The Epistemic Lightness of Truth: Deflationism and its Logic
Abstract
This book analyses and defends the deflationist claim that there is nothing deep about our notion of truth. According to this view, truth is a 'light' and innocent concept, devoid of any essence which could be revealed by scientific inquiry. Cezary Cieśliński considers this claim in light of recent formal results on axiomatic truth theories, which are crucial for understanding and evaluating the philosophical thesis of the innocence of truth. Providing an up-to-date discussion and original perspectives on this central and controversial issue, his book will be important for those with a background in logic who are interested in formal truth theories and in current philosophical debates about the deflationary conception of truth.
... By definition, the implicit commitment of a formal theory Th consists of sentences that are independent of the axioms of Th, but their acceptance is implicit in the acceptance of Th. In Cieśliński (2017Cieśliński ( , 2018, the phenomenon of implicit commitments was studied from the epistemological perspective through the lenses of the formal theory of believability. The current paper provides a comprehensive proof-theoretic analysis of this approach and compares it to other main theories of implicit commitments. ...
... We provide a detailed analysis of the proof theory of the internal theory of believability and some of its extensions, including a result on the feasible definability of this theory in the theory of iterated reflection over Th. We generalise our results to the theory of untyped truth and believability, answering an open problem posed by Cieśliński (2017). In Sect. ...
... Such a justification may be grounded e.g. in principles governing the use of the notion of acceptance (cf. Franzén, 2003;Cieśliński, 2017), or some other notion, such as truth. But some authors, most notably Horsten (2021), Horsten and Leigh (2017) and Fischer et al. (2021), claim that statements such as instances of reflection principles for Th need not be justified by any further evidence beyond the evidence used to justify the acceptance of Th itself. ...
By definition, the implicit commitment of a formal theory consists of sentences that are independent of the axioms of , but their acceptance is implicit in the acceptance of . In Cieśliński (2017, 2018), the phenomenon of implicit commitments was studied from the epistemological perspective through the lenses of the formal theory of believability. The current paper provides a comprehensive proof-theoretic analysis of this approach and compares it to other main theories of implicit commitments. We argue that the formal results presented in the paper favour the believability theory over its main competitors. However, there is still a fly in the ointment. We argue that in its current formulation, the theory cannot deliver all the goods for which it was defined. In particular, being amenable to a generalised conservativeness argument, it does not support the view that the notion of truth is epistemically light. At the end of the paper, we discuss possible ways out of the problem.
... 5 Indeed there is prima facie reason to be suspicious about this principle: for instance, it seems natural to assign probability 1 to propositions that express elementary observational results, which are obviously contingent. In any case, we now already see that the situation is dire for calculi of type-free subjective probability that do include Regularity (such as certain non-Archimedean theories of probability) 6 , for then, if we accept Necessitation, the Kaplan-Montague argument goes through. ...
... Proof. This follows from the proof of Theorem 5 in [7] (see also [6]). The point is that the theory RKf + CPr4 + "for every x, Pr(x) is either 1 or 0 or 1/2" is interpretable in the theory of the model (Q, B ), with B characterized as in Definition 6 of [7] (cf. ...
... The point is that the theory RKf + CPr4 + "for every x, Pr(x) is either 1 or 0 or 1/2" is interpretable in the theory of the model (Q, B ), with B characterized as in Definition 6 of [7] (cf. also Definition 13.4.5 of [6]). The interpretation is obtained by translating "Pr(x) = y" as " y = 1 ∧ B(x) ∨ y = 0 ∧ B(¬x) ∨ y = 1/2 ∧ ¬B(x) ∧ ¬B(¬x) ." ...
We formulate and explore two basic axiomatic systems of type-free subjective probability. One of them explicates a notion of finitely additive probability. The other explicates a concept of infinitely additive probability. It is argued that the first of these systems is a suitable background theory for formally investigating controversial principles about type-free subjective probability.
... This theorem imposes a very important limitation on any possible conservativeness proof for pure CT − : in general, we cannot hope to simply patch together some locally defined fragments of a compositional truth predicate in a coherent manner. A very closely related result from [15], so called nonstandard overspill, shows that if (M, T ) |= CT − and φ(v) is an arithmetical formula in the sense of M and T φ(n) holds for any standard numeral n, then it holds for some nonstandard a. 2 In this paper, we prove that CT − + PropSnd entails an analogue of the Smiths nonstandard overspill for arbitrary sequences of sentences, not necessarily substitutional instances of a single formula. It turns out that: ...
... This should not cause any confusion. The standard reference for axiomatic truth theories is [7], whereas [2] provides a detailed discussion of the Tarski Boundary programme. ...
... As we have just mentioned, the distinction between local and global principles is not firm, and the link between locality and conservativeness 7 A conservativeness proof can be found in [3], Section 4. 8 For the proof, see [2], Theorem 12.2.1. ...
It is an open question whether compositional truth with the principle of propositional soundness ,,all arithmetical sentences which are propositional tautologies are true'' is conservative over its arithmetical base theory. In this article, we show that the principle of propositional soundness imposes some saturation-like properties on the underlying model, thus showing significant limitations to the possible conservativity proof.
... We know that related principles such as "truth is closed under propositional logic" or "valid sentences of first-order logic are true" are not conservative and indeed are all equivalent to 0 -induction for the truth predicate. 2 In this article, we provide a partial answer to Cieśliński's question. We show that CT − extended with the principle expressing that propositional tautologies are true becomes nonconservative upon adding quantifier-free correctness principle QFC which states that T predicate agrees with partial arithmetical truth predicates on quantifier-free sentences. ...
... A comprehensive discussion of recent discoveries can be found in Cieśliński[2].2 The question was originally stated by Cieśliński in Cieśliński[3]. ...
... A comprehensive discussion of recent discoveries can be found in Cieśliński[2].2 The question was originally stated by Cieśliński in Cieśliński[3]. ...
In Cieśliński (J Philos Logic 39:325–337, 2010), Cieśliński asked whether compositional truth theory with the additional axiom that all propositional tautologies are true is conservative over Peano Arithmetic. We provide a partial answer to this question, showing that if we additionally assume that truth predicate agrees with arithmetical truth on quantifier-free sentences, the resulting theory is as strong as Δ0-induction for the compositional truth predicate, hence non-conservative. On the other hand, it can be shown with a routine argument that the principle of quantifier-free correctness is itself conservative.
... LPC ∞ is obtained from LPC by (essentially): replacing the axioms for ⊥ and with corresponding rules for arithmetical truth and falsity, and replacing (∀r) with an ω-rule. 10 By adapting the analysis in [37], it can be shown that LPC ∞ has nice proof-theoretical properties: weakening and contraction can be proved to be admissible in a way that preserves 9 See [31,33]. 10 Two more technical amendments are omitting free variables, and generalizing Cl and Cr to arbitrary terms which code formulae. ...
... 10 By adapting the analysis in [37], it can be shown that LPC ∞ has nice proof-theoretical properties: weakening and contraction can be proved to be admissible in a way that preserves 9 See [31,33]. 10 Two more technical amendments are omitting free variables, and generalizing Cl and Cr to arbitrary terms which code formulae. See [37] for details. the (possibly infinite) length of the derivation, its rules are invertible, and (crucially) cut is eliminable in it. ...
... However, (9) and (10) are interderivable in PKF by pure logic. ...
Substructural logics and their application to logical and semantic paradoxes have been extensively studied. In the paper, we study theories of naïve consequence and truth based on a non-reflexive logic. We start by investigating the semantics and the proof-theory of a system based on schematic rules for object-linguistic consequence. We then develop a fully compositional theory of truth and consequence in our non-reflexive framework.
... This is the new formulation of the theory of believability, proposed and discussed in (Cieśliński, 2017a). As it is mentioned in (Cieśliński, 2017a), in the presented version of the believability theory, (Ax 3 ) is considered to be an improvement to the following rule: 28 Cieśliński also considers a believability theory Bel(S) over S, which I do not discuss here. ...
... This is the new formulation of the theory of believability, proposed and discussed in (Cieśliński, 2017a). As it is mentioned in (Cieśliński, 2017a), in the presented version of the believability theory, (Ax 3 ) is considered to be an improvement to the following rule: 28 Cieśliński also considers a believability theory Bel(S) over S, which I do not discuss here. 29 In the interest of readability we are sloppy with the Gödel coding in what follows. ...
... As pointed out in(Cieśliński, 2017a), this newer version with (Ax 3 ) is preferred over the older version, which was also discussed by Horsten and myself in(Horsten and Zicchetti, 2021). ...
... Feferman later tried to simplify the presentation of these predicatively acceptable ordinals via a notion of truth [11], and via his framework for explicit mathematics. 2 The study of reflection principles and theories of truth led to a proliferation of studies in proof-theoretic ordinal analysis [27,14,3,4], and techniques in theories of truth [19,15,18,13,7]. 3 Despite this interest in frameworks that are motivated by implicit commitments, there hasn't been a direct logical analysis of the notion of implicit commitment itself. [20] carried out an 1 See for instance, [11, p.1]. 2 A comprehensive bibliography on explicit mathematics can be found at https://home.inf.unibe.ch//∼til/em_bibliography/. 3 More philosophical works have recently tackled the notion of implicit commitment directly. ...
... There is a formula ψ(y) such that x is the result of replacing P (y) with ψ(y) in ϕ. 9 It's important to notice that, according to this definition, Reflection Principles such as RFN(τ ) - 7 Here we are assuming that the expression s(t/x) stands for the code of the result of substituting, in the string coded by s, all occurrences of the string coded by x with the string coded by t. 8 See for instance [16,§V,Def. 4.2]. ...
... ψ n (v) the finitely many formulae occurring in the instances of UTB in a SC[τ ] proof D, together with the finitely many formulae instantiating (τ ⊆ T) in D. Consider the 12 See for instance, [2]. 13 For an overview of the systems, see [7] and [17]. ...
G\"odel's Incompleteness Theorems suggest that no single formal system can capture the entirety of one's mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit those \emph{implicit} assumptions. This notion of \emph{implicit commitment} motivates directly or indirectly several research programmes in logic and the foundations of mathematics; yet there hasn't been a direct logical analysis of the notion of implicit commitment itself. In a recent paper, \L elyk and Nicolai carried out an initial assessment of this project by studying necessary conditions for implicit commitments; from seemingly weak assumptions on implicit commitments of an arithmetical system S, it can be derived that a uniform reflection principle for S -- stating that all numerical instances of theorems of S are true -- must be contained in S's implicit commitments. This study gave rise to unexplored research avenues and open questions. This paper addresses the main ones. We generalize this basic framework for implicit commitments along two dimensions: in terms of iterations of the basic implicit commitment operator, and via a study of implicit commitments of theories in arbitrary first-order languages, not only couched in an arithmetical language.
... If there is an effective proof system for , this is equivalent to saying that A is The importance of ict for logic and the philosophy of mathematics is undeniable. It directly or indirectly motivates Turing's work on ordinal logics, Feferman's foundations of predicative mathematics, the extensions of Feferman's techniques to theories of truth and the ordinal analysis of mathematical systems (Beklemishev & Pakhomov, 2019;Cieśliński, 2017;Fischer et al., 2017;Franzén, 2004;Halbach, 2001;Horsten & Leigh, 2017). However, there are at least three problems with current analyses of implicit commitment for mathematical theories. ...
... We propose an epistemological analysis of the structure of implicit commitment based on such principles, according to which justified belief in a mathematical theory is transferred to its implicit commitments. Our analysis also sheds light on the recent approaches to the process of reflection in terms of epistemic entitlement (Horsten & Leigh, 2017), believability (Cieśliński, 2017) and truth (Nicolai & Piazza, 2019). The theory will also give new insights on Dean's non-uniformity objections to ict: the notion of epistemic stability, on which Dean's critique is based, is called into question. ...
... Philosophers and logicians have recently started to pay attention to the epistemology of proof-theoretic reflection principles (Cieśliński, 2017;Fischer, 2021;Fischer et al., 2019;Horsten, 2021;Horsten & Leigh, 2017). They focused on the difference between entitlement and justification in the context of reflection. ...
The notion of implicit commitment has played a prominent role in recent works in logic and philosophy of mathematics. Although implicit commitment is often associated with highly technical studies, it remains an elusive notion. In particular, it is often claimed that the acceptance of a mathematical theory implicitly commits one to the acceptance of a Uniform Reflection Principle for it. However, philosophers agree that a satisfactory analysis of the transition from a theory to its reflection principle is still lacking. We provide an axiomatization of the minimal commitments implicit in the acceptance of a mathematical theory. The theory entails that the Uniform Reflection Principle is part of one’s implicit commitments, and sheds light on why this is so. We argue that the theory has significant epistemological consequences in that it explains how justified belief in the axioms of a theory can be preserved to the corresponding reflection principle. The theory also improves on recent analyses of implicit commitment based on truth or epistemic notions.
... A different approach to characterize the epistemic commitments of a foundational theory Th was given in [Cieśliński 2017]. ...
... References: [Cieśliński 2017] The Epistemic Lightness of Truth, Cambridge University Press. [Feferman 1991] Reflecting on Incompleteness, Journal of Symbolic Logic, 56 (1) This symposium is predicated upon the assumption that one can distinguish between different scientific cultures. ...
... (Ketland 2005)). We will propose a different strategy, initiated in (Cieśliński 2017), which explains epistemic commitments in purely epistemic terms. In particular, the non-epistemic notion of truth will not play any essential part in the proposed explanation. ...
It is conventionally used to identify the beginning of the modern science with the scientific activity of Galileo Galilei.
Nevertheless, as is known thanks to copious studies about the Mathematics of the Renaissance, lots of intuitions of the Pisan
‘scientist’ were consequence of a lively scientific debate and a cultural milieu that marked the Sixteenth Century.
Among characteristics of modern science, surely the employ of the instrument to prove a theory was one of the most
important. However the protagonists of Sixteenth Century had already gained a certain awareness about the useful of
instrument to do science and as a good argument to defend their own thesis.
In this paper, I would like to show how into the controversy about the equilibrium conditions of a scale, a debate that
involved the main mathematicians of the time, Guidobaldo dal Monte, the patron of Galileo, often used experiments and
instruments to prove the indifferent equilibrium. This approach is really evident in Le mechaniche dell‘illustriss. sig. Guido
Ubaldo de‘ Marchesi del Monte: Tradotte in volgare dal sig. Filippo Pigafetta (1581), namely the Italian translation of
Mechanicorum Liber (1577), the first printed text entirely dedicated to mechanics.
... De este tipo de relaciones, entre términos, campos, formas polígonales, como de sus respectivas áreas internas, surgen diversas relaciones, como la que establece que el área del cuadrado proyectado de una hipotenusa es igual a la suma de las dos cuadrados proyectados de los lados de un triángulo rectángulo. Se trata pues de una versión geométrica del teorema que se le atribuye a Pitágoras 71 , aunque los babilónicos 72 por razones agrinométricas ya conocían tales relaciones trignonométricas y la prueba tendrá que esperar a "Los elementos" de Euclides (I.47) 73 . Es probable que el acercamiento de Pitágoras fuese más aritmético, que geométrico, partir de un triángulo con ángulo recto: 3: 4: 5, siendo 3 : 4, los lados: a, b y, 5 la hipotenusa c. ...
... El sujeto y los objetos, son los constituyentes del juicio, expresado en un orden específico que le da sentido. En breve, lo que hace a una creencias verdadera, es un hecho, que es independiente de la mente de la persona que tiene la creencia (72)(73)(74)(75). Para 1940 en Una investigación sobre el significado y la verdad, Russell ha modificado su teoría de la correspondencia, evitando los aspectos intensionales y relevandolos por aspectos extensionales. ...
... En la parte superior de la tablilla también aparece el número 30, que representa el tamaño del lado del cuadrado; mientras que en la inferior aparece 42; 25, 35 que en transcripción decimal sería: + + = , …. Entonces, de acuerdo con el teorema de Pitágoras, se puede calcular la diagonal del cuadrado, a partir de esos dos datos, es decir, multiplicando √ por el lado 30, de la siguiente manera: × √ = , … Demostrando un cálculo babilónico, que demuestra una aplicación de dicho teorema, más de mil años antes que Pitágoras. 73 La demostración del denominado como teorema de Pitágoras, esperará hasta la obra: "Elementos de 75 Aunque ha sido tema de debate, si el descubrimiento de la inconmensurabilidad (irracionalidad) era o no conocida en el pitagorismo primitivo, Kurt von Fritz (1945), sostiene que dado que la incomensurabilidad hacia finales del siglo V. a. e. c. ya era conocida y debatida en ámbitos filosóficos, su descubrimiento que se mantuvo como un secreto de la orden pitagórica, debió ser anterior, por lo que es posible remontarla a la comunidad primitiva (242)(243). Así se descubre en el Teeteto (147B) de Platón, escrito en el año -368/367 a.e.c., poco tiempo después de la muerte de este matemático. ...
En las alas de Ícaro. Filosofar como transgresión. Perspectiva transtextual, es un texto, que pretende ser un pretexto, para una iniciación a la Filosofía. Producto de la práctica docente del autor y por ende, tiene por insumos, diversas áreas disciplinares y académicas por una parte y por otra, innumerables experiencias directas con miles de discentes, una amplia cantidad de docentes, académicos y dilentantes de diversas ámbitos, de muy diversas instituciones y niveles. Por ello se desarrolla de manera paulatina desde sus inicios, aquellos temas fundamentales, requeridos en las primeras clases en un proceso de enseñanza-aprendizaje, sea para un curso inicial de filosofía, o para un discente o un diletante lego, independientemente de la metodéutica que puediese ser utilizada en dicho proceso.
... The principle of Regularity is widely rejected as a constraint on rational subjective probability. 5 Indeed there is prima facie reason to be suspicious about this principle: for instance, it seems natural to assign probability 1 to propositions that express elementary observational results, which are obviously contingent. In any case, we now already see that the situation is dire for calculi of typefree subjective probability that do include Regularity (such as certain non-Archimedean theories of probability) 6 , for then, if we accept Necessitation, the Kaplan-Montague argument goes through. ...
... Proof. This follows from the proof of Theorem 5 in [6], see also [5]. The point is that the theory RKf + CPr4 + "for every x, Pr(x) is either 1 or 0 or 1/2" is interpretable in the theory of the model (Q, B ω ), with B ω characterized as in Definition 6 of [6] (cf. also Definition 13.4.5 of [5]). ...
... This follows from the proof of Theorem 5 in [6], see also [5]. The point is that the theory RKf + CPr4 + "for every x, Pr(x) is either 1 or 0 or 1/2" is interpretable in the theory of the model (Q, B ω ), with B ω characterized as in Definition 6 of [6] (cf. also Definition 13.4.5 of [5]). The interpretation is obtained by translating "Pr(x) = y" as " y = 1 ∧ B(x) ∨ y = 0 ∧ B(¬x) ∨ y = 1/2 ∧ ¬B(x) ∧ ¬B(¬x) ". ...
We formulate and explore two basic axiomatic systems of typefree subjective probability. One of them explicates a notion of finitely additive probability. The other explicates a concept of infinitely additive probability. It is argued that the first of these systems is a suitable background theory for formally investigating controversial principles about typefree subjective probability.
... 3. We provide a new conservativity proof for CT 0 . Unlike in the first one from [15] we are able to show directly that CT 0 is arithmetically conservative over ω-iterations of uniform reflection over PA (denote this theory with REF ω (PA) 5 ). The proof is based on an essentially model-theoretic idea of prolonging a (partial) satisfaction class in an end-extension. ...
... The implication 3.⇒ 4. is established in [5]. The equivalence between 5. and 1. is demonstrated in [3]. ...
... For example the construction of a recursively saturated rather classless model of PA by Schmerl[14] employs them in a crucial way.4 Around 2012 a serious gap in the proof of Theorem 2.2 was discovered Richard Heck and Albert Visser.5 The direct conservativity argument for these theories is presented in[2] as well. ...
The current paper studies the formal properties of the Global Reflection Principle, to wit the assertion "All theorems of Th are true", where Th is a theory in the language of arithmetic and the truth predicate satisfies the usual Tarskian inductive conditions for formulae in the language of arithmetic. We fix the gap in Kotlarski's proof from [15], showing that the Global Reflection Principle for Peano Arithmetic is provable in the theory of compositional truth with bounded induction only (CT_0). Furthermore, we extend the above result showing that Σ_1-uniform reflection over a theory of uniform Tarski biconditionals (UTB −) is provable in CT_0 , thus answering the question of Beklemishev and Pakhomov [2]. Finally, we introduce the notion of a prolongable satisfaction class and use it to study the structure of models of CT_0. In particular, we provide a new model-theoretical characterization of theories of finite iterations of uniform reflection and present a new proof characterizing the arithmetical consequences of CT_0 .
... The last clause in the above axioms for CT − is called Regularity Axiom, REG. It is not included among the basic axioms in the standard presentations of this theory, like (Halbach, 2011) or (Cieśliński, 2017). The version of CT − with REG appears, e.g., in (Enayat et al., 2020) or (Łełyk and Wcisło, 2021). ...
... They worked again in purely relational languages. A version for functional languages can be found in (Cieśliński, 2017). A version of Enayat-Visser construction covering languages with functional symbols with the regularity axioms included is discussed in (Łełyk and Wcisło, 2021). ...
... For the proof of nonconservativity of CT0, see(Łełyk and Wcisło, 2017).3 One can find more information on the Tarski boundary in(Cieśliński, 2017). A more concise discussion is also contained in(Łełyk and Wcisło, 2016) and(Łełyk, 2019).4 ...
Ali Enayat had asked whether two halves of Disjunctive Correctness (DC) for the compositional truth predicate are conservative over Peano Arithmetic. In this article, we show that the principle "every true disjunction has a true disjunct" is equivalent to bounded induction for the compositional truth predicate and thus it is not conservative. On the other hand, the converse implication "any disjunction with a true disjunct is true" can be conservatively added to PA. The methods introduced here allow us to give a direct nonconservativeness proof for DC.
... Ainsi, des énoncés tels que « « La neige est blanche » est vrai » ou bien encore tels que « La proposition que la neige est blanche est vraie » sont à ses yeux peu usuels. Au contraire, les constructions typiques impliquant le prédicat de vérité sont celles où celui-ci apparaît dans des énoncés généraux, des expressions contenant des quantifications, comme par exemple 33. Ou plus précisément à la partie éliminative de la théorie de Ramsey, celle qui correspond à la seconde des deux étapes que nous avons isolées page 18. 34. ...
... i.e. F Con(F) → Con(F ∪ I) 33. Ainsi, par exemple, l'énoncé ∃xP rF∪I(x, ϕ) → ∃x P rF (x , ϕ) n'est pas Π 0 1 . ...
... Mais, nous dit Henri Galinon, pour chaque énoncé de notre langage, il existe un nom canonique : « quelque chose comme ce que l'on obtient par la mise entre guillemets de cet énoncé » 35 . Ce nom canonique est « véritablement33. dénotant un énoncé, soit qu'elle le nomme directement, soit qu'elle en constitue une description définie. ...
« Qu’est-ce que la vérité ? » À cette question, les déflationnistes aléthiques contemporains proposent une réponse originale : la propriété de vérité ne serait qu’un simple outil de décitation, indispensable pour formuler certaines généralisations mais dénué de tout pouvoir explicatif propre. Selon eux, elle ne jouerait donc pas de rôle important dans notre activité scientifique. L’objectif de cette thèse est d’évaluer la solidité de la position déflationniste en la confrontant à divers arguments avancés contre ce type de conceptions de la vérité. Après avoir précisé les doctrines centrales du déflationnisme actuel, notre travail se poursuit en deux parties, que l’on peut voir comme deux tentatives complémentaires de fournir un cadre méthodologique permettant d’examiner précisément les théories déflationnistes de la vérité. Dans un premier temps, nous analysons la thèse, souvent attribuée aux déflationnistes, selon laquelle le prédicat de vérité serait une sorte de notion logique. Dans un second temps nous examinons un célèbre argument anti-déflationniste appelé « argument de la conservativité ». Au final, si le déflationnisme ne nous paraît pas totalement désarmé face aux critiques dont il a fait l’objet, notre travail a néanmoins permis de montrer que certaines réponses majeures avancées pour sa défense ne sont plus tenables.
... The proofs in [10] mostly relied on the induction scheme in a crucial way, so recapturing the results in the setting of full satisfaction classes requires quite different arguments. 1 In this article, we investigate such full satisfaction classes with special model-theoretic properties. Among other things, we show that in a countable model of PA, every set is definable (without parametres) from a full satisfaction class. ...
... A classical textbook on axiomatic truth predicates is[6]. See also[1] for an overview of some more recent results in that field. ...
... See[2] for the result on disjunctions and[1] for the other proofs and an extensive discussion of the Tarski Boundary programme which investigates this phenomenon.11 We are grateful for this observation to Roman Kossak who also suggested to study automorphisms and definability properties of satisfaction classes. ...
We show that for every countable recursively saturated model M of Peano Arithmetic and every subset , there exists a full satisfaction class such that A is definable in without parametres. It follows that in every such model, there exists a full satisfaction class which makes every element definable and thus the expanded model is minimal and rigid. On the other hand, we show that for every full satisfaction class S there are two elements which have the same arithmetical type, but exactly one of them is in S. In particular, the automorphism group of a model expanded with a satisfaction class is never equal to the automorphism group of the original model. The analogue of many of the results proved here for full satisfaction classes were obtained by Roman Kossak for partial inductive satisfaction classes. However, most of the proofs relied heavily on the induction scheme in a crucial way, so recapturing the results in the setting of full satisfaction classes requires quite different arguments.
... MO is discussed in (Van Fraassen, 2022, p. 17) and (Cross, 2001). CI and WRef are discussed in (Cieśliński, 2017) and (Cieśliński, unpublished). In their discussion of these principles, most of these scholars interpret J not as justified belief but as some related notion; a philosophical discussion of many of these principles in terms of justification can be found in (Rosenkranz, 2018). ...
... The translation τ which distributes over logical connectives, is homophonic for purely mathematical statements, and which is such that τ (J φ) is set equal to Pr(τ (φ)) = 1 translates theorems of B into theorems of the system RKf, which is (by Theorem 3 of (Cieśliński et al., 2022)) proof-theoretically conservative over its background theory. 19 Cieśliński has developed a formal theory of 'believability' (Cieśliński, 2017), which is a notion which is closely related to that of justified belief. A few years after the publication of his book Cieśliński proposed an improved axiomatic theory of believability (Cieśliński, unpublished): we base our discussion on the later version of his theory. ...
Justified belief is a core concept in epistemology and there has been an increasing interest in its logic over the last years. While many logical investigations consider justified belief as an operator, in this paper, we propose a logic for justified belief in which the relevant notion is treated as a predicate instead. Although this gives rise to the possibility of liar-like paradoxes, a predicate treatment allows for a rich and highly expressive framework, which lives up to the universal ambitions of investigating epistemological concepts. We start with a base theory for justified belief, and then systematically present putative additional axioms for justified belief. We provide an overview of (in)consistency results when the additional principles are added to the base theory, and discuss their philosophical plausibility.
... They claim that HP may be taken as an implicit definition of the operator # ('the number of') in purely logical terms. 4 Hale and Wright's notion of implicit definition is deeply controversial. For our purposes, the main point is that Hale and Wright conceive of implicit definitions as true simply by stipulation [14, p. 117]. ...
... Semantic notions of conservativeness have also been studied in the literature on truth [4]. ...
Neo-Fregean logicists claim that Hume’s Principle ( HP ) may be taken as an implicit definition of cardinal number, true simply by fiat. A long-standing problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck’s Two-Sorted Frege Arithmetic ( 2FA ), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn’t. In fact, 2FA is not conservative over n -th order logic, for all . It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic.
... In a nutshell, a sequent is an expression of the form ⇒ , where , are finite sets of formulas. Informally, the formulas preceding the sequent arrow '⇒' are treated as assumptions, and the formulas in the succedent Footnote 4 continued more recent philosophical discussion of these principles see for instance Franzen (2004), Cieśliński (2017) and Horsten and Zicchetti (2021). 5 This principle was originally formulated by Kreisel and Lévy (1968). ...
... 44 However, such restriction is taken to be artificial. Halbach (2009), Horsten andLeigh (2017) and Cieśliński (2017Cieśliński ( , 2015 all independently argue that the theories of positive truth and falsity are well-motivated, via a careful analysis and diagnosis of the paradoxes of truth. 45 However, the force of the argument for positive truth still needs to be spelled out: for instance, one might (and should) ask how good the warrant for the acceptance of positive truth, by means of the analysis of the paradoxes, is. ...
The aim of this paper is twofold: first, I provide a cluster of theories of truth in classical logic that is (internally) consistent with global reflection principles: the theories of positive truth (and falsity). After that, I analyse the epistemic value of such theories. I do so employing the framework of cognitive projects introduced by Wright (Proc Aristot Soc 78:167–245, 2004), and employed—in the context of theories of truth—by Fischer et al. (Noûs 2019. 10.1111/nous.12292 ). In particular, I will argue that theories of positive truth are trustworthy , analogously to the theories of full disquotational truth. Moreover, I argue that, for a given cognitive project, if the acceptance of trustworthy theories is taken to be an epistemic norm of cognitive project, then one has good reasons to accept theories of positive truth over other rival theories of truth in classical logic. On the other hand, the latter theories are deemed epistemically unacceptable.
... The family of (Kalmár) elementary functions is a distinguished subfamily of the primitive recursive functions. 4 It is well-known that the provably recursive functions of EA are precisely the elementary functions; and that a function f is elementary iff f is computable by a Turing machine with a multiexponential time bound. Definition 3. We say that B is a base theory if B is formulated in L PA with B ⊇ EA. ...
... In CT5 above s and t denote finite tuples of terms; ands • ,t • refer to the corresponding valuations of s and t. The axiom CT5 is sometimes called generalized regularity, or generalized term-extensionality, and is not included in the accounts of CT − provided in the monographs of Halbach [10] and Cieśliński [4]. The conservativity of this particular version of CT − [PA] can be verified by a refinement of the model-theoretic method introduced in [6], as presented both in [7] and [12]. ...
We employ the lens provided by formal truth theory to study axiomatizations of PA (Peano Arithmetic). More specifically, let EA (Elementary Arithmetic) be the fragment I∆0 + Exp of PA, and CT − [EA] be the extension of EA by the commonly studied axioms of compositional truth CT −. We investigate both local and global properties of the family of first order theories of the form CT − [EA] + α, where α is a particular way of expressing "PA is true" (using the truth predicate). Our focus is dominantly on two types of axiomatizations, namely: (1) schematic axiomatizations that are deductively equivalent to PA, and (2) axiomatizations that are proof-theoretically equivalent to the canonical axiomatization of PA.
... See Halbach (2011). 9 For a critical and comprehensive survey, see Cieśliński (2017). ...
One of the goals of the natural sciences– for example biology– is to provide new information about certain phenomena with previously unknown nature. Their contribution to our knowledge is substantial. From this perspective, logic is seemingly not substantial. Sometimes, logic’s insubstantiality is taken for granted while explaining the alleged insubstantiality of other notions. For example, according to truth deflationism, truth is a non-substantial notion in the sense of being a logical property. However, it is not fully clear to what such an insubstantiality amounts. It is also debatable whether logic really is insubstantial. In this paper, we aim to clarify this issue by proposing a formal way of looking at it. In particular, we used the notion of conservativity, which has already been used by truth deflationism, for a similar aim. We show that if insubstantiality is read in terms of conservativity, then classical logic is substantial. We then argue that such a verdict of substantiality can be resisted if precise stances on certain prima facie unrelated issues of philosophy of logic are taken, or an anti-exceptionalist view is adopted.
... Presumably, proofs 14 This statement is often called the global reflection principle for S. See (Halbach 2014 , Theorem 8.39,104) for a sketch of the proof. For general philosophical discussions of reflection principles, see, for instance, (Franzén 2004 ) and (Cieśliński 2017 ). 15 A theory T is proof-theoretically non-conservative over its base theory S just in case S proves some statement p in the language of S, which is unprovable in S. Due to Gödel's Incompleteness, any consistent theory T that proves the consistency of S-in the language of S-must be non-conservative over S. 16 Although a discussion of deflationism would be interesting, it exceeds the scope of this investigation. ...
Soundness Arguments for the consistency of a (mathematical) theory S aim to show that S is consistent by first showing or employing the fact that S is sound, i.e., that all theorems of S are true. Although soundness arguments are virtually unanimously accepted as valid and sound for most of our accepted theories, philosophers disagree about their epistemic value, i.e., about whether such arguments can be employed to improve our epistemic situation concerning questions of consistency. This article provides a (partial) negative answer to this question and argues that soundness arguments cannot be employed to justify their conclusion. Additionally, soundness arguments are unconvincing; they cannot be employed to overcome reasonable open-mindedness about their conclusion.
... The theories UTB (Uniform Tarski Biconditionals) and CT (Compositional Truth) described below are well studied in the literature of axiomatic theories of truth (see, e.g., the monographs by Cieśliński [2] and Halbach [8]). ...
We investigate the theory Peano Arithmetic with Indiscernibles (PAI). Models of PAI are of the form (M,I), where M is a model of PA, I is an unbounded set of order indiscernibles over M, and (M,I) satisfies the extended induction scheme for formulae mentioning I. Our main results are Theorems A and B following. Theorem A.LetMbe a nonstandard model ofPA of any cardinality. Mhas an expansion to a model of PAIiffMhas an inductive partial satisfaction class. Theorem A yields the following corollary, which provides a new characterization of countable recursively saturated models of PA: Corollary.A countable modelM of PAis recursively saturated iff Mhas an expansion to a model of PAI. Theorem B.There is a sentence α in the language obtained by adding a unary predicateI(x) to the language of arithmetic such that given any nonstandard model MofPA of any cardinality, Mhas an expansion to a model of PAI+αiffMhas a inductive full satisfaction class.
... According to first-orderism, it is a proper subset. Since first-orderism holds that first-order PA is complete with respect to finitary, purely arithmetical truth, sentences that we notion that in accepting S, one is thereby warranted in accepting reflection principles for S (Cieśliński, 2010(Cieśliński, , 2017Feferman, 1962Feferman, , 1991Fischer, 2021;Fischer et al., 2021;Franzén, 2004;Horsten & Leigh, 2016;Ketland, 2005Ketland, , 2010Shapiro, 1998;Tennant, 2002Tennant, , 2005Turing, 1939). 56 But then the idea in (3) sits in tension with the idea of epistemic stability. ...
This paper ties together three threads of discussion about the following question: in accepting a system of axioms S, what else are we thereby warranted in accepting, on the basis of accepting S? First, certain foundational positions in the philosophy of mathematics are said to be epistemically stable, in that there exists a coherent rationale for accepting a corresponding system of axioms of arithmetic, which does not entail or otherwise rationally oblige the foundationalist to accept statements beyond the logical consequences of those axioms. Second, epistemic stability is said to be incompatible with the implicit commitment thesis, according to which accepting a system of axioms implicitly commits the foundationalist to accept additional statements not immediately available in that theory. Third, epistemic stability stands in tension with the idea that in accepting a system of axioms S, one thereby also accepts soundness principles for S. We offer a framework for analysis of sets of implicit commitment which reconciles epistemic stability with the latter two notions, and argue that all three ideas are in fact compatible.
... For the idea that theories come with a set of implicit commitments and/or entitlements, seeWright (2004),Dean (2015),Cieśliński (2017),Nicolai and Piazza (2019),Horsten (2021) andŁełyk and Nicolai (2022). ...
An old and well-known objection to non-classical logics is that they are too weak; in particular, they cannot prove a number of important mathematical results. A promising strategy to deal with this objection consists in proving so-called recapture results. Roughly, these results show that classical logic can be used in mathematics and other unproblematic contexts. However, the strategy faces some potential problems. First, typical recapture results are formulated in a purely logical language, and do not generalize nicely to languages containing the kind of vocabulary that usually motivates non-classical theories---for example, a language containing a naïve truth predicate. Second, proofs of recapture results typically employ classical principles that are not valid in the targeted non-classical system; hence, non-classical theorists do not seem entitled to those results. In this paper, we analyze these problems and provide solutions on behalf of non-classical theorists. To address the first problem, we provide a novel kind of recapture result, which generalizes nicely to a truth-theoretic language. As for the second problem, we argue that it relies on an ambiguity and that, once the ambiguity is removed, there are no reasons to think that non-classical logicians are not entitled to their recapture results.
... In order to introduce schemata of transfinite induction, we fix a standard notation system of ordinals up to Γ 0 . 10 We use a, b, c . . . to denote the code of our notation system whose value is α, β, γ · · · ∈ On (with the exception of ωand ε-numbers, for which we use the symbols 'ω' and 'ε' themselves), and we use ≺ to denote a standard primitive recursive ordering defined on codes of ordinals. Moreover, we let ...
This article investigates models of axiomatizations related to the semantic conception of truth presented by Kripke (J Philos 72(19):690–716, 1975), the so-called fixed-point semantics . Among the various proof systems devised as a proof-theoretic characterization of the fixed-point semantics, in recent years two alternatives have received particular attention: classical systems (i.e., systems based on classical logic) and nonclassical systems (i.e., systems based on some nonclassical logic). The present article, building on Halbach and Nicolai (J Philos Log 47(2):227–257, 2018), shows that there is a sense in which classical and nonclassical theories (in suitable variants) have the same models.
... The ever-increasing popularity of truth-theoretic deflationism (Cieśliński, 2017), together with a revived attention to the Liar paradox prompted by new technical tools (Field, 2008;Halbach, 2014;Horsten, 2012), led to a multiplication of formal systems extending some standard syntax theory with a primitive truth predicate governed by suitable axioms. These systems may have multiple aims: they may embody some conception of truth, including a solution to the Liar and related paradoxes; they may characterize the truth predicate as a logical tool whose formal properties witness the role that the notion of truth can play in (sustained) reasoning-e.g., in applied mathematics and in the formal sciences. ...
When are two formal theories of broadly logical concepts, such as truth, equivalent? The paper investigates a case study, involving two well-known variants of Kripke–Feferman truth. The first, KF + CONS , features a consistent but partial truth predicate. The second, KF + COMP , an inconsistent but complete truth predicate. It is known that the two truth predicates are dual to each other. We show that this duality reveals a much stricter correspondence between the two theories: they are intertraslatable. Intertranslatability, under natural assumptions, coincides with definitional equivalence, and is arguably the strictest notion of theoretical equivalence different from logical equivalence. The case of KF + CONS and KF + COMP raises a puzzle: the two theories can be proved to be strictly related, yet they appear to embody remarkably different conceptions of truth. We discuss the significance of the result for the broader debate on formal criteria of conceptual reducibility for theories of truth.
... In recent years, much attention was given to the study of axiomatic subtheories of Tar. See [Cie17] for a comprehensive account. Originally, the results were formulated in terms of satisfaction classes (binary), as we will do below. ...
This is a survey of results on definability and undefinability in models of arithmetic. The goal is to present a stark difference between undefinability results in the standard model and much stronger versions about expansions of nonstandard models. The key role is played by counting the number of automorphic images of subsets of countable resplendent models of Peano Arithmetic.
... See, e.g., Halbach (2014) orCieśliński (2017). 15 I have seen it remarked that a formalized theory, in particular for scientific theories, must always be stated directly in machine code. ...
A \emph{standard formalization} of a scientific theory is a system of axioms for that theory in a first-order language (possibly many-sorted; possibly with the membership primitive ). Patrick Suppes (\cite{sup92}) expressed skepticism about whether there is a ``simple or elegant method'' for presenting mathematicized scientific theories in such a standard formalization, because they ``assume a great deal of mathematics as part of their substructure''. The major difficulties amount to these. First, as the theories of interest are \emph{mathematicized}, one must specify the underlying \emph{applied mathematics base theory}, which the physical axioms live on top of. Second, such theories are typically \emph{geometric}, concerning quantities or trajectories in space/time: so, one must specify the underlying \emph{physical geometry}. Third, the differential equations involved generally refer to \emph{coordinate representations} of these physical quantities with respect to some implicit coordinate chart, not to the original quantities. These issues may be resolved. Once this is done, constructing standard formalizations is not so difficult---at least for the theories where the mathematics has been worked out rigorously. Here we give what may be claimed to be a simple and elegant means of doing that. This is for mathematicized scientific theories comprising differential equations for -valued quantities Q (that is, scalar fields), defined on n (``spatial'' or ``temporal'') dimensions, taken to be isomorphic to the usual Euclidean space . For illustration, I give standard (in a sense, ``text-book'') formalizations: for the simple harmonic oscillator equation in one-dimension and for the Laplace equation in two dimensions.
The seminal Krajewski–Kotlarski–Lachlan theorem (1981) states that every countable recursively saturated model of (Peano arithmetic) carries a full satisfaction class. This result implies that the compositional theory of truth over commonly known as is conservative over . In contrast, Pakhomov and Enayat (2019) showed that the addition of the so-called axiom of disjunctive correctness (that asserts that a finite disjunction is true iff one of its disjuncts is true) to axiomatizes the theory of truth that was shown by Wcisło and Łełyk (2017) to be nonconservative over . The main result of this paper (Theorem 3.12) provides a foil to the Pakhomov–Enayat theorem by constructing full satisfaction classes over arbitrary countable recursively saturated models of that satisfy arbitrarily large approximations of disjunctive correctness. This shows that in the Pakhomov–Enayat theorem the assumption of disjunctive correctness cannot be replaced with any of its approximations.
The aim of this paper is threefold. Firstly, sections 1 and 2 introduce the novel concept logical akrasia by analogy to epistemic akrasia. If successful, the initial sections will draw attention to an interesting akratic phenomenon which has not received much attention in the literature on akrasia (although it has been discussed by logicians in different terms). Secondly, sections 3 and 4 present a dilemma related to logical akrasia. From a case involving the consistency of Peano Arithmetic and Gödel's Second Incompleteness Theorem, it's shown that either we must be agnostic about the consistency of Peano Arithmetic or akratic in our arithmetical theorizing. If successful, these sections will underscore the pertinence and persistence of akrasia in arithmetic (by appeal to Gödel's seminal work). Thirdly, section 5 concludes by suggesting a way of translating the Dilemma of Arithmetical Akrasia into a case of regular epistemic akrasia; and further how one might try to escape the dilemma when it's framed this way.
According to deflationism, truth is insubstantial. Edwards (2018) argues that the deflationist thesis of insubstantiality is incoherent, regardless of how it is characterized. By clarifying the deflationist concepts of reference and truth (and their relations) and addressing the distinction between substantial properties and insubstantial properties within the deflationist framework, we will argue that Edwards’s self-defeating argument is problematic and ultimately unconvincing.
When accepting an axiomatic theory S, we are implicitly committed to various statements that are independent of its axioms. Examples of such implicit commitments include consistency statements and reflection principles for S. While foundational acceptance has received considerable attention in this context, the study of implicit commitments triggered by weaker notions remains underdeveloped. This article extends the analysis investigating implicit commitments inherent in instrumental acceptance, comparing them with the implicit commitments involved in foundational acceptance. Concentrating on Reinhardt’s instrumentalism vis-à-vis Kripke–Feferman theory of truth as a case study, we present a number of formal theories of acceptance motivated by Reinhardt’s program and we analyze their properties. We argue that, under reasonable assumptions, instrumental acceptance does entail non-trivial implicit commitments, yet weaker than those associated with foundational acceptance.
Gödel’s Incompleteness Theorems suggest that no single formal system can capture the entirety of one’s mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit those implicit assumptions. This notion of implicit commitment motivates directly or indirectly several research programmes in logic and the foundations of mathematics; yet there hasn’t been a direct logical analysis of the notion of implicit commitment itself. In a recent paper, we carried out an initial assessment of this project by studying necessary conditions for implicit commitments; from seemingly weak assumptions on implicit commitments of an arithmetical system S, it can be derived that a uniform reflection principle for S—stating that all numerical instances of theorems of S are true—must be contained in S’s implicit commitments. This study gave rise to unexplored research avenues and open questions. This paper addresses the main ones. We generalize this basic framework for implicit commitments along two dimensions: in terms of iterations of the basic implicit commitment operator, and via a study of implicit commitments of theories in arbitrary first-order languages, not only couched in an arithmetical language.
O presente estudo concentra-se nos fenômenos jurídicos derivados das relações
entre as ciências médicas e as ciências radiológicas, analisando os eventos
antropológicos, traumatológicos, infortunísticos e tanatológicos que resultam da
intersecção prática e conceitual entre médicos e pacientes. Tomamos como ponto de
partida as contribuições acadêmicas dos médicos e radiologistas forenses Byron
Gilliam Brogdon, MD (1925-2014), em sua obra "Forensic Radiology", e Michael J.
Thali, MD, em sua obra "The Virtopsy Approach: 3D Optical and Radiological
Scanning and Reconstruction in Forensic Medicine". Para atingir os objetivos
propostos, optamos por desenvolver um estudo exploratório com abordagem
qualitativa, com base nas principais publicações sobre o tema. A crise do paradigma
dominante pode ser a causa da tensão atual, pois a busca pela justiça, seja pela
ideia de proporcionalidade ou interseccionalidade de direitos, fica prejudicada pela
ausência de fontes normativas legítimas. Sabemos que a hiperfragmentação dos
objetos científicos no campo médico-legal afastou as percepções sobre as
consequências históricas e culturais da atuação médica, especialmente na radiologia.
A reflexão sobre os escritos que denunciam a atual ruptura permite deduzir a
seguinte conclusão: aparentemente, a Ciência Radiológica Legal é o método mais
adequado para estudar e classificar os fenômenos radiológicos, dinâmicos e
tecnológicos.
In this final chapter we will discuss connections between definability in first-order logic and in , as well as the number of symmetric images of functions and relations. There will be no proofs of general results, but we will see how they work with the aid of examples from previous sections.
We investigate the theory PAI (Peano Arithmetic with Indiscernibles). Models of PAI are of the form (M, I), where M is a model of PA, I is an unbounded set of order indiscernibles over M, and (M, I) satisfies the extended induction scheme for formulae mentioning I. Our main results are Theorems A and B below. Theorem A. Let M be a nonstandard model of PA of any cardinality. M has an expansion to a model of PAI iff M has an inductive partial satisfaction class. Theorem A yields the following corollary, which provides a new characterization of countable recursively saturated models of PA: Corollary. A countable model M of PA is recursively saturated iff M has an expansion to a model of PAI. Theorem B. There is a sentence s in the language obtained by adding a unary predicate I(x) to the language of arithmetic such that given any nonstandard model M of PA of any cardinality, M has an expansion to a model of PAI + s iff M has a inductive full satisfaction class.
We employ the lens provided by formal truth theory to study axiomatizations of Peano Arithmetic . More specifically, let Elementary Arithmetic be the fragment of , and let be the extension of by the commonly studied axioms of compositional truth . We investigate both local and global properties of the family of first order theories of the form , where is a particular way of expressing “ is true” (using the truth predicate). Our focus is dominantly on two types of axiomatizations, namely: (1) schematic axiomatizations that are deductively equivalent to and (2) axiomatizations that are proof-theoretically equivalent to the canonical axiomatization of .
As a response to the semantic and logical paradoxes, theorists often reject some principles of classical logic. However, classical logic is entangled with mathematics, and giving up mathematics is too high a price to pay, even for nonclassical theorists. The so-called recapture theorems come to the rescue. When reasoning with concepts such as truth/class membership/property instantiation, (These are examples of concepts that are taken to satisfy naive rules such as the naive truth schema and naive comprehension, and that therefore are compatible with a solution to paradox cast in the logics considered below. Other notions of similar kind can be added to the list.) if one is interested in consequences of the theory that only contain mathematical vocabulary, nothing is lost by reasoning in the nonclassical framework. This article shows that this claim is highly misleading, if not simply false. Under natural assumptions, some well-established approaches to recapture are incorrect.
Ali Enayat had asked whether two halves of Disjunctive Correctness (DC) for the compositional truth predicate are conservative over Peano Arithmetic (PA). In this paper, we show that the principle “every true disjunction has a true disjunct” is equivalent to bounded induction for the compositional truth predicate and thus it is not conservative. On the other hand, the converse implication “any disjunction with a true disjunct is true” can be conservatively added to PA. The methods introduced here allow us to give a direct nonconservativeness proof for DC.
The philosophical problem of implicit commitment can be roughly stated in the form: (*) What are we implicitly committed to in accepting a theory and what can we justifiably accept? As is well-known, (*) has its roots in the work by Kreisel (1958, 1970) (As rightly noticed by a referee, the problem has a much broader and longer history. But we here do not aim at a survey or a historical appreciation of the topics). Our aim is to consider two possible routes towards a solution, as given over the years by the late Solomon Feferman, starting already in the Sixties and the early Seventies with the work on Predicative Analysis (Feferman, 1968), thereafter with the investigation of self-referential truth in Feferman (1991), and eventually fully transformed by Feferman himself together with Thomas Strahm in Feferman and Strahm (2000). While the first route—reflecting—directly leads into the land of truth theories, the second one—unfolding, see Sect. 2—is more mathematical in spirit and hinges upon a point of view, which drives us to the very notion of operation. Our presentation consists in a survey of the two alternatives, while the implicit commitment issue is specifically dealt within the final section. (The proposals we consider are extensively and fully developed in Feferman [1991], Feferman and Strahm [2000, 2010], Buchholtz [2013], and Buchholtz et al. [2016]; as to their connections with the general theme of abstraction, they can also be supplemented by means of results in Cantini [1989, 1996, 2016]).KeywordsOperationTruthUnfoldingImplicit commitment2000 Mathematics Subject Classification03F0303F2503F353F4003A05
The current paper studies the formal properties of the Global Reflection Principle, to wit the assertion “All theorems of are true,” where is a theory in the language of arithmetic and the truth predicate satisfies the usual Tarskian inductive conditions for formulae in the language of arithmetic. We fix the gap in Kotlarski’s proof from [15], showing that the Global Reflection Principle for Peano Arithmetic is provable in the theory of compositional truth with bounded induction only ( ). Furthermore, we extend the above result showing that -uniform reflection over a theory of uniform Tarski biconditionals ( ) is provable in , thus answering the question of Beklemishev and Pakhomov [2]. Finally, we introduce the notion of a prolongable satisfaction class and use it to study the structure of models of . In particular, we provide a new model-theoretical characterization of theories of finite iterations of uniform reflection and present a new proof characterizing the arithmetical consequences of .
We consider extensions of the language of Peano arithmetic by transfinitely iterated truth definitions satisfying uniform Tarskian biconditionals. Without further axioms, such theories are known to be conservative extensions of the original system of arithmetic. Much stronger systems, however, are obtained by adding either induction axioms or reflection axioms on top of them. Theories of this kind can interpret some well-known predicatively reducible fragments of second-order arithmetic such as iterated arithmetical comprehension.
We obtain sharp results on the proof-theoretic strength of these systems using methods of provability logic. Reflection principles naturally define unary operators acting on the semilattice of axiomatizable extensions of our basic theory of iterated truth. The substructure generated by the top element of this algebra provides a canonical ordinal notation system for the class of theories under investigation.
Using these notations we obtain conservativity relationships for iterated reflection principles of different logical complexity levels corresponding to the levels of the hyperarithmetical hierarchy, i.e., the analogs of Schmerl's formulas. These relationships, in turn, provide proof-theoretic analysis of our systems and of some related predicatively reducible theories. In particular, we uniformly calculate the ordinals characterizing the standard measures of their proof-theoretic strength, such as provable well-orderings, classes of provably recursive functions, and Π10-ordinals.
Truth is often considered to be a logico-linguistic tool for expressing indirect endorsements and infinite conjunctions. In this article, I will point out another logico-linguistic function of truth: to enable and validate what I call a blind argument, namely, an argument that involves indirectly endorsed statements. Admitting this function among the logico-linguistic functions of truth has some interesting consequences. In particular, it yields a new type of so-called conservativeness argument, which poses a new type of threat to deflationism about truth.
According to a prominent objection, deflationist theories of truth can’t account for the explanatory connection between true belief and successful action [Putnam 1978 Putnam, Hilary 1978. Meaning and the Moral Sciences, London: Routledge & Kegan Paul. [Google Scholar]]. Canonical responses to the objection show how to reformulate truth-involving explanations of particular successful actions, so as to omit any mention of truth [Horwich 1998 Horwich, Paul 1998. Truth, New York: Oxford University Press.[Crossref] , [Google Scholar]]. According to recent critics, though, the canonical strategy misses the point. The deflated paraphrases lack the generality or explanatory robustness of the original explanatory appeals to truth [Kitcher 2002 Kitcher, Philip 2002. On the Explanatory Role of Correspondence Truth, Philosophy and Phenomenological Research 64/2: 346–64.[Crossref], [Web of Science ®] , [Google Scholar]; Lynch 2009 Lynch, Michael P. 2009. Truth as One and Many, Oxford: Clarendon Press.[Crossref] , [Google Scholar]; Gamester 2018 Gamester, Will 2018. Truth: Explanation, Success, and Coincidence, Philosophical Studies 175/5: 1243–65.[Crossref], [Web of Science ®] , [Google Scholar]]. This article diagnoses the canonical response’s failure and shows how deflationists can make sense of appeals to truth in explaining practical success, in all of their generality and robustness, without construing truth as a substantial property.
This paper outlines an account of number based on the numerical equivalence schema (NES), which consists of all sentences of the form #x.F x = n iff ∃ n x F x, where # is the number-of operator and ∃ n is defined in standard Russellian fashion. In the first part of the paper I point out some analogies between the NES and the T-schema for truth. In light of these analogies, I formulate a minimalist account of numbers, based on the NES, that strongly parallels the minimalist (deflationary) account of truth. One may be tempted to develop the minimalist account in a fictionalist direction, according to which arithmetic is useful but untrue, if taken at face value. In the second part I argue that this suggestion is not as attractive as it may appear at first. The NES suffers from a similar problem as the T-schema: it is deductively weak and does not enable the derivation of any non-trivial generalisations. In the third part of the paper I explore some strategies to deal with the generalisation problem, drawing again inspiration from the literature on truth. In closing this paper, I briefly compare the minimalist to some other accounts of number.
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