Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr(x) saying "x is true" and satisfying the "dequotation schema" $\varphi \equiv \text{Tr}(\bar{\varphi})$ for all sentences φ? This problem is investigated in the frame of Lukasiewicz infinitely valued logic.

This paper concerns the question of how to draw inferences common sensically from uncertain knowledge. Since the early work of Shore and Johnson (1980), Paris and Vencovská (1990), and Csiszár (1989), it has been known that the Maximum Entropy Inference Process is the only inference process which obeys certain common sense principles of uncertain reasoning. In this paper we consider the present status of this result and argue that within the rather narrow context in which we work this complete and (...) consistent mode of uncertain reasoning is actually characterised by the observance of just a single common sense principle (or slogan). (shrink)

A necessary and sufficient condition in terms of a de Finetti style representation is given for a probability function in Polyadic Inductive Logic to satisfy being part of a Language Invariant family satisfying Spectrum Exchangeability. This theorem is then considered in relation to the unary Carnap and Nix–Paris Continua.

Rational Pavelka logic extends Lukasiewicz infinitely valued logic by adding truth constants r̄ for rationals in [0, 1]. We show that this is a conservative extension. We note that this shows that provability degree can be defined in Lukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.

We give a brief account of some de Finetti style representation theorems for probability functions satisfying Spectrum Exchangeability in Polyadic Inductive Logic, together with applications to Non-splitting, Language Invariance, extensions with Equality and Instantial Relevance.

We investigate uncertain reasoning with quantified sentences of the predicate calculus treated as the limiting case of maximum entropy inference applied to finite domains.

Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr saying "x is true" and satisfying the "dequotation schema" $\varphi \equiv \text{Tr}$ for all sentences $\varphi$? This problem is investigated in the frame of Lukasiewicz infinitely valued logic.

This paper investigates the quantifier "there exist unboundedly many" in the context of first-order arithmetic. An alternative axiomatization is found for Peano arithmetic based on an axiom schema of regularity: The union of boundedly many bounded sets is bounded. We also obtain combinatorial equivalents of certain second-order theories associated with cuts in nonstandard models of arithmetic.

We propose and investigate an Analogy Principle in the context of Unary Inductive Logic based on a notion of support by structural similarity which is often employed to motivate scientific conjectures.

We prove that: • if there is a model of I∆₀ + ¬ exp with cofinal Σ₁-definable elements and a Σ₁ truth definition for Σ₁ sentences, then I∆₀ + ¬ exp +¬BΣ₁ is consistent, • there is a model of I∆₀ Ω₁ + ¬ exp with cofinal Σ₁-definable elements, both a Σ₂ and a ∏₂ truth definition for Σ₁ sentences, and for each n > 2, a Σ n truth definition for Σ n sentences. The latter result is obtained by (...) constructing a model with a recursive truth-preserving translation of Σ₁ sentences into boolean combinations of $\exists \sum {\begin{array}{*{20}{c}} h \\ 0 \\ \end{array} } $ sentences. We also present an old but previously unpublished proof of the consistency of I∆₀ + ¬ exp + ¬BΣ₁ under the assumption that the size parameter in Lessan's ∆₀ universal formula is optimal. We then discuss a possible reason why proving the consistency of I∆₀ + ¬ exp + ¬BΣ₁ unconditionally has turned out to be so difficult. (shrink)

We consider two formalizations of the notion of irrelevance as a rationality principle within the framework of (Carnapian) Inductive Logic: Johnson's Sufficientness Principle, JSP, which is classically important because it leads to Carnap's influential Continuum of Inductive Methods and the recently proposed Weak Irrelevance Principle, WIP. We give a complete characterization of the language invariant probability functions satisfying WIP which generalizes the Nix-Paris Continuum. We argue that the derivation of two very disparate families of inductive methods from alternative perceptions of (...) 'irrelevance' is an indication that this notion is imperfectly understood at present. (shrink)

We define certain properties of subsets of models of arithmetic related to their codability in end extensions and elementary end extensions. We characterize these properties using some more familiar notions concerning cuts in models of arithmetic.

We argue in favour of identifying one aspect of rational choice with the tendency to conform to the choice you expect another like-minded, but non-communicating, agent to make and study this idea in the very basic case where the choice is from a non-empty subset K of 2 A and no further structure or knowledge of A is assumed.

Rational Pavelka logic extends Lukasiewicz infinitely valued logic $by adding truth constants \bar{r} for rationals in [0, 1].$ We show that this is a conservative extension. We note that this shows that provability degree can be defined in Lukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.

We consider the version of Pure Inductive Logic which obtains for the language with equality and a single unary function symbol giving a complete characterization of the probability functions on this language which satisfy Constant Exchangeability.

Frank Plumpton Ramsey (1903–30) made seminal contributions to philosophy, mathematics and economics. Whilst he was acknowledged as a genius by his contemporaries, some of his most important ideas were not appreciated until decades later; now better appreciated, they continue to bear an influence upon contemporary philosophy. His historic significance was to usher in a new phase of analytic philosophy, which initially built upon the logical atomist doctrines of Bertrand Russell and Ludwig Wittgenstein, raising their ideas to a new level of (...) sophistication, but ultimately he became their successor rather than remain a mere acolyte. (shrink)

During two centuries of industrial revolution, history's most powerful ruling class has been produced, equipped, and armed to the teeth --not just with bullets but also with powerful media and an aggressive ideology of domination. Increasingly, the democratic institutions crafted at the dawn of capitalism are being undermined or overrun by corporate and financial overseers. Despite the fact that history gives ample reason to fear the worst for the future, social and political theory can be a form of resistance and (...) hope. The papers in this volume express this hope, exploring progressive and liberatory institutional conceptions; analyzing multiple experiences of alienation and culture; reconceiving gender, sexuality, and desire; and scrutinizing humanitarian intervention for both corrupted elements and future possibilities for the just defense of the defenseless. These papers were selected from among the best of those presented at the RPA's 4th biennial conference, held in November 2000 at Loyola University - Chicago. (shrink)

The series of conversations between Angela Y. Davis and Eduardo Mendieta entitled Abolition Democracy is a powerful investigation of the failed moral imagination of imperial democracies. After examining their discussion of how truncated political discourses enable abuses in both war and imprisonment, I look to the “exceptional” status of war prisons such as at Guantánamo and Abu Ghraib. I argue that domestic prisons, like international war prisons, are means for the paradigmatic functioning of the exception in modern democracy, as described (...) by Giorgio Agamben, and thus constitute no less of an “ultimate carceral threat.” Within the domestic prison, the legal status of inmates is virtually suspended and they are reduced to bare life. I conclude that we may yet share the hopes of Davis and Mendieta for an abolition democracy, and that such a democracy would bear the echoes of the unconditional sovereignty “to come” theorized by Jacques Derrida. (shrink)

The series of conversations between Angela Y. Davis and Eduardo Mendieta entitled Abolition Democracy is a powerful investigation of the failed moral imagination of imperial democracies. After examining their discussion of how truncated political discourses enable abuses in both war and imprisonment, I look to the “exceptional” status of war prisons such as at Guantánamo and Abu Ghraib. I argue that domestic prisons, like international war prisons, are means for the paradigmatic functioning of the exception in modern democracy, as described (...) by Giorgio Agamben, and thus constitute no less of an “ultimate carceral threat.” Within the domestic prison, the legal status of inmates is virtually suspended and they are reduced to bare life. I conclude that we may yet share the hopes of Davis and Mendieta for an abolition democracy, and that such a democracy would bear the echoes of the unconditional sovereignty “to come” theorized by Jacques Derrida. (shrink)

The logical consequence relations η▹η provide a very attractive way of inferring new facts from inconsistent knowledge bases without compromising standards of credibility. In this short note we provide proof theories and completeness theorems for these consequence relations which may have some applicability in small examples.