Decidability of General Extensional Mereology

The signature of the formal language of mereology contains only one binary predicate P which stands for the relation “being a part of”. Traditionally, P must be a partial ordering, that is, ${\forall{x}Pxx, \forall{x}\forall{y}((Pxy\land Pyx)\to x=y)}$ and ${\forall{x}\forall{y}\forall{z}((Pxy\land Pyz)\to Pxz))}$ are three basic mereological axioms. The best-known mereological theory is “general extensional mereology”, which is axiomatized by the three basic axioms plus the following axiom and axiom schema: (Strong Supplementation) ${\forall{x}\forall{y}(\neg Pyx\to \exists z(Pzy\land \neg Ozx))}$ , where Oxy means ${\exists z(Pzx\land Pzy)}$ , and (Fusion) ${\exists x\alpha \to \exists z\forall y(Oyz\leftrightarrow \exists x(\alpha \land Oyx))}$ , for any formula α where z and y do not occur free. In this paper, I will show that general extensional mereology is decidable, and will also point out that the decidability of the first-order approximation of the theory of complete Boolean algebras can be shown in the same way.     

Elusive Propositions

Journal of Philosophical Logic - David Kaplan observed in Kaplan (1995) that the principle $\forall p \Diamond \forall q (Qq \leftrightarrow q = p)$ cannot be verified at a world in a standard...     

Completeness theorems for some intermediate predicate calculi

We give completeness results — with respect to Kripke's semantic — for the negation-free intermediate predicate calculi: (1) $$\begin{gathered} BD = positive predicate calculus PQ + B:(\alpha \to \beta )v(\beta \to \alpha ) \hfill \\ + D:\forall x\left( {a\left( x \right)v\beta } \right) \to \forall xav\beta \hfill \\ \end{gathered}$$ (2) $$T_n D = PQ + T_n :\left( {a_0 \to a_1 } \right)v \ldots v\left( {a_n \to a_{n + 1} } \right) + D\left( {n \geqslant 0} \right)$$ and the superintuitionistic predicate calculus: (3) $$B^1 DH_2^ \urcorner = BD + intuitionistic negation + H_2^ \urcorner : \urcorner \forall xa \to \exists x \urcorner a.$$ The central point is the completeness proof for (1), which is obtained modifying Klemke's construction [3]. For a general account on negation-free intermediate predicate calculi — see Casari-Minari [1]; for an algebraic treatment of some superintuitionistic predicate calculi involving schemasB andD — see Horn [4] and Görnemann [2].     

Every world can see a reflexive world

Let be the class of frames satisfying the condition

The Place Of Geometry: Heidegger's Mathematical Excursus On Aristotle

'The Place of Geometry' discusses the excursus on mathematics from Heidegger's 1924–25 lecture course on Platonic dialogues, which has been published as Volume 19 of the Gesamtausgabe as Plato's Sophist , as a starting point for an examination of geometry in Euclid, Aristotle and Descartes. One of the crucial points Heidegger makes is that in Aristotle there is a fundamental difference between arithmetic and geometry, because the mode of their connection is different. The units of geometry are positioned, the units of arithmetic unpositioned. Following Heidegger's claim that the Greeks had no word for space, and David Lachterman's assertion that there is no term corresponding to or translatable as 'space' in Euclid's Elements , I examine when the term 'space' was introduced into Western thought. Descartes is central to understanding this shift, because his understanding of extension based in terms of mathematical co-ordinates is a radical break with Greek thought. Not only does this introduce this word 'space' but, by conceiving of geometrical lines and shapes in terms of numerical co-ordinates, which can be divided, it turns something that is positioned into unpositioned. Geometric problems can be reduced to equations, the length (i.e, quantity) of lines: a problem of number. The continuum of geometry is transformed into a form of arithmetic. Geometry loses position just as the Greek notion of 'place' is transformed into the modern notion of space.     

In Defense of the Notion of Truthlikeness

The notion of truthlikeness (verisimilitude, approximate truth), coined by Karl Popper, has very much fallen into oblivion, but the paper defends it. It can be regarded in two different ways. Either as a notion that is meaningful only if some formal measure of degree of truthlikeness can be constructed; or as a merely non-formal comparative notion that nonetheless has important functions to fulfill. It is the latter notion that is defended; it is claimed that such a notion is needed for both a reasonable backward-looking and a reasonable forward-looking view of science. On the one hand, it is needed in order to make sense of the history of science as containing a development; on the other, it is needed in order to understand present-day sciences as containing knowledge-seeking activities. The defense of truthlikeness requires also a defense of two other notions: quasi-comparisons and regulative ideas, which is supplied in this paper as well.     

Indiscriminability as Indiscernibility by Default

Most solutions to the sorites reject its major premise, i.e. the quantified conditional . This rejection appears to imply a discrimination between two elements that are supposed to be indiscriminable. Thus, the puzzle of the sorites involves in a fundamental way the notion of indiscriminability. This paper analyzes this relation and formalizes it, in a way that makes the rejection of the major premise more palatable. The intuitive idea is that we consider two elements indiscriminable by default, i.e. unless we know some information that discriminates between them. Specifically, following Rough Set Theory, two elements are defined to be indiscernible if they agree on the vague property in question. Then, a is defined to be indiscriminable from b if a is indiscernible by default from b. That is to say, a is indiscriminable from b if it is consistent to assume that a and b agree on the relevant vague property. Indiscernibility by default is formalized with the use of Default Logic, and is shown to have intuitively desirable properties: it is entailed by equality, is reflexive and symmetric. And while the relation is neither transitive nor substitutive, it is “almost” substitutive. This definition of indiscriminability is incorporated into three major theories of vagueness, namely the supervaluationist, epistemic, and contextualist views. Each one of these theories is reduced to a different strategy dealing with multiple extensions in Default Logic, and the rejection of the major premise is shown to follow naturally. Thus, while the proposed notion of indiscriminability does not solve the sorites by itself, it does make the unintuitive conclusion of many of its proposed solutions—the rejection of the major premise—a bit easier to accept.     

Kripke Semantics, Undecidability and Standard Completeness for Esteva and Godo's Logic MTL∀

The present paper deals with the predicate version MTL of the logic MTL by Esteva and Godo. We introduce a Kripke semantics for it, along the lines of Ono's Kripke semantics for the predicate version of FL_ew (cf. [O85]), and we prove a completeness theorem. Then we prove that every predicate logic between MTL and classical predicate logic is undecidable. Finally, we prove that MTL is complete with respect to the standard semantics, i.e., with respect to Kripke frames on the real interval [0,1], or equivalently, with respect to MTL-algebras whose lattice reduct is [0,1] with the usual order.     

Aspects of analytic deduction

Let ? be the ordinary deduction relation of classical first-order logic. We provide an “analytic” subrelation ?³ of ? which for propositional logic is defined by the usual “containment” criterion $$\Gamma \vdash ^a \varphi iff \Gamma \vdash \varphi and Atom(\varphi ) \subseteq Atom(\Gamma ),$$ whereas for predicate logic, ?^a is defined by the extended criterion $$\Gamma \vdash ^a \varphi iff \Gamma \vdash \varphi and Atom(\varphi ) \subseteq ' Atom(\Gamma ),$$ where Atom(?) $ \subseteq '$ Atom(Γ) means that every atomic formula occurring in ? “essentially occurs” also in Γ. If Γ, ? are quantifier-free, then the notions “occurs” and “essentially occurs” for atoms between Γ and ? coincide. If ? is formalized by Gentzen's calculus of sequents, then we show that ?^a is axiomatizable by a proper fragment of analytic inference rules. This is mainly due to cut elimination. By “analytic inference rule” we understand here a rule r such that, if the sequent over the line is analytic, then so is the sequent under the line. We also discuss the notion of semantic relevance as contrasted to the previous syntactic one. We show that when introducing semantic sequents as axioms, i.e. when extending the pure logical axioms and rules by mathematical ones, the property of syntactic relevance is lost, since cut elimination no longer holds. We conclude that no purely syntactic notion of analytic deduction can ever replace successfully the complex semantico-syntactic deduction we already possess.     

Generalization of Scott's formula for retractions from generalized Alexandroff's cube

In the paper [2] the following theorem is shown: Theorem (Th. 3,5, [2]), If α=0 or δ=∞ or α?δ, then a closure space X is an absolute extensor for the category of 〈α, δ〉 -closure spaces iff a contraction of X is the closure space of all 〈α, δ〉-filters in an 〈α, δ〉-semidistributive lattice. In the case when α=ω and δ=∞, this theorem becomes Scott's theorem: Theorem ([7]). A topological space X is an absolute extensor for the category of all topological spaces iff a contraction of X is a topological space of “Scott's open sets” in a continuous lattice. On the other hand, when α=0 and δ=ω, this theorem becomes Jankowski's theorem: Theorem ([4]). A closure space X is an absolute extensor for the category of all closure spaces satisfying the compactness theorem iff a contraction of X is a closure space of all filters in a complete Heyting lattice. But for separate cases of α and δ, the Theorem 3.5 from [2] is proved using essentialy different methods. In this paper it is shown that this theorem can be proved using, for retraction, one uniform formula. Namely it is proved that if α= 0 or δ= ∞ or α ? δ and \(F_{\alpha ,\delta } \left( L \right) \subseteq B_{\alpha ,\delta }^\mathfrak{n} \) and if L is an 〈α, δ〉-semidistributive lattice, then the function $$r:{\text{ }}B_{\alpha ,\delta }^\mathfrak{n} \to F_{\alpha ,\delta } \left( L \right)$$ such that for x ε ? ( \(\mathfrak{n}\) ): (*) $$r\left( x \right) = inf_L \left\{ {l \in L|\left( {\forall A \subseteq L} \right)x \in C\left( A \right) \Rightarrow l \in C\left( A \right)} \right\}$$ defines retraction, where C is a proper closure operator for \(B_{\alpha ,\delta }^\mathfrak{n} \) . It is also proved that the formula (*) defines retraction for all 〈α, δ〉, whenever L is an 〈α, δ〉 -pseudodistributive lattice. Moreover it is proved that when α=ω and δ=∞, the formula (*) defines identical retraction to the formula given in [7], and when α = 0 and δ=ω, the formula (*) defines identical retraction to the formula given in [4].     

The Second Nature of Human Beings: an Invitation for John McDowell to discuss Helmuth Plessner’s Philosophical Anthropology

Abstract

John McDowell argues for minimal empiricism via using the notion of second nature of human beings. I should like to invite him to discuss Helmuth Plessner's Philosophical Anthropology in order to elaborate a more substantial conception of second nature. McDowell seems to think that it is adequate for his more epistemological aim to remind us of second nature as though it were to be taken for granted. But I think, following Plessner, that this right reminder needs a therapeutic elaboration in Kant's sense of propaedeutics. What had been called our second nature found itself being questioned in order to limit the range of ways of treating the self we can authorize.     

Entailment and Bivalence

My purpose in this paper is to argue that the classical notion of entailment is not suitable for non-bivalent logics, to propose an appropriate alternative and to suggest a generalized entailment notion suitable to bivalent and non-bivalent logics alike. In classical two valued logic, one can not infer a false statement from one that is not false, any more than one can infer from a true statement a statement that is not true. In classical logic in fact preserving truth and preserving non-falsity are one and the same thing. They are not the same in non-bivalent logics however and I will argue that the classical notion of entailment that preserves only truth is not strong enough for such a logic. I will show that if we retain the classical notion of entailment in a logic that has three values, true, false and a third value in between, an inconsistency can be derived that can be resolved only by measures that seriously disable the logic. I will show this for a logic designed to allow for semantic presuppositions, then I will show that we get the same result in any three valued logic with the same value ordering. I will finally suggest how the notion of entailment should be generalized so that this problem may be avoided. The strengthened notion of entailment I am proposing is a conservative extension of the classical notion that preserves not only truth but the order of all values in a logic, so that the value of an entailed statement must alway be at least as great as the value of the sequence of statements entailing it. A notion of entailment this strong or stronger will, I believe, be found to be applicable to non-classical logics generally. In the opinion of Dana Scott, no really workable three valued logic has yet been developed. It is hard to disagree with this. A workable three valued logic however could perhaps be developed however, if we had a notion of entailment suitable to non-bivalent logics.     

The magical number seven: Making a “bit” of “sense”

The maximum number of categories distinguishable in making an absolute judgment was estimated by Miller to be “seven plus or minus two,” corresponding to about 3 bits of information transmitted per stimulus. Later work extended this range to include at least 2 to 4 bits of information, which reached 16 categories. In contrast, the number of distinguishable differences between two stimuli is in the order of 100. Why is this so? It is shown here that an answer to these questions can be obtained by constructing anentropy function, Hs, which is a measure of the uncertainty of a subject (or a sensory receptor) as it perceives the magnitude of an applied stimulus. Using this function, it is demonstrated that the ubiquitous 3 bits of information per stimulus can be approximated from the expression \(\log _2 \sqrt {{{\tau _2 } \mathord{\left/ {\vphantom {{\tau _2 } {\tau _1 }}} \right. \kern-0em} {\tau _1 }}} \) , where τ₁ and τ₂ are known time constants. The same entropy function can be used to derive various other psychophysical laws, such as the Weber-Fechner law, Stevens’ law, and the Bunsen-Roscoe law.     

The Undecidability of Propositional Adaptive Logic

We investigate and classify the notion of final derivability of two basic inconsistency-adaptive logics. Specifically, the maximal complexity of the set of final consequences of decidable sets of premises formulated in the language of propositional logic is described. Our results show that taking the consequences of a decidable propositional theory is a complicated operation. The set of final consequences according to either the Reliability Calculus or the Minimal Abnormality Calculus of a decidable propositional premise set is in general undecidable, and can be -complete. These classifications are exact. For first order theories even finite sets of premises can generate such consequence sets in either calculus.     

The perceptual organization of dot lattices

Bravais (1850/1949) demonstrated that there are five types of periodic dot patterns (or lattices): oblique, rectangular, centered rectangular, square, and hexagonal. Gestalt psychologists studied grouping by proximity in rectangular and square dot patterns. In the first part of the present paper, I (1) describe the geometry of the five types of lattices, and (2) explain why, for the study of perception, centered rectangular lattices must be divided into two classes (centered rectangular andrhombic). I also show how all lattices can be located in a two-dimensional space. In the second part of the paper, I show how the geometry of these lattices determines their grouping and their multistability. I introduce the notion ofdegree of instability and explain how to order lattices from most stable to least stable (hexagonal). In the third part of the paper, I explore the effect of replacing the dots in a lattice with less symmetric motifs, thus creating wallpaper patterns. When a dot pattern is turned into a wallpaper pattern, its perceptual organization can be altered radically, overcoming grouping by proximity. I conclude the paper with an introduction to the implications of motif selection and placement for the perception of the ensuing patterns.     

Local axioms in disguise: Hilbert on Minkowski diagrams

While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski??s Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating ??rigorously?? with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as ??drawn formulas??, and formulas as ??written diagrams??, thus suggesting that the former encapsulate propositional information which can be extracted and translated into formulas. In the case of Minkowski diagrams, local geometrical axioms were actually being produced, starting with the diagrams, by a process that was both constrained and fostered by the requirement, brought about by the axiomatic method itself, that geometry ought to be made independent of analysis. This paper aims at making a twofold point. On the one hand, it shows that Minkowski??s diagrammatic methods in number theory prompted Hilbert??s axiomatic investigations into the notion of a straight line as the shortest distance between two points, which start from his earlier work focused on the role of the triangle inequality property in the foundations of geometry, and lead up to his formulation of the 1900 Fourth Problem. On the other hand, it purports to make clear how Hilbert??s assessment of Minkowski??s diagram-based reasoning in number theory both raises and illuminates conceptual compatibility concerns that were crucial to his philosophy of mathematics.     

Sketch of a conversational society

Abstract

In this paper I consider what it might mean to see society as a kind of Rortian conversation. Although the idea of conversation is not always explicit in Rorty’s social thought, it is, I think, implicitly present. To therefore invoke it as a model is not to do an injustice to Rorty, but to bring out features of his own thought that he tends to underplay. In suggesting that we take seriously the notion of society as a kind of conversation, we should be careful not to overplay the aspect of talking, which is only a part of conversation. We should bear in mind that it also means living together. It must be admitted, nevertheless, that Rorty introduces the idea of conversation as a way of thinking about discourse, and so the notion as Rorty uses it prioritises the notion of talking. I would argue, however, that Rorty leaves his notion sufficiently vague and undefined to make it amenable to extension. In order to argue that we should look to the idea of conversation as a way of thinking about society more generally, I will proceed as follows. I will begin by considering the notion of conversation as discourse, focusing on two particularly prominent strands of criticism in response to this idea, namely that it ignores the role of argument and reason, and that it is a pointless sort of practice. Having rebutted these strands of criticism, I will outline a way in which we can extend the notion of conversation to society as a whole, and I will do this by debating with critics who see Rorty as privileging language over the more material and institutional aspects of society. Finally, I will argue that the conversational model is superior to the more entrenched deliberative model of democracy. Through an examination of one particular phenomenon, claro culture, which causes practical and theoretical difficulties for the deliberative model, I will offer a prima facie reason for suggesting conversation as a superior and more pragmatic alternative.     

动词体的球态语义学

本文介绍由塔斯基的立体几何导出的球态语义学,并将其应用于自然语言中的动词体现象。球态语义学特别适合应用于英语的进行体。这种方法有以下优点（i）它扩展了区间式语义,并同时避免了其缺陷,（ii）它解决了未完成体难题,（iii）它的解决方法无需诉诸最终结果策略。逻辑方法一般被认为难于处理自然语言的动词体问题。基于点的时间结构以及建立在该结构之上的经典普莱尔时态逻辑（[18]）太弱了。而基于区间的时态语义则缺乏足够的表达力,并且难以解释进行体（[4,8]）.本文给出一种新的基于球上整体－部分关系概念的模型和时态语义。这种球态语义学建基于塔斯基1927年引入的立体几何之上。与基于点和基于区间的语义不同,在球态语义学中很多动词体区分都由统一的逻辑方法刻画。在一个由封闭球构成的论域中,可达关系由相切性概念给出。相应地,我们可定义外切、内切、外径、内径以及同心等基本概念。与区间式语义不同,球是论域的初始概念,球态语义学不是在时间段而是在球中对事件赋值。因此,仅将时间区间作为初始概念而不承认其端点初性性的问题不复存在。英语中的进行体由球上的连续行动来刻画。行动是非终止的,只要球没有由外切相离。相应地,外切相离刻车动作完成。我们区分在均匀球和非均匀球中发生事件的整体-部分关系。非持续动作视为直径为零的同心球。球态语义学根据动作或执行完成的时刻来定义时间概念,其中不需要时间端点的概念。在保持与基于区间的时间模型类似的基础上,球态语义学暗示了一种关于可能世界的定性概念,并且它有利于解决时间的循环概念问题。     

Possible primitive notions for geometry of spine spaces

New systems of notions specific to the geometry of spine spaces, are introduced. In particular parallelism turns out to be a sufficient primitive notion to express the geometry of a spine space, and we show that structures related to projective closure are definitionally equivalent to spine spaces.