This article develops a theory of reasons that has strong similarities to Peter Klein's infinitism. The view it develops, Framework Reasons, upholds Klein's principles of avoiding arbitrariness (PAA) and avoiding circularity (PAC) without requiring an infinite regress of reasons. A view of reasons that holds that the “reason for” relation is constrained by PAA and that PAC can avoid an infinite regress if the “reason for” relation is contextual. Moreover, such a view of reasons can maintain that skepticism is false (...) by the maintaining that there is more to epistemic justification than can be expressed in any reasoning session. One crucial argument for Framework Reasons is that justification depends on a background of plausibility considerations. The final section of the article applies this view of reasons to Michael Bergmann's argument that any nonskeptical epistemology must embrace epistemic circularity. (shrink)

The notion of an ideal reasoner has several uses in epistemology. Often, ideal reasoners are used as a parameter of (maximum) rationality for finite reasoners (e.g. humans). However, the notion of an ideal reasoner is normally construed in such a high degree of idealization (e.g. infinite/unbounded memory) that this use is unadvised. In this dissertation, I investigate the conditions under which an ideal reasoner may be used as a parameter of rationality for finite reasoners. In addition, I present (...) and justify the research program of computational epistemology, which investigates the parameter of maximum rationality for finite reasoners using computer simulations. (shrink)

Nonmonotonic reasoning is a discipline of computer science, epistemology, and cognition: It models inferences where classical logic is inadequate in symbolic AI, defines normative models for reasoning with defeasible information in epistemology, and models human reasoning under information change in cognition. Its building blocks are defeasible rules formalised as DeFinetti conditionals. In this thesis, Christian Eichhorn examines qualitative and semi-quantitative inference relations on top said conditionals, using the conditional structure of the knowledge base and Spohn’s Ordinal Conditional (...) Functions, using established properties. Converting network approaches from probabilistics, he shows how to approach the relations with regard to implementation. (shrink)

In this paper I argue that de Finetti provided compelling reasons for rejecting countable additivity. It is ironical therefore that the main argument advanced by Bayesians against following his recommendation is based on the consistency criterion, coherence, he himself developed. I will show that this argument is mistaken. Nevertheless, there remain some counter-intuitive consequences of rejecting countable additivity, and one in particular has all the appearances of a full-blown paradox. I will end by arguing that in fact it is no (...) paradox, and that what it shows is that conditionalisation, often claimed to be integral to the Bayesian canon, has to be rejected as a general rule in a finitely additive environment. (shrink)

The negative zombie argument concludes that physicalism is false from the premises that p ∧¬q is ideally negatively conceivable and that what is ideally negatively conceivable is possible, where p is the conjunction of the fundamental physical truths and laws and q is a phenomenal truth (Chalmers 2002; 2010). A sentence φ is ideally negatively conceivable iff φ is not ruled out a priori on ideal rational reflection. In this paper, I argue that the negative zombie argument is neither a (...) priori nor conclusive. First, I argue that the premises of the argument are true only if there exists an adequate finite ideal reasoner R that believes ◊(p ∧ ¬q) on the basis of not believing p→q on a priori basis. Roughly, a finite reasoner is a reasoner with cognitive limitations (e.g. finite memory). I argue that R is finite only if R reasons nonmonotonically and only approach ideal reflection at the limit of a reasoning sequence. This would render the argument nonconclusive. Finally, I argue that, for some q, R does not believe ◊(p ∧ ¬q) on the basis of not believing p→q on a priori basis (e.g. for q =‘something is conscious’). This would render the choice of an adequate q dependent on empirical information (and the argument a posteriori). I conclude that the negative zombie argument (and, maybe, all zombie arguments) is neither a priori nor conclusive. (shrink)

Neo-Fregeanism contends that knowledge of arithmetic may be acquired by second-order logical reflection upon Hume's principle. Heck argues that Hume's principle doesn't inform ordinary arithmetical reasoning and so knowledge derived from it cannot be genuinely arithmetical. To suppose otherwise, Heck claims, is to fail to comprehend the magnitude of Cantor's conceptual contribution to mathematics. Heck recommends that finite Hume's principle be employed instead to generate arithmetical knowledge. But a better understanding of Cantor's contribution is achieved if it is (...) supposed that Hume's principle really does inform arithmetical practice. More generally, Heck's arguments misconceive the epistemological character of neo-Fregeanism. (shrink)

0.0. Theistic Ethics as a Challenge and a Diagnostic Tool. Naturalistic conceptions in metaethics come in many varieties. Many philosophers who have sought to situate moral reasoning in a naturalistic metaphysical conception have thought it necessary to adopt non-cognitivist, prescriptivist, projectivist, relativist, or otherwise deflationary conceptions. Recently there has been a revival of interest in non-deflationary moral realist approaches to ethical naturalism. Many non-deflationary approaches have exploited the resources of non-empiricist “causal” or “naturalistic” conceptions of reference and of kind (...) definitions in service of the “naturalistic” metaphilosophical conception that substantive moral questions, and questions about the metaphysics of morals, are broadly a posteriori questions, somewhat analogous to scientific questions, and are not amenable to a priori resolution by “conceptual analysis.”. (shrink)

0.0. Theistic Ethics as a Challenge and a Diagnostic Tool. Naturalistic conceptions in metaethics come in many varieties. Many philosophers who have sought to situate moral reasoning in a naturalistic metaphysical conception have thought it necessary to adopt non-cognitivist, prescriptivist, projectivist, relativist, or otherwise deflationary conceptions. Recently there has been a revival of interest in non-deflationary moral realist approaches to ethical naturalism. Many non-deflationary approaches have exploited the resources of non-empiricist “causal” or “naturalistic” conceptions of reference and of kind (...) definitions in service of the “naturalistic” metaphilosophical conception that substantive moral questions, and questions about the metaphysics of morals, are broadly a posteriori questions, somewhat analogous to scientific questions, and are not amenable to a priori resolution by “conceptual analysis.”. (shrink)

Finite-state methods are applied to determine the consequences of events, represented as strings of sets of fluents. Developed to flesh out events used in natural language semantics, the approach supports reasoning about action in AI, including the frame problem and inertia. Representational and inferential aspects of the approach are explored, centering on conciseness of language, context update and constraint application with bias.

By operations on models we show how to relate completeness with respect to permissivenominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissive-nominal models, so the construction hinges on generating from an instance of the latter, some instance of the former in which sufficiently many inequalities are preserved between elements. We do this using an infinite generalisation of nominal atoms-abstraction. The results are of interest in their (...) own right, but also, we factor the mathematics so as to maximise the chances that it could be used off-the-shelf for other nominal reasoning systems too. Models with infinite support can be easier to work with, so it is useful to have a semi-automatic theorem to transfer results from classes of infinitely-supported nominal models to the more restricted class of models with finite support. In conclusion, we consider different permissive-nominal syntaxes and nominal models and discuss how they relate to the results proved here. (shrink)

Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond NP ≠ coNP. These conjectures formally connect computational complexity with the difficulty of proving some sentences, which means that high computational complexity of a problem associated with a sentence implies that the sentence is not provable in a weak theory, or requires a long proof. Another reason for putting forward these conjectures is that some results in proof complexity seem to (...) be special cases of such general statements and we want to formalize and fully understand these statements. Roughly speaking, we are trying to connect syntactic complexity, by which we mean the complexity of sentences and strengths of the theories in which they are provable, with the semantic concept of complexity of the computational problems represented by these sentences. -/- We have introduced the most fundamental conjectures in our earlier works. Our aim in this article is to present them in a more systematic way, along with several new conjectures, and prove new connections between them and some other statements studied before. (shrink)

In the longstanding foundational debate whether to require that probability is countably additive, in addition to being finitely additive, those who resist the added condition raise two concerns that we take up in this paper. (1) _Existence_: Settings where no countably additive probability exists though finitely additive probabilities do. (2) _Complete Additivity_: Where reasons for countable additivity don’t stop there. Those reasons entail complete additivity—the (measurable) union of probability 0 sets has probability 0, regardless the cardinality of that union. Then (...) probability distributions are discrete, not continuous. We use Easwaran’s (Easwaran, Thought 2:53–61, 2013) advocacy of the _Comparative_ principle to illustrate these two concerns. Easwaran supports countable additivity, both for numerical probabilities and for finer, qualitative probabilities, by defending a condition he calls the _Comparative_ principle [ \({\mathcal{C}}\) ]. For numerical probabilities, principle \({\mathcal{C}}\) contrasts pairs, P 1 and P 2, defined over a common partition \(\prod\) = {a _i_ : _i_ ∈ I} of measurable events. \({\mathcal{C}}\) requires that no P 1 may be pointwise dominated, i.e., no (finitely additive) probability P 2 exists such that for each _i_ ∈ I, P 2 (a _i_ ) > P 1 (a _i_ ). By design, the cardinality of \(\prod\) is not limited in \({\mathcal{C}}\), which Easwaran asserts is important when arguing that the principle does not require more, or less, than that probability is countably additive. We agree that a numerical probability P satisfies principle \({\mathcal{C}}\) in all partitions just in case P is countably additive. However, we show that for numerical probabilities, by considering the size of the algebra of events to which probability is applied, principle \({\mathcal{C}}\) is subject to each of the above concerns, (1) and (2). Also, Easwaran considers principle \({\mathcal{C}}\) with non-numerical, qualitative probabilities, where a qualitative probability may be finer than an almost agreeing numerical probability P. A qualitative probability is regular if possible events are strictly more likely than impossible events. Easwaran motivates and illustrates regular qualitative probabilities using a continuous, almost agreeing quantitative probability that is uniform on the unit interval. We make explicit the conditions for applying principle \({\mathcal{C}}\) with qualitative probabilities and show that \({\mathcal{C}}\) restricts regular qualitative probabilities to those whose almost agreeing quantitative probabilities are completely additive. For instance, Easwaran’s motivating example of a regular qualitative probability is precluded by principle \({\mathcal{C}}\). (shrink)

We are concerned with the following problem. Suppose Γ and Σ are closed permutation groups on infinite sets C and W and ρ: Γ → Σ is a non-split, continuous epimorphism with finite kernel. Describe the possibilities for ρ. Here, we consider the case where ρ arises from a finite cover π: C → W. We give reasonably general conditions on the permutation structure W;Σ which allow us to prove that these covers arise in two possible ways. The (...) first way, reminiscent of covers of topological spaces, is as a covering of some Σ-invariant digraph on W. The second construction is less easy to describe, but produces the most familiar of these types of covers: a vector space covering its projective space. (shrink)

We agree with Hilbert’s assessment that the concept of ‘infinite’ stands in need of clarification – however, our proposed ‘solution’ is almost diammetrically opposed to that of Hilbert.

Moral particularists argue that because reasons for action are irreducibly context-dependent, the traditional quest in ethics for true and exceptionless moral principles is hopelessly misguided. In making this claim, particularists assume a general framework according to which reasons are the ground floor normative units undergirding all other normative properties and relations. They then argue that there is no cashing out in finite terms either (i) when a given non-normative feature gives rise to a reason for or against action, or (...) (ii) how the reasons that are present in a given context play off each other to determine one’s overall duties. However, the conjunction of these two theses leaves particularists without a coherent notion of a reason for action: posit too much irreducible context-dependence in the behavior of reasons, and the reasons-based framework breaks down. One upshot is that the particularists’ challenge to principle-based approaches to ethics has not, at present, been successfully made out; another upshot is that perhaps the best way to formulate that challenge involves renouncing the reasons-based framework all together. (shrink)

Our relationship to the infinite is controversial. But it is widely agreed that our powers of reasoning are finite. I disagree with this consensus; I think that we can, and perhaps do, engage in infinite reasoning. Many think it is just obvious that we can't reason infinitely. This is mistaken. Infinite reasoning does not require constructing infinitely long proofs, nor would it gift us with non-recursive mental powers. To reason infinitely we only need an ability to (...) perform infinite inferences. I argue that we have this ability. My argument looks to our best current theories of inference and considers examples of apparent infinite reasoning. My position is controversial, but if I'm right, our theories of truth, mathematics, and beyond could be transformed. And even if I'm wrong, a more careful consideration of infinite reasoning can only deepen our understanding of thinking and reasoning. -/- (Note for readers: the paper's brief discussion of uniform reflection and omega inconsistency is misleading. The imagined interlocutor's argument makes an assumption about the PA-provability of provability generalizations that, while true for the Godel sentence's instances, is unjustified, in general. This means my position is stronger against this objection than the paper suggests, since omega inconsistent theories are not automatically inconsistent with their uniform reflection principles, you also need to assume the arithmetically true Pi-2 sentences.). (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.1 (2003) 25-49 [Access article in PDF] Descartes's Nomic Concurrentism:Finite Causation and Divine Concurrence Andrew Pessin DESCARTES APPEARS TO HOLD the traditional view that God acts in the world via willing. 1 In recent papers on his successor Malebranche, who also holds that view, I have argued that since volitions are paradigm representational states, close attention to the representational content of God's (...) volitions has substantial exegetical rewards. 2 The same, I will argue, is true regarding Descartes. In particular, due to lack of such attention, scholars have largely failed to realize that Descartes retains [End Page 25] a version of the scholastic doctrine that God ordinarily concurs with the causal activities of both mental and material finite substances ("concurrentism"). 3 Since concurrentism implies the true causal powers of those finite substances, in exposing Descartes's adherence to it I aim to open up new ways of thinking about some old problems in Cartesian exegesis.The plan:(I) I will first sketch a partial "worldview," along with evidence that Descartes held something like it, then(II) argue, both on the basis of this worldview and other evidence, that we should interpret Descartes's account of divine and finite causality as a version of concurrentism, then(III) show how interpreting Descartes this way has significant implications for understanding his views on the causal powers of both mental and material finite substances, and finally(IV) contrast my interpretation with two recent relevant ones: Kenneth Clatterbaugh's and Daniel Garber's. 1. A Partial Worldview If Descartes's God acts on the world via willing, we may inquire into the contents of those volitions. I suggest that Descartes's God has, among others, distinguishable volitions with roughly the following contents. 4 These constitute a partial worldview (W), and will be discussed in turn:(V1) At all times beginning with original Creation matter exists everywhere.(V2) There is at original Creation a particular quantity q and particular distribution d of motion.(V3) At all times, the laws of motion apply.(V4) Mind m exists at time t (appropriately substituted).(V5) At all times, the laws of mind-body union apply.Our first task is to accommodate God's role in the Cartesian system as the continuous conserver of the material world. 5 One divine volition which would accomplish that might have the content, not that each individual body persists, but rather, simply, that [End Page 26](V1) at all times beginning with original Creation matter exists everywhere. 6Now (V1) may require us to rethink the way continuous conservation arguments are typically framed. In the texts of La Forge and Malebranche, for example, what God conserves are individual bodies,and Cartesian texts seem sometimes to suggest the same. 7 Such framing might suggest, contra (V1), that God's conservational volitional contents make explicit reference to each individual body. Nevertheless, there are reasons to think that Descartes's God's conserving activity is and ought to be framed as in (V1):(i) God's willing that (V1) provides an easy way to reconcile Cartesian mind —> body causal interaction with continuous conservation. 8 Many think that there is a tension between these two doctrines, since God, in conserving bodies, would seem to be responsible for all their modes including motion, leaving nothing for minds to contribute to them. Now that might be true where God's conserving volition explicitly represents each individual body, particularly given the role that motion plays in distinguishing individual Cartesian bodies. 9 (V1), however, in making no reference to individual bodies, is perfectly consistent with minds affecting either the quantity or distribution of motion in the material world.(ii) Similarly in Physiologia, Des Chene convincingly defends the claim that for Descartes, "since the existence of motion... is not entailed by the existence of matter, the creation of motion [is] independent of the creation of matter" (329). In light, again, of Descartes's conception of individual bodies (n. 9, below), this independence is best modeled by distinguishing God's conserving volition from those volitions... (shrink)

Reasoning about change is a central issue in research on human and robot planning. We study an approach to reasoning about action and change in a dynamic logic setting and provide a solution to problems which are related to the Frame problem. Unlike most work on the frame problem the logic described in this paper is monotonic. It (implicitly) allows for the occurrence of actions of multiple agents by introducing non-stationary notions of waiting and test. The need to (...) state a large number of frame axioms is alleviated by introducing a concept of chronological preservation to dynamic logic. As a side effect, this concept permits the encoding of temporal properties in a natural way. We compare the relative merits of our approach and non-monotonic approaches as regards different aspects of the frame problem. Technically, we show that the resulting extended systems of propositional dynamic logic preserve (weak) completeness, finite model property and decidability. (shrink)

In this note we shall assume acquaintance with [T4] and the parts of [T1] which deal with intuitionistic arithmetic in all finite types. The bibliography just continues the bibliography of [T4].The principal purpose of this note is the discussion of two models for intuitionistic finite type arithmetic with fan functional. The first model is needed to correct an oversight in the proof of Theorem 6 [T4, §5]: the model ECF+as defined there cannot be shown to have the required (...) properties inEL+ QF-AC, the reason being that a change in the definition ofW12alone does not suffice—if one wishes to establish closure under the operations of HAωthe definitions ofW1σfor other σ have to be adopted as well. It is difficult to see how to do this directly in a uniform way — but we succeed via a detour, which is described in §2.For a proper understanding, we should perhaps note already here thaton the assumption of the fan theorem, ECF+as defined in [T4] and the new model of this note coincide ; but inELit is impossible to prove this. (shrink)

The collapse models of arithmetic are inconsistent, nontrivial models obtained from ℕ and set out in the Logic of Paradox (LP). They are given a general treatment by Priest (Priest, 2000). Finite collapse models are decidable, and thus axiomatizable, because finite. LP, however, is ill-suited to normal axiomatic reasoning, as it invalidates Modus Ponens, and almost all other usual conditional inferences. I set out a logic, A3, first given by Avron (Avron, 1991), and give a first order (...) axiom system for the finite collapse models. I present some standard arithmetical axioms in addition to a cyclic axiom and prove that these axioms are sound and complete for the cyclic models, reporting a similar result for the heap models. The state of the situation for the each of the kinds of infinite collapse model is, however, left an open question. (shrink)

In the current paper we consider theories with vocabulary containing a number of binary and unary relation symbols. Binary relation symbols represent labeled edges of a graph and unary relations represent unique annotations of the graph's nodes. Such theories, which we call annotation theories^ can be used in many applications, including the formalization of argumentation, approximate reasoning, semantics of logic programs, graph coloring, etc. We address a number of problems related to annotation theories over finite models, including satisfiability, (...) querying problem, specification of preferred models and model checking problem. We show that most of considered problems are NPTime- or co-NPTime-complete. In order to reduce the complexity for particular theories, we use second-order quantifier elimination. To our best knowledge none of existing methods works in the case of annotation theories. We then provide a new second-order quantifier elimination method for stratified theories, which is successful in the considered cases. The new result subsumes many other results, including those of [2, 28, 21]. (shrink)

A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information.

We analyze common reasoning about admissibility in the strategic and extensive form of a game. We define a notion of sequential proper admissibility in the extensive form, and show that, in finite extensive games with perfect recall, the strategies that are consistent with common reasoning about sequential proper admissibility in the extensive form are exactly those that are consistent with common reasoning about admissibility in the strategic form representation of the game. Thus in such games the (...) solution given by common reasoning about admissibility does not depend on how the strategic situation is represented. We further explore the links between iterated admissibility and backward and forward induction. (shrink)

Probability logics have been an active topic of investigation of beliefs in type spaces in game theoretical economics. Beliefs are expressed as subjective probability measures. Savage’s postulates in decision theory imply that subjective probability measures are not necessarily countably additive but finitely additive. In this paper, we formulate a probability logic Σ + that is strongly complete with respect to this class of type spaces with finitely additive probability measures, i.e. a set of formulas is consistent in Σ + iff (...) it is satisfied in a finitely additive type space. Although we can characterize Σ + -theories satisfiable in the class as maximally consistent sets of formulas, we prove that any canonical model of maximally consistent sets is not universal in the class of type spaces with finitely additive measures, and, moreover, it is not a type space. At the end of this paper, we show that even a minimal use of probability indices causes the failure of compactness in probability logics. (shrink)

One striking feature of Finite and Eternal Being is Edith Stein’s exceedingly rare use of the term “metaphysics.” She uses the term “formal ontology” numerous times, but the term “metaphysics” only appears a handful of times in the body of the text, and even those references are themselves a bit surprising. This could be explained in several ways, some of which may be quite innocent and have nothing to do with whether she understands her project as metaphysical. In the (...) following, however, I would like to explore a differing explanation and argue that (at least, in part) her reason for avoiding describing her work as metaphysical is connected with the type of philosophical critique she wants to make of traditional metaphysics. I will not argue that Finite and Eternal Being should ultimately be read as a phenomenological analysis of being rather than any sort of metaphysical treatise, but I will argue that Stein has explicitly phenomenological reasons for being cautious about using the term “metaphysics.”. (shrink)

Solid-state processing of metal material is a very complex physical and chemical process, which is coupled by a series of variations including heat transfer, momentum transfer, mass transfer, and phase change. Applying three-dimensional finite element numerical method to the simulation of solid-state processing can perform analysis of metal material’s forging processes before production trial production, can obtain their relevant information such as material flow law, temperature field, and strain field under the minimum physical test conditions, thereby predicting metal material’s (...) forming defects and improving their forging quality. On the basis of summarizing and analyzing previous research works, this paper expounded the current status and significance of solid-state processing of metal materials, elaborated the development background, current status, and future challenges of 3D finite element numerical simulation, introduced the discrimination method and free surface solution method of numerical simulation calculation, conducted finite element model’s geometric assumptions, material selection, element division, model establishment, parameter selection, and initial and boundary condition determination, and simulated and analyzed rheological casting, remelting heating, thixoforming, and rotary piercing processes of metal materials. The results show that the 3D finite element numerical method can not only simulate various processes of flow field, temperature field, stress field, and microstructure in solid-state processing but also can provide a reliable basis for effectively obtaining a reasonable description and finding a more optimized design plan for metal material processing in a short time, which plays an important role in understanding and analyzing solid metal forming process, controlling and optimizing process parameters, guiding and mastering rheological casting, and secondary heating and rotary piercing of metal materials. (shrink)

A primary problem in the area of natural language processing has been semantic analysis. This book looks at the semantics of natural languages in context. It presents an approach to the computational processing of English text that combines current theories of knowledge representation and reasoning in Artificial Intelligence with the latest linguistic views of lexical semantics. The book will interest postgraduates and researchers in computational linguistics as well as industrial research groups specializing in natural language processing.

I explain the difficulty of making various concepts of and relating to probability precise, rigorous and physically significant when attempting to apply them in reasoning about objects living in infinite-dimensional spaces, working through many examples from cosmology. I focus on the relation of topological to measure-theoretic notions of and relating to probability, how they diverge in unpleasant ways in the infinite-dimensional case, and are even difficult to work with on their own. Even in cases where an appropriate family of (...) spacetimes is finite-dimensional, and so admits a measure of the relevant sort, however, it is always the case that the family is not a compact topological space, and so does not admit a physically significant, well behaved probability measure. Problems of a different but still deeply troubling sort plague arguments about likelihood in that context, which I also discuss. I conclude that most standard forms of argument used in cosmology to estimate the likelihood of the occurrence of various properties or behaviors of spacetimes have serious mathematical, physical and conceptual problems. (shrink)

This paper considers the existence of finite equational axiomatisations of behavioural equivalences over a calculus of finite state processes. To express even simple properties such as μxE = μxE[E/x] some notation for substitutions is required. Accordingly, the calculus is embedded in a simply typed lambda calculus, allowing such schemas to be expressed as equations between terms containing first order variables. A notion of first order trace congruence over such terms is introduced and used to show that no (...) class='Hi'>finite set of such equations is sound and complete for any reasonable equivalence finer than trace equivalence. The intermediate results are then applied to give two nonaxiomatisability results over calculi of regular expressions. (shrink)

We provide first-order axioms for the theories of finite trees with bounded branching and finite trees with arbitrary (finite) branching. The signature is chosen to express, in a natural way, those properties of trees most relevant to linguistic theories. These axioms provide a foundation for results in linguistics that are based on reasoning formally about such properties. We include some observations on the expressive power of these theories relative to traditional language complexity classes.

The article explores the idea that according to Spinoza finite thought and substantial thought represent reality in different ways. It challenges “acosmic” readings of Spinoza's metaphysics, put forth by readers like Hegel, according to which only an infinite, undifferentiated substance genuinely exists, and all representations of finite things are illusory. Such representations essentially involve negation with respect to a more general kind. The article shows that several common responses to the charge of acosmism fail. It then argues that (...) we must distinguish the well-founded ideality of representations of finite things from mere illusoriness, insofar as for Spinoza we can have true knowledge of what is known only abstractly. Finite things can be seen as well-founded beings of reason. The article also proposes that within Spinoza's framework it is possible to represent a finite thing without drawing on representations of mind-dependent entities. (shrink)

Perhaps no other aspect of Hans-Georg Gadamer's Wahrheit und Methode has generated more controversy and caustic criticism than his attempt to defend the role of ‘prejudice’ in human understanding. Gadamer's goal in challenging what he calls ‘the Enlightenment's prejudice against prejudice’ is not to defend irresponsible, idiosyncratic, parochial or otherwise self-willed understanding in the human sciences, but to argue that all human cognition is ‘finite’ and ‘limited’ in the sense that it always involves, to borrow Polanyi's phrase, a ‘tacit (...) dimension’ of implicit judgments, concerns, or commitments which shape definitively our grasp of the subject matter in ways we cannot anticipate or control. This implies a ‘finite’ or ‘limited’ view of human understanding in at least two ways. First, Gadamer seems to use the word ‘finite’ in conscious opposition to Hegel's notion of an ‘infinite’ intellect. For Hegel the ‘infinite’ was the ‘unconditioned’ and ‘self-determining’. An ‘infinite’ intellect would be its own master, an autonomous, spontaneous source of all its essential activities and contents. (shrink)

Defeasible conditionals are statements of the form ‘if A then normally B’. One plausible interpretation introduced in nonmonotonic reasoning dictates that ) is true iff B is true in ‘most’ A-worlds. In this paper, we investigate defeasible conditionals constructed upon a notion of ‘overwhelming majority’, defined as ‘truth in a cofinite subset of \’, the first infinite ordinal. One approach employs the modal logic of the frame \\), used in the temporal logic of discrete linear time. We introduce and (...) investigate conditionals, defined modally over \\); several modal definitions of the conditional connective are examined, with an emphasis on the nonmonotonic ones. An alternative interpretation of ‘majority’ as sets cofinal ) rather than cofinite ) is examined. For these modal approaches over \\), a decision procedure readily emerges, as the modal logic \ of this frame is well-known and a translation of the conditional sentences can be mechanically checked for validity; this allows also for a quick proof of \-completeness of the satisfiability problem for these logics. A second approach employs the conditional version of Scott-Montague semantics, in the form of \-many possible worlds, endowed with neighborhoods populated by collections of cofinite subsets of \. This approach gives rise to weak conditional logics, as expected. The relative strength of the conditionals introduced is compared to KLM logics and other conditional logics in the literature. (shrink)

A measure of coherence is said to be reliability conducive if and only if a higher degree of coherence (as measured) results in a higher likelihood that the witnesses are reliable. Recently, it has been proved that several coherence measures proposed in the literature are reliability conducive in a restricted scenario (Olsson and Schubert 2007, Synthese 157:297–308). My aim is to investigate which coherence measures turn out to be reliability conducive in the more general scenario where it is any (...) class='Hi'>finite number of witnesses that give equivalent reports. It is shown that only the so-called Shogenji measure is reliability conducive in this scenario. I take that to be an argument for the Shogenji measure being a fruitful explication of coherence. (shrink)

Automated reasoning issues are addressed for a finite lattice-valued propositional logic LnP(X) with truth-values in a finite lattice-valued logical algebraic structure—lattice implication algebra. We investigate extended strategies and rules from classical logic to LnP(X) to simplify the procedure in the semantic level for testing the satisfiability of formulas in LnP(X) at a certain truth-value level α (α-satisfiability) while keeping the role of truth constant formula played in LnP(X). We propose a lock resolution method at a certain truth-value (...) level α (α-lock resolution) in LnP(X) and have proved its theorems of soundness and weak completeness, respectively. We provide more efficient resolution based automated reasoning in LnP(X) and key supports for α-resolution-based automated reasoning approaches and algorithms in lattice based linguistic truth-valued logic. (shrink)

This paper begins with Hegel’s critique of Kant in the Encyclopaedia’s examination of three positions on objectivity. According to this critique, Kant’s philosophy is flawed because it reduces objectivity to a relation isolated within the subjectivity of the knower, does not integrate the contingent into its understanding of the rational, and does not acknowledge the reality status of contradiction. The second section of the paper examines Hegel’s analysis of dialectical proof procedure in the introductory essays of his major works. The (...) rest of the paper examines the way the Phenomenology proves that rationality is a common ground governing both independent thought and the independence of the natural world, that the contradictory otherness of nature requires an irrational element, which neither observational nor practical thought can overcome, and that truth is an infinite spirit that both transcends and dwells within the finite reality of the human spirit. (shrink)

We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schröder . Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very complex in the following sense: (...) for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 < n < ω. Completely analogous statement holds for the case n ω. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCAn + k. We prove that the complementation-free subreducts of RCAn do not form a variety. We also investigate the reason for the above “non-finite axiomatizability” behaviour of RCAn. We look at all the possible reducts of RCAn and investigate which are finitely axiomatizable. We obtain several positive results in this direction. Finally, we summarize the results and remaining questions in a figure. We carry through the same programme for RPEAn and for RRA. By looking into the reducts we also investigate what other kinds of natural algebras of relations are possible with more positive behaviour than that of the well known ones. Our investigations have direct consequences for the logical properties of the n-variable fragment Ln of first order logic. The reason for this is that RCAn and RPEAn are the natural algebraic counterparts of Ln while the varieties SNrnCAn + k are in connection with the proof theory of Ln. This paper appears in two parts. The first part contains the non-finite axiomatizability results. The present second part contains finite axiomatizations of some fragments together with a figure summarizing the finite and non-finite axiomatizability results in this area and the problems left open. (shrink)

We propose a rudimentary formal framework for reasoning about recursion equations over inductively generated data. Our formalism admits all equational programs , and yet singles out none. While being simple, this framework has numerous extensions and applications. Here we lay out the basic concepts and definitions; show that the deductive power of our formalism is similar to that of Peano's Arithmetic; prove a strong normalization theorem; and exhibit a mapping from natural deduction derivations to an applied λ -calculus, à (...) la Curry–Howard, which provides a transparent proof of Gödel's result that the provably recursive functions of PA are definable by primitive recursion in finite types. Also of interest is the extent, uncommon among similar constructions, to which our mapping is oblivious to first order terms, including first order quantification and equality. Additional applications and extensions of the method will be presented in sequel papers. In particular, the methodology introduced here lends itself to systematic links between natural sub-formalisms and computational complexity classes. Since these sub-formalisms are largely transparent, allowing one to reason formally in the full system while leaving to automatic software tools the task of verifying that restrictions are complied with, the framework is particularly attractive for the development of Feasible Mathematics. (shrink)

This paper presents examples of infinite diagrams (as well as infinite limits of finite diagrams) whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a "pre" form of this thesis that every proof can be presented in everyday statements-only form.

Godel's and Tarski's theorems were inspired by paradoxes: the Richard paradox, the Liar. Godel, in the 1951 Gibbs lecture argued from his metatheoretical results for a metaphysical claim: the impossibility of reducing, both, mathematics to the knowable by the human mind and the human mind to a finite machine (e.g. the brain). So Godel reasoned indirectly from paradoxes for metaphysical theses. I present four metaphysical theses concerning mechanism, reductive physicalism and time for the only purpose of suggesting how it (...) could be argued for them directly from paradoxical sentences. (shrink)