coherence theory of truth that it is not assumed from the outset that truth is definable.

Deflationists about truth usually don’t assume that truth is definable, while proponents of substantial theories such as the correspondence theory often believe that truth can be defined away in terms of correspondence.

‘Believe only what is true!’ is a useless epistemic norm. It’s correct, but it cannot guide us, because what we need are criteria we can actually apply; truth, however, isn’t a criterion we can directly test. ‘Believe only what is consistent with your entire belief system!’ is similarly useless. By Church’s theorem we cannot test a given set of axioms for its consistency. In general, this is beyond the computational powers of the best ideal computer.

Volker Halbach works in the philosophy of Logic, Philosophy of Mathematics, Philosophy of Language and Epistemology. With talk of ‘post-truth’ and ‘fake news’ all around us it seems right to talk to an expert about the nature of truth. So here he talks about the difference between axiomatic and semantic approaches to truth, why he thinks the axiomatic approach is superior, what conditions this approach must satisfy, the interplay between the two approaches, whether his approach is a kind of deflationary theory of truth, whether deflationists should be conservative, why consistency is out of reach of any knowledge theory, on voluntarism and whether Descartes was a voluntarist, what is meant when it is said that arithmetic is a computational structure, and self reference in mathematics.

3:AM: What made you become a philosopher?

Volker Halbach: till at school I became fascinated by Nietzsche, Schopenhauer and other German classics. I went to university in Munich to study them; butbecame quickly disappointed by the way classical German philosophy wastaught. Analytic philosophy looked more open minded and outwardlooking. I very quickly switched to logic and analytic philosophy. Eversince I have been enjoying doing logic and have been lucky enough to getpaid for it.

3:AM: From the beginning you have been investigating the nature of truth. One of the big distinctions that you’ve been working with is approaches to truth that are axiomatic and approaches that are semantic. So can we begin by asking you to sketch what the distinction is, and what is at stake?

VH: The axiomatic approach is very simple: We stipulate axioms for the truth predicate that look plausible and that avoid the paradoxes. Truth is taken to be a primitive notion. The axiomatic approach differs from traditional definitional approaches such as the correspondence or coherence theory of truth that it is not assumed from the outset that truth is definable.

Semantic theories of truth provide methods for defining semantics or models for a language with a truth predicate. The semantic definition of a model is usually carried out in set theory for a language that is essentially weaker. Semantic theories cannot provide models for the language in which the model is defined. In philosophical logic this is a standard approach that has been applied to many other notions: Toy languages with symbols for necessity, knowledge, or the like are given a suitable semantics. In semantic theories of truth the same strategy is applied to the truth predicate. Kripke’s theory and the revision semantics of truth by Gupta and Herzberger belong into this category, but also Tarski’s extremely
successful model-theoretic account of truth that is the starting point for all later accounts.

3:AM: The axiomatic theories seem to have been a central focus of your work. Is this the type of theory of truth that you favour and if so what do you think are its advantages over the semantic alternative?

VH: In semantic theories the ultimate framework is usually set theory. The goal is to define a model for a toy language containing a truth predicate. In axiomatic theories we aim at a notion of truth for our overall language without the need of ascending to a stronger metalanguage. For philosophical purposes ranging from epistemology or the definition of logical consequence to moral philosophy, this is the kind of notion of truth that is needed. For instance, when we say that knowledge implies truth, we don’t restrict ourselves to beliefs that can be formulated in some restricted object language; we want to claim factivity of knowledge without any restriction. Moreover, in semantic theories, truth is defined relative to a model, not truth simpliciter. But truth simpliciter is required and used in philosophy all the time.

The motivation behind semantic theories of truth is often the belief that a notion is only well understood if we can define a semantics for it in set theory. I don’t share that belief. In the case of truth the approach is especially strange: Truth is the central notion of semantics. To think that it can only be understood if we can define a set-theoretic semantics for it over and beyond the central notion of semantics strikes me as implausible. This is not to say that semantic approaches aren’t useful. I only reject semantic theories as ultimate analyses of truth.

3:AM: What are the conditions that any axiomatic truth theory must satisfy?

VH: Axiomatic theories of truth have been designed for various purposes. They can be used to provide foundations for mathematical theories or frameworks for thinking about the Gödel incompleteness phenomena. Which theory is the best may depend on the purpose. My preferred theory of truth must be able to serve the purposes for which truth is used in philosophy as much as possible. Part of this purpose is the expression of infinite generalizations. To return to an example mentioned earlier, an epistemologist may assert the factivity of knowledge by saying: ‘Everything that is known is true’ and reason with this claim. A good theory of truth should support this reasoning. Our theory of truth should also prove generalizations such as ‘a conjunction is true if and only if both conjuncts are true’. Only then the theory of truth can serve its purpose in the definition of logical consequence, as given in an introductory logic class.

3:AM: Is there an interplay between the two types of theory? How can an axiomatic theory capture a semantic construction? Is it important that it can?

VH: There are many links between axiomatic and semantic theories. One way to go about constructing an axiomatic theory is to look at a semantic theory and try to ‘capture’ the semantic construction. By thinking about how certain features of the semantic construction can be captured by the axiomatization, we can find new axiomatic theories. In many cases semantic constructions can give us confidence that certain axioms are consistent.

3:AM: Are all deflationist theories of truth axiomatic, and if they’re not, do you think they are nevertheless more appropriately applied to a deflationist theory? Are you a type of deflationist?

VH: It’s not easy to say what deflationism is, but I’d say yes, all deflationist theories of truth rely on axiomatic theories. I don’t say that they *are* axiomatic theories of truth, because there is often more to a deflationist theory than just axioms for truth. Deflationist theories often contain claims about the status of these axioms and their purpose.

Deflationists about truth usually don’t assume that truth is definable, while proponents of substantial theories such as the correspondence theory often believe that truth can be defined away in terms of correspondence. So deflationists don’t assume that truth can be eliminated by a definition. In this sense the deflationists’ notion of truth is more ‘expensive’ than that of many correspondence theorists.

Stewart Shapiro once told me: ‘You are trying to be a deflationist, but you aren’t.’ He may have been right. I share the deflationists’ view that truth should be treated as a primitive axiomatized notion, but I don’t share the belief of many deflationists that truth is a harmless notion. All the logical research over the last decade has shown that truth is a very powerful notion.

3:AM: Should the deflationist be conservative in that its theory shouldn’t add anything non-trivial to what we already know or is being conservative just a suggestion from the shadows that the deflationist doesn’t have to agree to? Are there deflationist theories that actually yield important new facts?

VH: Some deflationists expect that truth doesn’t yield any new insights outside semantics. Technically speaking, this amounts to the following requirement: All new sentences that can be proved with the deflationist axioms of truth must contain the truth predicate; no new purely mathematical, linguistic, physical, epistemological, and so on claims must become provable. However, logicians realized early on that adding certain plausible axioms for truth allows us to prove new mathematical sentences such as the Gödel sentence or the consistency of the starting theory. They aren’t provable without using the truth axioms or other additional assumptions. Over the last decades a lot of highly sophisticated work has been done that explores which new sentences become provable, depending on which axioms are added.

I don’t believe that the conservativeness claim should be taken as part of deflationism. Starting with the assumption that there are useful truth theories that are conservative strikes me as unnecessarily restrictive. The deflationist should specify truth axioms and then see what happens and not stick to the conservativeness requirement for ideological reasons.

3:AM: In modern epistemology there are philosophers who say we can’t have direct knowledge of the world – all knowledge is mediated. There are also those who say that consistency is similarly out of reach because defining or proving it is beyond our intellectual capacities. Is it true that consistency can’t be proved and is out of reach?

VH: Yes, I would say that consistency of our overall theory isn’t within our reach. Under fairly general conditions, the consistency of a system isn’t provable within the system. This is Gödel’s second incompleteness theorem. I don’t think that our mathematical powers transcend that of a formal system.

In epistemology the problem is often ignored. ‘Believe only what is true!’ is a useless epistemic norm. It’s correct, but it cannot guide us, because what we need are criteria we can actually apply; truth, however, isn’t a criterion we can directly test. ‘Believe only what is consistent with your entire belief system!’ is similarly useless. By Church’s theorem we cannot test a given set of axioms for its consistency. In general, this is beyond the computational powers of the best ideal computer. Nevertheless consistency is often thought to be a necessary condition for epistemic justification. Of course, if we hit upon an inconsistency, we should do something about it. But as a general criterion consistency cannot so easily be applied.

3:AM: There’s a conflict between two intuitions in epistemology: we tend to think that epistemic behaviour such as believing or judging is voluntary – we choose to do it – but we also tend to think that epistemic behaviour is not subject to voluntary control. Descartes has been thought of by some to be an epistemic voluntarist hasn’t he – but you don’t think he was in all cases do you? How do you understand Descartes in respect of this issue – and where do you stand?

VH: Very often Descartes is used as a bad example. He is often claimed to have held silly views that had to be corrected by other philosophers. A reason why these silly views are ascribed to Descartes is that he is highly readable, but using old and sometimes idiosyncratic terminology. The old-fashioned terminology cannot have been the only reason why people misunderstood him; it started during his lifetime. Hobbes was one of the first to do so. Descartes did not do much to avoid misunderstanding by his often dismissive reactions.

His epistemic voluntarism is an example where a caricature of Descartes’views was used by philosophers from Hobbes to Alston and Goldman to attack a silly view. I cannot directly choose what to believe. I cannot simply choose to believe that Trump is not president. Many philosophers claimed that Descartes thought we can directly choose our beliefs. Descartes didn’t hold that view. According to Descartes, we can, however, influence our beliefs indirectly and thereby have voluntary control over them. For instance, I can pay attention to certain things and thereby indirectly influence my beliefs. I can choose to double check my calculations and thereby avoid some false beliefs. Descartes thought we should pay attention to clear and distinct ideas and that we can thereby indirectly influence our beliefs. This is actually very close to the views of many contemporary epistemologists, for instance, Goldman with the difference that they would recommend us to pay attention to other aspects.

3:AM: What’s philosophically interesting about the claim that some arithmetic is a computational structure? What difference does it make?

VH: It is tempting to say that mathematical objects are given by axioms. In particular, it might be thought that any structure satisfying the axioms of arithmetic can serve as the structure of natural numbers. If these
axioms are taken to be the usual axioms of arithmetic in first-order logic, then they will be satisfied also by pathological structures; these are structures that are uncountable or structures in which the Gödel sentence and other true sentences are false. Structuralists about mathematics have tried to find ways to rule out such structures, for instance, by using second-order logic. This does not appeal to me because I agree with Quine that second-order logic is set theory is sheep’s clothing. However, if we require that addition and multiplication are computable in the structure, then the deviant structures can be ruled out as well without using second-order logic. The assumption of computability strikes me as highly possible: Whatever the natural numbers are, we must be able to carry out calculations with them.

3:AM: So generally, what do you think it means for a sentence in arithmetic to ascribe to itself a property?

VH: Usually declarative sentences are about objects. Some sentences – such as the preceding sentence – are about sentences; and one of the sentences may be that sentence itself. Some sentences are only about themselves. The best-known sentences of this kind are liar sentences. For instance, [the sentence in square brackets is not true] is about itself, and it says about itself that it is not true. Sentences in arithmetic are about numbers. We can assign to each sentence of arithmetic a number, for instance, by listing all sentence in alphabetical order and then numbering them. Then there is a first sentence, a second sentence, and so on. If a sentence of arithmetic is about the number 12122, then it is about the 12122th sentence. Gödel showed how to construct a sentence of arithmetic that is about itself in this indirect way and that says about itself that it is not provable. There are many other sentences in arithmetic that make claims about themselves. The Henkin sentence, for instance, says about itself that it is provable.

That’s the idea at least. However, it’s not very clear what it means for a sentence to be about something. Linguists, philosophers of language, and logicians have struggled with the notion of aboutness. It is a very elusive notion. Consequently it is not very clear what self-reference is, not only in arithmetic. However, even if it’s difficult to come up with a general definition of self-reference, there are paradigmatic self-referential sentences.

What interests me is the following: For a given property there may be different sentences ascribing to themselves the property and these sentences may have very different properties. For instance, a sentence may say about itself that it is true and be true; another sentence may also say about itself that it is true, but be false. Hence the loose way of talking about *the* sentence ascribing to itself a certain property has to be used with caution. The properties of self-referential sentences can be very sensitive to the way these sentences are constructed.

3:AM: And finally have you five books you could recommend that would take us further into your philosophical world?

VH: Sure.
The following book provides an introduction to deflationism. It is well
written, thought provoking, and completely non-technical:

Horwich, Paul (1998), Truth, Basil Blackwell, Oxford, second edition.

To those interested in axiomatic theories of truth I recommend the
following two books.

Leon Horsten (2011), The Tarskian Turn: Deflationism and AxiomaticTruth, MIT Press, Cambridge, Mass.

Volker Halbach (2014), Axiomatic Theories of Truth, Cambridge University
Press, revised edition

The following two are classics in the theory of the paradoxes:

Vann McGee (1991), Truth, Vagueness, and Paradox: An Essay on the Logicof Truth, Hackett Publishing, Indianapolis and Cambridge.

Belnap, Nuel and Anil Gupta (1993), The Revision Theory of Truth, MIT
Press, Cambridge, Mass.

ABOUT THE INTERVIEWER
Richard Marshall is still biding his time.

Buy his new book here or his first book here to keep him biding!

First published in 3:AM Magazine: Sunday, July 30th, 2017.