On Stratified Truth

Is there a consistent axiomatization of a stratified form of the Tarskian hierarchy, where stratification is meant in the sense of Quine’s New Foundations \(\textsf{NF}\)? In the following we propose a system of truth and abstraction, which might be regarded as an answer to the problem.

1. 1.

   In his talk at the Princeton Conference Pillars of Truth, April 8–10, 2011.

2. 2.

   See Russell’s Principles of Mathematics, appendix B; the paradox is also known as the Russell–Myhill paradox.

3. 3.

   Of course, one might simply postulate β-conversion at the object level, i.e. if A is stratified,

   $$pred( [x|A],u)=[A[x:=u]]$$

   Then the schemata would be trivially derivable. The reason is that we do not know how to prove its consistency.

4. 4.

   Henceforth IH stands for induction hypothesis in short.

5. 5.

   We underline that our formalization does not literally represent the paradox of the final section 500 of russell03. For the reader’s sake here is Russell’s text:

   If m be a class of propositions, the proposition “every m is true” may or not be itself an m. But there is a one-one-relation of this proposition to m: if n be different from m, “every n is true” is not the same proposition as as “every m is true”. Consider now the whole class of propositions of the form Italy“every m is true”, and having the property of not being members of their respective m’s. Let this class be w, and let p be the proposition “every w is true”. If p is a w, it must possess the defining property of w; but this property demands that p should not be a w. On the other hand, if p is not a w, then p does possess the defining property of w, and therefore is a w. Thus the contradiction appears unavoidable.

6. 6.

   Individual constants included; these can be given any type compatible with the clauses below.

7. 7.

   If the abstraction operator is assumed as primitive, the extended logic contains the schema

   $$\forall u (\varphi(u)\leftrightarrow\psi(u))\rightarrow \{ x \,|\, \varphi(x)\}= \{ x \,|\, \psi(x)\}$$
8. 8.

   We recall that a formula \(A(x,a)\) is positive in a if every free occurrence of a in the negation normal form of A is located in atoms of the form \(t\in a\), which are prefixed by an even number of negations and where \(a\notin FV(t)\).

9. 9.

   For a thorough critical discussion of the distinction between typed and type-free theories of truth, we send the reader to Halbach 2011, especially part II, and Chaps. 10–11 in part III.

Authors and Affiliations

1. Dipartimento di Lettere e Filosofia, Sezione di Filosofia,, Università degli studi di Firenze, via Bolognese, 52, 50139, Firenze, Italy

   Andrea Cantini

Corresponding author

Correspondence to Andrea Cantini .

Editors and Affiliations

1. Philosophy, ILLC & AUC - University of Amsterdam, Amsterdam, The Netherlands

   Theodora Achourioti

2. PHIER, Université Blaise Pascal, Clermont Ferrrand, France

   Henri Galinon

3. Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona, Barcelona, Barcelona, Spain

   José Martínez Fernández

4. Department of Mathematics and Department of Philosophy, University of Bristol, Bristol, United Kingdom

   Kentaro Fujimoto

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