INTRODUCTION to LOGIC Lecture 1 Validity Introduction to Sets And
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Dialetheists' Lies About the Liar
PRINCIPIA 22(1): 59–85 (2018) doi: 10.5007/1808-1711.2018v22n1p59 Published by NEL — Epistemology and Logic Research Group, Federal University of Santa Catarina (UFSC), Brazil. DIALETHEISTS’LIES ABOUT THE LIAR JONAS R. BECKER ARENHART Departamento de Filosofia, Universidade Federal de Santa Catarina, BRAZIL [email protected] EDERSON SAFRA MELO Departamento de Filosofia, Universidade Federal do Maranhão, BRAZIL [email protected] Abstract. Liar-like paradoxes are typically arguments that, by using very intuitive resources of natural language, end up in contradiction. Consistent solutions to those paradoxes usually have difficulties either because they restrict the expressive power of the language, orelse because they fall prey to extended versions of the paradox. Dialetheists, like Graham Priest, propose that we should take the Liar at face value and accept the contradictory conclusion as true. A logical treatment of such contradictions is also put forward, with the Logic of Para- dox (LP), which should account for the manifestations of the Liar. In this paper we shall argue that such a formal approach, as advanced by Priest, is unsatisfactory. In order to make contradictions acceptable, Priest has to distinguish between two kinds of contradictions, in- ternal and external, corresponding, respectively, to the conclusions of the simple and of the extended Liar. Given that, we argue that while the natural interpretation of LP was intended to account for true and false sentences, dealing with internal contradictions, it lacks the re- sources to tame external contradictions. Also, the negation sign of LP is unable to represent internal contradictions adequately, precisely because of its allowance of sentences that may be true and false. -
Relations on Semigroups
International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 Relations on Semigroups 1D.D.Padma Priya, 2G.Shobhalatha, 3U.Nagireddy, 4R.Bhuvana Vijaya 1 Sr.Assistant Professor, Department of Mathematics, New Horizon College Of Engineering, Bangalore, India, Research scholar, Department of Mathematics, JNTUA- Anantapuram [email protected] 2Professor, Department of Mathematics, SKU-Anantapuram, India, [email protected] 3Assistant Professor, Rayalaseema University, Kurnool, India, [email protected] 4Associate Professor, Department of Mathematics, JNTUA- Anantapuram, India, [email protected] Abstract: Equivalence relations play a vital role in the study of quotient structures of different algebraic structures. Semigroups being one of the algebraic structures are sets with associative binary operation defined on them. Semigroup theory is one of such subject to determine and analyze equivalence relations in the sense that it could be easily understood. This paper contains the quotient structures of semigroups by extending equivalence relations as congruences. We define different types of relations on the semigroups and prove they are equivalence, partial order, congruence or weakly separative congruence relations. Keywords: Semigroup, binary relation, Equivalence and congruence relations. I. INTRODUCTION [1,2,3 and 4] Algebraic structures play a prominent role in mathematics with wide range of applications in science and engineering. A semigroup -
“The Church-Turing “Thesis” As a Special Corollary of Gödel's
“The Church-Turing “Thesis” as a Special Corollary of Gödel’s Completeness Theorem,” in Computability: Turing, Gödel, Church, and Beyond, B. J. Copeland, C. Posy, and O. Shagrir (eds.), MIT Press (Cambridge), 2013, pp. 77-104. Saul A. Kripke This is the published version of the book chapter indicated above, which can be obtained from the publisher at https://mitpress.mit.edu/books/computability. It is reproduced here by permission of the publisher who holds the copyright. © The MIT Press The Church-Turing “ Thesis ” as a Special Corollary of G ö del ’ s 4 Completeness Theorem 1 Saul A. Kripke Traditionally, many writers, following Kleene (1952) , thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing ’ s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis — what Turing (1936) calls “ argument I ” — has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing himself in what he calls “ argument II. ” The idea is that computation is a special form of math- ematical deduction. Assuming the steps of the deduction can be stated in a first- order language, the Church-Turing thesis follows as a special case of G ö del ’ s completeness theorem (first-order algorithm theorem). I propose this idea as an alternative foundation for the Church-Turing thesis, both for human and machine computation. Clearly the relevant assumptions are justified for computations pres- ently known. -
A Compositional Analysis for Subset Comparatives∗ Helena APARICIO TERRASA–University of Chicago
A Compositional Analysis for Subset Comparatives∗ Helena APARICIO TERRASA–University of Chicago Abstract. Subset comparatives (Grant 2013) are amount comparatives in which there exists a set membership relation between the target and the standard of comparison. This paper argues that subset comparatives should be treated as regular phrasal comparatives with an added presupposi- tional component. More specifically, subset comparatives presuppose that: a) the standard has the property denoted by the target; and b) the standard has the property denoted by the matrix predi- cate. In the account developed below, the presuppositions of subset comparatives result from the compositional principles independently required to interpret those phrasal comparatives in which the standard is syntactically contained inside the target. Presuppositions are usually taken to be li- censed by certain lexical items (presupposition triggers). However, subset comparatives show that presuppositions can also arise as a result of semantic composition. This finding suggests that the grammar possesses more than one way of licensing these inferences. Further research will have to determine how productive this latter strategy is in natural languages. Keywords: Subset comparatives, presuppositions, amount comparatives, degrees. 1. Introduction Amount comparatives are usually discussed with respect to their degree or amount interpretation. This reading is exemplified in the comparative in (1), where the elements being compared are the cardinalities corresponding to the sets of books read by John and Mary respectively: (1) John read more books than Mary. |{x : books(x) ∧ John read x}| ≻ |{y : books(y) ∧ Mary read y}| In this paper, I discuss subset comparatives (Grant (to appear); Grant (2013)), a much less studied type of amount comparative illustrated in the Spanish1 example in (2):2 (2) Juan ha leído más libros que El Quijote. -
The Validities of Pep and Some Characteristic Formulas in Modal Logic
THE VALIDITIES OF PEP AND SOME CHARACTERISTIC FORMULAS IN MODAL LOGIC Hong Zhang, Huacan He School of Computer Science, Northwestern Polytechnical University,Xi'an, Shaanxi 710032, P.R. China Abstract: In this paper, we discuss the relationships between some characteristic formulas in normal modal logic with their frames. We find that the validities of the modal formulas are conditional even though some of them are intuitively valid. Finally, we prove the validities of two formulas of Position- Exchange-Principle (PEP) proposed by papers 1 to 3 by using of modal logic and Kripke's semantics of possible worlds. Key words: Characteristic Formulas, Validity, PEP, Frame 1. INTRODUCTION Reasoning about knowledge and belief is an important issue in the research of multi-agent systems, and reasoning, which is set up from the main ideas of situational calculus, about other agents' states and actions is one of the most important directions in Al^^'^\ Papers [1-3] put forward an axiom scheme in reasoning about others(RAO) in multi-agent systems, and a rule called Position-Exchange-Principle (PEP), which is shown as following, was abstracted . C{(p^y/)^{C(p->Cy/) (Ce{5f',...,5f-}) When the length of C is at least 2, it reflects the mechanism of an agent reasoning about knowledge of others. For example, if C is replaced with Bf B^/, then it should be B':'B]'{(p^y/)-^iBf'B]'(p^Bf'B]'y/) 872 Hong Zhang, Huacan He The axiom schema plays a useful role as that of modus ponens and axiom K when simplifying proof. Though examples were demonstrated as proofs in papers [1-3], from the perspectives both in semantics and in syntax, the rule is not so compellent. -
Binary Relations and Functions
Harvard University, Math 101, Spring 2015 Binary relations and Functions 1 Binary Relations Intuitively, a binary relation is a rule to pair elements of a sets A to element of a set B. When two elements a 2 A is in a relation to an element b 2 B we write a R b . Since the order is relevant, we can completely characterize a relation R by the set of ordered pairs (a; b) such that a R b. This motivates the following formal definition: Definition A binary relation between two sets A and B is a subset of the Cartesian product A×B. In other words, a binary relation is an element of P(A × B). A binary relation on A is a subset of P(A2). It is useful to introduce the notions of domain and range of a binary relation R from a set A to a set B: • Dom(R) = fx 2 A : 9 y 2 B xRyg Ran(R) = fy 2 B : 9 x 2 A : xRyg. 2 Properties of a relation on a set Given a binary relation R on a set A, we have the following definitions: • R is transitive iff 8x; y; z 2 A :(xRy and yRz) =) xRz: • R is reflexive iff 8x 2 A : x Rx • R is irreflexive iff 8x 2 A : :(xRx) • R is symmetric iff 8x; y 2 A : xRy =) yRx • R is asymmetric iff 8x; y 2 A : xRy =):(yRx): 1 • R is antisymmetric iff 8x; y 2 A :(xRy and yRx) =) x = y: In a given set A, we can always define one special relation called the identity relation. -
Interpretations and Mathematical Logic: a Tutorial
Interpretations and Mathematical Logic: a Tutorial Ali Enayat Dept. of Philosophy, Linguistics, and Theory of Science University of Gothenburg Olof Wijksgatan 6 PO Box 200, SE 405 30 Gothenburg, Sweden e-mail: [email protected] 1. INTRODUCTION Interpretations are about `seeing something as something else', an elusive yet intuitive notion that has been around since time immemorial. It was sys- tematized and elaborated by astrologers, mystics, and theologians, and later extensively employed and adapted by artists, poets, and scientists. At first ap- proximation, an interpretation is a structure-preserving mapping from a realm X to another realm Y , a mapping that is meant to bring hidden features of X and/or Y into view. For example, in the context of Freud's psychoanalytical theory [7], X = the realm of dreams, and Y = the realm of the unconscious; in Khayaam's quatrains [11], X = the metaphysical realm of theologians, and Y = the physical realm of human experience; and in von Neumann's foundations for Quantum Mechanics [14], X = the realm of Quantum Mechanics, and Y = the realm of Hilbert Space Theory. Turning to the domain of mathematics, familiar geometrical examples of in- terpretations include: Khayyam's interpretation of algebraic problems in terms of geometric ones (which was the remarkably innovative element in his solution of cubic equations), Descartes' reduction of Geometry to Algebra (which moved in a direction opposite to that of Khayyam's aforementioned interpretation), the Beltrami-Poincar´einterpretation of hyperbolic geometry within euclidean geom- etry [13]. Well-known examples in mathematical analysis include: Dedekind's interpretation of the linear continuum in terms of \cuts" of the rational line, Hamilton's interpertation of complex numbers as points in the Euclidean plane, and Cauchy's interpretation of real-valued integrals via complex-valued ones using the so-called contour method of integration in complex analysis. -
Church's Thesis and the Conceptual Analysis of Computability
Church’s Thesis and the Conceptual Analysis of Computability Michael Rescorla Abstract: Church’s thesis asserts that a number-theoretic function is intuitively computable if and only if it is recursive. A related thesis asserts that Turing’s work yields a conceptual analysis of the intuitive notion of numerical computability. I endorse Church’s thesis, but I argue against the related thesis. I argue that purported conceptual analyses based upon Turing’s work involve a subtle but persistent circularity. Turing machines manipulate syntactic entities. To specify which number-theoretic function a Turing machine computes, we must correlate these syntactic entities with numbers. I argue that, in providing this correlation, we must demand that the correlation itself be computable. Otherwise, the Turing machine will compute uncomputable functions. But if we presuppose the intuitive notion of a computable relation between syntactic entities and numbers, then our analysis of computability is circular.1 §1. Turing machines and number-theoretic functions A Turing machine manipulates syntactic entities: strings consisting of strokes and blanks. I restrict attention to Turing machines that possess two key properties. First, the machine eventually halts when supplied with an input of finitely many adjacent strokes. Second, when the 1 I am greatly indebted to helpful feedback from two anonymous referees from this journal, as well as from: C. Anthony Anderson, Adam Elga, Kevin Falvey, Warren Goldfarb, Richard Heck, Peter Koellner, Oystein Linnebo, Charles Parsons, Gualtiero Piccinini, and Stewart Shapiro. I received extremely helpful comments when I presented earlier versions of this paper at the UCLA Philosophy of Mathematics Workshop, especially from Joseph Almog, D. -
Against Logical Form
Against logical form Zolta´n Gendler Szabo´ Conceptions of logical form are stranded between extremes. On one side are those who think the logical form of a sentence has little to do with logic; on the other, those who think it has little to do with the sentence. Most of us would prefer a conception that strikes a balance: logical form that is an objective feature of a sentence and captures its logical character. I will argue that we cannot get what we want. What are these extreme conceptions? In linguistics, logical form is typically con- ceived of as a level of representation where ambiguities have been resolved. According to one highly developed view—Chomsky’s minimalism—logical form is one of the outputs of the derivation of a sentence. The derivation begins with a set of lexical items and after initial mergers it splits into two: on one branch phonological operations are applied without semantic effect; on the other are semantic operations without phono- logical realization. At the end of the first branch is phonological form, the input to the articulatory–perceptual system; and at the end of the second is logical form, the input to the conceptual–intentional system.1 Thus conceived, logical form encompasses all and only information required for interpretation. But semantic and logical information do not fully overlap. The connectives “and” and “but” are surely not synonyms, but the difference in meaning probably does not concern logic. On the other hand, it is of utmost logical importance whether “finitely many” or “equinumerous” are logical constants even though it is hard to see how this information could be essential for their interpretation. -
Truth-Bearers and Truth Value*
Truth-Bearers and Truth Value* I. Introduction The purpose of this document is to explain the following concepts and the relationships between them: statements, propositions, and truth value. In what follows each of these will be discussed in turn. II. Language and Truth-Bearers A. Statements 1. Introduction For present purposes, we will define the term “statement” as follows. Statement: A meaningful declarative sentence.1 It is useful to make sure that the definition of “statement” is clearly understood. 2. Sentences in General To begin with, a statement is a kind of sentence. Obviously, not every string of words is a sentence. Consider: “John store.” Here we have two nouns with a period after them—there is no verb. Grammatically, this is not a sentence—it is just a collection of words with a dot after them. Consider: “If I went to the store.” This isn’t a sentence either. “I went to the store.” is a sentence. However, using the word “if” transforms this string of words into a mere clause that requires another clause to complete it. For example, the following is a sentence: “If I went to the store, I would buy milk.” This issue is not merely one of conforming to arbitrary rules. Remember, a grammatically correct sentence expresses a complete thought.2 The construction “If I went to the store.” does not do this. One wants to By Dr. Robert Tierney. This document is being used by Dr. Tierney for teaching purposes and is not intended for use or publication in any other manner. 1 More precisely, a statement is a meaningful declarative sentence-type. -
Semiring Orders in a Semiring -.:: Natural Sciences Publishing
Appl. Math. Inf. Sci. 6, No. 1, 99-102 (2012) 99 Applied Mathematics & Information Sciences An International Journal °c 2012 NSP Natural Sciences Publishing Cor. Semiring Orders in a Semiring Jeong Soon Han1, Hee Sik Kim2 and J. Neggers3 1 Department of Applied Mathematics, Hanyang University, Ahnsan, 426-791, Korea 2 Department of Mathematics, Research Institute for Natural Research, Hanyang University, Seoal, Korea 3 Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, U.S.A Received: Received May 03, 2011; Accepted August 23, 2011 Published online: 1 January 2012 Abstract: Given a semiring it is possible to associate a variety of partial orders with it in quite natural ways, connected with both its additive and its multiplicative structures. These partial orders are related among themselves in an interesting manner is no surprise therefore. Given particular types of semirings, e.g., commutative semirings, these relationships become even more strict. Finally, in terms of the arithmetic of semirings in general or of some special type the fact that certain pairs of elements are comparable in one of these orders may have computable and interesting consequences also. It is the purpose of this paper to consider all these aspects in some detail and to obtain several results as a consequence. Keywords: semiring, semiring order, partial order, commutative. The notion of a semiring was first introduced by H. S. they are equivalent (see [7]). J. Neggers et al. ([5, 6]) dis- Vandiver in 1934, but implicitly semirings had appeared cussed the notion of semiring order in semirings, and ob- earlier in studies on the theory of ideals of rings ([2]). -
On Tarski's Axiomatization of Mereology
On Tarski’s Axiomatization of Mereology Neil Tennant Studia Logica An International Journal for Symbolic Logic ISSN 0039-3215 Volume 107 Number 6 Stud Logica (2019) 107:1089-1102 DOI 10.1007/s11225-018-9819-3 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Nature B.V.. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Neil Tennant On Tarski’s Axiomatization of Mereology Abstract. It is shown how Tarski’s 1929 axiomatization of mereology secures the re- flexivity of the ‘part of’ relation. This is done with a fusion-abstraction principle that is constructively weaker than that of Tarski; and by means of constructive and relevant rea- soning throughout. We place a premium on complete formal rigor of proof. Every step of reasoning is an application of a primitive rule; and the natural deductions themselves can be checked effectively for formal correctness. Keywords: Mereology, Part of, Reflexivity, Tarski, Axiomatization, Constructivity.