Steps Toward a Computational Metaphysics

In this paper, the authors describe their initial investigations in computational metaphysics. Our method is to implement axiomatic metaphysics in an automated reasoning system. In this paper, we describe what we have discovered when the theory of abstract objects is implemented in prover9 (a first-order automated reasoning system which is the successor to otter). After reviewing the second-order, axiomatic theory of abstract objects, we show (1) how to represent a fragment of that theory in prover9’s first-order syntax, and (2) how prover9 then finds proofs of interesting theorems of metaphysics, such as that every possible world is maximal. We conclude the paper by discussing some issues for further research.     

Inductive Incompleteness

Nelson Goodman cast the ‘problem of induction’ as the task of articulating the principles and standards by which to distinguish valid from invalid inductive inferences. This paper explores some logical bounds on the ability of a rational reasoner to accomplish this task. By a simple argument, either an inductive inference method cannot admit its own fallibility, or there exists some non-inferable hypothesis whose non-inferability the method cannot infer (violating the principle of ‘negative introspection’). The paper discusses some implications of this limited self-knowledge for the justifiability of inductive inferences, auto-epistemic logic, and the epistemic foundations of game theory.     

Fictionalism and Incompleteness

The modal fictionalist faces a problem due to the fact that her chosen story seems to be incomplete—certain things are neither fictionally true nor fictionally false. The significance of this problem is not localized to modal fictionalism, however, since many fictionalists will face it too. By examining how the fictionalist should analyze the notion of truth according to her story, and, in particular, the role that conditionals play for the fictionalist, I develop a novel and elegant solution to the incompleteness problem.     

Modal Incompleteness Revisited

In this paper, we are going to analyze the phenomenon of modal incompleteness from an algebraic point of view. The usual method of showing that a given logic L is incomplete is to show that for some L and some cannot be separated from by a suitably wide class of complete algebras — usually Kripke algebras. We are going to show that classical examples of incomplete logics, e.g., Fine logic, are not complete with respect to any class of complete BAOs. Even above Grz it is possible to find a continuum of such logics, which immediately implies the existence of a continuum of neighbourhood-incomplete Grz logics. Similar results can be proved for Löb logics. In addition, completely incomplete logics above Grz may be found uniformly as a result of failures of some admissible rule of a special kind.     

包容多样性的理论建构

Ye和Stam (2012)的评论中将Hong, Chao, Yang和Rosner (2010)标签成逻辑经验主义派, 他们认为这个派别常常“限制”了理论的应用和其本身的价值。为了向读者们提供一个对理论建立过程的更完整的理解, 同时也为了从这些迥异的观点中汲取更多知识, 在这篇文章中, 我们比较 Hong等人(2010)与Ye和Stam (2012)所建议的两种不同方法论之间的差异, 并将其放入更广阔的历史背景下进行论述。我们的主张是：每一种方法论都是有它自己的适用范围和局限性。对研究者来说, 保持在对待不同论点上开明和宽容的态度, 并避免本质化不同方法论间的差异也是十分重要的。     

The Incompleteness of RGL

RGLis a version of the modal logic GLbased on the relevant logic R. It is shown that the class of RKframes that verify all theorems of RGLalso verify a scheme that we call (!). If RGLhas (!) as a theorem, however, it is not a relevant logic. I go on to show that not all instances of (!) are theorems of RGL, hence this logic is not complete over any class of RKframes.     

Rorty's Debt to Sellarsian Metaphysics

Rorty regards himself as furthering the project of the Enlightenment by separating Enlightenment liberalism from Enlightenment rationalism. To do so, he rejects the very need for explicit metaphysical theorizing. Yet his commitments to naturalism, nominalism, and the irreducibility of the normative come from the metaphysics of Wilfrid Sellars. Rorty's debt to Sellars is concealed by his use of Davidsonian arguments against the scheme/content distinction and the nonsemantic concept of truth. The Davidsonian arguments are used for Deweyan ends: to advance secularization and anti‐authoritarianism. However, Rorty's conflation of theology and metaphysics conceals the possibility of post‐theological metaphysics. The key distinction lies between “metaphysics” and “Metaphysics.” The former provisionally models the relations between different vocabularies; the latter continues theology by other means. Sellars shows how to do metaphysics without Metaphysics. This approach complements Rorty's prioritization of cultural politics over ontology and his vision of Enlightenment liberalism without Enlightenment rationalism.     

Kant's Commitment to Metaphysics of Morals

A definitive feature of Kant's moral philosophy is its rationalism. Kant insists that moral theory, at least at its foundation, cannot take account of empirical facts about human beings and their circumstances in the world. This is the core of Kant's commitment to ‘metaphysics of morals’, and it is what he sees as his greatest contribution to moral philosophy. The paper clarifies what it means to be committed to metaphysics of morals, why Kant is committed to it, and where he thinks empirical considerations may enter moral theory. The paper examines recent work of contemporary Kantians (Barbara Herman, Allen Wood, and Christine Korsgaard) who argue that there is a central role for empirical considerations in Kant's moral theory. Either these theorists interpret Kant himself as permitting empirical considerations to enter, or they propose to extend Kant's theory so as to allow them to enter. With some qualifications, I argue that these interpretive trends are not supported by the texts, and that the proposed extensions are not plausibly Kantian. Kant's insistence on the exclusion of empirical considerations from the foundations of moral theory is not an incidental feature of his thought which might be modified while the rest remains unchanged. Rather, it is the very centre of his endeavours in moral philosophy. If we disagree with it, I argue, we have grounds for moving to a distinctly different theoretical framework.     

The Incompleteness of Ideal Theory

Can one give an account of a perfectly just society without invoking principles governing our responses to injustice? My claim is that addressing this question puts us in a position to reveal ambiguities and problems with the way in which Rawls draws the ideal/nonideal theory distinction that have so far gone unnoticed. In the first part of my paper, I demonstrate that Rawls’s original definition of the ideal/nonideal theory distinction is ambiguous as it is composed of two different conceptual distinctions, before clarifying the distinctions involved, paying particular attention to the unfamiliar distinction between primary and secondary principles. I then show that we can best account for what Rawls is actually doing at the level of ideal and nonideal theory by invoking this distinction between primary and secondary principles. This result sets the stage for my argument in the second part. I first explain why Rawls does not have access to an understanding of the strict compliance condition that can account for the irrelevance of secondary principles for a complete account of the principles regulating a perfectly just basic structure. I then point out that there is a tension between what Rawls claims to be doing at the level of ideal theory and what he is actually doing at the level of ideal theory. On this basis, I argue that Rawls’s ideal (domestic and international) conceptions of justice are incomplete because they do not encompass secondary principles. The Conclusion unpacks the contributions this article makes to the ideal/nonideal theory debate.     

Towards a Theological Metaphysics of Death

Death is an inescapable event. It defines our existence as mortal. It is both a marker of our finitude and a portal to a deeper mystery: what awaits us in, through, and beyond this moment of ending? In thinking about death in the context of a theological dialogue, I identify two key questions: the phenomenological ‘What is death?’ and the philosophical ‘What is the meaning of death?’ as the beginning points of reflection. In general, it is the function of religion to provide some sort of answer to these questions, to give death both coherent definition and adequate understanding. The primary religious response is to articulate appropriate beliefs. But in their intellectual formation and construction, beliefs draw on metaphysics—the structures of logic, language, conceptuality and general worldview presuppositions by which we render all things intelligible and communicable. In this article I shall undertake an exploration of Christian viewpoints on death, which will require noting antecedents and corollaries in Judaism, then discuss possible metaphysical readings of death, that is, philosophical understandings of death that lie within and under, as it were, the otherwise stated religious beliefs about death. The aim of this article is not to present a fully developed ‘theological metaphysics of death’ as such; rather, in the context of an inter-religious dialogical engagement, to raise issues and perspectives from a Christian point of view that might contribute to a wider, more encompassing, theistically oriented understanding of death.     

The Incompleteness of Theories of Games

We first state a few previously obtained results that lead to general undecidability and incompleteness theorems in axiomatized theories that range from the theory of finite sets to classical elementary analysis. Out of those results we prove several incompleteness theorems for axiomatic versions of the theory of noncooperative games with Nash equilibria; in particular, we show the existence of finite games whose equilibria cannot be proven to be computable.     

Incompleteness and the Barcan formula

A (normal) system of prepositional modal logic is said to be complete iff it is characterized by a class of (Kripke) frames. When we move to modal predicate logic the question of completeness can again be raised. It is not hard to prove that if a predicate modal logic is complete then it is characterized by the class of all frames for the propositional logic on which it is based. Nor is it hard to prove that if a propositional modal logic is incomplete then so is the predicate logic based on it. But the interesting question is whether a complete propositional modal logic can have an incomplete extension. In 1967 Kripke announced the incompleteness of a predicate extension of S4. The purpose of the present article is to present several such systems. In the first group it is the systemswith the Barcan Formula which are incomplete, while those without are complete. In the second group it is thosewithout the Barcan formula which are incomplete, while those with the Barcan Formula are complete. But all these are based on propositional systems which are characterized by frames satisfying in each case a single first-order sentence.     

On Interpreting Chaitin's Incompleteness Theorem

The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin's famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of the strength of the theory.I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental.     

Incompleteness and the Halting Problem

Studia Logica - We present an abstract framework in which we give simple proofs for Gödel’s First and Second Incompleteness Theorems and obtain, as consequences, Davis’,...