Propositional Quantifiers in Modal Logic1
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Chapter 2 Introduction to Classical Propositional
CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC 1 Motivation and History The origins of the classical propositional logic, classical propositional calculus, as it was, and still often is called, go back to antiquity and are due to Stoic school of philosophy (3rd century B.C.), whose most eminent representative was Chryssipus. But the real development of this calculus began only in the mid-19th century and was initiated by the research done by the English math- ematician G. Boole, who is sometimes regarded as the founder of mathematical logic. The classical propositional calculus was ¯rst formulated as a formal ax- iomatic system by the eminent German logician G. Frege in 1879. The assumption underlying the formalization of classical propositional calculus are the following. Logical sentences We deal only with sentences that can always be evaluated as true or false. Such sentences are called logical sentences or proposi- tions. Hence the name propositional logic. A statement: 2 + 2 = 4 is a proposition as we assume that it is a well known and agreed upon truth. A statement: 2 + 2 = 5 is also a proposition (false. A statement:] I am pretty is modeled, if needed as a logical sentence (proposi- tion). We assume that it is false, or true. A statement: 2 + n = 5 is not a proposition; it might be true for some n, for example n=3, false for other n, for example n= 2, and moreover, we don't know what n is. Sentences of this kind are called propositional functions. We model propositional functions within propositional logic by treating propositional functions as propositions. -
12 Propositional Logic
CHAPTER 12 ✦ ✦ ✦ ✦ Propositional Logic In this chapter, we introduce propositional logic, an algebra whose original purpose, dating back to Aristotle, was to model reasoning. In more recent times, this algebra, like many algebras, has proved useful as a design tool. For example, Chapter 13 shows how propositional logic can be used in computer circuit design. A third use of logic is as a data model for programming languages and systems, such as the language Prolog. Many systems for reasoning by computer, including theorem provers, program verifiers, and applications in the field of artificial intelligence, have been implemented in logic-based programming languages. These languages generally use “predicate logic,” a more powerful form of logic that extends the capabilities of propositional logic. We shall meet predicate logic in Chapter 14. ✦ ✦ ✦ ✦ 12.1 What This Chapter Is About Section 12.2 gives an intuitive explanation of what propositional logic is, and why it is useful. The next section, 12,3, introduces an algebra for logical expressions with Boolean-valued operands and with logical operators such as AND, OR, and NOT that Boolean algebra operate on Boolean (true/false) values. This algebra is often called Boolean algebra after George Boole, the logician who first framed logic as an algebra. We then learn the following ideas. ✦ Truth tables are a useful way to represent the meaning of an expression in logic (Section 12.4). ✦ We can convert a truth table to a logical expression for the same logical function (Section 12.5). ✦ The Karnaugh map is a useful tabular technique for simplifying logical expres- sions (Section 12.6). -
Propositional Calculus
CSC 438F/2404F Notes Winter, 2014 S. Cook REFERENCES The first two references have especially influenced these notes and are cited from time to time: [Buss] Samuel Buss: Chapter I: An introduction to proof theory, in Handbook of Proof Theory, Samuel Buss Ed., Elsevier, 1998, pp1-78. [B&M] John Bell and Moshe Machover: A Course in Mathematical Logic. North- Holland, 1977. Other logic texts: The first is more elementary and readable. [Enderton] Herbert Enderton: A Mathematical Introduction to Logic. Academic Press, 1972. [Mendelson] E. Mendelson: Introduction to Mathematical Logic. Wadsworth & Brooks/Cole, 1987. Computability text: [Sipser] Michael Sipser: Introduction to the Theory of Computation. PWS, 1997. [DSW] M. Davis, R. Sigal, and E. Weyuker: Computability, Complexity and Lan- guages: Fundamentals of Theoretical Computer Science. Academic Press, 1994. Propositional Calculus Throughout our treatment of formal logic it is important to distinguish between syntax and semantics. Syntax is concerned with the structure of strings of symbols (e.g. formulas and formal proofs), and rules for manipulating them, without regard to their meaning. Semantics is concerned with their meaning. 1 Syntax Formulas are certain strings of symbols as specified below. In this chapter we use formula to mean propositional formula. Later the meaning of formula will be extended to first-order formula. (Propositional) formulas are built from atoms P1;P2;P3;:::, the unary connective :, the binary connectives ^; _; and parentheses (,). (The symbols :; ^ and _ are read \not", \and" and \or", respectively.) We use P; Q; R; ::: to stand for atoms. Formulas are defined recursively as follows: Definition of Propositional Formula 1) Any atom P is a formula. -
A Formal Mereology of Potential Parts
Not Another Brick in the Wall: an Extensional Mereology for Potential Parts Contents 0. Abstract .................................................................................................................................................................................................. 1 1. An Introduction to Potential Parts .......................................................................................................................................................... 1 2. The Physical Motivation for Potential Parts ............................................................................................................................................ 5 3. A Model for Potential Parts .................................................................................................................................................................... 9 3.1 Informal Semantics: Dressed Electron Propagator as Mereological Model ...................................................................................... 9 3.2 Formal Semantics: A Join Semi-Lattice ............................................................................................................................................ 10 3.3 Syntax: Mereological Axioms for Potential Parts ............................................................................................................................ 14 4. Conclusions About Potential Parts ....................................................................................................................................................... -
Why Would You Trust B? Eric Jaeger, Catherine Dubois
Why Would You Trust B? Eric Jaeger, Catherine Dubois To cite this version: Eric Jaeger, Catherine Dubois. Why Would You Trust B?. Logic for Programming, Artificial In- telligence, and Reasoning, Nov 2007, Yerevan, Armenia. pp.288-302, 10.1007/978-3-540-75560-9. hal-00363345 HAL Id: hal-00363345 https://hal.archives-ouvertes.fr/hal-00363345 Submitted on 23 Feb 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Why Would You Trust B? Eric´ Jaeger12 and Catherine Dubois3 1 LIP6, Universit´eParis 6, 4 place Jussieu, 75252 Paris Cedex 05, France 2 LTI, Direction centrale de la s´ecurit´edes syst`emes d’information, 51 boulevard de La Tour-Maubourg, 75700 Paris 07 SP, France 3 CEDRIC, Ecole´ nationale sup´erieure d’informatique pour l’industrie et l’entreprise, 18 all´ee Jean Rostand, 91025 Evry Cedex, France Abstract. The use of formal methods provides confidence in the cor- rectness of developments. Yet one may argue about the actual level of confidence obtained when the method itself – or its implementation – is not formally checked. We address this question for the B, a widely used formal method that allows for the derivation of correct programs from specifications. -
An Introduction to Formal Methods for Philosophy Students
An Introduction to Formal Methods for Philosophy Students Thomas Forster February 20, 2021 2 Contents 1 Introduction 13 1.1 What is Logic? . 13 1.1.1 Exercises for the first week: “Sheet 0” . 13 2 Introduction to Logic 17 2.1 Statements, Commands, Questions, Performatives . 18 2.1.1 Truth-functional connectives . 20 2.1.2 Truth Tables . 21 2.2 The Language of Propositional Logic . 23 2.2.1 Truth-tables for compound expressions . 24 2.2.2 Logical equivalence . 26 2.2.3 Non truth functional connectives . 27 2.3 Intension and Extension . 28 2.3.1 If–then . 31 2.3.2 Logical Form and Valid Argument . 33 2.3.3 The Type-Token Distinction . 33 2.3.4 Copies . 35 2.4 Tautology and Validity . 36 2.4.1 Valid Argument . 36 2.4.2 V and W versus ^ and _ .................... 40 2.4.3 Conjunctive and Disjunctive Normal Form . 41 2.5 Further Useful Logical Gadgetry . 46 2.5.1 The Analytic-Synthetic Distinction . 46 2.5.2 Necessary and Sufficient Conditions . 47 2.5.3 The Use-Mention Distinction . 48 2.5.4 Language-metalanguage distinction . 51 2.5.5 Semantic Optimisation and the Principle of Charity . 52 2.5.6 Inferring A-or-B from A . 54 2.5.7 Fault-tolerant pattern-matching . 54 2.5.8 Overinterpretation . 54 2.5.9 Affirming the consequent . 55 3 4 CONTENTS 3 Proof Systems for Propositional Logic 57 3.1 Arguments by LEGO . 57 3.2 The Rules of Natural Deduction . 57 3.2.1 Worries about reductio and hypothetical reasoning . -
Michael Jraven
MICHAEL J RAVEN Department of Philosophy | University of Victoria P.O. BOX 1700 STN CSC | Victoria, BC V8W 2Y2 | CANADA [email protected] • raven.site RESEARCH INTERESTS metaphysics • ground; fundamentality; essence; social items; time; change language/mind • de re thought; mental states; aesthetic judgment epistemology • explanation; epistemic relativism ACADEMIC POSITIONS Professor, Philosophy, University of Victoria 2021— Affiliate Professor, Philosophy, University of Washington 2018— Associate Professor, Philosophy, University of Victoria 2013—2021 Visiting Scholar, Philosophy, University of Washington 2016-2017, 2012-2013 Assistant Professor, Philosophy, University of Victoria 2009-2013 Instructor, Philosophy, New York University 2008-2009 EDUCATION PHD, Philosophy, New York University 2009 [Kit Fine (chair), Ted Sider, Crispin Wright] MA, Philosophy, New York University 2007 [Ned Block (adviser) BA, Philosophy, Reed College 2002 [Mark Hinchliff (adviser)] GRANTS & FELLOWSHIPS Connection Grant ($15,640), Social Science & Humanities Research Council 2020 Humanities Faculty Fellowship (course release), University of Victoria 2020 Insight Grant ($152,589), Social Science & Humanities Research Council 2018-2022 [principal investigator: 2021-2022, collaborator: 2018-2020; co-applicant with Kathrin Koslicki] Connection Grant ($15,965), Social Science & Humanities Research Council 2018 Internal Research Grants ($29,428 total), University of Victoria 2010-2014,2016,2020 Conference Grant ($2000 - declined), Canadian Journal of Philosophy -
Matter and Mereology∗
Matter and mereology∗ Jeremy Goodman Draft of March 22, 2015 I am a material thing. But I am not the same thing as the matter out of which I am composed, since my matter, unlike me, could have existed as scattered interstellar dust. The distinction between matter and objects that are merely composed of matter is central to our ordinary conception of the material world. According to that conception, material objects have a hierarchical structure with matter at its foundation. This paper shows how matter and material constitution can be understood in terms of the part-whole relation.1 I present a novel mereology and apply it to debates about the persistence and plenitude of material objects, and compare my view to more familiar hylomorphic ones. A formal model of the theory is given in an appendix. 1 Anti-symmetry Consider the following principle: weak supplementation: If x is a proper part of y, then y has a part that does not overlap x. where x is a proper part of y =df x is a part of y and x 6= y. x overlaps y =df some part of x is part of y. Everyday cases of material coincidence are counterexamples to weak supple- mentation. A statue s composed of some clay c. c is distinct from s, since squashing c would destroy s but would not destroy c.2 c is part of s, since any ∗Thanks to Andrew Bacon, Cian Dorr, Maegan Fairchild, Kit Fine, Peter Fritz, John Hawthorne, Harvey Lederman, Gabriel Uzquiano, Tim Williamson and an anonymous referee for No^us for comments on previous versions of this paper, and to an audience at the University of Cambridge. -
Vagueness at Every Order
Vagueness at every order Andrew Bacon* May 4, 2020 Abstract There are some properties, like being bald, for which it is vague where the boundary between the things that have it, and the things that do not, lies. A number of argument threaten to show that such properties can still be associated with determinate and knowable boundaries: not between the things that have it and those that don't, but between the things such that it is borderline at some order whether they have it, and the things for which it is not. I argue that these arguments, if successful, turn on a contentious prin- ciple in the logic of determinacy: Brouwer's Principle, that every truth is determinately not determinately false. Other paradoxes which do not appear to turn on this principle often tacitly make assumptions about assertion, knowledge and higher order vagueness. In this paper I'll show how one can avoid sharp higher-order boundaries by rejecting these as- sumptions. I used to be a child, but now I am not. It follows, given classical logic, that I stopped being a child at some point. Indeed, there was a first nanosecond at which I stopped: a nanosecond before which I was a child (or unborn) but at which I was not a child. Although such a nanosecond exists, it is a vague matter when it occurred: it happened during a period when it was borderline whether I was a child. Different theories provide different accounts of what borderlineness consists in | perhaps it's ignorance, or semantic indecision, or something else. -
The Question of Realism Ity
My aim in this paper is to help lay the conceptual and meth- odological foundations for the study of realism. I come to two main conclusions: first, that there is a primitive meta- physical concept of reality, one that cannot be understood in fundamentally different terms; and second, that questions of what is real are to be settled upon the basis of considerations of ground. The two conclusions are somewhat in tension with one another, for the lack of a definition of the concept of reality would appear to stand in the way of developing a sound methodology for determining its application; and one of my main concerns has been to show how the tension be- tween the two might be resolved. The paper is in two main parts. In the first, I point to the difficulties in making out a metaphysical conception of real- The Question of Realism ity. I begin by distinguishing this conception from the ordi- nary conception of reality (§1) and then show how the two leading contenders for the metaphysical conception—the factual and the irreducible—both appear to resist formula- tion in other terms. This leads to the quietist challenge, that Kit Fine questions of realism are either meaningless or pointless (§4); and the second part of the paper (§§5-10) is largely devoted to showing how this challenge might be met. I begin by in- troducing the notion of ground (§5) and then show how it can be used as a basis for resolving questions both of factual- ity (§§6-7) and of irreducibility (§§8-9). -
Categorical Proof Theory of Classical Propositional Calculus
Categorical Proof Theory of Classical Propositional Calculus Gianluigi Bellin Martin Hyland Edmund Robinson Christian Urban Queen Mary, University of London, UK University of Cambridge, UK Technical University of Munich (TUM), D Abstract We investigate semantics for classical proof based on the sequent calculus. We show that the propositional connectives are not quite well-behaved from a traditional categorical per- spective, and give a more refined, but necessarily complex, analysis of how connectives may be characterised abstractly. Finally we explain the consequences of insisting on more familiar categorical behaviour. Key words: classical logic, proof theory, category theory 1 Introduction In this paper we describe the shape of a semantics for classical proof in accord with Gentzen’s sequent calculus. For constructive proof we have the familiar correspondence between deductions in minimal logic and terms of a typed lambda calculus. Deductions in minimal logic (as in most constructive systems) reduce to a unique normal form, and around 1970 Per Martin-L¨of (see [18]) suggested using equality of normal forms as the identity criterion for proof objects in his constructive Type Theories: normal forms serve as the semantics of proof. But ¬ -normal forms for typed lambda calculus give maps in a free cartesian closed category; so we get a whole range of categorical models of constructive proof. This is the circle of connections surrounding the Curry-Howard isomorphism. We seek analogues of these ideas for classical proof. There are a number of immediate problems. The established term languages for classical proofs are either incompatible with the symmetries apparent in the sequent calculus (Parigot [16]) or in reconciling themselves to that symmetry at least make evaluation deterministic (cf Danos et al [5,21]). -
Logical Derivability of the Concept of Meaning, Formulation I
IOSR Journal of Humanities and Social Science (IOSRJHSS) ISSN: 2279-0845 Volume 1, Issue 5 (Sep-Oct 2012), PP 01-04 www.iosrjournals.org Logical Derivability of the Concept of Meaning, Formulation I Jonathan Chimakonam Okeke, Department of Philosophy,University of Calabar, Calabar, P. M. B 1115, Cross River State Nigeria ABSTRACT: Expressions, words and symbols without reference to something else which could be called their meanings are semantically helpless. But not all expressions and words refer; some even come with ambiguities and equivocations like Golden Mountain, Chimera etc., however, any symbol which does not refer could not properly be called a symbol. So because every symbol necessarily refers to something definite, it is not the case that ambiguities and equivocations would sneak into symbolic expressions. Hence, logic becomes that science which prefers symbolic or artificial or formal language to natural language. Therefore, since “meaning” or semantics is a central focus of logic together with syntax, we attempted in this work to obtain a logical derivation of it in the symbolic language of logic devoid of the ambiguities and equivocations of natural language. Keywords- language, logic, meaning, semantics, symbol, syntax. I. Introduction Meaning can be defined as having a fixed sense or conveying a fixed idea or a fixed thought. This means that having a transitory or tentative sense does not qualify as having meaning. In philosophy of language meaning is very crucial for our words and sentences will be useless if they contain no meaning. But meaning is not something peculiar to words or sentences alone, it can in similar way be extended to symbols or whatever stands as symbol.