Temporal Reference in Linear Tense Logic

The paper introduces a first-order theory in the language of predicate tense logic which contains a single simple axiom. It is shewn that this theory enables times to be referred to and sentences involving ‘now’ and ‘then’ to be formalised. The paper then compares this way of increasing the expressive capacity of predicate tense logic with other mechanisms, and indicates how to generalise the results to other modal and tense systems.     

Modal Logic for Other-World Agnostics: Neutrality and Halldén Incompleteness

The logic of ‘elsewhere,’ i.e., of a sentence operator interpretable as attaching to a formula to yield a formula true at a point in a Kripke model just in case the first formula is true at all other points in the model, has been applied in settings in which the points in question represent spatial positions (explaining the use of the word ‘elsewhere’), as well as in the case in which they represent moments of time. This logic is applied here to the alethic modal case, in which the points are thought of as possible worlds, with the suggestion that its deployment clarifies aspects of a position explored by John Divers un-der the name ‘modal agnosticism.’ In particular, it makes available a logic whose Halldén incompleteness explicitly registers the agnostic element of the position – its neutrality as between modal realism and modal anti-realism.     

SUPERVALUATION FIXED-POINT LOGICS OF TRUTH

Michael Kremer defines fixed-point logics of truth based on Saul Kripke’s fixed point semantics for languages expressing their own truth concepts. Kremer axiomatizes the strong Kleene fixed-point logic of truth and the weak Kleene fixed-point logic of truth, but leaves the axiomatizability question open for the supervaluation fixed-point logic of truth and its variants. We show that the principal supervaluation fixed point logic of truth, when thought of as consequence relation, is highly complex: it is not even analytic. We also consider variants, engendered by a stronger notion of ‘fixed point’, and by variant supervaluation schemes. A ‘logic’ is often thought of, not as a consequence relation, but as a set of sentences – the sentences true on each interpretation. We axiomatize the supervaluation fixed-point logics so conceived.     

Actuality, Necessity, and Logical Truth

The traditional view that all logical truths are metaphysically necessary has come under attack in recent years. The contrary claim is prominent in David Kaplan’s work on demonstratives, and Edward Zalta has argued that logical truths that are not necessary appear in modal languages supplemented only with some device for making reference to the actual world (and thus independently of whether demonstratives like ‘I’, ‘here’, and ‘now’ are present). If this latter claim can be sustained, it strikes close to the heart of the traditional view. I begin this paper by discussing and refuting Zalta’s argument in the context of a language for propositional modal logic with an actuality connective (section 1). This involves showing that his argument in favor of real world validity his preferred explication of logical truth, is fallacious. Next (section 2) I argue for an alternative explication of logical truth called general validity. Since the rule of necessitation preserves general validity, the argument of section 2 provides a reason for affirming the traditional view. Finally (section 3) I show that the intuitive idea behind the discredited notion of real world validity finds legitimate expression in an object language connective for deep necessity. Earlier versions of this paper were read at the universities of Graz, Maribor, and Salzburg, and at a workshop on the philosophy of logic at the National Autonomous University of Mexico (UNAM) in Mexico City. My thanks to those present at these events for many helpful suggestions. Thanks are also due to an anonymous referee for Philosophical Studies.     

A defense of contingent logical truths

A formula is a contingent logical truth when it is true in every model M but, for some model M, false at some world of M. We argue that there are such truths, given the logic of actuality. Our argument turns on defending Tarski’s definition of truth and logical truth, extended so as to apply to modal languages with an actuality operator. We argue that this extension is the philosophically proper account of validity. We counter recent arguments to the contrary presented in Hanson’s ‘Actuality, Necessity, and Logical Truth’ (Philos Stud 130:437–459, 2006).     

Consciousness and Modal Empiricism

David Chalmers supports his contention that there is a possible world populated by our zombie twins by arguing for the assumption that conceivability entails possibility. But, I argue, the modal epistemology he sets forth, ‘modal rationalism,’ ignores the problem of incompleteness and relies on an idealized notion of conceivability. As a consequence, this epistemology can’t justify our quotidian judgments of possibility, let alone those judgments that concern the mind/body connection. Working from the analogy that the imagination is to the possible as perception is to the actual, I set forth a competing epistemology, ‘modal empiricism.’ This epistemology survives the incompleteness objection and allows some of our everyday modal judgments to be justified. But this epistemology can’t justify the claim that Zombie World is possible, which leaves Chalmers’s property dualism without the support it needs.

On the Mind Dependence of Truth

The claim that truth is mind dependent has some initial plausibility only if truth bearers are taken to be mind dependent entities such as beliefs or statements. Even on that assumption, however, the claim is not uncontroversial. If it is spelled out as the thesis that “in a world devoid of mind nothing would be true”, then everything depends on how the phrase ‘true in world w’ is interpreted. If ‘A is true in w’ is interpreted as ‘A is true of w’ (i.e. ‘w satisfies A’s truth conditions’, the claim need not be true. If on the other hand it is interpreted as ‘A is true of w and exists in w’ then the claim is trivially true, though devoid of any antirealistic efficacy. Philosophers like Heidegger and Rorty, who hold that truth is mind dependent but reality is not, must regard such principles as “A if and only if it is true that A” as only contingently true, which may be a good reason to reject the mind dependence of truth anyway.     

Against Stepping Back: A Critique of Contextualist Approaches to the Semantic Paradoxes

A number of philosophers have argued that the key to understanding the semantic paradoxes is to recognize that truth is essentially relative to context. All of these philosophers have been motivated by the idea that once a liar sentence has been uttered we can ‘step back’ and, from the point of view of a different context, judge that the liar sentence is true. This paper argues that this ‘stepping back’ idea is a mistake that results from failing to relativize truth to context in the first place. Moreover, context-relative liar sentences, such as ‘This sentence is not true in any context’ present a paradox even after truth has been relativized to context. Nonetheless, the relativization of truth to context may offer us the means to avoid paradox, if we can justifiably deny that a sentence about a context can be true in the very context it is about.     

From a Phono-Logical Point of View: Neutralizing Quine’s Argument Against Analyticity

Though largely unnoticed, in “Two Dogmas” Quine (1951, Two Dogmas of Empiricism, Philosophical Review 60, 20–43. Reprinted in From a Logical Point of View, 20–46) himself invokes a distinction: a distinction between logical and analytic truths. Unlike analytic statements equating ‘bachelor’ with ‘unmarried man’, strictly logical tautologies relating two word-tokens of the same word-type, e.g., ‘bachelor’ and ‘bachelor’ are true merely in virtue of basic phonological form, putatively an exclusively non-semantic function of perceptual categorization or brute stimulus behavior. Yet natural language phonemic categorization is not entirely free of interpretive semantic considerations. “Phonemic reductionism” in both its linguistic (Bloch 1953, Contrast, Language 29, 59–61) and behavioral (Quine 1990, The Phoneme’s Long Shadow, Emics and Etics: The Insider/Outsider Debate, T. Headland, K. Pike and M. Harris, (eds.), Newbury Park, CA, Sage Publications, 164–167) guise is false. The semantic basis of phonological equivalence, however, has repercussions vis-à-vis Quine’s critique of analyticity. A consistent rejection of meaning-based equivalencies eliminates not only analyticity, but imposes a form of phonological eliminativism too. Phonological eliminativism is the reductio result of applying Quinean meaning skepticism to the phonological typing of natural language. But unlike analyticity, phonology is presumably not subject to philosophical dismissal. The semantic basis of natural language phonology serves to neutralize Quine’s argument against analyticity: without the semantics of meaning, more than just synonymy is lost; basic phonology must also be forfeited. Let’s begin with the fact that even Quine has to admit that it is possible for two tokens of the same orthographic type to be synonymous, for that much is presupposed by his own account of logical truth. Paul Boghossian (1999, 343)     

Free Semantics

Free Semantics is based on normalized natural deduction for the weak relevant logic DW and its near neighbours. This is motivated by the fact that in the determination of validity in truth-functional semantics, natural deduction is normally used. Due to normalization, the logic is decidable and hence the semantics can also be used to construct counter-models for invalid formulae. The logic DW is motivated as an entailment logic just weaker than the logic MC of meaning containment. DW is the logic focussed upon, but the results extend to MC. The semantics is called ‘free semantics’ since it is disjunctively and existentially free in that no disjunctive or existential witnesses are produced, unlike in truth-functional semantics. Such ‘witnesses’ are only assumed in generality and are not necessarily actual. The paper sets up the free semantics in a truth-functional style and gives a natural deduction interpetation of the meta-logical connectives. We then set out a familiar tableau-style system, but based on natural deduction proof rather than truth-functional semantics. A proof of soundness and completeness is given for a reductio system, which is a transform of the tableau system. The reductio system has positive and negative rules in place of the elimination and introduction rules of Brady’s normalized natural deduction system for DW. The elimination-introduction turning points become closures of threads of proof, which are at the points of contradiction for the reductio system.     

信息哲学与未来中国科技哲学的基础

The research programme of the philosophy of information (PI) proposed in 2002 made it an independent area or discipline in philosophical research. The scientific concept of ‘information’ is formally accepted in philosophical inquiry. Hence a new and tool-driven philosophical discipline of PI with its interdisciplinary nature has been established. Philosophy of information is an ‘orientative’ rather than ‘cognitive’ philosophy. When PI is under consideration in the history of Western philosophy, it can be regarded as a shift of large tradition. There are three large traditions at large, known as Platonic, Kantian and Leibniz-Russellian. In the discussion of the position of the possible worlds, we have modal Platonism and modal realism, but both of the theories are made in the framework of Western philosophy. In this essay, it is argued that possible worlds could be seen as worlds in information, which is then an interpretation of modal information theory (MIT). Our interpretation is made on the basis of Leibniz’s lifelong connection with China, a fact often overlooked by the Western philosophers. Possible world theory was influenced by the Neo-Confucianism flourishing since the Song Dynasty of China, the foundation of which is Yijing. It could be argued that Leibniz’s possible world theory was formulated in respect to the impact of the thoughts reflected in Yijing, in that one of the prominent features is the model-theoretic construction of theories. There are two approaches to theory construction, i.e., axiom-theoretic and model-theoretic. The origin of the former is from ancient Greece and the latter from ancient China. And they determined the different features of theoretic structures between the oriental and occidental traditions of science and technology. The tendency of the future development of science and technology is changing from the axiom-theoretic to the model-theoretic orientation, at least the two approaches being complementary each other. To some extent, this means the retrospective of tradition in the turning point of history, and some of the China’s cultural traditions might become the starting points in formulating the future Chinese philosophy of science and technology.     

Truth Values,Neither-true-nor-false,and Supervaluations

The first section (§1) of this essay defends reliance on truth values against those who, on nominalistic grounds, would uniformly substitute a truth predicate. I rehearse some practical, Carnapian advantages of working with truth values in logic. In the second section (§2), after introducing the key idea of auxiliary parameters (§2.1), I look at several cases in which logics involve, as part of their semantics, an extra auxiliary parameter to which truth is relativized, a parameter that caters to special kinds of sentences. In many cases, this facility is said to produce truth values for sentences that on the face of it seem neither true nor false. Often enough, in this situation appeal is made to the method of supervaluations, which operate by “quantifying out” auxiliary parameters, and thereby produce something like a truth value. Logics of this kind exhibit striking differences. I first consider the role that Tarski gives to supervaluation in first order logic (§2.2), and then, after an interlude that asks whether neither-true-nor-false is itself a truth value (§2.3), I consider sentences with non-denoting terms (§2.4), vague sentences (§2.5), ambiguous sentences (§2.6), paradoxical sentences (§2.7), and future-tensed sentences in indeterministic tense logic (§2.8). I conclude my survey with a look at alethic modal logic considered as a cousin (§2.9), and finish with a few sentences of “advice to supervaluationists” (2.10), advice that is largely negative. The case for supervaluations as a road to truth is strong only when the auxiliary parameter that is “quantified out” is in fact irrelevant to the sentences of interest—as in Tarski’s definition of truth for classical logic. In all other cases, the best policy when reporting the results of supervaluation is to use only explicit phrases such as “settled true” or “determinately true,” never dropping the qualification.     

On the Possibility of Inference to the Best Explanation

Various proposals have suggested that an adequate explanatory theory should reduce the number or the cardinality of the set of logically independent claims that need be accepted in order to entail a body of data. A (and perhaps the only) well-formed proposal of this kind is William Kneale’s: an explanatory theory should be finitely axiomatizable but it’s set of logical consequences in the data language should not be finitely axiomatizable. Craig and Vaught showed that Kneale theories (almost) always exist for any recursively enumerable but not finitely axiomatizable set of data sentences in a first order language with identity. Kneale’s criterion underdetermines explanation even given all possible data in the data language; gratuitous axioms may be “tacked on.” Define a Kneale theory, T, to be logically minimal if it is deducible from every Kneale theory (in the vocabulary of T) that entails the same statements in the data language as does T. If they exist, minimal Kneale theories are candidates for best explanations: they are “bold” in a sense close to Popper’s; some minimal Kneale theory is true if any Kneale theory is true; the minimal Kneale theory that is data equivalent to any given Kneale theory is unique; and no Kneale theory is more probable than some minimal Kneale theory. I show that under the Craig-Vaught conditions, no minimal Kneale theories exist.     

Moore’s Paradox is not just another pragmatic paradox

One version of Moore’s Paradox is the challenge to account for the absurdity of beliefs purportedly expressed by someone who asserts sentences of the form ‘p & I do not believe that p’ (‘Moorean sentences’). The absurdity of these beliefs is philosophically puzzling, given that Moorean sentences (i) are contingent and often true; and (ii) express contents that are unproblematic when presented in the third-person. In this paper I critically examine the most popular proposed solution to these two puzzles, according to which Moorean beliefs are absurd because Moorean sentences are instances of pragmatic paradox; that is to say, the propositions they express are necessarily false-when-believed. My conclusion is that while a Moorean belief is a pragmatic paradox, it is not just another pragmatic paradox, because this diagnosis does not explain all the puzzling features of Moorean beliefs. In particularly, while this analysis is plausible in relation to the puzzle posed by characteristic (i) of Moorean sentences, I argue that it fails to account for (ii). I do so in the course of an attempt to formulate the definition of a pragmatic paradox in more precise formal terms, in order to see whether the definition is satisfied by Moorean sentences, but not by their third-person transpositions. For only an account which can do so could address (ii) adequately. After rejecting a number of attempted formalizations, I arrive at a definition which delivers the right results. The problem with this definition, however, is that it has to be couched in first-person terms, making an essential use of ‘I’. Thus the problem of accounting for first-/third-person asymmetry recurs at a higher order, which shows that the Pragmatic Paradox Resolution fails to identify the source of such asymmetry highlighted by Moore’s Paradox.     

Space Cannot Be Cut—Why Self-Identity Naturally Includes Neighbourhood

Psychology is not alone in its struggle with conceptualizing the dynamic relationship between space and individual or collective identity. This general epistemological issue haunts biology where it has a specific focus in evolutionary arguments. It arises because of the incompatibility between definitive logical systems of ‘contradiction or unity’, which can only apply to inert material systems, and natural evolutionary processes of cumulative energetic transformation. This incompatibility makes any attempt to apply definitive logic to evolutionary change unrealistic and paradoxical. It is important to recognise, because discrete perceptions of self and group, based on the supposition that any distinguishable identity can be completely cut free, as an ‘independent singleness’, from the space it inescapably includes and is included in, are a profound but unnecessary source of psychological, social and environmental conflict. These perceptions underlie Darwin’s definition of ‘natural selection’ as ‘the preservation of favoured races in the struggle for life’. They result in precedence being given to striving for homogeneous supremacy, through the competitive suppression of others, instead of seeking sustainable, co-creative evolutionary relationship in spatially and temporally heterogeneous communities. Here, I show how ‘natural inclusion’, a new, post-dialectic understanding of evolutionary process, becomes possible through recognising space as a limitless, indivisible, receptive (non-resistive) ‘intangible presence’ vital for movement and communication, not as empty distance between one tangible thing and another. The fluid boundary logic of natural inclusion as the co-creative, fluid dynamic transformation of all through all in receptive spatial context, allows all form to be understood as flow-form, distinctive but dynamically continuous, not singularly discrete. This simple move from regarding space and boundaries as sources of discontinuity and discrete definition to sources of continuity and dynamic distinction correspondingly enables self-identity to be understood as a dynamic inclusion of neighbourhood, through the inclusion of space throughout and beyond all natural figural forms as configurations of energy. Fully to appreciate and communicate the significance of this move, it is necessary to widen the linguistic, mathematical and imaginative remit of conventional scientific argument and explication so as to include more poetic, fluid and artistic forms of expression.     

Chomsky On The ‘Ordinary Language’ View Of Language

There is a common-sense view of language, which is held by Wittgenstein, Strawson Dummett, Searle, Putnam, Lewis, Wiggins, and others. According to this view a language consists of conventions, it is rule-governed, rules are conventionalised, a language is learnt, there are general learning mechanisms in the brain, and so on. I shall call this view the ‘ordinary language’ view of language. Chomsky’s attitude towards this view of language has been rather negative, and his rejection of it is a major motivation for the development of his own theory. In this paper I shall review Chomsky’s long-standing criticisms. I shall show that (1)Chomsky’s argument does not constitute a dismissal of the ‘ordinarylanguage’ view of language, (2) Chomsky’s conclusions about language do not follow from his argument, and (3) the ‘ordinary language’ view actually points to a promising way for us to understand the true nature of language and mind. This revised version was published online in August 2006 with corrections to the Cover Date.     

Formalizing common sense: an operator-based approach to the Tibbles–Tib problem

The paper argues, that a direct formalization of the way common sense thinks about the numerical identity of enduring entities, requires that traditional predicate logic is developed. If everyday language mirrors the world, then persons, organisms, organs, cells, and ordinary material things can lose some parts but nonetheless remain numerically exactly the same entity. In order to formalize this view, two new logical operators are introduced; and they bring with them some non-standard syntax. One of the operators is called ‘the instantiation operator’; it is needed because the existential quantifier and its traditional relatives cannot do the job required. The other operator is called ‘the form-on-matter operator’, and it allows an individual (an instance of a form) to stay the same even though some of its parts (its constituting matter) is taken away from it. Also, a certain kind of predicates, called ‘nature terms’, is needed in order to represent what gives a particular its kind of identity. Both the operators and the nature terms introduced can be used in constructions of formal languages and formal systems, but no such constructions are made in the paper. The paper is structured as a comment on the philosophical problem called ‘the problem of the cats Tibbles and Tib’.     

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.     

Traditional Logic,Modern Logic and Natural Language

In a recent paper Johan van Benthem reviews earlier work done by himself and colleagues on ‘natural logic’. His paper makes a number of challenging comments on the relationships between traditional logic, modern logic and natural logic. I respond to his challenge, by drawing what I think are the most significant lines dividing traditional logic from modern. The leading difference is in the way logic is expected to be used for checking arguments. For traditionals the checking is local, i.e. separately for each inference step. Between inference steps, several kinds of paraphrasing are allowed. Today we formalise globally: we choose a symbolisation that works for the entire argument, and thus we eliminate intuitive steps and changes of viewpoint during the argument. Frege and Peano recast the logical rules so as to make this possible. I comment also on the traditional assumption that logical processing takes place at the top syntactic level, and I question Johan’s view that natural logic is ‘natural’.