Type-Free Truth
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Tarski's Threat to the T-Schema
Tarski’s Threat to the T-Schema MSc Thesis (Afstudeerscriptie) written by Cian Chartier (born November 10th, 1986 in Montr´eal, Canada) under the supervision of Prof. dr. Frank Veltman, and submitted to the Board of Examiners in partial fulfillment of the requirements for the degree of MSc in Logic at the Universiteit van Amsterdam. Date of the public defense: Members of the Thesis Committee: August 30, 2011 Prof. Dr. Frank Veltman Prof. Dr. Michiel van Lambalgen Prof. Dr. Benedikt L¨owe Dr. Robert van Rooij Abstract This thesis examines truth theories. First, four relevant programs in philos- ophy are considered. Second, four truth theories are compared according to a range of criteria. The truth theories are categorised according to Leitgeb’s eight criteria for a truth theory. The four truth theories are then compared with each-other based on three new criteria. In this way their relative use- fulness in pursuing some of the aims of the four programs is evaluated. This presents a springboard for future work on truth: proposing ideas for differ- ent truth theories that advance unambiguously different programs on the basis of the four truth theories proposed. 1 Acknowledgements I first thank my supervisor Prof. Dr. Frank Veltman for regularly read- ing and providing constructive feedback, recommending some papers along the way. This thesis has been much improved by his corrections and input, including extensive discussions and suggestions of the ideas proposed here. His patience in working with me on the thesis has been much appreciated. Among the other professors in the ILLC, Dr. Robert van Rooij also provided helpful discussion. -
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ON NUMBERS by LINDA ELIZABETH WETZEL B.A. City College of New York (1975) Submitted to the Department of Linguistics and Philosophy in Partial Fulfillment of the Requirements of the Degree of DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 1984 @ Linda E. Wetzel The author hereby grants to M.I.T. permission to reproduce and to distribute copies of this thesis document in whole or in part, Signature of Author: Department of Linguistics and Philosophy January 13, 1984 Certified by : .-_ - Richard L, Cartwright Thesis Supervisor Accepted by: v Richard L. Cartwright Chairman, Departmental Graduate Committee ON NUMBERS Linda E, Wetzel Submitted to the Department of Linguistics and Philosophy of October 21, 1983 in partial fulfillment of the requirements for the degree of Doctor of Philosophy ABSTRACT We talk as though there are numbers. The view I defend, the "popularn view, has it that there -are numbers. However, since they clearly are not physical objects, we reason that they must be abstract ones. This suggests a realm of non-spatial non-temporal objects standing in numerical relations; arithmetic knowledge is then knowledge of this realm. But how do spatio~temporal creatures like ourselves come to have knowledge of this realm? The problem ("Benacerrafls probleni") can be avoided by arguing that there are no numbers. In "What Numbers Could Not Bew Benacerraf himself took such a route. In chapter one, I discuss three of Benacerrafls arguments, showing that the first is circular, that the second involves a consideration that can be explained by less drastic means than supposing there are no numbers, and that the third would, if successful, show that neither sets nor expressions exist either. -
Intrinsic Explanation and Field's Dispensabilist Strategy Sydney-Tilburg Conference on Reduction and the Special Sciences Russ
Intrinsic Explanation and Field’s Dispensabilist Strategy Sydney-Tilburg Conference on Reduction and the Special Sciences Russell Marcus Department of Philosophy, Hamilton College 198 College Hill Road Clinton NY 13323 [email protected] (315) 859-4056 (office) (315) 381-3125 (home) September 2007 ~2880 words Abstract: Philosophy of mathematics for the last half-century has been dominated in one way or another by Quine’s indispensability argument. The argument alleges that our best scientific theory quantifies over, and thus commits us to, mathematical objects. In this paper, I present new considerations which undermine the most serious challenge to Quine’s argument, Hartry Field’s reformulation of Newtonian Gravitational Theory. Intrinsic Explanation, Page 1 §1: Introduction Quine argued that we are committed to the existence mathematical objects because of their indispensable uses in scientific theory. In this paper, I defend Quine’s argument against the most popular objection to it, that we can reformulate science without reference to mathematical objects. I interpret Quine’s argument as follows:1 (QIA) QIA.1: We should believe the theory which best accounts for our empirical experience. QIA.2: If we believe a theory, we must believe in its ontic commitments. QIA.3: The ontic commitments of any theory are the objects over which that theory first-order quantifies. QIA.4: The theory which best accounts for our empirical experience quantifies over mathematical objects. QIA.C: We should believe that mathematical objects exist. An instrumentalist may deny either QIA.1 or QIA.2, or both. Regarding QIA.1, there is some debate over whether we should believe our best theories. -
Robert C. Koons
ROBERT C. KOONS ADDRESSES Department of Philosophy 1, University Station C3500 University of Texas at Austin Austin, Texas 78712-3500 (512) 471-5530 [email protected] EDUCATION 1979 B.A., Philosophy, Michigan State University, Summa cum laude 1981 B.A., Philosophy and Theology, Oxford University First Class Honours 1987 Ph.D., Philosophy, UCLA AREAS OF SPECIALIZATION Metaphysics and Epistemology Philosophical Logic Philosophy of Religion PROFESSIONAL EXPERIENCE Sept. 1, 2000 Professor, University of Texas at Austin 1993-2000 Associate Professor, University of Texas at Austin 1987-1993 Assistant Professor, University of Texas at Austin HONORS Visiting Scholar, Nanjing University, April/May 2007 Philosopher-in-Residence, Valparaiso University, Spring 2001 Gustave O. Arlt Award (Council of Graduate Schools) 1992 Carnap Prize (UCLA) 1987 Richard M. Weaver Fellow, 1985-87 Danforth Fellow, l979-85 Dillistone Scholar (Oriel College, Oxford), l980 Marshall Scholar, l979-1981 ROBERT C. KOONS PAGE 2 RESEARCH GRANTS National Science Foundation, Division of Information, Robotics and Intelligent Systems, "The Logic and Representation of Properties and Propositions for Computer Natural Language Processing," with Kamp, Bonevac, Asher, and C. Smith, 1988-1989. National Research Council Travel Grant for Attendance of the Ninth International Congress on Logic, Methodology and Philosophy of Science, Uppsala, Sweden, 1991. Faculty Research Assignment, "The Logic of Causation and Teleological Function," Spring 1997. Visiting Scholar, Institute for Advanced -
Realism and Anti-Realism in Mathematics
REALISM AND ANTI-REALISM IN MATHEMATICS The purpose of this essay is (a) to survey and critically assess the various meta- physical views - Le., the various versions of realism and anti-realism - that people have held (or that one might hold) about mathematics; and (b) to argue for a particular view of the metaphysics of mathematics. Section 1 will provide a survey of the various versions of realism and anti-realism. In section 2, I will critically assess the various views, coming to the conclusion that there is exactly one version of realism that survives all objections (namely, a view that I have elsewhere called full-blooded platonism, or for short, FBP) and that there is ex- actly one version of anti-realism that survives all objections (namely, jictionalism). The arguments of section 2 will also motivate the thesis that we do not have any good reason for favoring either of these views (Le., fictionalism or FBP) over the other and, hence, that we do not have any good reason for believing or disbe- lieving in abstract (i.e., non-spatiotemporal) mathematical objects; I will call this the weak epistemic conclusion. Finally, in section 3, I will argue for two further claims, namely, (i) that we could never have any good reason for favoring either fictionalism or FBP over the other and, hence, could never have any good reason for believing or disbelieving in abstract mathematical objects; and (ii) that there is no fact of the matter as to whether fictionalism or FBP is correct and, more generally, no fact of the matter as to whether there exist any such things as ab- stract objects; I will call these two theses the strong epistemic conclusion and the metaphysical conclusion, respectively. -
Jared Warren
JARED WARREN Curriculum Vitae B [email protected] T 904.866.8439 www.jaredwarren.org Education 2015 NEW YORK UNIVERSITY PhD in Philosophy COMMITTEE: David Chalmers, Hartry Field (chair), Crispin Wright Areas of Specialization Mind and Language, Metaphysics and Epistemology, Mathematics and Logic Areas of Competence History of Analytic Philosophy, Metaethics, Philosophy of Science Publications (1) “The Possibility of Truth by Convention”(2015) The Philosophical Quarterly 65(258): 84-93. (2) “Quantifier Variance and the Collapse Argument” (2015) The Philosophical Quarterly 65(259): 241-253. (3) “Conventionalism, Consistency, and Consistency Sentences” (2015) Synthese 192(5): 1351-1371. (4) “Talking with Tonkers” (2015) Philosophers’ Imprint 15(24): 1-24. (5) “Trapping the Metasemantic Metaphilosophical Deflationist?” (2016) Metaphilosophy 47(1): 108-121. (6) “Sider on the Epistemology of Structure” (2016) Philosophical Studies 173(9): 2417-2435. (7) “Epistemology vs Non-Causal Realism” (2017) Synthese 194(5): 1643-1662. (8) “Revisiting Quine on Truth by Convention” (2017) The Journal of Philosophical Logic 46(2): 119-139. 1 (9) “Internal and External Questions Revisited” (2016) The Journal of Philosophy 113(4): 177-209. (10) “Change of Logic, Change of Meaning” (forthcoming) Philosophy & Phenomenological Research. (11) “Quantifier Variance and Indefinite Extensibility” (2017) The Philosophical Review 126(1): 81-122. (12) “A Metasemantic Challenge for Mathematical Determinacy” (forthcoming) Synthese. (with Daniel Waxman) (13) “Quantifier Variance -
First-Order Logic
Chapter 10 First-Order Logic 10.1 Overview First-Order Logic is the calculus one usually has in mind when using the word ‘‘logic’’. It is expressive enough for all of mathematics, except for those concepts that rely on a notion of construction or computation. However, dealing with more advanced concepts is often somewhat awkward and researchers often design specialized logics for that reason. Our account of first-order logic will be similar to the one of propositional logic. We will present ² The syntax, or the formal language of first-order logic, that is symbols, formulas, sub-formulas, formation trees, substitution, etc. ² The semantics of first-order logic ² Proof systems for first-order logic, such as the axioms, rules, and proof strategies of the first-order tableau method and refinement logic ² The meta-mathematics of first-order logic, which established the relation between the semantics and a proof system In many ways, the account of first-order logic is a straightforward extension of propositional logic. One must, however, be aware that there are subtle differences. 10.2 Syntax The syntax of first-order logic is essentially an extension of propositional logic by quantification 8 and 9. Propositional variables are replaced by n-ary predicate symbols (P , Q, R) which may be instantiated with either variables (x, y, z, ...) or parameters (a, b, ...). Here is a summary of the most important concepts. 101 102 CHAPTER 10. FIRST-ORDER LOGIC (1) Atomic formulas are expressions of the form P c1::cn where P is an n-ary predicate symbol and the ci are variables or parameters. -
A Subsystem of Classical Propositional Logic
VALUATIONS FOR DIRECT PROPOSITIONAL LOGIC In (1969) a subsystem of classical propositional logic - called direct logic since all indirect inferences are excluded from it - was formulated as a generalized sequential calculus. With its rules the truth-value (true (t) or false (0) of the consequent can be determined from the truth-values of the premisses. In this way the semantical stipulations are formulated construc• tively as in a calculus of natural deduction, i.e. semantics itself is a formal system of rules. The rules for propositional composita define an extension of basic calculi containing only truth rules for atomic sentences. The sen• tences provable in the extensions of all basic calculi, i.e. in the proposi• tional calculus itself, are the logically true sentences. The principle of bi- valence is not presupposed for the basic calculi. That it holds for a specific calculus - and then also in its propositional extension - is rather a theorem to be proved by metatheoretic means. In this approach a semantics in the usual sense, i.e. a semantics based on a concept of valuation, has no place since the calculus of direct logic itself is already conceived of as a system of semantical rules. Nevertheless it is of some interest to see that there is an adequate and intuitively plau• sible valuation-semantics for this calculus. The language L used in what follows is the usual language of proposi• tional logic with —i, A and n> as basic operators.1 A classical or total valuation of L is a function V mapping the set of all sentences of L into {t, f} so that K(-i A) = t iff V(A) = f, V(A A B) = t iff V(A) = V(B) = t. -
Statistical Parsing and Unambiguous Word Representation in Opencog’S Unsupervised Language Learning Project
Statistical parsing and unambiguous word representation in OpenCog’s Unsupervised Language Learning project Master’s thesis in MPALG and MPCAS CLAUDIA CASTILLO DOMENECH and ANDRES SUAREZ MADRIGAL Department of Computer Science and Engineering CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG Gothenburg, Sweden 2018 Master’s thesis 2018 Statistical parsing and unambiguous word representation in OpenCog’s Unsupervised Language Learning project CLAUDIA CASTILLO DOMENECH ANDRES SUAREZ MADRIGAL Department of Computer Science and Engineering Chalmers University of Technology University of Gothenburg Gothenburg, Sweden 2018 Statistical parsing and unambiguous word representation in OpenCog’s Unsupervised Language Learning project CLAUDIA CASTILLO DOMENECH ANDRES SUAREZ MADRIGAL « CLAUDIA CASTILLO DOMENECH and ANDRES SUAREZ MADRIGAL, 2018. Supervisor: Luis Nieto Piña, Faculty of Humanities Advisor: Ben Goertzel, Hanson Robotics Ltd. Examiner: Morteza Haghir Chehreghani, Data Science division Master’s Thesis 2018 Department of Computer Science and Engineering Chalmers University of Technology and University of Gothenburg SE-412 96 Gothenburg Telephone +46 31 772 1000 Typeset in LATEX Gothenburg, Sweden 2018 iv Statistical parsing and unambiguous word representation in OpenCog’s Unsupervised Language Learning project CLAUDIA CASTILLO DOMENECH and ANDRES SUAREZ MADRIGAL Department of Computer Science and Engineering Chalmers University of Technology and University of Gothenburg Abstract The work presented in the current thesis is an effort -
14 the Semantics of Classical First- Order Logic
The Semantics of Classical First- 14 Order Logic 1. Introduction...........................................................................................................................2 2. Semantic Evaluations............................................................................................................2 3. Semantic Items and their Categories.....................................................................................2 4. Official versus Conventional Identifications of Semantic Items ...........................................3 5. The Categorial Correspondence Rule...................................................................................3 6. The Categorial Composition Principle .................................................................................5 7. The Sub-Functions of a Semantic Evaluation........................................................................6 8. Interpretations.......................................................................................................................6 9. Truth and Falsity for Simple Atomic Formulas.....................................................................7 10. Pronouns – Variables and Constants.....................................................................................9 11. Multiple Pronouns ..............................................................................................................10 12. Assignment Functions – Constants ......................................................................................12 -
Propositional Logic
Propositional Logic Andreas Klappenecker Propositions A proposition is a declarative sentence that is either true or false (but not both). Examples: • College Station is the capital of the USA. • There are fewer politicians in College Station than in Washington, D.C. • 1+1=2 • 2+2=5 Propositional Variables A variable that represents propositions is called a propositional variable. For example: p, q, r, ... [Propositional variables in logic play the same role as numerical variables in arithmetic.] Propositional Logic The area of logic that deals with propositions is called propositional logic. In addition to propositional variables, we have logical connectives such as not, and, or, conditional, and biconditional. Syntax of Propositional Logic Approach We are going to present the propositional logic as a formal language: - we first present the syntax of the language - then the semantics of the language. [Incidentally, this is the same approach that is used when defining a new programming language. Formal languages are used in other contexts as well.] Formal Languages Let S be an alphabet. We denote by S* the set of all strings over S, including the empty string. A formal language L over the alphabet S is a subset of S*. Syntax of Propositional Logic Our goal is to study the language Prop of propositional logic. This is a language over the alphabet ∑=S ∪ X ∪ B , where - S = { a, a0, a1,..., b, b0, b1,... } is the set of symbols, - X = {¬,∧,∨,⊕,→,↔} is the set of logical connectives, - B = { ( , ) } is the set of parentheses. We describe the language Prop using a grammar. Grammar of Prop ⟨formula ⟩::= ¬ ⟨formula ⟩ | (⟨formula ⟩ ∧ ⟨formula ⟩) | (⟨formula ⟩ ∨ ⟨formula ⟩) | (⟨formula ⟩ ⊕ ⟨formula ⟩) | (⟨formula ⟩ → ⟨formula ⟩) | (⟨formula ⟩ ↔ ⟨formula ⟩) | ⟨symbol ⟩ Example Using this grammar, you can infer that ((a ⊕ b) ∨ c) ((a → b) ↔ (¬a ∨ b)) both belong to the language Prop, but ((a → b) ∨ c does not, as the closing parenthesis is missing. -
Propositional Logic
Propositional Logic LX 502 - Semantics September 19, 2008 1. Semantics and Propositions Natural language is used to communicate information about the world, typically between a speaker and an addressee. This information is conveyed using linguistic expressions. What makes natural language a suitable tool for communication is that linguistic expressions are not just grammatical forms: they also have content. Competent speakers of a language know the forms and the contents of that language, and how the two are systematically related. In other words, you can know the forms of a language but you are not a competent speaker unless you can systematically associate each form with its appropriate content. Syntax, phonology, and morphology are concerned with the form of linguistic expressions. In particular, Syntax is the study of the grammatical arrangement of words—the form of phrases but not their content. From Chomsky we get the sentence in (1), which is grammatical but meaningless. (1) Colorless green ideas sleep furiously Semantics is the study of the content of linguistic expressions. It is defined as the study of the meaning expressed by elements in a language and combinations thereof. Elements that contribute to the content of a linguistic expression can be syntactic (e.g., verbs, nouns, adjectives), morphological (e.g., tense, number, person), and phonological (e.g., intonation, focus). We’re going to restrict our attention to the syntactic elements for now. So for us, semantics will be the study of the meaning expressed by words and combinations thereof. Our goals are listed in (2). (2) i. To determine the conditions under which a sentence is true of the world (and those under which it is false) ii.