Reflection Principles are commonly thought to produce only strong axioms of infinity consistent with V = L. It would be desirable to have some notion of strong reflection to remedy this, and we have proposed Global Reflection Principles based on a somewhat Cantorian view of the universe. Such principles justify the kind of cardinals needed for, inter alia , Woodin’s Ω-Logic.

Contextualist approaches to the Liar Paradox postulate the occurrence of a context shift in the course of the Liar reasoning. In particular, according to the contextualist proposal advanced by Charles Parsons and Michael Glanzberg, the Liar sentence L doesn’t express a true proposition in the initial context of reasoning c, but expresses a true one in a new, richer context c', where more propositions are available for expression. On the further assumption that Liar sentences involve propositional quantifiers whose domains may (...) vary with context, the Liar reasoning is blocked. But why should context shift? We argue that the paradox involves principles of contextualist reflection that explain, by analogy with well-known reflection principles for arithmetic, why context must shift from c to c' in the course of the Liar reasoning. This provides a diagnosis of the Liar Paradox—one that equally applies to two revenge arguments against contextualist approaches, one recently advanced by Andrew Bacon, the other mentioned by Charles Parsons and more recently revived by Cory Juhl. (shrink)

First, Mahtani argues that both in the game The Mug and in the Sleeping Beauty we should not defer to a trusted person under a particular designation if they do not self-identify under this designation. This invites a more complex Reflection Principle. I respond that there are more parsimonious ways to avoid the challenges posed to the Reflection Principle. Second, Mahtani argues that preferences create a hyperintensional context, which poses a challenge to the Ex-Ante Pareto (...) class='Hi'>Principle that can be averted by supervaluation. I respond that such an appeal to supervaluation would block randomization as a fair allocation device. (shrink)

We study reflection principles in Kelley-Morse set theory with urelements (KMU). We first show that First-Order Reflection Principle is not provable in KMU with Global Choice. We then show that KMU + Limitation of Size + Second-Order Reflection Principle is mutually interpretable with KM + Second-Order Reflection Principle. Furthermore, these two theories are also shown to be bi-interpretable with parameters. Finally, assuming the existence of a κ+-supercompact cardinal κ in KMU, we construct a (...) model of KMU + Second-Order Reflection Principle where Limitation of Size fails. (shrink)

Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified. Taking as our starting point Tait’s work on general reflection principles, we prove a series of limitative results concerning this approach. These results collectively show that general reflection principles are either weak ) or inconsistent. The philosophical (...) significance of these results is discussed. (shrink)

A diagonal version of the strong reflection principle is introduced, along with fragments of this principle associated with arbitrary forcing classes. The relationships between the resulting principles and related principles, such as the corresponding forcing axioms and the corresponding fragments of the strong reflection principle, are analyzed, and consequences are presented. Some of these consequences are “exact” versions of diagonal stationary reflection principles of sets of ordinals. We also separate some of these diagonal strong (...) reflection principles from related axioms. (shrink)

We introduce three families of diagonal reflection principles for matrices of stationary sets of ordinals. We analyze both their relationships among themselves and their relationships with other known principles of simultaneous stationary reflection, the strong reflection principle, and the existence of square sequences.

We present a variety of (ω 1 ,∞)-distributive forcings which when applied to models of Martin's Maximum separate certain well known reflection principles. In particular, we do this for the reflection principles SR, SR α (α ≤ ω 1 ), and SRP.

This article introduces a new reflection principle. It is based on the idea that whatever is true in all entities of some kind is also true in a set-sized collection of them. Unlike standard reflection principles, it does not re-interpret parameters or predicates. This allows it to be both consistent in all higher-order languages and remarkably strong. For example, I show that in the language of second-order set theory with predicates for a satisfaction relation, it is consistent (...) relative to the existence of a 2-extendible cardinal (Theorem 7.12) and implies the existence of a proper class of 1-extendible cardinals (Theorem 7.9). (shrink)

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each (...) express a form of width reflection in contrast to the usual height reflection of the Lévy–Montague reflection theorem. They are each equiconsistent with ZFC and indeed \Pi_2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles. (shrink)

The probabilities a Bayesian agent assigns to a set of events typically change with time, for instance when the agent updates them in the light of new data. In this paper we address the question of how an agent's probabilities at different times are constrained by Dutch-book coherence. We review and attempt to clarify the argument that, although an agent is not forced by coherence to use the usual Bayesian conditioning rule to update his probabilities, coherence does require the agent's (...) probabilities to satisfy van Fraassen's [1984] reflection principle (which entails a related constraint pointed out by Goldstein [1983]). We then exhibit the specialized assumption needed to recover Bayesian conditioning from an analogous reflection-style consideration. Bringing the argument to the context of quantum measurement theory, we show that "quantum decoherence" can be understood in purely personalist terms---quantum decoherence (as supposed in a von Neumann chain) is not a physical process at all, but an application of the reflection principle. From this point of view, the decoherence theory of Zeh, Zurek, and others as a story of quantum measurement has the plot turned exactly backward. (shrink)

Classically, any structure for a signature ${\rm{\Sigma }}$ may be completed to a model of a desired regular theory ${T}}$ by means of the chase construction or small object argument. Moreover, this exhibits ${\rm{Mod}}\left$ as weakly reflective in ${\rm{Str}}\left$.We investigate this in the constructive setting. The basic construction is unproblematic; however, it is no longer a weak reflection. Indeed, we show that various reflectivity principles for models of regular theories are equivalent to choice principles in the ambient set theory. (...) However, the embedding of a structure into its chase-completion still satisfies a conservativity property, which suffices for applications such as the completeness of regular logic with respect to Tarski models.Unlike most constructive developments of predicate logic, we do not assume that equality between symbols in the signature is decidable. While in this setting, we also give a version of one classical lemma which is trivial over discrete signatures but more interesting here: the abstraction of constants in a proof to variables. (shrink)

The weakly compact reflection principle\\) states that \ is a weakly compact cardinal and every weakly compact subset of \ has a weakly compact proper initial segment. The weakly compact reflection principle at \ implies that \ is an \-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that \ is \\)-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at (...) \ then there is a forcing extension preserving this in which \ is the least \-weakly compact cardinal. Along the way we generalize the well-known result which states that if \ is a regular cardinal then in any forcing extension by \-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if \ is a weakly compact cardinal then after forcing with a ‘typical’ Easton-support iteration of length \ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal. (shrink)

A well-known result states that, over basic Kalmar elementary arithmetic EA, the induction schema for ∑n formulas is equivalent to the uniform reflection principle for ∑n + 1 formulas . We show that fragments of arithmetic axiomatized by various forms of induction rules admit a precise axiomatization in terms of reflection principles as well. Thus, the closure of EA under the induction rule for ∑n formulas is equivalent to ω times iterated ∑n reflection principle. Moreover, (...) for k < ω, k times iterated ∑n reflection principle over EA precisely corresponds to the extension of EA by k nested applications of ∑n induction rule.The above relationship holds in greater generality than just stated. In fact, we give general formulas characterizing in terms of iterated reflection principles the extension of any given theory by k nested applications of ∑n or Πn induction rules. In particular, the closure of a theory T under just one application of ∑1 induction rule is equivalent to T together with ∑1 reflection principle for each finite Π2 axiomatized subtheory of T.These results have closely parallel ones in the theory of subrecursive function classes. The rules under study correspond, in a canonical way, to natural closure operators on the classes of provably recursive functions. Thus, ∑1 induction rule precisely corresponds to the primitive recursive closure operator, and ∑1 collection rule, introduced below, corresponds to the elementary closure operator. (shrink)

We investigate the theory IΔ 0 + Ω 1 and strengthen [Bu86. Theorem 8.6] to the following: if NP ≠ co-NP. then Σ-completeness for witness comparison formulas is not provable in bounded arithmetic. i.e. $I\delta_0 + \Omega_1 + \nvdash \forall b \forall c (\exists a(\operatorname{Prf}(a.c) \wedge \forall = \leq a \neg \operatorname{Prf} (z.b))\\ \rightarrow \operatorname{Prov} (\ulcorner \exists a(\operatorname{Prf}(a. \bar{c}) \wedge \forall z \leq a \neg \operatorname{Prf}(z.\bar{b})) \urcorner)).$ Next we study a "small reflection principle" in bounded arithmetic. We prove (...) that for all sentences φ $I\Delta_0 + \Omega_1 \vdash \forall x \operatorname{Prov}(\ulcorner \forall y \leq \bar{x} (\operatorname{Prf} (y. \overline{\ulcorner \varphi \urcorner}) \rightarrow \varphi)\urcorner).$ The proof hinges on the use of definable cuts and partial satisfaction predicates akin to those introduced by Pudlak in [Pu86]. Finally, we give some applications of the small reflection principle, showing that the principle can sometimes be invoked in order to circumvent the use of provable Σ-completeness for witness comparison formulas. (shrink)

The current paper studies the formal properties of the Global Reflection Principle, to wit the assertion “All theorems of$\mathrm {Th}$are true,” where$\mathrm {Th}$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 (...) ($\mathrm {CT}_0$). Furthermore, we extend the above result showing that$\Sigma _1$-uniform reflection over a theory of uniform Tarski biconditionals ($\mathrm {UTB}^-$) is provable in$\mathrm {CT}_0$, 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$\mathrm {CT}_0$. 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$\mathrm {CT}_0$. (shrink)

The “puzzle of the unmarked clock” derives from a conflict between the following: (1) a plausible principle of epistemic modesty, and (2) “Rational Reflection”, a principle saying how one’s beliefs about what it is rational to believe constrain the rest of one’s beliefs. An independently motivated improvement to Rational Reflection preserves its spirit while resolving the conflict.

Moore introduced the Mapping Reflection Principle and proved that the Bounded Proper Forcing Axiom implies that the size of the continuum is ℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _2$$\end{document}. The Mapping Reflection Principle follows from the Proper Forcing Axiom. To show this, Moore utilized forcing notions whose conditions are countable objects. Chodounský–Zapletal introduced the Y-Proper Forcing Axiom that is a weak fragments of the Proper Forcing Axiom but implies some important conclusions from (...) the Proper Forcing Axiom, for example, the P-ideal Dichotomy. In this article, it is proved that the Y-Proper Forcing Axiom implies the Mapping Reflection Principle by introducing forcing notions whose conditions are finite objects. (shrink)

This article shows that there is a liar-like paradox that arises for rational credence that relies only on very weak logical and credal principles. The paradox depends on a weak rational reflection principle, logical principles governing conjunction, and principles governing the relationship between rational credence and proof. To respond to this paradox, we must either reject even very weak rational reflection principles or reject some highly plausible logical or credal principle.

In 5, Tait identifies a set of reflection principles called equation image-reflection principles which Peter Koellner has shown to be consistent relative to the existence of κ, the first ω-Erdős cardinal 1. Tait also defines a set of reflection principles called equation image-reflection principles; however, Koellner has shown that these are inconsistent when m > 2, but identifies restricted versions of them which he proves consistent relative to κ 2. In this paper, we introduce a new (...) large-cardinal property, the α-reflective cardinals. Their definition is motivated by Tait's remarks on parameters of third or higher order in reflection principles. We prove that if κ is ℶκ + α + 1-supercompact and 0 < α < κ then κ is α-reflective. Furthermore we show that α-reflective cardinals relativize to L, and that if κ exists then the set of cardinals λ < κ, such that λ is α-reflective for all α such that 0 < α < λ, is a stationary subset of κ. We show that an ω-reflective cardinal satisfies some restricted versions of equation image-reflection, as well as all the reflection properties proved consistent in 2. (shrink)

One recent topic of debate in Bayesian epistemology has been the question of whether imprecise credences can be rational. I argue that one account of imprecise credences, the orthodox treatment as defended by James M. Joyce, is untenable. Despite Joyce’s claims to the contrary, a puzzle introduced by Roger White shows that the orthodox account, when paired with Bas C. van Fraassen’s Reflection Principle, can lead to inconsistent beliefs. Proponents of imprecise credences, then, must either provide a compelling (...) reason to reject Reflection or admit that the rational credences in White’s case are precise. (shrink)

Gödel once offered an argument for the general reflection principle in set theory that took the form of an analogy with Leibniz' Monadology. I discuss the mathematical and philosophical background to Gödel's argument, reconstruct the proposed analogy in detail, and argue that it has no justificatory force.

We will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of their Hilbert-style deductive (...) proofs. Our axiom systems will not be strong enough to recognize their Canonical Reflection principle, but they will be capable of recognizing an approximation of it, called the "Tangibility Reflection Principle". We will also prove some new versions of the Second Incompleteness Theorem stating essentially that it is not possible to extend our exceptions to the Incompleteness Theorem much further. (shrink)

Are there plausible synchronic constraints on how a subject thinks of herself extended over time? At first glance, Bas van Fraassen’s principle of Reflection seems to prescribe the sort of epistemic authority one’s future self should be taken by one to have over one’s current epistemic states. (The gist of this principle is that I should now believe what I’m convinced I will believe tomorrow.) There has been a general consensus that, as a principle concerning epistemic (...) authority, Reflection does not apply to epistemically non-ideal agents. I agree with this, but argue here that it misses the point of Reflection. Rather than an epistemic principle concerning reasons for belief, Reflection concerns the semantics of belief avowal. I present a non-factual interpretation of Reflection, argue that the principle provides a constraint on the ways in which one can reflectively endorse one’s future epistemic self, and say something about the logic governing such an interpretation. (shrink)

For an arbitrary forcing class Γ, the Γ-fragment of Todorčević’s strong reflection principle SRPis isolated in such a way that the forcing axiom for Γ implies the Γ-fragment of SRP, the sta...

We give simple proofs of the Singular Cardinal Hypothesis from the Weak Reflection Principle and the Fodor-type Reflection Principle which do not use better scales.

This open access book provides the first explicit case study for an application of the method of reflective equilibrium (RE), using it to develop and defend a precautionary principle. It thereby makes an important and original contribution to questions of philosophical method and methodology. The book shows step-by-step how RE is applied, and develops a methodological framework which will be useful for everyone who wishes to use reflective equilibrium. With respect to precautionary principles, the book demonstrates how a rights-based (...) precautionary principle can be constructed and defended. The case study succeeds in demonstrating that RE can be successfully applied and puts real constraints on the justification process. This is all the more remarkable as the case study was designed as an open-ended process, without presupposing any specific results. This book will be highly relevant both to people interested in philosophical methodology and epistemology, as well as to researchers who are interested in using philosophical methods and tools and applying them to practical problems. (shrink)

We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant ‘true’ by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform reflection schemata in arithmetic, possibly of unrestricted logical complexity. We formulate an arithmetically complete calculus with modalities labeled by natural numbers and ω, where ω corresponds to the full uniform reflection schema, whereas n<ω corresponds to its restriction to arithmetical (...) Πn+1-formulas. This calculus is shown to be complete w.r.t. a suitable class of finite Kripke models and to be decidable in polynomial time. (shrink)

. The term inversion principle goes back to Lorenzen who coined it in the early 1950s. It was later used by Prawitz and others to describe the symmetric relationship between introduction and elimination inferences in natural deduction, sometimes also called harmony. In dealing with the invertibility of rules of an arbitrary atomic production system, Lorenzen’s inversion principle has a much wider range than Prawitz’s adaptation to natural deduction. It is closely related to definitional reflection, which is a (...) principle for reasoning on the basis of rule-based atomic definitions, proposed by Hallnäs and Schroeder-Heister. After presenting definitional reflection and the inversion principle, it is shown that the inversion principle can be formally derived from definitional reflection, when the latter is viewed as a principle to establish admissibility. Furthermore, the relationship between definitional reflection and the inversion principle is investigated on the background of a universalization principle, called the ω- principle, which allows one to pass from the set of all defined substitution instances of a sequent to the sequent itself. (shrink)

The aim of this essay is to reflect on the Millian, utilitarian argument from truth that is held as one of the most conspicuous answers to the question Why tolerate? This argument postulates that only in a free market of ideas may the truth be discovered. Even the most unpopular idea may contain some truth in it and may contribute to the advancement of knowledge. It further commands us to contest those opinions which are believed to be true vigorously and (...) earnestly to explore some further truth, and to acknowledge their false aspects. The results of free exchange of opinions will either confirm our idea, or bring some doubt to our attention, to cast accounts and maybe change our opinion, at least to some extent. After reflecting on some of the common arguments that are frequently mentioned in this connection, I discuss Mill's formulation of the Truth Principle as it came into expression in his writings. Here I reflect on his article "Law of Libel and Liberty of the Press" which did not receive adequate attention by scholars. I show that it is difficult to reconcile what Mill had to say in this article with what he later said in On Liberty. I further maintain that the Millian view of truth as superior to all other social values might endanger the very ground which the Truth Principle is intended to safeguard, i.e., tolerance. The holding of truth as the most important value might result in harming individuals, and in generating an atmosphere of intolerance. (shrink)

This book offers a comprehensive account of logic that addresses fundamental issues concerning the nature and foundations of the discipline. The authors claim that these foundations can not only be established without the need for strong metaphysical assumptions, but also without hypostasizing logical forms as specific entities. They present a systematic argument that the primary subject matter of logic is our linguistic interaction rather than our private reasoning and it is thus misleading to see logic as revealing "the laws of (...) thought". In this sense, fundamental logical laws are implicit to our "language games" and are thus more similar to social norms than to the laws of nature. Peregrin and Svoboda also show that logical theories, despite the fact that they rely on rules implicit to our actual linguistic practice, firm up these rules and make them explicit. By carefully scrutinizing the project of logical analysis, the authors demonstrate that logical rules can be best seen as products of the so called reflective equilibrium. They suggest that we can profit from viewing languages as "inferential landscapes" and logicians as "geographers" who map them and try to pave safe routes through them. This book is an essential resource for scholars and researchers engaged with the foundations of logical theories and the philosophy of language. (shrink)

We establish the thesis that in moral education, particularly in the first years of the child’s development, unreflexive acts or unreflexiveness in certain behaviours of adults is a condition for the development of the personality structure and virtues that enable autonomous ethical reflection and a relation to the Other. With the notion of unreflexiveness we refer to resolvedness in the response of adults when it is necessary to establish a limit, or cut, in the child’s demand for pleasure, as (...) well as to resolvedness as one of the structurally essential elements in behaviours with which the adult subordinates the child to the symbolic order or language. In philosophy, a symptomatic position in this context is represented by Kant’s theory of education. On the basis of the “traditional” concept of “negative education”, and through Freud’s and Lacan’s psychoanalytic concepts, we identify two principles that should, in contemporary times, be an essential part of the moral educational behaviours of adults: the principle of a delimited response to the child’s demand for the satisfaction of pleasure, and the principle of reasoned, reflected, but nevertheless certain, persevering, resolved unreflexiveness in the subordination of the child’s desire to the symbolic order . On the basis of the preceding analysis, we highlight certain consequences of this structure of the subject for the contexts of particular theoretical discussions and for school practice. (shrink)

After discussing the limitations inherent to all set-theoretic reflection principles akin to those studied by A. Lévy et. al. in the 1960s, we introduce new principles of reflection based on the general notion of Structural Reflection and argue that they are in strong agreement with the conception of reflection implicit in Cantor’s original idea of the unknowability of the Absolute, which was subsequently developed in the works of Ackermann, Lévy, Gödel, Reinhardt, and others. We then present (...) a comprehensive survey of results showing that different forms of the new principle of Structural Reflection are equivalent to well-known large cardinal axioms covering all regions of the large-cardinal hierarchy, thereby justifying the naturalness of the latter. (shrink)