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 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.

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.

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.

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)

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.

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)

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

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.

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.

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.

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.

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.

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 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)

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.

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)

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)

This book traces a dialectic relationship between “politics” and “antipolitics,” the first, as used here, being akin to philosophy as an activity of open inquiry, plural democracy, and truth-finding, and the latter in the realm of ideology, technocracy, and presupposed certainties. It returns back to the emergence of a New Left movement in the 1960s in order to follow the history of this relationship since then. It addresses contemporary debates by looking to the fall of the Berlin Wall and the (...) Soviet Bloc, and asking in the wake of that: what is a revolution? Finally, it draws on these analyses to examine the age of terrorism after the tragedy of September 11, 2001, and resounds with a call to pursue democracy and real politics in the face of new forms of antipolitics. (shrink)

Animals in Disasters is a comprehensive book on animal rescue written by Dr. Dick Green who shares his experiences, best practices and lessons learned from well over 125 domestic and international disasters. It provides a step-by-step process for communities and states to more effectively address animal issues and enhance their animal response capabilities. Sections include an overview of the history of animal rescue, where we are today, and the steps needed to better prepare for tomorrow. This how-to book for (...) emergency managers who want to develop programs, craft policy, and build response capability/capacity is an ideal companion to their work. Clearly identifies the components of building a resilient community Introduces the Community Preparedness Checklist Helps readers develop and deliver effective animal response training. (shrink)

In earlier work on limitarianism, I argued that setting an upper limit to the amount of wealth that people can permissibly have is justified when decision-makers are unaware of or disagree about the appropriate distributive criterion or if they are unaware of people’s relevant features (or both). Robert Huseby has raised several powerful objections to this presumptive argument for limitarianism. Some of these objections call for a revision of my defence of presumptive limitarianism while others call for clarification, both of (...) which I aim to do in this chapter. I will argue that unless decision-makers have substantive reasons to suggest otherwise, they must act as if there is an upper limit to the amount of wealth that people can permissibly have. (shrink)

Following A.N. Whitehead’s rhythm of education, the author provides a guide for parents and educators on raising children to thrive in times of tempestuous change. Each chapter presents exemplary educational events rich in context, and then draws on seminal research to ground her recommendations in a robust theoretical foundation.

Excerpt from The Principle of Synthetic Unity in Berkeley and Kant This little volume was prepared as a thesis for the degree of Doctor of Philosophy in the University of Michigan. By the advice of Dr. John Dewey I have undertaken to interpret the Metaphysical Notes of Berkeley's Commonplace Book, and as far as possible discover the Principle of Unity which occasionally manifests itself in Berkeley's works and which formed a basis for a Treatise on the Will which Berkeley contemplated (...) but never produced. I wish to express my indebtedness to Dr. Dewey for his assistance in the selection of collateral reading and for his suggestions in the development of the thesis. No literature could be secured bearing upon the interpretation of the Notes, hence the Notes have been classified and such as bear upon the theme under discussion have been used. Often the phraseology has been preserved but where that could not be done the thought has been expressed in phraseology as nearly Berkeleian as the author could select so as to preserve the unity that runs through the Notes. This principle of unity found in Berkely has been compared and contrasted with the Unity of Kant. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

The first part of this study is devoted to an attempt to provide a systematic philosophical argument for the political movement from Marx to Kant. The second half proposes analyses of the New Left of the 60s and the potential birth of a new New Left in the 80s.

Looks at how powerful businesspeople and politicians are valuing their own greed over the welfare of future generations by blocking climate change remediation.

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)

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)

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)

Identifying scientism as religion’s secular counterpart, this collection studies contemporary contestations of the authority of science. These controversies suggest that what we are witnessing today is not an increase in the authority of science at the cost of religion, but a dual decline in the authorities of religion and science alike. This entails an erosion of the legitimacy of universally binding truth claims, be they religiously or scientifically informed. Approaching the issue from a cultural-sociological perspective and building on theories from (...) the sociology of religion, the volume unearths the cultural mechanisms that account for the headwind faced by contemporary science. The empirical contributions highlight how the field of academic science has lost much of its former authority vis-à-vis competing social realms; how political and religious worldviews define particular research findings as favorites while dismissing others; and how much of today’s distrust of science is directed against scientific institutions and academic scientists rather than against science per se. (shrink)