The Tarskian Turn: Deflationism and Axiomatic Truth

Leon Horsten
Pages: 178
https://www.jstor.org/stable/j.ctt5hhj92

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1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xii)
(pp. 1-10)

Horwich’s bookTruthfirst appeared in 1990. It established deflationism about truth as a central subject of philosophical investigation. In the first decade after the appearance of this book, deflationism was mostly associated with an axiomatic theory of truth that is known as the disquotational theory. This theory has its origins in Tarski’s work: It is the earliest axiomatic theory of truth that meets minimal adequacy conditions.

In recent years, the attention of philosophers is gradually shifting to stronger axiomatic theories of truth. There is a growing consensus that truth theories stronger than the disquotational theory are needed for the...

5. 2 Axiomatic Theories of Truth
(pp. 11-26)

In this chapter, contemporary theories of truth will first be distinguished from more traditional, substantial truth theories. Subsequently, a distinction is drawn, within contemporary theories, between semantic and axiomatic approaches. It will be argued that the axiomatic approach ought to be preferred over the semantic approach. Thus we will acquire a strong motivation for embarking on a search for adequate axiomatic theories of truth.

What is truth? (John 18:38)

The philosophical debate about truth was dominated until recently by deep and substantial theories of truth. No attempt is made here to do justice to the complexity of these theories. We...

6. 3 On the Shoulders of Giants
(pp. 27-46)

Our first results concerning axiomatic truth theories are of a negative nature. Gödel proved that no sufficiently strong, consistent mathematical theory proves its own consistency. Using Gödel’s techniques, Tarski’s theorem can be proved. This theorem says that no sufficiently expressive language can define its own truth predicate. Tarski’s theorem on the undefinability of truth in turn has negative consequences for axiomatic theories of truth. It implies that no consistent truth theory implies all the Tarski-biconditionals.

Tarski’s theorem on the undefinability of truth is the starting point of contemporary axiomatic theories of truth. Versions of Tarski’s theorem are reviewed in this...

7. 4 The Disquotational Theory
(pp. 47-58)

Tarski wanted to define truth. He saw that truth could not be defined in the object language: It has to be done in an essentially stronger metalanguage. In his definition, Tarski assigned a pivotal role to the Tarski-biconditionals. Nowadays, many a deflationist philosopher takes a suitable class of Tarski-biconditionals as theaxiomsof her theory of truth.

Tarski’s theory, as set out in [Tarski 1935], is the point of departure for most, if not all, recent publications on truth. Rather than giving a historically precise account of Tarski’s account, I elaborate on those features that are most important for later...

8. 5 Deflationism
(pp. 59-68)

Deflationism is the most popular philosophical view of truth these days. Many philosophers believe that the view is essentially correct. But there is surprisingly little agreement on what the view is committed to. This is to some extent due to the fact that just as mechanism is not really a physical theory, deflationism is not a theory of the laws of truth. It is rather a view on the nature and role of the concept of truth.

In this chapter, we look at what philosophers mean when they say that truth should be approached in a deflationary way. We outline...

9. 6 The Compositional Theory
(pp. 69-78)

In this chapter, we investigate a second axiomatic theory that is directly inspired by Tarski’s work on truth. Tarski’s definition of truth in a model inspires an axiomatic theory of truth that explicates the compositional nature of truth. We see that the compositional theory has several advantages when compared with the disquotational theory.

The following two propositions are typical illustrations of the proof-theoretic strength of the disquotational theory.

Proposition 24For all$\phi\in{{\cal L}_{PA}}: DT\vdash T (\phi)\vee T(\neg\phi)$

ProofAlready propositional logic alone proves$\phi\vee\neg\phi$.Two restricted Tarskibiconditionals are$T(\phi)\leftrightarrow\phi$and$T(\neg\phi)\leftrightarrow\neg\phi$.Combining these facts yields the desired result.

Proposition 25 (Tarski)$DT \nvdash \forall \phi \in {{\cal L}_{PA}}:T(\phi )\vee T(\neg \phi )$

ProofTake any...

10. 7 Conservativeness and Deflationism
(pp. 79-102)

In this chapter, we explore the relation between deflationism and the arithmetical nonconservativeness of the compositional theory of truth. In this context, we also describe the role that the concept of truth plays in specific philosophical disciplines such as epistemology, philosophy of language, and metaphysics.

Deflationist theories hold that truth is somehow an “insubstantial” notion, that it does not carry much ontological weight. This claim is sometimes combined with the view that the purpose of the notion of truth is toexpresscertain things (infinite conjunctions) that we could not otherwise express, but not toprovethings about matters not...

11. 8 Maximizing Classical Compositionality
(pp. 103-116)

We now investigate to what extent Tarski’s compositional theory of truth can be consistently extended. In particular, we look at a theory of truth that was formulated by Harvey Friedman and Michael Sheard, and we discuss its relation with the revision theory of truth.

We have seen that Tarski’s compositional and disquotational theories fail to validate certain seemingly unproblematic sentences containing iterations of the truth predicate. Remember, for example, that bothDTandTCfail to proveT(T(0 = 0)). We know that the naive attempt to solve this by lifting the restriction inDTof the Tarski-biconditionals...

12. 9 Kripke's Theory Axiomatized
(pp. 117-140)

Kripke formulated what is currently probably the most influential semantical theory of truth [Kripke 1975]. In this chapter, we see that, like Tarski’s semantical notion of a (classical) model, Kripke’s semantical theory has been a rich source of inspiration for proof-theoretic approaches to truth. Some of our strongest and best axiomatic theories of truth result from attempts to axiomatize some of his models for the language of truth.

Kripke has constructed a semantical theory of self-referential truth. He is not occupied with formulating an axiomatic theory of truth. For Kripke, models come first. His aim is to construct particularly nice...

13. 10 Truth and Philosophy
(pp. 141-152)

In this final chapter, we reflect on the relationship between the concept of truth and philosophy. We have seen that there are sound theories of truth that are considerably stronger even than the typed compositional theory of truth. First, we want to know what becomes of deflationism in the light of the existence of proof-theoretically strong truth theories. Second, we speculate about the extent to which even some of the stronger theories of truth that we have discussed play an essential role in philosophical discussions.

As philosophers, we seek a theory of truth for our entire language. In such a...

14. Bibliography
(pp. 153-158)
15. Glossary
(pp. 159-160)
16. Author Index
(pp. 161-162)
17. Subject Index
(pp. 163-165)