Reason and Value collects fifteen brand-new papers by leading contemporary philosophers on themes from the moral philosophy of Joseph Raz. The subtlety and power of Raz's reflections on ethical topics - including especially his explorations of the connections between practical reason and the theory of value - make his writings a fertile source for anyone working in this area. The volume honours Raz's accomplishments in the area of ethical theorizing, and will contribute to an enhanced appreciation of the significance of (...) his work for the subject. (shrink)

The UN has to date not been effective in preventing genocide, and has had only a slightly better record in stopping it. There have been occasions when its interventions has occurred only after a genocide has taken place, and even then its major focus has been on facilitating the provision of aid by non-governmental agencies rather than on the task of tracking down the perpetrators and bringing them to justice. The exceptions of the ICTY and the ICTR are so stark, (...) in this regard, that they only serve to throw light on the many other genocidal events where the UN has not initiated measures against those responsible for carrying them out. In short, as a body the UN has no—until very recently—even approached the fulfillment of its mandate as articulated in the Convention on the Prevention and Punishment of the Crime of Genocide, and only rarely invoked Chapter VII of the UN Charter in order to intervene physically for the purpose of countering threats to peace or stopping conflict. Its strengths, so far as there have been any, have focused on balancing great power interests with demands to intervene more forcefully. While in the years up to 1989 this could be seen as a way of maintaining the peace (albeit over the broken bodies of victims of genocide in places such as Biafra, Cambodia, and East Timor), since then the UN has been required to act with greater resolve and purpose. The failures of Bosnia, Rwanda, and Kosovo must thus be seen as having been brought on by a transition from one international regime to another; from a Cold War regime in which the UN—s main role was one of preventing a third (and possibly nuclear) World War from breaking out, to a post-Cold War regime which appears increasingly to be characterized by the UN searching for a new role in which humanitarian issues are to assume a higher priority than they once did. Whether or not this will continue, of course, will depend on an extremely wide variety of circumstances—and at this time it is likely that only a few of these can accurately be anticipated. (shrink)

This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth. The chapters are arranged so that the two introductory articles come first; (...) these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science. (shrink)

Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic.

Let T be a first-order theory. A T-normal Kripke structure is one in which every world is a classical model of T. This paper gives a characterization of the intuitionistic theory T of sentences intuitionistically valid in all T-normal Kripke structures and proves the corresponding soundness and completeness theorems. For Peano arithmetic , the theory PA is a proper subtheory of Heyting arithmetic , so HA is complete but not sound for PA-normal Kripke structures.

The bounded arithmetic theory S2 is finitely axiomatized if and only if the polynomial hierarchy provably collapses. If T2i equals S2i + 1 then T2i is equal to S2 and proves that the polynomial time hierarchy collapses to ∑i + 3p, and, in fact, to the Boolean hierarchy over ∑i + 2p and to ∑i + 1p/poly.

In his warranted christian belief, Alvin Plantinga launches a forceful attack on apophaticism, the view that God is in some sense or other beyond description. This paper explores his attack before searching for a Plantinga-proof formulation of apophaticism.

This paper discusses lower bounds for proof length, especially as measured by number of steps (inferences). We give the first publicly known proof of Gödel's claim that there is superrecursive (in fact. unbounded) proof speedup of (i + 1)st-order arithmetic over ith-order arithmetic, where arithmetic is formalized in Hilbert-style calculi with + and · as function symbols or with the language of PRA. The same results are established for any weakly schematic formalization of higher-order logic: this allows all tautologies as (...) axioms and allows all generalizations of axioms as axioms. Our first proof of Gödel's claim is based on self-referential sentences: we give a second proof that avoids the use of self-reference based loosely on a method of Statman. (shrink)

Samuel R. Buss and Aleksandar Ignjatović. Unprovability of Consistency Statements in Fragments of Bounded Arithmetic.

We study the long-standing open problem of giving$\forall {\rm{\Sigma }}_1^b$separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the$\forall {\rm{\Sigma }}_1^b$Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole (...) principle for FPNPfunctions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of$T_2^1$augmented with the surjective weak pigeonhole principle for polynomial time functions. (shrink)

Partial consistency statements can be expressed as polynomial-size propositional formulas. Frege proof systems have polynomial-size partial self-consistency proofs. Frege proof systems have polynomial-size proofs of partial consistency of extended Frege proof systems if and only if Frege proof systems polynomially simulate extended Frege proof systems. We give a new proof of Reckhow's theorem that any two Frege proof systems p-simulate each other. The proofs depend on polynomial size propositional formulas defining the truth of propositional formulas. These are already known to (...) exist since the Boolean formula value problem is in alternating logarithmic time; this paper presents a proof of this fact based on a construction which is somewhat simpler than the prior proofs of Buss and of Buss-Cook- Gupta-Ramachandran. (shrink)

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

The paper proves refined feasibility properties for the disjunction property of intuitionistic propositional logic. We prove that it is possible to eliminate all cuts from an intuitionistic proof, propositional or first-order, without increasing the Horn closure of the proof. We obtain a polynomial time, interactive, realizability algorithm for propositional intuitionistic proofs. The feasibility of the disjunction property is proved for sequents containing Harrop formulas. Under hardness assumptions for NP and for factoring, it is shown that the intuitionistic propositional calculus does (...) not always have polynomial size proofs and that the classical propositional calculus provides a superpolynomial speedup over the intuitionistic propositional calculus. The disjunction property for intuitionistic propositional logic is proved to be P-hard by a reduction to the circuit value problem. (shrink)

In this paper we argue that the moral value of an agent is determined solely by their dispositions to act intentionally and freely. We then put this conclusion to work. It resolves a putative moral paradox first posed by Saul Smilansky, and it undermines a prominent line of argument for a variety of Trinitarian theology. Finally, we derive our conclusion about the moral worth of agents not only from our initial series of thought experiments, but also from Abrahamic theism itself. (...) This means that Smilansky’s paradox can only possibly be rehabilitated by an atheist, and that the aforementioned line of argumentation for the Trinity is radically self-undermining, since it relies upon the denial of a corollary of Abrahamic theism. (shrink)

We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depthd Frege proofs ofm lines can be transformed into depthd proofs ofO(m d+1) symbols. We show that renaming (...) Frege proof systems are p-equivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed. (shrink)

The author reviews the planned withdrawal of healthcare from the primary public hospital, Grady Memorial Hospital, of Atlanta, Georgia. Prior to passage of the Affordable Care Act of 2010, at least half of the patients had no public or private health insurance and their care was financially supported by State and County funding as well as supplementation from Emory University. New administration in the elected positions of the State and County and at the University reached agreement to decrease care. The (...) University President stated that the Hippocratic Oath was no longer valid for salaried physicians who must follow the direction of their business manager. The author, a tenured Professor, was forcibly transferred to the Veterans Hospital, which also was withholding standard medical care, including life-saving medications. Organizing multiple concerned organizations including members of the King Center, the Carter Center, the Open Door Community Center, Gay Rights, etc., (Grady Coalition) multiple public protests were successful in raising the necessary funds to continue adequate care at Grady and begin an investigation of the Veterans Affairs Hospital. (shrink)

Originating from work in operations research the cutting plane refutation systemCP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the non-existence of boolean solutions to associated families of linear inequalities. Polynomial sizeCP proofs are given for the undirecteds-t connectivity principle. The subsystemsCP q ofCP, forq≥2, are shown to be polynomially equivalent toCP, thus answering problem 19 from the list of open problems of [8]. We present a normal form theorem forCP (...) 2-proofs and thereby for arbitraryCP-proofs. As a corollary, we show that the coefficients and constant terms in arbitrary cutting plane proofs may be exponentially bounded by the number of steps in the proof, at the cost of an at most polynomial increase in the number of steps in the proof. The extensionCPLE +, introduced in [9] and there shown top-simulate Frege systems, is proved to be polynomially equivalent to Frege systems. Lastly, since linear inequalities are related to threshold gates, we introduce a new threshold logic and prove a completeness theorem. (shrink)

A pool resolution proof is a dag-like resolution proof which admits a depth-first traversal tree in which no variable is used as a resolution variable twice on any branch. The problem of determining whether a given dag-like resolution proof is a valid pool resolution proof is shown to be NP-complete.

We present sharpened lower bounds on the size of cut free proofs for first-order logic. Prior lower bounds for eliminating cuts from a proof established superexponential lower bounds as a stack of exponentials, with the height of the stack proportional to the maximum depth d of the formulas in the original proof. Our results remove the constant of proportionality, giving an exponential stack of height equal to d — 0(1). The proof method is based on more efficiently expressing the Gentzen-Solovay (...) cut formulas as low depth formulas. (shrink)

We refine the constructions of Ferrante-Rackoff and Solovay on iterated definitions in first-order logic and their expressibility with polynomial size formulas. These constructions introduce additional quantifiers; however, we show that these extra quantifiers range over only finite sets and can be eliminated. We prove optimal upper and lower bounds on the quantifier complexity of polynomial size formulas obtained from the iterated definitions. In the quantifier-free case and in the case of purely existential or universal quantifiers, we show that Ω quantifiers (...) are necessary and sufficient. The last lower bounds are obtained with the aid of the Yao-Håstad switching lemma. (shrink)

The classical interpretation of mathematical statements can be seen as comprising two separate but related aspects: a domain and a truth-schema. L. E. J. Brouwer’s intuitionistic project lays the groundwork for an alternative conception of the objects in this domain, as well as an accompanying intuitionistic truth-schema. Drawing on the work of Arend Heyting and Michael Dummett, I present two objections to classical mathematical semantics, with the aim of creating an opening for an alternative interpretation. With this accomplished, I then (...) make the case for intuitionism as a suitable candidate to fill this void. (shrink)

Throughout the world, female students are less likely than males to take advanced physics courses. This mixed-methods study uses a concurrent, nested design to study an online homework intervention designed to address choice and achievement. A choice of three different contexts (biological, sports, and traditional) were offered to students for each physics problem, intending to stimulate females’ interest and enhance achievement. Informed by aspects of Artino’s social-cognitive model of academic motivation and emotion, we investigated: Which context of physics problems do (...) males and females select?; What explanations do students give for their choices?; Are there differences in the achievement of males and females?; and Is there a relationship between student achievement and the context selected? Fifty-two high school physics students from five U.S. states participated. Data included pre and post Force Concept Inventory scores, homework context choices and achievement, and rationales for choices. Findings indicate that females were most likely to select biology contexts; males, traditional. All students made more attempts on video questions over word questions, although females did not score as well. For all questions, students generally persisted until they answered them correctly, with females taking fewer attempts on problems. Context choice was mostly driven by interest, for males, and perceptions of difficulty level for females; however, rationales were indistinguishable by gender. On their first homework question attempt, females scored significantly better than the males. Initially, males had significantly higher FCI scores; post homework intervention, females increased their mean scores significantly on the FCI, erasing the initial gender gap, with no growth nor decline in males’ scores. Females with FCI growth were equally as likely to choose biology contexts as traditional contexts; males were more likely to choose biology contexts. Findings from this study suggest that modest changes to homework problems that provide choice and make the physics problems more contextually interesting - even without changes in classroom instruction - could increase interest and motivation students and increase achievement for both male and female students. Recommendations will be discussed. (shrink)

Ever since Plato's Republic, a persistent problem and dilemma in Western thought has been the relation of the love of wisdom and political power, especially the role that the intellectual does or ought to play in the world of action. This volume includes both theoretical studies and case studies of modern intellectuals. Most of the articles have been published before but several, including T. Parson's "'The Intellectual': A Social Role Category" and J. Netl's "Ideas, Intellectuals, and Structures of Dissent" were (...) written for this volume. Other contributors are Shils, Dahrendorf, Berlin, Bushnell, Samuels, Comte, Nisbet, and Rieff. The most eloquent and moving essay is Isaiah Berlin's sympathetic study of Moses Hess. Through the great diversity of approaches and issues discussed, a common theme emerges--the fragility of the role of the intellectual, who is frequently duped, sometimes subtly corrupted or persecuted, but who, on occasion can shape and humanize the vision of his fellowmen, even though he may appear as the "fool" to his contemporaries. One striking lack here is a "case study" of any representative intellectual of the "New Left," but the total effect of this intelligently chosen collection is to provide us with a needed perspective for assessing the possibilities and dangers open to intellectuals in our time.--R. J. B. (shrink)