In “Some Remarks on Extending and Interpreting Theories with a Partial Truth Predicate”, Reinhardt [21] famously proposed an instrumentalist interpretation of the truth theory Kripke–Feferman ( $\mathrm {KF}$ ) in analogy to Hilbert’s program. Reinhardt suggested to view $\mathrm {KF}$ as a tool for generating “the significant part of $\mathrm {KF}$ ”, that is, as a tool for deriving sentences of the form $\mathrm{Tr}\ulcorner {\varphi }\urcorner $. The constitutive question of Reinhardt’s program was whether it was possible “to justify the (...) use of nonsignificant sentences entirely within the framework of significant sentences”. This question was answered negatively by Halbach & Horsten [10] but we argue that under a more careful interpretation the question may receive a positive answer. To this end, we propose to shift attention from $\mathrm {KF}$ -provably true sentences to $\mathrm {KF}$ -provably true inferences, that is, we shall identify the significant part of $\mathrm {KF}$ with the set of pairs $\langle {\Gamma, \Delta }\rangle $, such that $\mathrm {KF}$ proves that if all members of $\Gamma $ are true, at least one member of $\Delta $ is true. In way of addressing Reinhardt’s question we show that the provably true inferences of suitable $\mathrm {KF}$ -like theories coincide with the provable sequents of matching versions of the theory Partial Kripke–Feferman ( $\mathrm {PKF}$ ). (shrink)

The uniform reflection principle for the theory of uniform T-sentences is added to PA. The resulting system is justified on the basis of a disquotationalist theory of truth where the provability predicate is conceived as a special kind of analyticity. The system is equivalent to the system ACA of arithmetical comprehension. If the truth predicate is also allowed to occur in the sentences that are inserted in the T-sentences, yet not in the scope of negation, the system with the reflection (...) schema for these T-sentences assumes the strength of the Kripke-Feferman theory KF, and thus of ramified analysis up to go. (shrink)

The logic Kf of the modalities of finite, devised to capture the notion of 'there exists a finite number of accessible worlds such that . . . is true', was introduced and axiomatized by Fattorosi. In this paper we enrich the logical framework of Kf: we give consistency properties and a tableau system (which yields the decidability) explicitly designed for Kf, and we introduce a shorter and more natural axiomatization. Moreover, we show the strong and suggestive relationship between Kf and (...) the much older logic of the physical modalities of Burks. (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)

We investigate axiomatizations of Kripke's theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman's axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke's semantics in the straightforward sense: only the sentences that can be proved to be true in KF are valid in Kripke's partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in (...) KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt's Conjecture, and provide a direct axiomatization PKF of Kripke's theory in partial logic. We argue that any natural axiomatization of Kripke's theory in Strong Kleene logic has the same proof-theoretic strength as PKF, namely the strength of the system RA< ωω ramified analysis or a system of Tarskian ramified truth up to ωω. Thus any such axiomatization is much weaker than Feferman's axiomatization KF in classical logic, which is equivalent to the system RA<ε₀ of ramified analysis up to ε₀. (shrink)

Solutions to semantic paradoxes often involve restrictions of classical logic for semantic vocabulary. In the paper we investigate the costs of these restrictions in a model case. In particular, we fix two systems of truth capturing the same conception of truth: of the system KF of Feferman formulated in classical logic, and the system PKF of Halbach and Horsten, formulated in basic De Morgan logic. The classical system is known to be much stronger than the nonclassical one. We assess the (...) reasons for this asymmetry by showing that the truth theoretic principles of PKF cannot be blamed: PKF with induction restricted to non-semantic vocabulary coincides in fact with what the restricted version of KF proves true. (shrink)

What requirements must deflationary formal theories of truth satisfy? This chapter argues against the widely accepted view that compositional and Tarskian theories of truth are substantial or otherwise unacceptable to deflationists. First, two purposes that a formal truth theory can serve are distinguished: one descriptive, the other logical (i.e., to characterise the correctness of inferences involving ‘true’). The chapter argues that the most compelling arguments for the incompatibility of compositional and Tarskian theories concern descriptive theories only. -/- Second, two requirements (...) that any deflationist truth theory intended to serve a logical purpose must satisfy are put forward. These requirements, it is argued, suggest that (i) many well-known compositional and Tarskian theories are acceptable from a deflationist standpoint (including CT); (ii) certain other popular theories of truth (including KF and FS) are not similarly acceptable; (iii) there are no conclusive reasons to impose a conservativeness requirement on deflationary theories of truth. (shrink)

Questions concerning the proof-theoretic strength of classical versus nonclassical theories of truth have received some attention recently. A particularly convenient case study concerns classical and nonclassical axiomatizations of fixed-point semantics. It is known that nonclassical axiomatizations in four- or three-valued logics are substantially weaker than their classical counterparts. In this paper we consider the addition of a suitable conditional to First-Degree Entailment—a logic recently studied by Hannes Leitgeb under the label HYPE. We show in particular that, by formulating the theory (...) PKF over HYPE, one obtains a theory that is sound with respect to fixed-point models, while being proof-theoretically on a par with its classical counterpart KF. Moreover, we establish that also its schematic extension—in the sense of Feferman—is as strong as the schematic extension of KF, thus matching the strength of predicative analysis. (shrink)

Der Beschreibung der zeitlichen Entwicklung lebender Systeme kann eine reine Differentialanalyse nicht genügen. In solchen Fällen muss man sich an Stelle der gewöhnlichen Differentialgleichungen der integraldifferentiellen, bezw. der Integralgleichungen bedienen. Zur leichteren Veranschaulichung der mathematischen Darstellung betrachtet Verfasser zuerst diejenigen Systeme, deren innerer Zustand sich durch ein einziges Parameterc bestimmen lässt. Die zeitliche Entwicklung eines leblosen Systems dieser Klasse werde durch die Differentialgleichung $$\frac{{dc}}{{dt}} = kf...$$ dargestellt, wot=Zeit, undk eine Funktion der äusseren Parameterα, Β, γ. ist. Im Falle eines jeden (...) Systems sind die äusseren Parameterα, Β, γ. charakteristische individuelle Funktionen der Zeit. Misst manc am. Zeitpunktt, so lässt sich die Zukunft des Systems vermittelst der Gleichung berechnen, wenn die zukünftigen Werte der äusseren Parameter gegeben sind. Solche Systeme nennt Verfasser nicht-historische Systeme. Im Falle der lebenden Systeme werde die zeitliche Entwicklung durch die Integral-differentialgleichung $$\frac{{dc}}{{dt}} = k_1 f_1 + \mathop \smallint \limits_{t - \lambda }^t k_2 f_2 dt........$$ dargestellt, wof 1 undf 2 Funktionen des Zustandsparametersc, undk 1 undk 2 Funktionen der äusseren Parameter sind. Das Zeitintegral erstreckt sich auf das vergangene Zeitintervalt-λ →t, woλ einen positiven endlichen konstanten Wert besitzt. Diese Systeme nennt Verfasser historische Systeme. Setzt manH für das darin enthaltene Integralglied, so verwandelt sich die Gleichung in: $$\frac{{dc}}{{dt}} = k_1 f_1 + H..........$$ Misst man nunc am Zeitpunktt, so kann man in diesem Fall die Zukunft des Systems ohne Kenntnis des historischen GliedsH nicht berechnen. Das Vorkommen des historischen IntegralgliedsH nennt Verfasser Chronoholismus. Man kann deshalb die historischen Systeme auch chronoholistische Systeme nennen. Die biologische Interpretation dieser Betrachtungsweise wird vom Verfasser diskutiert, und die Gleichung auf Systeme mit erblichen Eigenschaften ausgedehnt. Es werden zuletzt Systeme mit vielen Zustandsparametern in gewissen einfachen Fällen kurz behandelt. L'analyse différentielle ne suffit pas, selon l'avis de de l'auteur, pour la description de l'évolution des systèmes vivants au cours des temps. Il faut utiliser dans ces cas les équations intégro-différentielles ou intégrales. Pour faciliter le traÎtement mathématique, on peut considérer d'abord les systèmes dont l'état intérieur est défini par un seul paramètrec. L'évolution d'un système inanimé de cette classe se laisse décrire par l'équation différentielle $$\frac{{dc}}{{dt}} = k f,...........$$ oùt=temps, etk est une fonction déterminée des paramètres extérieurs α,Β, γ. Si l'on connaÎtc à un temps donnét, l'état du système à un temps futurt′ se laisse prévoir au moyen de l'équation, quand les valeurs des paramètres extérieurs dans l'intervallet → t∼' sont fixées. L'auteur appelle les systèmes de cette classe des systèmes non-historiques. Pour les systèmes vivants, il faut se servir, selon l'avis de l'auteur, des équations intégro-différentielles. Pour les systèmes définis par un seul paramètrec, on peut se servir, par exemple, de l'équation $$\frac{{dc}}{{dt}} = k_1 f_1 + \mathop \smallint \limits_{t - \lambda }^t k_2 f_2 dt........$$ où k1 et k2 sont des fonctions déterminées des paramètres extérieurs. L'intégrale s'étend sur l'intervallet-λ→ t où λ est positif et constant. En indiquant le terme intégral par le symboleH, on peut écrire $$\frac{{dc}}{{dt}} = k_1 f_1 + H..........$$ La connaissance de c à un temps donnét ne laisse pas prévoir l'état du système à un temps futurt′ sans connaissance du terme historiqueH, quand mÊme les paramètres α, Β, γ,..., seraient des fonctions déterminées du temps dans l'intervallet→t′. L'auteur appelle l'existence du terme intégral historiqueH le chronoholisme. On peut appeler les systèmes de cette classe des systèmes historiques ou chronoholistiques. L'interprétation biologique de cette faÇon de traÎter les systèmes vivants est discutée par l'auteur, qui étend l'équation pour le faire représenter aussi les propriétés héréditaires des Êtres vivants. Quelques cas de systèmes dont l'état intérieur est défini par plusieurs paramètres et plusieurs phases sont traités brièvement. (shrink)

Weakening classical logic is one of the most popular ways of dealing with semantic paradoxes. Their advocates often claim that such weakening does not affect non-semantic reasoning. Recently, however, Halbach and Horsten have shown that this is actually not the case for Kripke’s fixed-point theory based on the Strong Kleene evaluation scheme. Feferman’s axiomatization $\textsf{KF}$ in classical logic is much stronger than its paracomplete counterpart $\textsf{PKF}$, not only in terms of semantic but also in arithmetical content. This paper compares the (...) proof-theoretic strength of an axiomatization of Kripke’s construction based on the paraconsistent evaluation scheme of $\textsf{LP}$, formulated in classical logic with that of an axiomatization directly formulated in $\textsf{LP}$, extended with a consistency operator. The ultimate goal is to find out whether paraconsistent solutions to the paradoxes that employ consistency operators fare better in this respect than paracomplete ones. (shrink)

We axiomatize Leitgeb's (2005) theory of truth and show that this theory proves all arithmetical sentences of the system of ramified analysis up to $\epsilon_0$. We also give alternative axiomatizations of Kripke's (1975) theory of truth (Strong Kleene and supervaluational version) and show that they are at least as strong as the Kripke-Feferman system KF and Cantini's VF, respectively.

This article considers claims that biology should seek general theories similar to those found in physics but argues for an alternative framework for biological theories as collections of prototypical interlevel models that can be extrapolated by analogy to different organisms. This position is exemplified in the development of the Hodgkin‐Huxley giant squid model for action potentials, which uses equations in specialized ways. This model is viewed as an “emergent unifier.” Such unifiers, which require various simplifications, involve the types of heuristics (...) discussed in Wimsatt’s writings on reduction, but with a twist. Here, the heuristics are used to generate emergent rather than reductive explanations. †To contact the author, please write to: Department of History and Philosophy of Science, University of Pittsburgh, 1017 Cathedral of Learning, Pittsburgh, PA 15260; e‐mail: [email protected]. (shrink)

The uniform reflection principle for the theory of uniform T-sentences is added to PA. The resulting system is justified on the basis of a disquotationalist theory of truth where the provability predicate is conceived as a special kind of analyticity. The system is equivalent to the system ACA of arithmetical comprehension. If the truth predicate is also allowed to occur in the sentences that are inserted in the T-sentences, yet not in the scope of negation, the system with the reflection (...) schema for these T-sentences assumes the strength of the Kripke-Feferman theory KF, and thus of ramified analysis up to $\varepsilon_0$. (shrink)

In this article we study the systems KF and VF of truth over set theory as well as related systems and compare them with the corresponding systems over arithmetic.

With the widespread application of electric vehicles, the study of the power lithium-ion battery has broad prospects and great academic significance. The state of charge is one of the key parts in battery management system, which is used to provide guarantee for the safe and efficient operation of LIB. To obtain the reliable SOC estimation result under the influence of simple model and measurement noise, a novel estimation method with adaptive feedback compensator is presented in this paper. The simplified dynamic (...) external electrical characteristic of LIB is represented by the one-order Thevenin equivalent circuit model and then the ECM parameters are identified by the forgetting factor recursive least squares method. Fully taking into account the feedback effect of terminal voltage innovation, the combination of adaptive extended Kalman filter and innovation vector-based proportional-integral-derivative feedback is proposed to estimate the LIB SOC. The common single proportional feedback of Kalman filter is replaced by the innovation vector-based PID feedback, which means that the multiple prior terminal voltage innovation is used in the measurement correction step of KF. The results reveal that the AEKF with PID feedback compensation strategy can improve the SOC estimation performance compared with the common AEKF, and it reveals good robust capability and rapid convergence speed for initial SOC errors. The maximum absolute error and average absolute error for SOC estimation are close to 4% and 2.6%, respectively. (shrink)

In this paper we shall introduce two types of contextual-hierarchical approaches to the strengthened liar problem. These approaches, which we call the ‘standard’ and the ‘alternative’ ch-reconstructions of the strengthened liar problem, differ in their philosophical view regarding the nature of truth and the relation between the truth predicates T r n and T r n+1 of different hierarchy-levels. The basic idea of the standard ch-reconstruction is that the T r n+1-schema should hold for all sentences of \. In contrast, (...) the alternative ch-reconstruction, for which we shall argue in section four, is motivated by the idea that T r n and T r n+1 are coherent in the sense that the same sentences of \ should be true according to T r n and T r n+1. We show that instances of the standard ch-reconstruction can be obtained by iterating Kripke’s strong Kleene jump operator. Furthermore, we will demonstrate how instances of the alternative ch-reconstruction can be obtained by a slight modification of the iterated axiom system KF and of the iterated strong Kleene jump operator. (shrink)

When are two formal theories of broadly logical concepts, such as truth, equivalent? The paper investigates a case study, involving two well-known variants 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 well-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. (shrink)

Bocheński's lucid, unpartisan, and judiciously critical discussion of Soviet Marxism-Leninism in his book Der sowjetrussische dialektische Materialismus (1950) filled a major gap in our understanding of that influential movement. Prior to its publication there had been only two works on the subject in English, John Somerville's Soviet Philosophy (1946) and the Handbook of Philosophy (1949), edited and adapted by Howard Selsam from the Kratkij filosofskij slovar' (2nd ed. 1940). Both are marked by strong partisanship and ideological bias. Somerville is uncritically (...) pro-Soviet and abjectly Stalinist. Selsam, although he tones down the adulation of Marx, Stalin et al. of the KFS, retains that work's abuse of such "reactionary" and "idealist" thinkers as Plato and such "reactionary" and "bourgeois" thinkers as Hegel. The benign influence of Bocheński's work increased with the publication of the English translation, Soviet Russian Dialectical Materialism, in 1963. (shrink)