This paper introduces a new kind of fixed-point semantics, filling a gap within approaches to Liar-like paradoxes involving fixed-point models à la Kripke (1975). The four-valued models presented below, (i) unlike the three-valued, consistent fixed-point models defined in Kripke (1975), are able to differentiate between paradoxical and pathological-but-unparadoxical sentences, and (ii) unlike the four-valued, paraconsistent fixed-point models first studied in Visser (1984) and Woodruff (1984), preserve consistency and groundedness of truth.

Michael Kremer defines fixed-point logics of truth based on Saul Kripke’s fixed point semantics for languages expressing their own truth concepts. Kremer axiomatizes the strong Kleene fixed-point logic of truth and the weak Kleene fixed-point logic of truth, but leaves the axiomatizability question open for the supervaluation fixed-point logic of truth and its variants. We show that the principal supervaluation fixed point logic of truth, when thought of as consequence relation, is highly complex: it is not even analytic. We also (...) consider variants, engendered by a stronger notion of ‘fixed point’, and by variant supervaluation schemes. A ‘logic’ is often thought of, not as a consequence relation, but as a set of sentences – the sentences true on each interpretation. We axiomatize the supervaluation fixed-point logics so conceived. (shrink)

In the context of his theory of numberings, Ershov showed that Kleene's recursion theorem holds for any precomplete numbering. We discuss various generalizations of this result. Among other things, we show that Arslanov's completeness criterion also holds for every precomplete numbering, and we discuss the relation with Visser's ADN theorem, as well as the uniformity or nonuniformity of the various fixed point theorems. Finally, we base numberings on partial combinatory algebras and prove a generalization of Ershov's theorem in this context.

The problem of Uniqueness and Explicit Definability of Fixed Points for Interpretability Logic is considered. It turns out that Uniqueness is an immediate corollary of a theorem of Smoryski.

This paper considers standard completeness for fixed-pointed involutive micanorm-based logics. For this, we first discuss fixed-pointed involutive micanorm-based logics together with their algebraic semantics. Next, after introducing some examples of fixed-pointed involutive micanorms, we provide standard completeness results for those logics.

In this paper it is shown that the intuitionistic fixed point theory equation image for α times iterated fixed points of strictly positive operator forms is conservative for negative arithmetic and equation image sentences over the theory equation image for α times iterated arithmetic comprehension without set parameters. This generalizes results previously due to Buchholz [5] and Arai [2].

In this paper, we study when a set-continuous operator has a fixed-point that is the intersection of a directed family. The framework of our study is the Kelley-Morse theory KMC– and the Gödel-Bernays theory GBC–, both theories including an Axiom of Choice and excluding the Axiom of Foundation. On the one hand, we prove a result concerning monotone operators in KMC– that cannot be proved in GBC–. On the other hand, we study conditions on directed superclasses in GBC– in order (...) that their intersection is a fixed-point of a set-continuous operator. Finally, we illustrate our results with a solution to the liar paradox and a construction of maximal bisimulations. (shrink)

In response to the liar’s paradox, Kripke developed the fixed-point semantics for languages expressing their own truth concepts. Kripke’s work suggests a number of related fixed-point theories of truth for such languages. Gupta and Belnap develop their revision theory of truth in contrast to the fixed-point theories. The current paper considers three natural ways to compare the various resulting theories of truth, and establishes the resulting relationships among these theories. The point is to get a sense of the lay of (...) the land amid a variety of options. Our results will also provide technical fodder for the methodological remarks of the companion paper to this one. (shrink)

The problem of Uniqueness and Explicit Definability of Fixed Points for Interpretability Logic is considered. It turns out that Uniqueness is an immediate corollary of a theorem of Smoryński.

In Aczel [1], the existence of largest fixed points of set continuous operators is proved assuming the schema version of dependent choices in Zermelo-Fraenkel set theory without the axiom of Foundation. In the present paper, we study whether the existence of largest fixed points of set continuous operators is provable without the schema version of dependent choices, using Boffa's weak antifoundation axioms.

When we are faced with a choice among acts, but are uncertain about the true state of the world, we may be uncertain about the acts’ “choiceworthiness”. Decision theories guide our choice by making normative claims about how we should respond to this uncertainty. If we are unsure which decision theory is correct, however, we may remain unsure of what we ought to do. Given this decision-theoretic uncertainty, meta-theories attempt to resolve the conflicts between our decision theories...but we may be (...) unsure which meta-theory is correct as well. This reasoning can launch a regress of ever-higher-order uncertainty, which may leave one forever uncertain about what one ought to do. There is, fortunately, a class of circumstances under which this regress is not a problem. If one holds a cardinal understanding of subjective choiceworthiness, and accepts certain other criteria, one’s hierarchy of metanormative uncertainty ultimately converges to precise definitions of “subjective choiceworthiness” for any finite set of acts. If one allows the metanormative regress to extend to the transfinite ordinals, the convergence criteria can be weakened further. Finally, the structure of these results applies straightforwardly not just to decision-theoretic uncertainty, but also to other varieties of normative uncertainty, such as moral uncertainty. (shrink)

We prove an analogue of the fixed-point theorem for the case of definably amenable groups.

In this paper I argue that it’s impossible for there to be a single universal theory of meaning for a language. First, I will consider some minimal expressiveness requirements a language must meet to be able to express semantic claims. Then I will argue that in order to have a single unified theory of meaning, these expressiveness requirements must be satisfied by a language which the semantic theory itself applies to. That is, we would need a language which can express (...) its own meaning. It has been well-known since Tarski that theories of meaning whose central notion is truth can’t be expressed in a language which they apply to. Here, I develop Quine’s formulation of the Liar Paradox in grammatical terms and use this to extend Tarski’s result to all theories of meaning. This general version of the paradox can be formalised as a special case of the Lawvere Fixed-Point Theorem applied to a categorial grammar. Taken together with the initial arguments, I infer that a universal theory of meaning is impossible and conclude the paper with a brief discussion on what alternatives are available. (shrink)

Jäger, G., Fixed points in Peano arithmetic with ordinals, Annals of Pure and Applied Logic 60 119-132. This paper deals with some proof-theoretic aspects of fixed point theories over Peano arithmetic with ordinals. It studies three such theories which differ in the principles which are available for induction on the natural numbers and ordinals. The main result states that there is a natural theory in this framework which is a conservative extension of Peano arithmeti.

The relation between least and diagonal fixed points is a well known and completely studied question for a large class of partially ordered models of the lambda calculus and combinatory logic. Here we consider this question in the context of algebraic recursion theory, whose close connection with combinatory logic recently become apparent. We find a comparatively simple and rather weak general condition which suffices to prove the equality of least fixed points with canonical (corresponding to those produced by the Curry (...) combinator in lambda calculus) diagonal fixed points in a class of partially ordered algebras which covers both combinatory spaces of Skordev and operative spaces of Ivanov. Especially, this yields an essential improvement of the axiomatization of recursion theory via combinatory spaces. (shrink)

This manuscript investigates fixed point of single-valued Hardy-Roger’s type F -contraction globally as well as locally in a convex b -metric space. The paper, using generalized Mann’s iteration, iterates fixed point of the abovementioned contraction; however, the third axiom of the F -contraction is removed, and thus the mapping F is relaxed. An important approach used in the article is, though a subset closed ball of a complete convex b -metric space is not necessarily complete, the convergence of the Cauchy (...) sequence is confirmed in the subset closed ball. The results further lead us to some important corollaries, and examples are produced in support of our main theorems. The paper most importantly presents application of our results in finding solution to the integral equations. (shrink)

One of the problems in economics that economists have devoted a considerable amount of attention in prevalent years has been to ensure consistency in the models they employ. Assuming markets to be generally in some state of equilibrium, it is asked under what circumstances such equilibrium is possible. The fundamental mathematical tools used to address this concern are fixed point theorems: the conditions under which sets of assumptions have a solution. This book gives the reader access to the mathematical techniques (...) involved and goes on to apply fixed point theorems to proving the existence of equilibria for economics and for co-operative and noncooperative games. Special emphasis is given to economics and games in cases where the preferences of agents may not be transitive. The author presents topical proofs of old results in order to further clarify the results. He also proposes fresh results, notably in the last chapter, that refer to the core of a game without transitivity. This book will be useful as a text or reference work for mathematical economists and graduate and advanced undergraduate students. (shrink)

This paper deals with the modal logics associated with (possibly nonstandard) provability predicates of Peano Arithmetic. One of our goals is to present some modal systems having the fixed point property and not extending the Gödel-Löb system GL. We prove that, for every has the explicit fixed point property. Our main result states that every complete modal logic L having the Craig's interpolation property and such that , where and are suitable modal formulas, has the explicit fixed point property.

In this paper we show that the intuitionistic fixed point theory FiXi over set theories T is a conservative extension of T if T can manipulate finite sequences and has the full foundation schema.

We study the fixed point property and the Craig interpolation property for sublogics of the interpretability logic \(\textbf{IL}\). We provide a complete description of these sublogics concerning the uniqueness of fixed points, the fixed point property and the Craig interpolation property.

We prove a definable analogue to Brouwer's Fixed Point Theorem for o-minimal structures of real closed field expansions: A continuous definable function mapping from the unit simplex into itself admits a fixed point, even though the underlying space is not necessarily topologically complete. Our proof is direct and elementary; it uses a triangulation technique for o-minimal functions, with an application of Sperner's Lemma.

This article constitutes the new fixed point results of dynamic process D through FIC-integral contractions of the Ciric kind and investigates the said contraction to iterate a fixed point of set-valued mappings in the module of metric space. To do so, we use the dynamic process instead of the conventional Picard sequence. The main results are examined by tangible nontrivial examples which display the motivation for such investigation. The work is completed by giving an application to Liouville‐Caputo fractional differential equations.

Recent years have seen a certain impatience with John Rawls's approach to political philosophy and calls for the discipline to move beyond it. One source of dissatisfaction is Rawls's idea of a well-ordered society. In a recent article, Alex Schaefer has tried to give further impetus to this movement away from Rawlsian theorizing by pursuing a question about well-ordered societies that he thinks other critics have not thought to ask. He poses that question in the title of his article: “Is (...) Justice a Fixed Point?.” Though Schaefer is critical of Rawlsian political theorizing, I shall contend that his arguments also suggest two paths forward for those who would follow the Rawlsian approach. First, the intellectual devices Schaefer deploys help those who would continue the Rawlsian project to see, and precisely to chart, the next step that that project needs to take—a step necessitated by a concession Rawls himself made late in the development of political liberalism. Second, the clarity and economy with which Schaefer lays out his alternative to the Rawlsian approach make it possible to state some fundamental Rawlsian challenges to a form of theorizing that has considerable appeal to many critics of that approach. (shrink)

Let T be Kripke–Platek set theory with infinity extended by the axiom (Beta) plus the schema that claims that every set-bounded Σ-definable monotone operator from the collection of all sets to Pow(a) for some set a has a fixed point. Then T proves that every such operator has a least fixed point. This result is obtained by following the proof of an analogous result for von Neumann–Bernays–Gödel set theory in an earlier work by Sato, with some minor modifications.

We study the system IFP of intuitionistic fixed point logic, an extension of intuitionistic first-order logic by strictly positive inductive and coinductive definitions. We define a realizability interpretation of IFP and use it to extract computational content from proofs about abstract structures specified by arbitrary classically true disjunction free formulas. The interpretation is shown to be sound with respect to a domain-theoretic denotational semantics and a corresponding lazy operational semantics of a functional language for extracted programs. We also show how (...) extracted programs can be translated into Haskell. As an application we extract a program converting the signed digit representation of real numbers to infinite Gray code from a proof of inclusion of the corresponding coinductive predicates. (shrink)

The aim of this research work is to find out some results in fixed point theory for a pair of families of multivalued mappings fulfilling a new type of U -contractions in modular-like metric spaces. Some new results in graph theory for multigraph-dominated contractions in modular-like metric spaces are developed. An application has been presented to ensure the uniqueness and existence of a solution of families of nonlinear integral equations.

We present a quite simple proof of the fixed point theorem for GL. We also use this proof to show that Sambin's algorithm yields a fixed point.

Ingram (2015) has argued that Cuneo and Shafer-Landau’s (2014) ‘moral fixed points’ theory entails that error theorists are conceptually deficient with moral concepts. They are conceptually deficient with moral concepts because they do not grasp moral fixed points (e.g. ‘Torture for fun is pro tanto wrong’). Ingram (2015) concluded that moral fixed points theory cannot substantiate the conceptual deficiency charge and, therefore, the theory is defeated. In defense of moral fixed points theory, Kyriacou (2017a) argued that the theory is coherent (...) with the error theorists being conceptually competent with moral concepts, while reflectively denying moral fixed points theory. This is because error theorists could be conceptually competent with moral concepts but, due to intellectual vices, be swayed by unsound arguments against moral fixed points theory. Ingram (2018) replied to Kyriacou (2017a) that the intellectual vice charge against error theorists cannot be substantiated and, therefore, the moral fixed points theory is still defeated. In this paper, I argue that the possibility of intellectual vice is independently plausible and sufficient to undermine Ingram’s argument against the moral fixed points theory. (shrink)

In this article we show how to use the result in Jäger and Probst [7] to adapt the technique of pseudo-hierarchies and its use in Avigad [1] to subsystems of set theory without foundation. We prove that the theory KPi0 of admissible sets without foundation, extended by the principle (Σ-FP), asserting the existence of fixed points of monotone Σ operators, has the same proof-theoretic ordinal as KPi0 extended by the principle (Σ-TR), that allows to iterate Σ operations along ordinals. By (...) Jäger and Probst [6] we conclude that the metapredicative Mahlo ordinal φ ω00 is also the ordinal of KPi0+(Σ-FP). Hence the relationship between fixed points and iteration persists in the framework of set theory without foundation. (shrink)

The article is about one of the vital problem for analytic philosophy which is how to define truth value for sentences which include their own truth predicate. The aim of the article is to determine Saul Kripke’s approach to widen epistemological truth to create a systemic model of truth. Despite a lot of work on the subject, the theme of truth is no less relevant to modern philosophy. With the help of S. Kripke’s article “Outline of the Theory of Truth” (...) and R. L. Kirkham’s work “Theories of Truth: A Critical Introduction” the author tried to represent a non-classical approach which is now known as Saul Kripke’s truth value gaps theory. Considering his solving of the Liar Paradox, as well as the Strenghened Liar, it is obvious that there could be the circumstances (or facts) that make sentences have no truth value. In that case the solving of the paradox attracts Kleene’s non-classical three-valued logic which contains the third way and the solution falls into the gap between truth and falsity. Hence there are no relevant and full satisfactory ways to solve the Liar’s Paradox by the language-levels ap- proach. The task consists in an extension of the approach to natural languages and elimination of an ad hoc element as well. Besides that, the language-levels approach refers to the speaker that means that only the last one is allowed to define the hierarchy of the levels and truth value of sentence, but often he does not know what level he is speaking of. As Kripke stated, contingent facts can lead to a paradox which isn’t allowed for a logical point of view. However usage of mathematical apparatus allows the theory in question to be universalized and moreover paves the way to futher epistemological develop- ment of the problem in question. As Kripke’s theory is formalized and presented as a functional series, it gives access to a system model of truth whose elements would be epistemological theories (concepts) of truth. Such a model, for instance, confirms the validity of the deflationary concept of truth. Refs 7. (shrink)

We clarify the respective roles fixed points, diagonalization, and self- reference play in proofs of Gödel’s first incompleteness theorem.

The most efficient known method for solving certain computational problems is to construct an iterated map whose fixed points are by design the problem's solution. Although the origins of this idea go back at least to Newton, the clearest expression of its logical basis is an example due to Mermin. A contemporary application in image recovery demonstrates the power of the method.

This is indeed a very nice draft that I have read with great pleasure, and that has helped me to better understand the completeness proof for LCC. Modal fixed point logic allows for an illuminating new version (and a further extension) of that proof. But still. My main comment is that I think the perspective on substitutions in the draft paper is flawed. The general drift of the paper is that relativization, (predicate) substitution and product update are general operations on (...) models, and that it is important to check whether given logical languages are closed under these operations. FO logic is closed under relativization, predicate substitution and product constructions (such as those involved in relative interpretation). The minimal modal logic is closed under relativization, which explains the reduction of epistemic logic (withhout common knowledge) + public announcement to epistemic logic simpliciter (as observed in Van Benthem, [2]). The reduction breaks down as soon as one adds common knowledge. The minimal modal logic is also closed under substitution, which explains the reduction of epistemic logic plus (publicly observable) factual change to epistemic logic simpliciter, via the following reduction axioms (I use !p := φ for the operation of publicly changing the truth value of p to φ): [!p := φ]p ↔ φ [!p := φ]q ↔ q (p and q syntactically different ) [!p := φ]¬ψ ↔ ¬[!p := φ]ψ [!p := φ](ψ1 ∧ ψ2) ↔ [!p := φ]ψ1 ∧ [!p := φ]ψ2 [!p := φ][i]ψ ↔ [i][!p := φ]ψ Unlike the case of relativisation, this can be extended to the case of epistemic logic with common knowledge, by means of: [!p := φ]CGψ ↔ CG[!p := φ]ψ We get the following.. (shrink)

We argue that probability mistakes indicate that at least some of us often do not adequately possess the concept of probability (and its cognates) and that the digital dissemination of such misinfo...

Enkrasia and the Fixed Point Thesis are equivalent.

We consider fixed point logics, i.e., extensions of first order predicate logic with operators defining fixed points. A number of such operators, generalizing inductive definitions, have been studied in the context of finite model theory, including nondeterministic and alternating operators. We review results established in finite model theory, and also consider the expressive power of the resulting logics on infinite structures. In particular, we establish the relationship between inflationary and nondeterministic fixed point logics and second order logic, and we consider (...) questions related to the determinacy of games associated with alternating fixed points. (shrink)

We study a system, μŁΠ, obtained by an expansion of ŁΠ logic with fixed points connectives. The first main result of the paper is that μŁΠ is standard complete, i.e., complete with regard to the unit interval of real numbers endowed with a suitable structure. We also prove that the class of algebras which forms algebraic semantics for this logic is generated, as a variety, by its linearly ordered members and that they are precisely the interval algebras of real closed (...) fields. This correspondence is extended to a categorical equivalence between the whole category of those algebras and another category naturally arising from real closed fields. Finally, we show that this logic enjoys implicative interpolation. (shrink)

We develop a variant of Least Fixed Point logic based on First Orderlogic with a relaxed version of guarded quantification. We develop aGame Theoretic Semantics of this logic, and find that under reasonableconditions, guarding quantification does not reduce the expressibilityof Least Fixed Point logic. But we also find that the guarded version ofa least fixed point algorithm may have a greater time complexity thanthe unguarded version, by a linear factor.

We give a characterization of the fixed points and of the lattices of fixed points of fuzzy Galois connections. It is shown that fixed points are naturally interpreted as concepts in the sense of traditional logic.

Our purpose is to present some connections between modal incompleteness andmodal logics related to the Gödel-Löb logic GL. One of our goals is to prove that for all m, n, k, l ∈ ℕ the logic K + equation image□i □jp ↔ p) → equation image□ip is incomplete and does not have the fixed point property. As a consequence we shall obtain that the Boolos logic KH does not have the fixed point property.

This article defends the Fixed Point Thesis: that it is always a rational mistake to have false beliefs about the requirements of rationality. The Fixed Point Thesis is inspired by logical omniscience requirements in formal epistemology. It argues to the Fixed Point Thesis from the Akratic Principle: that rationality forbids having an attitude while believing that attitude is rationally forbidden. It then draws out surprising consequences of the Fixed Point Thesis, for instance that certain kinds of a priori justification are (...) indefeasible and that misleading all-things-considered evidence about rational requirements is impossible. Finally, the Fixed Point Thesis is applied to defend the Right Reasons position on peer disagreement, according to which an agent who has drawn the correct conclusion from her evidence should retain belief in that conclusion even in the face of disagreeing peers. (shrink)

Minimal predicates P satisfying a given first-order description φ(P) occur widely in mathematical logic and computer science. We give an explicit first-order syntax for special first-order ‘PIA conditions’ φ(P) which guarantees unique existence of such minimal predicates. Our main technical result is a preservation theorem showing PIA-conditions to be expressively complete for all those first-order formulas that are preserved under a natural model-theoretic operation of ‘predicate intersection’. Next, we show how iterated predicate minimization on PIA-conditions yields a language MIN(FO) equal (...) in expressive power to LFP(FO), first-order logic closed under smallest fixed-points for monotone operations. As a concrete illustration of these notions, we show how our sort of predicate minimization extends the usual frame correspondence theory of modal logic, leading to a proper hierarchy of modal axioms: first-order-definable, first-order fixed-point definable, and beyond. (shrink)

In a recent paper by M. Rathjen and the present author it has been shown that the statement “every normal function has a derivative” is equivalent to $\Pi ^1_1$ -bar induction. The equivalence was proved over $\mathbf {ACA_0}$, for a suitable representation of normal functions in terms of dilators. In the present paper, we show that the statement “every normal function has at least one fixed point” is equivalent to $\Pi ^1_1$ -induction along the natural numbers.

The semantic paradoxes are associated with self-reference or referential circularity. However, there are infinitary versions of the paradoxes, such as Yablo's paradox, that do not involve this form of circularity. It remains an open question what relations of reference between collections of sentences afford the structure necessary for paradoxicality -- these are the so-called "dangerous" directed graphs. Building on Rabern, et. al (2013) we reformulate this problem in terms of fixed points of certain functions, thereby boiling it down to get (...) a purely mathematical problem. (shrink)

Cuneo and Shafer-Landau (2014) argued that there are moral conceptual truths that are substantive in content, what they called ‘moral fixed points’. I argue that insofar as we have some reason to postulate moral fixed points, we have equal reason to postulate epistemic fixed points (e.g. the factivity condition). To this effect, I show that the two basic reasons Cuneo and Shafer-Landau (2014) offer in support of moral fixed points naturally carry over to epistemic fixed points. In particular, epistemic fixed (...) points exhibit the four ‘marks’ of conceptual truths that they identify and can be utilized to address important challenges to epistemic realism. I conclude that insofar as we have some reason to postulate moral fixed points, we have equal reason to postulate epistemic fixed points. -/- . (shrink)