We define a class of functions, the descent recursive functions, relative to an arbitrary elementary recursive system of ordinal notations. By means of these functions, we provide a general technique for measuring the proof-theoretic strength of a variety of systems of first-order arithmetic. We characterize the provable well-orderings and provably recursive functions of these systems, and derive various conservation and equiconsistency results.

In a modal system of arithmetic, a theory S has the modal disjunction property if whenever $S \vdash \square\varphi \vee \square\psi$ , either $S \vdash \square\varphi$ or $S \vdash \square\psi. S$ has the modal numerical existence property if whenever $S \vdash \exists x\square\varphi(x)$ , there is some natural number n such that $S \vdash \square\varphi(\mathbf{n})$ . Under certain broadly applicable assumptions, these two properties are equivalent.

In a language for arithmetic with a predicate T, intended to mean “ x is the Gödel number of a true sentence”, a set S of axioms and rules of inference has the truth disjunction property if whenever S ⊢ T ∨ T, either S ⊢ T or S ⊢ T. Similarly, S has the truth existence property if whenever S ⊢ ∃χ T ), there is some n such that S ⊢ T ). Continuing previous work, we establish whether (...) these properties hold or fail for a large collection of possible axiomatic systems. (shrink)

This paper compares the roles classical and intuitionistic logic play in restricting the free use of truth principles in arithmetic. We consider fifteen of the most commonly used axiomatic principles of truth and classify every subset of them as either consistent or inconsistent over a weak purely intuitionistic theory of truth.

In this investigation we explore a general strategy for constructing modal theories where the modal notion is conceived as a predicate. The idea of this strategy is to develop modal theories over axiomatic theories of truth. In this first paper of our two part investigation we develop the general strategy and then apply it to the axiomatic theory of truth Friedman-Sheard. We thereby obtain the theory Modal Friedman-Sheard. The theory Modal Friedman-Sheard is then discussed (...) from three different perspectives. First, we show that Modal Friedman-Sheard preserves theoremhood modulo translation with respect to modal operator logic. Second, we turn to semantic aspects and develop a modal semantics for the newly developed theory. Third, we investigate whether the modal predicate of Modal Friedman-Sheard can be understood along the lines of a proposal of Kripke, namely as a truth predicate modified by a modal operator. (shrink)

Following the finitist’s rejection of the complete totality of the natural numbers, a finitist language allows only propositional connectives and bounded quantifiers in the formula-construction but not unbounded quantifiers. This is opposed to the currently standard framework, a first-order language. We conduct axiomatic studies on the notion of truth in the framework of finitist arithmetic in which at least smash function $\#$ is available. We propose finitist variants of Tarski ramified truth theories up to rank $\omega $, of Kripke–Feferman truth (...) theory and of Friedman–Sheard truth theory, and show that all of these have the same strength as the finitist arithmetic of one higher level along Grzegorczyk hierarchy. On the other hand, we also show that adding Burgess-style groundedness schema, adjusted to the finitist setting, makes Kripke–Feferman truth theory as strong as primitive recursive arithmetic. Meanwhile, we obtain some basic results on finitist theories of (full and hat) inductive definitions and on the second order axiom of hat inductive definitions for positive operators. (shrink)

Let ${\cal T}$ be any of the three canonical truth theories CT^− (compositional truth without extra induction), FS^− (Friedman–Sheard truth without extra induction), or KF^− (Kripke–Feferman truth without extra induction), where the base theory of ${\cal T}$ is PA. We establish the following theorem, which implies that ${\cal T}$ has no more than polynomial speed-up over PA. Theorem.${\cal T}$is feasibly reducible to PA, in the sense that there is a polynomial time computable function f such that for every (...) ${\cal T}$-proof π of an arithmetical sentence ϕ, f(π) is a PA-proof of ϕ. (shrink)

In this volume, the author investigates and argues for, a particular answer to the question: What is the right way to logically analyze modalities from natural language within formal languages? The answer is: by formalizing modal expressions in terms of predicates. But, as in the case of truth, the most intuitive modal principles lead to paradox once the modal notions are conceived as predicates. -/- The book discusses the philosophical interpretation of these modal paradoxes and argues that any satisfactory approach (...) to modality will have to face the paradoxes independently of the grammatical category of the modal notion. By systematizing modal principles with respect to their joint consistency and inconsistency, Stern provides an overview of the options and limitations of the predicate approach to modality that may serve as a useful starting point for future work on predicate approaches to modality. Stern also develops a general strategy for constructing philosophically attractive theories of modal notions conceived as predicates. The idea is to characterize the modal predicate by appeal to its interaction with the truth predicate. This strategy is put to use by developing the modal theories Modal Friedman-Sheard and Modal Kripke-Feferman. (shrink)

In this second and last paper of the two part investigation on "Modality and Axiomatic Theories of Truth" we apply a general strategy for constructing modal theories over axiomatic theories of truth to the theory Kripke-Feferman. This general strategy was developed in the first part of our investigation. Applying the strategy to Kripke-Feferman leads to the theory Modal Kripke-Feferman which we discuss from the three perspectives that we had already considered in the first paper, where we discussed the theory Modal (...) Friedman-Sheard. That is, we first show that Modal Kripke-Feferman preserves theoremhood modulo translation with respect to modal operator logic. Second, we develop a modal semantics fitting the newly developed theory. Third, we investigate whether the modal predicate of Modal Kripke-Feferman can be understood along the lines of a proposal of Kripke, namely as a truth predicate modified by a modal operator. (shrink)

As is mentioned in Leigh :845-865, 2015), it is an open problem whether for several axiomatic theories of truth, including Friedman–Sheard theory \ and Kripke–Feferman theory \ :690-716, 1976), there exist cut-elimination arguments that give the upper bounds of their proof-theoretic strengths. In this paper, we give complete cut-elimination results for several well-known axiomatic theories of truth. In particular, we treat the systems \, and \ \\) of Friedman and Sheard’s theories and \.

En este artículo, tengo dos objetivos distintos. En primer lugar, mostrar que no es una buena idea tener una teoría de la verdad que, aunque consistente, sea omega-inconsistente. Para discutir este punto, considero un caso particular: la teoría de Friedman-Sheard FS. Argumento que en los lenguajes de primer orden omega inconsistencia implica que la teoría de la verdad no tiene modelo estándar. Esto es, no hay un modelo cuyo dominio sea el conjunto de los números naturales en el (...) cual esta teoría de la verdad pueda tener una interpretación consistente. En ese sentido, la introducción del predicado veritativo no mantiene la ontología estándar. Además, cuando se considera un lenguaje de orden superior, la situación es aun peor. En teorías de segundo orden con semántica estándar, la misma introducción produce una teoría que no tiene modelo. Por eso, si la omega-inconsistencia es un mal síntoma, la insatisfacibilidad de una teoría es aun peor. En segundo lugar, propongo abandonar el principio de unión de teorías FSn y aceptar una extensibilidad indefinida de teorías FS0, FS1, FS2, FS3,.... De acuerdo a mi punto de vista, la secuencia de teorías tiene las mismas virtudes que FS sin sus desagradables consecuencias. In this paper, I have two different purposes. Firstly, I want to show that it's not a good idea to have a theory of truth that is consistent but omega-inconsistent. In order to bring out this point, it is useful to consider a particular case: FS. I argue that in First-order languages omega-inconsistency implies that a theory of truth has not standard model. Then, there is no model whose domain is the set of natural numbers in which this theory of truth could acquire a consistent interpretation. So, in theories of truth without standard models, the introduction of the truth-predicate to a first order theory does not maintain the standard ontology. I add that in Higher-order languages the situation is even worst. In second order theories with standard semantic the same introduction produces a theory that doesn't have a model. So, if an omega-inconsistent theory of truth is bad, an unsatisfiable theory is really bad. Secondly, I propose to give up the union principle of theories FSn and accept an indefinite extensibility of theories FS0, FS1, FS2, FS3,... According to my view, the sequence of theories has the same virtues of FS without its disgusting consequences. (shrink)

The main aim of our paper was to present three formal tools for comparing various axiomatic theories of truth. In Section 2 we aimed at showing that there are indeed many different approaches to defining a set of axioms for the notion of truth. In Section 3 we introduced three different \measures of strength" of axiomatic theories of truth, i.e. three reflexive and transitive relations on the set of axiomatic theories of truth. We have explained the intuition behind each of (...) them. The three relations were called : Fujimoto definability, model-theoretical strength, proof-theoretical strength. Then in the last section we described how they order the truth theories introduced in Section 2. We observed that theories made equivalent by the coarser relation can be strictly ordered by the next one. (shrink)

We develop an untyped framework for the multiverse of set theory. $\mathsf {ZF}$ is extended with semantically motivated axioms utilizing the new symbols $\mathsf {Uni}(\mathcal {U})$ and $\mathsf {Mod}(\mathcal {U, \sigma })$, expressing that $\mathcal {U}$ is a universe and that $\sigma $ is true in the universe $\mathcal {U}$, respectively. Here $\sigma $ ranges over the augmented language, leading to liar-style phenomena that are analyzed. The framework is both compatible with a broad range of multiverse conceptions and suggests its (...) own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle that the background theory does not hold a privileged position over the theories of its internal universes. Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. We apply this to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory $\mathsf {ZF}$, and thus not seriously limiting to the diversity of the set-theoretic multiverse. We end with case studies applying the framework to two multiverse conceptions of set theory: arithmetic absoluteness and Joel D. Hamkins’ multiverse theory. (shrink)

What is the property of being true like? To answer this question, begin with a Canberra-plan analysis of the concept of truth. That is, assemble the platitudes for the concept of truth, and then investigate which property might satisfy them. This project is aided by Friedman and Sheard’s groundbreaking analysis of twelve logical platitudes for truth. It turns out that, because of the paradoxes like the liar, the platitudes for the concept of truth are inconsistent. Moreover, there are (...) so many distinct paradoxes that only small subsets of platitudes for truth are consistent. The result is that there is no property of being true. The failure of the Canberra plan analysis of the concept of truth, points the way toward a new methodology: a conceptual engineering project for the concept of truth. Conceptual engineering is assessing the quality of our concepts, and when they are found defective, offering new and better concepts to replace them for certain purposes. Still, there are many aletheic properties, which are properties satisfied by reasonably large subsets of platitudes for the concept of truth. We can treat these aletheic properties as a guide to the multitude of new aletheic concepts, which are concepts similar to, but distinct from, the concept of truth. Any new aletheic concept or team of concepts might be called on to replace the concept of truth. In particular, the concepts of ascending truth and descending truth are recommended, but the most important point is that we need a full-scale investigation into the space of aletheic properties and new aletheic concepts—that is, we need an Aletheic Principles Project. (shrink)

The first attempt at a systematic approach to axiomatic theories of truth was undertaken by Friedman and Sheard (Ann Pure Appl Log 33:1–21, 1987). There twelve principles consisting of axioms, axiom schemata and rules of inference, each embodying a reasonable property of truth were isolated for study. Working with a base theory of truth conservative over PA, Friedman and Sheard raised the following questions. Which subsets of the Optional Axioms are consistent over the base theory? What (...) are the proof-theoretic strengths of the consistent theories? The first question was answered completely by Friedman and Sheard; all subsets of the Optional Axioms were classified as either consistent or inconsistent giving rise to nine maximal consistent theories of truth.They also determined the proof-theoretic strength of two subsets of the Optional Axioms. The aim of this paper is to continue the work begun by Friedman and Sheard. We will establish the proof-theoretic strength of all the remaining seven theories and relate their arithmetic part to well-known theories ranging from PA to the theory of ${\Sigma^1_1}$ dependent choice. (shrink)

To the axioms of Peano arithmetic formulated in a language with an additional unary predicate symbol T we add the rules of necessitation and conecessitation T and axioms stating that T commutes with the logical connectives and quantifiers. By a result of McGee this theory is -inconsistent, but it can be approximated by models obtained by a kind of rule-of-revision semantics. Furthermore we prove that FS is equivalent to a system already studied by Friedman and Sheard and give (...) an analysis of its proof theory. (shrink)

A diverse group of scholars reflect on the scholarship of Robert M. Veatch, the breadth of which is unmatched in modern day bioethics. Essays were written by both philosophers and clinician-philosophers, by contemporaries and mentees. They span the breadth of Bob’s work and include analyses of his ideas about death, dying and organ transplantation, human experimentation and research ethics, disability, equality and justice, the doctor-patient relationship, the history of bioethics, as well as his pedagogical approach to teaching bioethics to clinicians (...) across the health care spectrum. Recognition of Bob’s influence in the modern field of bioethics and the challenges that persist are clearly identified. (shrink)

The advanced donation program was proposed in 2014 to allow an individual to donate a kidney in order to provide a voucher for a kidney in the future for a particular loved one. In this article, we explore the logistical and ethical issues that such a program raises. We argue that such a program is ethical in principle but there are many logistical issues that need to be addressed to ensure that the actual program is fair to both those who (...) do and do not participate in this program. (shrink)

The title of this Festschrift to Stephen Brown points to the understanding of medieval philosophy and theology in the longue durée of their traditions and discourses.

Hayek developed two contradictory epistemologies. The epistemology for which he is famous attributed dispersed knowledge to economic actors and credited the price system for aggregating and communicating this knowledge. The other epistemology attributed to human and non-human organisms alike the error-prone interpretation of stimuli, which could never truly be said to be “knowledge.” Several of the paradoxes of Hayek's economic and political thought that are explored in this symposium can be explained by the triumph of the first epistemology over the (...) second, including his historical interpretation of socialism as a planning mentality; his tendentious definitions of “liberty” and “justice”; and his opposition to economic redistribution even as he endorsed all manner of economic and social regulations. (shrink)

Using almost disjoint coding we prove the consistency of the existence of a definable ω-mad family of infinite subsets of ω together with.

We explore the possibilities for elementary embeddings $j : M \to N$, where M and N are models of ZFC with the same ordinals, $M \subseteq N$, and N has access to large pieces of j. We construct commuting systems of such maps between countable transitive models that are isomorphic to various canonical linear and partial orders, including the real line ${\mathbb R}$.

The "best interest of the child" standard is central to both pediatric bioethics and the child rights community. In pediatric bioethics in the United States, the best interest of the child standard is cited as the guidance principle for parental decision-making.1 Likewise, in the child rights community, the best interest of the child standard is "of paramount consideration" ). Both approaches also recognize parental rights and responsibilities and support a role for the maturing child in the decision-making process. Why, then, (...) do members of the pediatric bioethics community and the child rights communities not speak the same language?This article was developed out of... (shrink)

In thinking and talking about rights, including property rights, it seems natural to put the argument in either moral or legal terms. From the former viewpoint, rights are part of a description of what actions are right or wrong. The fact that I have a right to do something is an argument, although not necessarily a sufficient argument, that someone who prevents me from doing it is acting wrongly.

We completely characterize the simple majority weighted voting game achievable hierarchies, and, in doing so, show that a problem about representative government, noted by J. Banzhaf [Rutgers Law Review 58, 317–343 (1965)] cannot be resolved using the simple majority quota. We also demonstrate that all hierarchies achievable by any quota can be achieved if the simple majority quota is simply incremented by one.

If the bounded proper forcing axiom BPFA holds and ω 1 = ${\mathrm{\omega }}_{1}^{\mathrm{L}}$ , then there is a lightface ${\mathrm{\Sigma }}_{3}^{1}$ well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of "David's trick." We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface ${\mathrm{\Sigma }}_{4}^{1}$ , for many "consistently locally certified" relations R on $\mathrm{\mathbb{R}}$ . (...) This is accomplished through a use of David's trick and a coding through the ∑ 2 stable ordinals of L. (shrink)

We explore Woodin's Universality Theorem and consider to what extent large cardinal properties are transferred into HOD (and other inner models). We also separate the concepts of supercompactness, supercompactness in HOD and being HOD-supercompact. For example, we produce a model where a proper class of supercompact cardinals are not HOD-supercompact but are supercompact in HOD. Additionally we introduce a way to measure the degree of HOD-supercompactness of a supercompact cardinal, and we develop methods to control these degrees simultaneously for a (...) proper class of supercompact cardinals. Finally, we also produce a model in which the unique supercompact cardinal is also the only strongly compact cardinal, no cardinal is supercompact up to an inaccessible cardinal, level by level inequivalence holds and the unique supercompact cardinal is not HOD-supercompact. (shrink)

The consensus recommendations by Salter and colleagues (2023) regarding pediatric decision-making intentionally omitted adolescents due to the additional complexity their evolving autonomy presented. Using two case studies, one focused on truth-telling and disclosure and one focused on treatment refusal, this article examines medical decision-making with and for adolescents in the context of the six consensus recommendations. It concludes that the consensus recommendations could reasonably apply to older children.

We isolate a condition on a class A of ordinals sufficient to Δ1-code it by a real in a class-generic extension of L. We then apply this condition to show that the class of ordinals of L-cofinality ω is Δ1 in a real of L-degree strictly below O#.