Propositional Logic
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Propositional Logic - Basics
Outline Propositional Logic - Basics K. Subramani1 1Lane Department of Computer Science and Electrical Engineering West Virginia University 14 January and 16 January, 2013 Subramani Propositonal Logic Outline Outline 1 Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Subramani Propositonal Logic (i) The Law! (ii) Mathematics. (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox). Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation Why Logic? Subramani Propositonal Logic (ii) Mathematics. (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox). Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation Why Logic? (i) The Law! Subramani Propositonal Logic (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox). Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation Why Logic? (i) The Law! (ii) Mathematics. Subramani Propositonal Logic (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. -
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. -
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). -
Adding the Everywhere Operator to Propositional Logic
Adding the Everywhere Operator to Propositional Logic David Gries∗ and Fred B. Schneider† Computer Science Department, Cornell University Ithaca, New York 14853 USA September 25, 2002 Abstract Sound and complete modal propositional logic C is presented, in which 2P has the interpretation “ P is true in all states”. The interpretation is already known as the Carnapian extension of S5. A new axiomatization for C provides two insights. First, introducing an inference rule textual substitution allows seamless integration of the propositional and modal parts of the logic, giving a more practical system for writing formal proofs. Second, the two following approaches to axiomatizing a logic are shown to be not equivalent: (i) give axiom schemes that denote an infinite number of axioms and (ii) write a finite number of axioms in terms of propositional variables and introduce a substitution inference rule. 1Introduction Logic gives a syntactic way to derive or certify truths that can be expressed in the language of the logic. The expressiveness of the language impacts the logic’s utility —the more expressive the language, the more useful the logic (at least if the intended use is to prove theorems, as opposed to, say, studying logics). We wish to calculate with logic, much as one calculates in algebra to derive equations, and we find it useful for 2P to be a formula of the logic and have the meaning “ P is true in all states”. When a propositional logic extended with 2 is further extended to predicate logic and then to other theories, the logic can be used for proving theorems that could otherwise be ∗Supported by NSF grant CDA-9214957 and ARPA/ONR grant N00014-91-J-4123. -
On the Logic of Left-Continuous T-Norms and Right-Continuous T-Conorms
On the Logic of Left-Continuous t-Norms and Right-Continuous t-Conorms B Llu´ıs Godo1( ),Mart´ın S´ocola-Ramos2, and Francesc Esteva1 1 Artificial Intelligence Research Institute (IIIA-CSIC), Campus de la UAB, Bellaterra, 08193 Barcelona, Spain {godo,esteva}@iiia.csic.es 2 Juniv¨agen 13, 27172 K¨opingebro, Sweden [email protected] Abstract. Double residuated lattices are expansions of residuated lat- tices with an extra monoidal operator, playing the role of a strong dis- junction operation, together with its dual residuum. They were intro- duced by Orlowska and Radzikowska. In this paper we consider the subclass of double residuated structures that are expansions of MTL- algebras, that is, prelinear, bounded, commutative and integral residu- ated lattices. MTL-algebras constitute the algebraic semantics for the MTL logic, the system of mathematical fuzzy logic that is complete w.r.t. the class of residuated lattices on the real unit interval [0, 1] induced by left-continuous t-norms. Our aim is to axiomatise the logic whose intended semantics are commutative and integral double residu- ated structures on [0, 1], that are induced by an arbitrary left-continuous t-norm, an arbitrary right-continuous t-conorm, and their corresponding residual operations. Keywords: Mathematical fuzzy logic · Double residuated lattices · MTL · DMCTL · Semilinear logics · Standard completeness 1 Introduction Residuated lattices [7] are a usual choice for general algebras of truth-values in systems of mathematical fuzzy logic [1–3,9], as well as for valuation structures in lattice-valued fuzzy sets (L-fuzzy sets) [8] and in multiple-valued generalisations of information relations [11,13,14]. -
Lecture 4 1 Overview 2 Propositional Logic
COMPSCI 230: Discrete Mathematics for Computer Science January 23, 2019 Lecture 4 Lecturer: Debmalya Panigrahi Scribe: Kevin Sun 1 Overview In this lecture, we give an introduction to propositional logic, which is the mathematical formaliza- tion of logical relationships. We also discuss the disjunctive and conjunctive normal forms, how to convert formulas to each form, and conclude with a fundamental problem in computer science known as the satisfiability problem. 2 Propositional Logic Until now, we have seen examples of different proofs by primarily drawing from simple statements in number theory. Now we will establish the fundamentals of writing formal proofs by reasoning about logical statements. Throughout this section, we will maintain a running analogy of propositional logic with the system of arithmetic with which we are already familiar. In general, capital letters (e.g., A, B, C) represent propositional variables, while lowercase letters (e.g., x, y, z) represent arithmetic variables. But what is a propositional variable? A propositional variable is the basic unit of propositional logic. Recall that in arithmetic, the variable x is a placeholder for any real number, such as 7, 0.5, or −3. In propositional logic, the variable A is a placeholder for any truth value. While x has (infinitely) many possible values, there are only two possible values of A: True (denoted T) and False (denoted F). Since propositional variables can only take one of two possible values, they are also known as Boolean variables, named after their inventor George Boole. By letting x denote any real number, we can make claims like “x2 − 1 = (x + 1)(x − 1)” regardless of the numerical value of x. -
2 First-Order Logic
prev next 2 First-Order Logic First-order logic (also called predicate logic) is an extension of propositional logic that is much more useful than propositional logic. It was created as a way of formalizing common mathematical reasoning. In first-order logic, you start with a nonempty set of values called the domain of discourse U. Logical statements talk about properties of values in U and relationships among those values. 2.1 Predicates In place of propositional variables, first-order logic uses predicates. Definition 2.1. A predicate P takes zero or more parameters x1, x2, ::: , xn and yields either true or false. First-order formula P (x1, ::: , xn) is the value of predicate P with parameters x1, ::: , xn. A predicate with no parameters is a propositional variable. Suppose that the domain of discourse U is the set of all integers. Here are some examples of predicates. There is no standard collection of predicates that are always used. Rather, each of these is like a function definition in a computer program; different programs contain different functions. • We might define even(n) to be true if n is even. For example even(4) is true and even(5) is false. • We might define greater(x; y) to be true if x > y. For example, greater(7; 3) is true and greater(3; 7) is false. • We might define increasing(x; y; z) to be true if x < y < z. For example, increasing(2; 4; 6) is true and increasing(2; 4; 2) is false. 1 2.2 Terms A term is an expression that stands for a particular value in U. -
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. -
Major Results: (Wrt Propositional Logic) • How to Reason Correctly. • How To
Major results: (wrt propositional logic) ² How to reason correctly. ² How to reason efficiently. ² How to determine if a statement is true or false. Fuzzy logic deal with statements that are somewhat vague, such as: this paint is grey. Probability deals with statements that are possibly true, such as whether I will win the lottery next week. Propositional logic deals with concrete statements that are either true true or false. Predicate Logic (first order logic) allows statements to contain variables, such as if X is married to Y then Y is married to X. 1 We need formal definitions for: ² legal sentences or propositions (syntax) ² inference rules, for generated new sentence. (entailment) ² truth or validity. (semantics) Example Syntax: (x > y)&(y > z) ! (x > z) Interpretation: x = John’s age y= Mary’s age z= Tom’s age If John is older than Mary and Mary is older than Tom, then we can conclude John is older than Tom. (semantics) Conclusion is not approximately true or probably true, but absolutely true. 2 1 Propositional Logic Syntax: logical constants True and False are sentences. propositional symbols P,Q, R, etc are sentences These will either be true (t) or false (f) Balanced Parenthesis if S is a sentence, so is (S). logical operators or connectives : _; ^; »;;;; ); ::: Actually NAND (Shaeffer stroke) is the only connective that is needed. Sentences legal combinations of connectives and variables, i.e. propositional let- ters are sentences and if s1 and s2 are sentences so are » s1; (s1); s1 ) s2; Note: Instead of ) one can use _ and ^ since p ) q ,» p _ q Semantics: ² Each propositional variable can be either True or False. -
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