Some Notes on Truths and Comprehension

Abstract

In this paper we study several translations that map models and formulae of the language of second-order arithmetic to models and formulae of the language of truth. These translations are useful because they allow us to exploit results from the extensive literature on arithmetic to study the notion of truth. Our purpose is to present these connections in a systematic way, generalize some well-known results in this area, and to provide a number of new results. Sections 3 and 4 contain some recursion-and proof-theoretic results about Kripke-style fixed-point theories of truth. Section 5 shows how to derive full second-order arithmetic from principles of truth. Section 6 investigates the proof-theoretic strength of disquotation without an arithmetical base theory.

Higher-Order Logic and Disquotational Truth

“…Note that there are no rules or axioms for non-logical operators in the calculus, so FOL imposes no special constraints on the truth values formulae of the form lϕ may take. 28 If l is an n-place operator, we have that }l} w M `Ď D n M . 29 In general, if l is an n-place operator, M, w, g ( lpϕ 1 .…”

“…The results in the present section do not only establish that inferences can be preserved as well, but also that UTBrτ s doesn't license any more inferences than SOL. Moreover, in[28] it hasn't been noticed that essentially no arithmetical or syntax axioms are needed for that result.…”

Una teoría no transitiva de la verdad sobre PA

“…We can then show that STAT[S] proves ¬∀x (T ⌜λ⌝ ∧ x = x) and T ⌜λ⌝ ∧ n = n for every n ∈ ω: ◻ Now in order to show how we can easily construct very strong nontransitive theories of truth, consider the strongest disquotational theory of truth known. (Schindler, 2018;Picollo & Schindler, 2018) define the axiomatic theory of truth UTB(Z 2 ) as PAT plus all instances of the uniform T-schema ∀x 1 ,...,x n (T ⌜ϕ(x 1 ,...,x n )⌝↔ϕ(x 1 ,...,x n )), where ϕ is a translation of a formula of second-order Arithmetic into our firstorder language containing T (the details of translation do not matter here) and the quantifiers of the T-schema are restricted to translated formulae.…”

Classes, why and how

“…Moreover, Parsons (1983b) observes that the notion of satisfaction can be seen a means to generalize predicate places as well, and that the usual predicative theories of satisfaction and classes are mutually interpretable. In Schindler (2015Schindler ( , 2017 it is shown that even impredicative theories of classes can be interpreted in (type-free) theories of satisfaction. Given the similar functions of the notions of truth and class, and the mentioned interpretability results, this suggests that someone who already has a broadly deflationary understanding of the notions of truth and satisfaction should probably have a deflationary understanding of the notion of class as well.…”