A Note on Algebraic Semantics for S5 with Propositional Quantifiers
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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. -
Intuitionistic Modal Logic IS5 Formalizations, Interpretation, Analysis
Uniwersytet Wroclawski Wydzia lMatematyki i Informatyki Instytut Informatyki Intuitionistic Modal Logic IS5 Formalizations, Interpretation, Analysis Agata Murawska A Master's Thesis written under the supervision of Ma lgorzataBiernacka Wroclaw, May 2013 1 2 Abstract IS5 is an intuitionistic variant of S5 modal logic, one of the normal modal logics, with accessibility relation defined as an equivalence. In this thesis we formalize two known variants of natural deduction systems for IS5 along with their corresponding languages. First, the syntactically pure IS5LF vari- ant that does not name the modal worlds, is due to Galmiche and Sahli. The second one, IS5L, using world names (labels) in inference rules and in terms of its language, is due to Tom Murphy et al. For each of the languages accompanying these logics we prove standard properties such as progress and preservation. We show the connection be- tween these languages via a series of type-preserving transformations of terms and contexts. Finally, we give a proof of termination for the call-by-name strategy of evaluation using logical relations method. The formalization of all of the above properties is done in Coq1 proof assistant. In particular, the proof of termination allows { via Curry-Howard isomorphism { to extract the evaluator in OCaml from the proof. Our contributions include providing a term language LIS5-LF for IS5LF, as well as creating an in-between logic IS5Hyb and the corresponding language LIS5-Hyb, used when showing the connection between LIS5-L and LIS5-LF. For the former language we formalize the termination of call-by-name evaluation strategy. 1The Coq development is available at author's github page, http://github.com/ Ayertienna/IS5. -
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. -
On the Logics with Propositional Quantifiers Extending S5
On the Logics with Propositional Quantifiers Extending S5Π Yifeng Ding (voidprove.com) Aug. 27, 2018 @ AiML 2018 UC Berkeley Group of Logic and the Methodology of Science • Kit Fine systematically studied a few modal logic systems with propositional quantifers based on S5. • We provide an analogue of Scroggs's theorem for modal logics with propositional quantifiers using algebraic semantics. • More generally, it is interesting to see how classical results generalize when using algebraic semantics. Introduction • We have expressions that quantifies over propositions: \Everything I believe is true." (Locally) 2 • We provide an analogue of Scroggs's theorem for modal logics with propositional quantifiers using algebraic semantics. • More generally, it is interesting to see how classical results generalize when using algebraic semantics. Introduction • We have expressions that quantifies over propositions: \Everything I believe is true." (Locally) • Kit Fine systematically studied a few modal logic systems with propositional quantifers based on S5. 2 • More generally, it is interesting to see how classical results generalize when using algebraic semantics. Introduction • We have expressions that quantifies over propositions: \Everything I believe is true." (Locally) • Kit Fine systematically studied a few modal logic systems with propositional quantifers based on S5. • We provide an analogue of Scroggs's theorem for modal logics with propositional quantifiers using algebraic semantics. 2 Introduction • We have expressions that quantifies over propositions: \Everything I believe is true." (Locally) • Kit Fine systematically studied a few modal logic systems with propositional quantifers based on S5. • We provide an analogue of Scroggs's theorem for modal logics with propositional quantifiers using algebraic semantics. • More generally, it is interesting to see how classical results generalize when using algebraic semantics. -
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. -
Kripke Completeness Revisited
Kripke completeness revisited Sara Negri Department of Philosophy, P.O. Box 9, 00014 University of Helsinki, Finland. e-mail: sara.negri@helsinki.fi Abstract The evolution of completeness proofs for modal logic with respect to the possible world semantics is studied starting from an analysis of Kripke’s original proofs from 1959 and 1963. The critical reviews by Bayart and Kaplan and the emergence of Henkin-style completeness proofs are detailed. It is shown how the use of a labelled sequent system permits a direct and uniform completeness proof for a wide variety of modal logics that is close to Kripke’s original arguments but without the drawbacks of Kripke’s or Henkin-style completeness proofs. Introduction The question about the ultimate attribution for what is commonly called Kripke semantics has been exhaustively discussed in the literature, recently in two surveys (Copeland 2002 and Goldblatt 2005) where the rˆoleof the precursors of Kripke semantics is documented in detail. All the anticipations of Kripke’s semantics have been given ample credit, to the extent that very often the neutral terminology of “relational semantics” is preferred. The following quote nicely summarizes one representative standpoint in the debate: As mathematics progresses, notions that were obscure and perplexing become clear and straightforward, sometimes even achieving the status of “obvious.” Then hindsight can make us all wise after the event. But we are separated from the past by our knowledge of the present, which may draw us into “seeing” more than was really there at the time. (Goldblatt 2005, section 4.2) We are not going to treat this issue here, nor discuss the parallel development of the related algebraic semantics for modal logic (Jonsson and Tarski 1951), but instead concentrate on one particular and crucial aspect in the history of possible worlds semantics, namely the evolution of completeness proofs for modal logic with respect to Kripke semantics. -
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. -
Fixpointed Idempotent Uninorm (Based) Logics
mathematics Article Fixpointed Idempotent Uninorm (Based) Logics Eunsuk Yang Department of Philosophy & Institute of Critical Thinking and Writing, Colleges of Humanities & Social Science Blvd., Chonbuk National University, Rm 417, Jeonju 54896, Korea; [email protected] Received: 8 December 2018; Accepted: 18 January 2019; Published: 20 January 2019 Abstract: Idempotent uninorms are simply defined by fixpointed negations. These uninorms, called here fixpointed idempotent uninorms, have been extensively studied because of their simplicity, whereas logics characterizing such uninorms have not. Recently, fixpointed uninorm mingle logic (fUML) was introduced, and its standard completeness, i.e., completeness on real unit interval [0, 1], was proved by Baldi and Ciabattoni. However, their proof is not algebraic and does not shed any light on the algebraic feature by which an idempotent uninorm is characterized, using operations defined by a fixpointed negation. To shed a light on this feature, this paper algebraically investigates logics based on fixpointed idempotent uninorms. First, several such logics are introduced as axiomatic extensions of uninorm mingle logic (UML). The algebraic structures corresponding to the systems are then defined, and the results of the associated algebraic completeness are provided. Next, standard completeness is established for the systems using an Esteva–Godo-style approach for proving standard completeness. Keywords: substructural fuzzy logic; algebraic completeness; standard completeness; fixpoint; idempotent uninorm 1. Introduction Uninorms are a special kind of associative and commutative aggregation operators allowing an identity in unit interval [0, 1]. They were introduced by Yager and Rybalov in Reference [1], characterizing their structure along with Fodor [2]. Since then, these kinds of operators have been extensively investigated due to their special interest from a theoretical and applicative point of view. -
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.