The overall pattern of the evolution of information in dissipative,material systems

I have argued for the reality of a second law of infodynamics. That is, that the information carrying capacity in a system, as viewed from within, must continue to increase as long as the system supports internal observation. I will be concerned here with matters of scale, as well as with system senescence (rigidity produced by the asymptotic approach to maximum information storage within a system). My major categories derive from a view of system development, and are: immaturity, maturity and senescence. As an immature system grows in size, It deploys information over ever smaller scales as it transforms from the vagueness of immaturity toward ever more definite embodiment. This eventually results in an information overload, gradually eroding system adaptability as it increasingly inhibits the variety of system responses by limiting its ability to further qualify stored information;     

A Model of Tolerance

According to the naive theory of vagueness, the vagueness of an expression consists in the existence of both positive and negative cases of application of the expression and in the non-existence of a sharp cut-off point between them. The sorites paradox shows the naive theory to be inconsistent in most logics proposed for a vague language. The paper explores the prospects of saving the naive theory by revising the logic in a novel way, placing principled restrictions on the transitivity of the consequence relation. A lattice-theoretical framework for a whole family of (zeroth-order) “tolerant logics” is proposed and developed. Particular care is devoted to the relation between the salient features of the formal apparatus and the informal logical and semantic notions they are supposed to model. A suitable non-transitive counterpart to classical logic is defined. Some of its properties are studied, and it is eventually shown how an appropriate regimentation of the naive theory of vagueness is consistent in such a logic.     

On Evans's Vague Object from Set Theoretic Viewpoint

Gareth Evans proved that if two objects are indeterminately equal then they are different in reality. He insisted that this contradicts the assumption that there can be vague objects. However we show the consistency between Evans's proof and the existence of vague objects within classical logic. We formalize Evans's proof in a set theory without the axiom of extensionality, and we define a set to be vague if it violates extensionality with respect to some other set. There exist models of set theory where the axiom of extensionality does not hold, so this shows that there can be vague objects.     

Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis

A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis (‘SIA’), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis (‘CA’) without resort to the method of limits. Formally, however, unlike Robinsonian ‘nonstandard analysis’, SIA conflicts with CA, deriving, e.g., ‘not every quantity is either = 0 or not = 0.’ Internally, consistency is maintained by using intuitionistic logic (without the law of excluded middle). This paper examines problems of interpretation resulting from this ‘change of logic’, arguing that standard arguments based on ‘smoothness’ requirements are question-begging. Instead, it is suggested that recent philosophical work on the logic of vagueness is relevant, especially in the context of a Hilbertian structuralist view of mathematical axioms (as implicitly defining structures of interest). The relevance of both topos models for SIA and modal-structuralism as appled to this theory is clarified, sustaining this remarkable instance of mathematical pluralism.     

A Generic Framework for Adaptive Vague Logics

In this paper, we present a generic format for adaptive vague logics. Logics based on this format are able to (1) identify sentences as vague or non-vague in light of a given set of premises, and to (2) dynamically adjust the possible set of inferences in accordance with these identifications, i.e. sentences that are identified as vague allow only for the application of vague inference rules and sentences that are identified as non-vague also allow for the application of some extra set of classical logic rules. The generic format consists of a set of minimal criteria that must be satisfied by the vague logic in casu in order to be usable as a basis for an adaptive vague logic. The criteria focus on the way in which the logic deals with a special ⊡-operator. Depending on the kind of logic for vagueness that is used as a basis for the adaptive vague logic, this operator can be interpreted as completely true, definitely true, clearly true, etc. It is proven that a wide range of famous logics for vagueness satisfies these criteria when extended with a specific ⊡-operator, e.g. fuzzy basic logic and its well known extensions, cf. [7], super- and subvaluationist logics, cf. [6], [9], and clarity logic, cf. [13]. Also a fuzzy logic is presented that can be used for an adaptive vague logic that can deal with higher-order vagueness. To illustrate the theory, some toy-examples of adaptive vague proofs are provided.     

A defence of informational structural realism

This is the revised version of an invited keynote lecture delivered at the 1st Australian Computing and Philosophy Conference (CAP@AU; the Australian National University in Canberra, 31 October–2 November, 2003). The paper is divided into two parts. The first part defends an informational approach to structural realism. It does so in three steps. First, it is shown that, within the debate about structural realism (SR), epistemic (ESR) and ontic (OSR) structural realism are reconcilable. It follows that a version of OSR is defensible from a structuralist-friendly position. Second, it is argued that a version of OSR is also plausible, because not all relata (structured entities) are logically prior to relations (structures). Third, it is shown that a version of OSR is also applicable to both sub-observable (unobservable and instrumentally-only observable) and observable entities, by developing its ontology of structural objects in terms of informational objects. The outcome is informational structural realism, a version of OSR supporting the ontological commitment to a view of the world as the totality of informational objects dynamically interacting with each other. The paper has been discussed by several colleagues and, in the second half, ten objections that have been moved to the proposal are answered in order to clarify it further.     

How (not) to specify normal conditions for response-dependent concepts

The extensions of response-dependent concepts are a priori connected with the subjective responses that competent users of that concept have in normal conditions. There are two strategies for specifying normal conditions for response-dependent concepts: topic-specific and topic-neutral. On a topic-specific specification, a characterization of normal conditions would be given separately for each response-dependent concept (or a non-trivial subset of response-dependent concepts, such as our colour concepts), whereas a topic-neutral specification would be given in a uniform way for all response-dependent concepts. In this paper I argue, using a thought experiment, that only topic-neutral specifications will deliver the a priori knowledge constitutive of response-dependence.     

Chain-Arguments and the Sorites Paradox

A finite chain of valid arguments can never lead from truth to falsehood. Call this the concatenation principle, or CP. Some propose to reject CP in response to the sorites paradox. I offer a justification of a restricted version of CP based on intuitively-appealing principles, all of which are already assumed in mathematics. The restricted version excludes soritical cases, but covers intuitively correct applications of CP in mathematics and elsewhere. The upshot is that the cost of rejecting the unrestricted CP is much lower than is often assumed.     

Paradox lost: Understanding vague predicates

Abstract

The paper is concerned with the status of vague predicates. It is argued that they are for the most part ‘classifiers’, which are covertly comparatives and name not monadic properties but relations. The Sorites Paradox, it is claimed, is thus defused and a verdict theory of vague predicates is presented. Our practice in using vague words is described and it is contended that in our use of these predicates we always have a permanent possibility of independent demarcation. Wittgenstein's picture of the wall and the swamp is deployed to avoid the Transition Problem and it is argued against Fregeans that we need vague language for the advancement of knowledge and understanding.     

Vagueness and Blurry Sets

This paper presents a new theory of vagueness, which is designed to retain the virtues of the fuzzy theory, while avoiding the problem of higher-order vagueness. The theory presented here accommodates the idea that for any statement S ₁ to the effect that Bob is bald is x true, for x in [0,1], there should be a further statement S ₂ which tells us how true S ₁ is, and so on – that is, it accommodates higher-order vagueness – without resorting to the claim that the metalanguage in which the semantics of vagueness is presented is itself vague, and without requiring us to abandon the idea that the logic – as opposed to the semantics – of vague discourse is classical. I model the extension of a vague predicate P as a blurry set, this being a function which assigns a degree of membership or degree function to each object o, where a degree function in turn assigns an element of [0,1] to each finite sequence of elements of [0,1]. The idea is that the assignment to the sequence 0.3,0.2, for example, represents the degree to which it is true to say that it is 0.2 true that o is P to degree 0.3. The philosophical merits of my theory are discussed in detail, and the theory is compared with other extensions and generalisations of fuzzy logic in the literature.