Online research in philosophy

This paper has two main purposes. First, it will provide an introductory discussion of hyperset theory, and show that it is useful for modeling complex systems. Second, it will use hyperset theory to analyze Robert Rosen’s metabolismrepair systems and his claim that living things are closed to efficient cause. It will also briefly compare closure to efficient cause to two other understandings of autonomy, operational closure and catalytic closure.

In this paper we show that some standard topological constructions may be fruitfully used in the theory of closure spaces (see [5], [4]). These possibilities are exemplified by the classical theorem on the universality of the Alexandroff's cube for T 0-closure spaces. It turns out that the closure space of all filters in the lattice of all subsets forms a generalized Alexandroff's cube that is universal for T 0-closure spaces. By this theorem we obtain the following characterization of the consequence operator of the classical logic: If is a countable set and C: P() P() is a closure operator on X, then C satisfies the compactness theorem iff the closure space ,C is homeomorphically embeddable in the closure space of the consequence operator of the classical logic.We also prove that for every closure space X with a countable base such that the cardinality of X is not greater than 2 there exists a subset X of irrationals and a subset X of the Cantor's set such that X is both a continuous image of X and a continuous image of X.

The main result of this paper is the following theorem: a closure space X has an , , Q-regular base of the power iff X is Q-embeddable in It is a generalization of the following theorems:(i) Stone representation theorem for distributive lattices ( = 0, = , Q = ), (ii) universality of the Alexandroff's cube for T 0-topological spaces ( = , = , Q = 0), (iii) universality of the closure space of filters in the lattice of all subsets for , -closure spaces (Q = 0). By this theorem we obtain some characterizations of the closure space given by the consequence operator for the classical propositional calculus over a formalized language of the zero order with the set of propositional variables of the power . In particular we prove that a countable closure space X is embeddable with finite disjunctions preserved into F iff X is a consistent closure space satisfying the compactness theorem and X contains a 0, -base consisting of -prime sets.

Causal closure arguments against interactionist dualism are currently popular amongst physicalists. Such an argument appeals to some principles of the causal closure of the physical, together with certain other premises, to conclude that at least some mental events are identical with physical events. However, it is crucial to the success of any such argument that the physical causal closure principle to which it appeals is neither too strong nor too weak by certain standards. In this paper, it is argued that various forms of naturalistic dualism, of an emergentist character, are consistent with the strongest physical causal closure principles that can plausibly be advocated.

The question whether epistemological concepts are closed under deduction is an important one since many skeptical arguments depend on closure. Such skepticism can be avoided if closure is not true of knowledge (or justification). This response to skepticism is rejected by Peter Klein and others. Klein argues that closure is true, and that far from providing the skeptic with a powerful weapon for undermining our knowledge, it provides a tool for attacking the skeptic directly. This paper examines various arguments in favor of closure and Klein's attempted use of closure to refute skepticism. Such a refutation of skepticism is mistaken. But the closure principle is in any case false, so the skepticism that depends on it is undermined. The appeal of the closure principle derives from a failure to recognize an important feature of our epistemological concepts, namely, their context relativity.

Epistemic closure, the idea that knowledge is closed under known implication, plays a central role in current discussions of skepticism and the semantics of knowledge reports. Contextualists in particular rely heavily on the truth of epistemic closure in staking out their distinctive response to the so-called "skeptical paradox." I argue that contextualists should re-think their commitment to closure. Closure principles strong enough to force the skeptical paradox on us are too strong, and closure principles weak enough to express unobjectionable epistemic principles are too weak to generate the skeptical paradox. I briefly consider how the contextualist might live without (strong) closure.

Graham and Maitzen think my CORNEA principle is in trouble because it entails “intolerable violations of closure under known entailment.” I argue that the trouble arises from current befuddlement about closure itself, and that a distinction drawn by Rudolph Carnap, suitably extended, shows how closure, when properly understood, works in tandem with CORNEA. CORNEA does not obey Closure because it shouldn’t: it applies to “dynamic” epistemic operators, whereas closure principles hold only for “static” ones. What the authors see as an intolerable vice of CORNEA is actually a virtue, helping us see what closure principles should—and shouldn’t—themselves be about.

Graham and Maitzen think my CORNEA principle is in trouble because it entails “intolerable violations of closure under known entailment.” I argue that the trouble arises from current befuddlement about closure itself, and that a distinction drawn by Rudolph Carnap, suitably extended, shows how closure, when properly understood, works in tandem with CORNEA. CORNEA does not obey Closure because it shouldn’t: it applies to “dynamic” epistemic operators, whereas closure principles hold only for “static” ones. What the authors see as an intolerable vice of CORNEA is actually a virtue, helping us see what closure principles should—and shouldn’t—themselves be about.

Closure is the principle that a person, who knows a proposition p and knows that p entails q, also knows q. Closure is usually regarded as expressing the commonplace assumption that persons can increase their knowledge through inference from propositions they already know. In this paper, I will not discuss whether closure as a general principle is true. The aim of this paper is to explore the various relations between closure and knowledge through inference. I will show that closure can hold for two propositions p and q for numerous different reasons. The standard reason that S knows q through inference from p, if S knows p and knows that p entails q, is only one of them. Therefore, the relations between closure and inferential knowledge are more complex than one might suspect.

How should the contrastivist formulate closure? That is, given that knowledge is a ternary contrastive state Kspq (s knows that p rather than q), how does this state extend under entailment? In what follows, I will identify adequacy conditions for closure, criticize the extant invariantist and contextualist closure schemas, and provide a contrastive schema based on the idea of extending answers. I will conclude that only the contrastivist can adequately formulate closure.