Abstract
The paper draws attention to an important, but apparently neglected distinction relating to axiomatic theories of truth, viz. the distinction between weakly and strongly truth-compositional theories of truth. The paper argues that the distinction might be helpful in classifying weak axiomatic theories of truth and examines some of them with respect to it.
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Notes
In order to avoid paradox one has to restrict the T-scheme in one way or the other. This can be done by restricting the metavariable \(\phi \) in (T) to sentences not containing the truth-predicate.
In the following \(L_{PA}^{atom}\) stands for the class of atomic arithmetical sentences (i.e. equations) and \(T_0(x)\) for the truth-predicate restricted to atomic sentences, which is definable even in Peano Arithmetic PA and for which Tarski’s ‘Convention T’ can be shown to be satisfied in PA. Note also that \(\forall x T(\phi (x)\) is to be understood as \(\forall x T(\dot {sub}(\textbf {n},\textbf {m}, \dot {nu}(x)))\), where n is the Gödel number of the formula \(\phi (x)\), m is the Gödel number of the variable x, \(\dot {nu}(x)\) is the function mapping every natural number to the Gdel number of it’s numeral and \(sub(x,y,z)\) is the substitution function. In order to improve the readability, similar simplifications concerning notation have been made throughout this paper.
This is strictly speaking not true, for Tarski defined the more general notion of satisfaction recursively in order to define the concept of truth.
To see this we just look at the negation-case: assume \(DT \vdash \forall \phi \in \mathcal L_{PA}: \sim T(\phi ) \leftrightarrow T(\sim \phi )\); Now a proof of this statement can use only finitely many axioms \(DT_0\) of DT, i.e. \(DT_0 \vdash \forall \phi \in \mathcal L_{PA}: \sim T(\phi ) \leftrightarrow T(\sim \phi )\). Let \(T(\phi _1) \leftrightarrow \phi _1 , \ldots . T(\phi _n) \leftrightarrow \phi _n\) be a complete list of the T-Biconditionals used in the proof. Now define a model \((\mathbb N, E_T)\) (based on the standard-model of arithmetic \(\mathbb N\)) by including all (codes of) true sentences \(\phi _i\) and the negations of all false sentences \(\phi _i\) (for \(1 \leq i \leq n\)) into the extension \(E_T\) of the truth predicate. Then \((\mathbb N, E_T)\) makes all arithmetical axioms used in the proof true as well as all the T-Biconditionals. But in this model the quantified sentence \(\forall \phi \in \mathcal L_{PA}: \sim T(\phi ) \leftrightarrow T(\sim \phi )\) is clearly false. Hence \(DT_0 \nvDash \forall \phi \in \mathcal L_{PA}: \sim T(\phi ) \leftrightarrow T(\sim \phi )\) and therefore \(DT_0 \nvdash \forall \phi \in \mathcal L_{PA}: \sim T(\phi ) \leftrightarrow T(\sim \phi )\). Contradiction.
I do not want to suggest that Horsten and Halbach are not aware of this fact. But they seem to underestimate the significance of this point. DT as well as its uniform version UDT are discussed in more detail in Halbach’s [2]. Although in his [2] Halbach does not adress the issue under consideration explicitly, the point just made is implicit in his Theorem 7.6. (p. 57), which states that if DT proves \(\forall x(\psi (x) \rightarrow T(x))\), then the base theory PA proves that there are only finitely many objects satisfying \(\psi (x)\).
A semantic theory may be called strictly compositional or simply compositional, if analogue conditions for the satisfaction relation \(Sat(x,y)\) are derivable. Note also that although in this paper attention is restricted to typed theories of truth, there is no reason not to apply the distinction between wtc and stc theories of truth to untyped theories of truth as well.
The theory comprising of DT plus all instances of the schema \(TC_S 4\) seems to be all too ad hoc to count as a well-motivated ‘interesting’ theory of truth.
Note also, that it does not seem to be a trivial matter whether the theory consisting of the schemes corresponding to TC1–TC4, call it \(TC_S\), is ‘interesting’. The problem with \(TC_S\) is that it is not obvious if it proves every instance of the T-scheme, thereby meeting Convention T. Call the schematic theory one gets from \(TC_S\) by allowing free variables to occur in its instances \(TC_S^P\). Clearly \(TC_S^P\) proves every instance of the T-scheme \(T(\phi ) \leftrightarrow \phi \). This can be shown by proving the stronger claim that from \(TC_S^P\) every instance of the uniform T-scheme \(\forall x (T(\phi (x) \leftrightarrow \phi (x))\), is derivable. This in turn is proved by a straightforward induction on the complexity of \(\phi (x)\). But it is not clear if this can be done for \(TC_S\), where only sentences are allowed to occur in the schemes corresponding to TC1–TC3.
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Eder, G. Remarks on Compositionality and Weak Axiomatic Theories of Truth. J Philos Logic 43, 541–547 (2014). https://doi.org/10.1007/s10992-013-9279-1
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DOI: https://doi.org/10.1007/s10992-013-9279-1