We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property.

We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property.

Solovay's 1976 completeness result for modal provability logic employs the recursion theorem in its proof. It is shown that the uses of the recursion theorem can in this proof replaced by the diagonalization lemma for arithmetic and that, in effect, the proof neatly fits the framework of another, enriched, system of modal logic so that any arithmetical system for which this logic is sound is strong enough to carry out the proof, in particular $\text{I}\Delta _{0}+\text{EXP}$ . The method is adapted (...) to obtain a similar completeness result for the Rosser logic. (shrink)

A definition is given for formulae $A_1,\ldots,A_n$ in some theory $T$ which is formalized in a propositional calculus $S$ to be (in)dependent with respect to $S$. It is shown that, for intuitionistic propositional logic $\mathbf{IPC}$, dependency (with respect to $\mathbf{IPC}$ itself) is decidable. This is an almost immediate consequence of Pitts' uniform interpolation theorem for $\mathbf{IPC}$. A reasonably simple infinite sequence of $\mathbf{IPC}$-formulae $F_n(p, q)$ is given such that $\mathbf{IPC}$-formulae $A$ and $B$ are dependent if and only if at least (...) on the $F_n(A, B)$ is provable. (shrink)

Solovay's 1976 completeness result for modal provability logic employs the recursion theorem in its proof. It is shown that the uses of the recursion theorem can in this proof be replaced by the diagonalization lemma for arithmetic and that, in effect, the proof neatly fits the framework of another, enriched, system of modal logic (the so-called Rosser logic of Gauspari-Solovay, 1979) so that any arithmetical system for which this logic is sound is strong enough to carry out the proof, in (...) particular I0+EXP. The method is adapted to obtain a similar completeness result for the Rosser logic. (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 Smoryski.

To the standard propositional modal system of provability logic constants are added to account for the arithmetical fixed points introduced by Bernardi-Montagna in [5]. With that interpretation in mind, a system LR of modal propositional logic is axiomatized, a modal completeness theorem is established for LR and, after that, a uniform arithmetical completeness theorem with respect to PA is obtained for LR.

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.

The modal completeness proofs of Guaspari and Solovay (1979) for their systems R and R – are improved and the relationship between R and R – is clarified.

This first chapter contains an introduction to modal logic. In section 1.1 the syntactic side of the matter is discussed, and in section 1.2 the subject is approached from a semantic point of view.

This paper contains a completeness proof for the system ILW, a rather bewildering axiom system belonging to the family of interpretability logics. We have treasured this little proof for a considerable time, keeping it just for ourselves. Johan’s ftieth birthday appears to be the right occasion to get it out of our wine cellar.

To the standard propositional modal system of provability logic constants are added to account for the arithmetical fixed points introduced by Bernardi-Montagna in [5]. With that interpretation in mind, a system LR of modal propositional logic is axiomatized, a modal completeness theorem is established for LR and, after that, a uniform arithmetical (Solovay-type) completeness theorem with respect to PA is obtained for LR.

It is shown that for arithmetical interpretations that may include free variables it is not the Guaspari-Solovay system R that is arithmetically complete, but their system R –. This result is then applied to obtain the nonvalidity of some rules under arithmetical interpretations including free variables, and to show that some principles concerning Rosser orderings with free variables cannot be decided, even if one restricts oneself to usual proof predicates.

Standard unary modal logic and binary modal logic, i.e. modal logic with one binary operator, are shown to be definitional extensions of one another when an additional axiom |$U$| is added to the basic axiomatization of the binary side. This is a strengthening of our previous results. It follows that all unary modal logics extending Classical Modal Logic, in other words all unary modal logics with a neighborhood semantics, can equivalently be seen as binary modal logics. This in particular applies (...) to standard modal logics, which can be given simple natural axiomatizations in binary form. We illustrate this in the logic K. We call such logics binary expansions of the unary modal logics. There are many more such binary expansions than the ones given by the axiom |$U$|⁠. We initiate an investigation of the properties of these expansions and in particular of the maximal binary expansions of a logic. Our results directly imply that all sub- and superintuitionistic logics with a standard modal companion also have binary modal companions. The latter also applies to the weak subintuitionistic logic WF of our previous papers. This logic doesn’t seem to have a unary modal companion. (shrink)

It is shown that for arithmetical interpretations that may include free variables it is not the Guaspari-Solovay system R that is arithmetically complete, but their system R⁻. This result is then applied to obtain the nonvalidity of some rules under arithmetical interpretations including free variables, and to show that some principles concerning Rosser orderings with free variables cannot be decided, even if one restricts onself to "usual" proof predicates.

In this paper the system IL for relative interpretability is studied.

In 1995 Visser, van Benthem, de Jongh, and Renardel de Lavalette introduced NNIL-formulas, showing that these are exactly the formulas preserved under taking submodels of Kripke models. In this article we show that NNIL-formulas are up to frame equivalence the formulas preserved under taking subframes of frames, that NNIL-formulas are subframe formulas, and that subframe logics can be axiomatized by NNIL-formulas. We also define a new syntactic class of ONNILLI-formulas. We show that these are the formulas preserved in monotonic (...) images of frames and that ONNILLI-formulas are stable formulas as introduced by Bezhanishvili and Bezhanishvili in 2013. Thus, ONNILLI is a syntactically defined set of formulas axiomatizing all stable logics. This resolves a problem left open in 2013. (shrink)

We give alternative characterizations of exact, extendible and projective formulas in intuitionistic propositional calculus IPC in terms of n -universal models. From these characterizations we derive a new syntactic description of all extendible formulas of IPC in two variables. For the formulas in two variables we also give an alternative proof of Ghilardi’s theorem that every extendible formula is projective.

We give alternative characterizations of exact, extendible and projective formulas in intuitionistic propositional calculus IPC in terms of n-universal models. From these characterizations we derive a new syntactic description of all extendible formulas of IPC in two variables. For the formulas in two variables we also give an alternative proof of Ghilardi’s theorem that every extendible formula is projective.

We study the modal properties of intuitionistic modal logics that belong to the provability logic or the preservativity logic of Heyting Arithmetic. We describe the □-fragment of some preservativity logics and we present fixed point theorems for the logics iL and iPL, and show that they imply the Beth property. These results imply that the fixed point theorem and the Beth property hold for both the provability and preservativity logic of Heyting Arithmetic. We present a frame correspondence result for the (...) preservativity principle Wp that is related to an extension of Löb's principle. (shrink)

In this paper, we study IL(), the interpretability logic of . As is neither an essentially reflexive theory nor finitely axiomatizable, the two known arithmetical completeness results do not apply to : IL() is not or . IL() does, of course, contain all the principles known to be part of IL, the interpretability logic of the principles common to all reasonable arithmetical theories. In this paper, we take two arithmetical properties of and see what their consequences in the modal logic (...) IL() are. These properties are reflected in the so-called Beklemishev Principle , and Zambella’s Principle , neither of which is a part of IL. Both principles and their interrelation are submitted to a modal study. In particular, we prove a frame condition for . Moreover, we prove that follows from a restricted form of . Finally, we give an overview of the known relationships of IL() to important other interpretability principles. (shrink)

Semantic Automata Johan van Ben them. INTRODUCTION An attractive, but never very central idea in modern semantics has been to regard linguistic expressions ...

We give a systematic method of constructing extensions of the Kuznetsov-Gerčiu logic KG without the finite model property (fmp for short), and show that there are continuum many such. We also introduce a new technique of gluing of cyclic intuitionistic descriptive frames and give a new simple proof of Gerčiu’s result [9, 8] that all extensions of the Rieger-Nishimura logic RN have the fmp. Moreover, we show that each extension of RN has the poly-size model property, thus improving on [9]. (...) Furthermore, for each function f: omega -> omega, we construct an extension Lf of KG such that Lf has the fmp, but does not have the f-size model property. We also give a new simple proof of another result of Gerčiu [9] characterizing the only extension of KG that bounds the fmp for extensions of KG. We conclude the paper by proving that RN.KC = RN + (¬p vee ¬¬p) is the only pre-locally tabular extension of KG, introduce the internal depth of an extension L of RN, and show that L is locally tabular if and only if the internal depth of L is finite. (shrink)

The issue of the relation between financial derivatives, money and crisis remains one of on-going debate within Marxism. This paper takes issue with a recent contribution to this debate by Tony Norfield. We contend that the relationship between financial derivatives and the concept of ‘money’ needs to be framed in the context of a changing understanding of liquidity, and that issues of crisis and renewed accumulation are better understood though this path than via debates about speculative versus real investment and (...) productive versus unproductive capital. Indeed these latter taxonomies are being superseded by current developments within finance, and Marxian analysis needs to be attuned to these current developments. (shrink)

In this rethinking of Marxism and its blind spots, Dick Howard argues that the collapse of European communism in 1989 should not be identified with a victory for capitalism and makes possible a wholesale reevaluation of democratic politics in the U.S. and abroad. The author turns to the American and French Revolutions to uncover what was truly "revolutionary" about those events, arguing that two distinct styles of democratic life emerged, the implications of which were misinterpreted in light of the (...) rise of communism. Howard uses a critical rereading of Marx as a theorist of democracy to offer his audience a new way to think about this political ideal. He argues that it is democracy, rather than Marxism, that is radical and revolutionary, and that Marx could have seen this but did not. In Part I, Howard explores the attraction Marxism held for intellectuals, particularly French intellectuals, and he demonstrates how the critique of totalitarianism from a Marxist viewpoint allowed these intellectuals to see the radical nature of democracy. Part II examines two hundred years of democratic political life--comparing America's experience as a democracy to that of France. Part III offers a rethinking of Marx's contribution to democratic politics. Howard concludes that Marx was attempting a "philosophy by other means," and that paradoxically, just because he was such an astute philosopher, Marx was unable to see the radical political implications of his own analyses. The philosophically justified "revolution" turns out to be the basis of an anti-politics whose end was foreshadowed by the fall of European communism in 1989. (shrink)

In this article, I assess the prospects for the limitarian thesis that someone has too much wealth if they exceed a specific wealth threshold. Limitarianism claims that there are good political and/or ethical reasons to prevent people from having such ‘surplus wealth’, for example, because it has no moral value for the holder or because allowing people to have surplus wealth has less moral value than redistributing it. Drawing on recent literature on distributive justice, I defend two types of limitarian (...) principles of justice. First, limitarian midlevel principles draw on the limitarian thesis to specify normative commitments for guiding institutional design and individual actions. Second, the limitarian presumption draws on that thesis to specify what a just allocation of wealth requires under epistemic constraints. Such a presumption says that without substantive reasons to the contrary, we should regard a distribution as unjust if some people’s wealth exceeds the limitarian threshold. Furthermore, I will argue that we must reject a possible but implausible interpretation of limitarianism as an ideal distributive pattern. Yet both as a midlevel principle and as a presumption, limitarianism can play an important role in theorizing about justice in the real world. (shrink)

A central question in contemporary ethics and political philosophy concerns which entities have moral status. In this article, I provide a detailed analysis of the view that moral status comes in degrees. I argue that degrees of moral status can be specified along two dimensions: (i) the weight of the reason to protect an entity’s morally significant rights and interests; and/or (ii) the rights and interests that are considered morally significant. And I explore some of the complexities that arise when (...) we link degrees of moral status to the weight of reasons to protect bundles and/or the content and size of these bundles, in particular regarding the attribution of moral status to nonparadigmatic entities (e.g. people with severe cognitive disabilities, children, embryos and foetuses, robots, non-human animals, ecosystems, and rivers). (shrink)

Recent instances of corporate misconduct and examples of blatant leader self-serving behavior have rekindled interest in leader personality traits as antecedents of negative leader behavior. The current research builds upon that work, and examines the relationship between leader psychopathy and leader self-serving behavior. Moreover, we investigate whether follower self-esteem affects the occurrence of self-serving behavior in leaders with psychopathic tendencies. We predict that self-serving behaviors by psychopathic leaders are more likely to occur in the interaction with followers low in self-esteem. (...) We first conducted an experimental study (N = 156), in which we manipulated follower self-esteem, measured leader psychopathy, and assessed their combined effect on leader self-serving behavior using an ultimatum game. We then conducted a multi-source field study (N = 124 leader-follower dyads) using questionnaires to assess leader psychopathy, follower self-esteem, and perceived leader self-serving behavior. Across both studies, we found that leader psychopathy was positively related to their self-serving behavior, but only when followers had low rather than high self-esteem. As expected, our studies showed that the degree to which (perceived) psychopathic traits of leaders are reflected in their behavior depends on the characteristics of their followers. Apparently, the behavioral expression of negative leader traits is not only a matter of the trait strength, but instead is the result of the interplay between leader and follower in a certain context. (shrink)

One of the new ways used by companies to demonstrate their social responsibility is to encourage employee volunteering, whereby employees engage in socially beneficial activities on company time, while being paid by the company. The reasoning is that it is good for employee motivation (internal effects) and good for the company reputation (external effects). This article reports an empirical investigation of the internal effects of employee volunteering conducted amongst employees of the Dutch ABN-AMRO bank. The study showed that (a) socio-demographic (...) characteristics from employee volunteers markedly differ from those of non-volunteers and community volunteers and (b) employee volunteering seems to have positive effects on attitudes and behavior towards the organization. (shrink)

The commons have emerged as a key notion and underlying experience of many efforts around the world to promote justice and democracy. A central question for political theories of the commons is whether the visions of social order and regimes of political economy they propose are complementary or opposed to public goods that are backed up by governmental coordination and compulsion. This essay argues that the post-Marxist view, which posits an inherent opposition between the commons as a sphere of inappropriable (...) usage and statist public infrastructure, is mistaken, because justice and democracy are not necessarily furthered by the institution of inappropriability. I articulate an alternative pluralist view based on James Tully’s work, which discloses the dynamic interplay between public and common modes of provision and enjoyment, and their civil and civic orientations respectively. Finally, the essay points to the Janus-faced character of the commons and stresses the co-constitutive role of public goods and social services for just and orderly social life while remaining attentive to the dialectic of empowerment and tutelage that marks provision by government. (shrink)

Clifford, Dick The world outlook is rather grim. Greece is bankrupt, the efforts to cure the problem by making new loans to the banks and cutting living standards is likely only to postpone the date when bankruptcy is declared. Italy and Spain are in a similar position. Britain, Europe and the USA are loaded with debt, only a few countries like Iceland are adopting methods which are the reverse of what conventional economics requires and seem to be recovering from (...) their problems slowly. (shrink)

Carl Knight argues that lexical sufficientarianism, which holds that sufficientarian concerns should have lexical priority over other distributive goals, is ‘excessive’ in many distinct ways and that sufficientarians should either defend weighted sufficientarianism or become prioritarians. In this article, I distinguish three types of weighted sufficientarianism and propose a weighted sufficientarian view that meets the excessiveness objection and is preferable to both Knight’s proposal and prioritarianism. More specifically, I defend a multi-threshold view which gives weighted priority to benefits directly above (...) and below its thresholds, but gives benefits below the lowest threshold lexical priority over benefits above the highest threshold. (shrink)