DOCSLIB.ORG

COMP6463 Overview of Logic and Computation: Propositional Logic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
COMP6463 Overview of Logic and Computation: Propositional Logic COMP6463 Overview of Logic and Computation: Propositional Logic Alwen Tiu The Australian National University 2011 Alwen Tiu (ANU) COMP6463 2011 1 / 35 Outline Lecture 1 Introduction to propositional logic: syntax and semantics. Lecture 2 Normal forms and the resolution proof method for unsatisfiability checking. Lecture 3 Efficient representation of boolean functions: Binary Decision Diagram (BDD). Lecture 4 Satisfiability checking: the DP and DPLL algorithms. Lecture 5 Methods for validity checking: the tableaux method and the compactness theorem. Slides will be made available at: http://users.rsise.anu.edu.au/~tiu/teaching/comp6463 Alwen Tiu (ANU) COMP6463 2011 2 / 35 References Most standard textbooks on logic include a chapter on propositional logic,e.g., Michael Huth and Mark Ryan. Logic in Computer Science, Cambridge University Press, 2nd Edition, 2004. A. Aho and J.D. Ullman. Foundations of Computer Science (Chapter 12). Available online for free. http://infolab.stanford.edu/~ullman/focs.html For details of implementation of automated reasoning tools for propositional logic: John Harrison. Handbook of Practical Logic and Automated Reasoning. Cambridge University Press, 2009. Alwen Tiu (ANU) COMP6463 2011 3 / 35 What is propositional logic? Propositional logic is concerned with propositions, i.e., statements which can be either true or false, and compositions of their truth values. Atomic propositions can be any sentences, e.g., I It is raining. I Joe takes his umbrella. I x < 0: I x = 0: These sentences are considered atomic, i.e., the particular subjects/objects they mention are irrelevant. Atomic sentences are denoted by letters, such as, a, b, c, etc. These are called propositional variables. Alwen Tiu (ANU) COMP6463 2011 4 / 35 Composing propositions Atomic propositions can be composed to form complex sentences, e.g., It is raining and Joe takes his umbrella. Or x ≤ 0 or x = 0: We are interested in studying the truth value of the combined propositions, and how their truth can be systematically computed. That is, assuming we know the truth values of atomic propositions, how do we compute or reason about the truth values of their compositions. Alwen Tiu (ANU) COMP6463 2011 5 / 35 Some problems in propositional logic The satisfiability (SAT) problem: given a (complex) formula, how do we assign truth values to the atomic propositions in the formula so that the formula becomes true? The validity problem: is a formula always true no matter how we assign truth values to atomic propositions? E.g., statements like \it is raining or it is not raining" is always true. The unsatisfiability problem: converse of the validity problem. Some statements cannot be made true, e.g., logical paradoxes like 1 \The following statement is true". 2 \The preceding statement is false". Alwen Tiu (ANU) COMP6463 2011 6 / 35 Some applications of propositional logic Propositional logic has many applications in different areas of computer science (apart from digitial circuits design). Complexity: the notion of NP-completeness was motivated by the satisfiability problem. Hardware and software verification: model state transitions of a complex system and its properties as propositional formulae, and use SAT solvers to prove its properties. Automated reasoning: many problems in automated reasoning can be translated into satisfiability problems in propositional logic. Artificial intelligence: knowledge representation, planning and diagnoses. Alwen Tiu (ANU) COMP6463 2011 7 / 35 Syntax vs Semantics One important feature of logic is the separation of the form (syntax) of a symbolic expression and its meaning (semantics). Typically, we are concerned with (i) some particular set of allowed formal expressions, and (ii) their corresponding meanings. The two are connected via an interpretation: Interpretation Expression / Meaning A big part of formal symbolic logic is the study of syntactic manipulation of the forms, independently of the semantics. Alwen Tiu (ANU) COMP6463 2011 8 / 35 The syntax of propositional logic Assume a countably infinite set of propositional variables, denoted by lower case letters, e.g., a, b, p, q, etc., representing atomic propositions. The language of formulae is defined inductively as follows: I > (`true') and ? (`false') are formulae. I Every propositional variable is a formula (also called an atomic formula). I If A is a formula then :A (`not A') is a formula. I If A and B are formulae then so are: I A ^ B (A `and' B), I A _ B (A `or' B), I and A ! B (A `implies' B). A literal is an atomic formula or its negation, e.g., a, :a, c, :c, etc. Assume : has the highest precedence, so :x _ y means (:x) _ y: Alwen Tiu (ANU) COMP6463 2011 9 / 35 Formulae as an inductive data type The set of propositional formulae can be defined as an inductive data type in functional programming languages. A typical definition in Haskell-like language would look like this: data Formula a = False | True | Atom a | Not (Formula a) | And (Formula a) (Formula a) | Or (Formula a) (Formula a) | Imp (Formula a) (Formula a) Alwen Tiu (ANU) COMP6463 2011 10 / 35 Formulae as expression trees A formula can also be viewed as a tree whose leaf nodes are propositional variables and ? and >, and whose internal nodes are logical connectives. Example: the expression tree for (p !:q) _? _ nn @ nn @@ nnn @ nnn @@ nnn @ ! n ? ~ AA ~~ AA ~~ AA ~~ AA p ~ : q Alwen Tiu (ANU) COMP6463 2011 11 / 35 The structural induction principle for formulae A useful proof principle to prove properties of a formula is by structural induction on the formation of the formula. To prove a property P holds for every formula F : Base cases: prove P for atomic formula, > and ?: Inductive cases: I Assuming P holds for A (the induction hypothesis), show that P holds for :A: I Assuming P holds for A and B, show that P holds for A ^ B, A _ B, and A ! B: Alwen Tiu (ANU) COMP6463 2011 12 / 35 Subformulae A formula A is an immediate subformula of B if one of the following holds: I B is :A; I B is A ^ C, or C ^ A, or A _ C, or C _ A, A ! C or C ! A, for some formula C: We write A ≺ B if A is an immediate subformula of B. Let ≺∗ be the transitive closure of ≺ : We say that A is a subformula of B if A ≺∗ B: View as expression trees, A is a subformula of B if A appears as a subtree in B. Alwen Tiu (ANU) COMP6463 2011 13 / 35 Strong induction principle Just as induction on natural numbers, we also have a strong induction principle on formulae, using the subformula relation. To prove a property P holds for every formula F : Base cases: if F is either an atomic formula or > or ?, then prove P for F directly. Inductive cases: otherwise, assuming P holds for every A such that A ≺∗ F , show that P holds for F : Alwen Tiu (ANU) COMP6463 2011 14 / 35 Semantics The meaning of a propositional formula is simply true or false. The truth value of a formula depends on the interpretation of its propositional variables (as truth values). A boolean valuation is a function from the set of propositional variables to truth values, i.e., the set f0; 1g (denoting `false' and `true'). Given a boolean valuation M and a formula F , we write M j= F to denote that M is true under the valuation M, and M 6j= F if F is false under the valuation M: Alwen Tiu (ANU) COMP6463 2011 15 / 35 Semantics The relation j= is defined inductively on the structure of formulae as follows: M j= > ; M 6j= ?: M j= p iff (if and only if) M(p) = 1: M j= :A iff M 6j= A: M j= A ^ B iff M j= A and M j= B: M j= A _ B iff M j= A or M j= B: M j= A ! B iff M 6j= A or M j= B: When M j= F holds, we say that M is a model for F : Alwen Tiu (ANU) COMP6463 2011 16 / 35 Finite model property Let Var(F ) bet the set of propositional variables in F . Then the truth value of F is dependent only on the valuations of Var(F ). Lemma Let F be a formula and let M1 and M2 be two boolean valuations such that M1(p) = M2(p) for every p 2 Var(F ): Then M1 j= F if and only if M2 j= F : Proof. Exercise: prove by structural induction on F . A consequence of this is that we can represent finitely all models of F . Alwen Tiu (ANU) COMP6463 2011 17 / 35 Representing models of a formula Since the truth value of a formula F depends only on the valuation of its propositional variables, a model M of F can be represented by listing the finite mappings on Var(F ). For example, if F = (a ^ b) _ (c ! d) then M = fa 7! 0; b 7! 1; c 7! 1; d 7! 1g is a model of F . Alwen Tiu (ANU) COMP6463 2011 18 / 35 Boolean functions and truth tables An easy way to characterise the semantics of logical connectives is to view them as boolean functions. A boolean variable x is a variable ranging over f0; 1g. We use x; y; z, possibly with subscripts, to denote boolean variables. So a propositional variable is a boolean variable. A boolean function f of n arguments is a function from f0; 1gn to f0; 1g: A boolean function can be defined via a truth table: I the columns correspond to the variables and the output of the function, I the rows of the tables correspond to all possible combination of input and their output.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • A Note on Algebraic Semantics for S5 with Propositional Quantifiers
    A Note on Algebraic Semantics for S5 with Propositional Quantifiers Wesley H. Holliday University of California, Berkeley Preprint of April 2017. Forthcoming in Notre Dame Journal of Formal Logic. Abstract In two of the earliest papers on extending modal logic with propositional quanti- fiers, R. A. Bull and K. Fine studied a modal logic S5Π extending S5 with axioms and rules for propositional quantification. Surprisingly, there seems to have been no proof in the literature of the completeness of S5Π with respect to its most natural algebraic semantics, with propositional quantifiers interpreted by meets and joins over all ele- ments in a complete Boolean algebra. In this note, we give such a proof. This result raises the question: for which normal modal logics L can one axiomatize the quantified propositional modal logic determined by the complete modal algebras for L? Keywords: modal logic, propositional quantifiers, algebraic semantics, monadic algebras, MacNeille completion MSC: 03B45, 03C80, 03G05 1 Introduction The idea of extending the language of propositional modal logic with propositional quan- tifiers 8p and 9p was first investigated in Kripke 1959, Bull 1969, Fine 1970, and Kaplan 1970.1 The language LΠ of quantified propositional modal logic is given by the grammar ' ::= p j :' j (' ^ ') j ' j 8p'; where p comes from a countably infinite set Prop of propositional variables. The other connectives _, !, and $ are defined as usual, and we let ♦' := ::' and 9p' := :8p:'. A focus of the papers cited above was on extending the modal logic S5 with propositional quantification. As usual, one can think about natural extensions syntactically or seman- tically.
    [Show full text]
  • Section 6.2 Propositional Calculus Propositional Calculus Is the Language of Propositions (Statements That Are True Or False)
    Section 6.2 Propositional Calculus Propositional calculus is the language of propositions (statements that are true or false). We represent propositions by formulas called well-formed formulas (wffs) that are constructed from an alphabet consisting of Truth symbols: T (or True) and F (or False) Propositional variables: uppercase letters. Connectives (operators): ¬ (not, negation) ∧ (and, conjunction) ∨ (or, disjunction) → (conditional, implication) Parentheses symbols: ( and ). A wff is either a truth symbol, a propositional variable, or if V and W are wffs, then so are ¬ V, V ∧ W, V ∨ W, V → W, and (W). Example. The expression A ¬ B is not a wff. But each of the following three expressions is a wff: A ∧ B → C, (A ∧ B) → C, and A ∧ (B → C). Truth Tables. The connectives are defined by the following truth tables. P Q ¬P P "Q P #Q P $ Q T T F T T T T F F F T F F T T F T T F F T F F T 1 ! Semantics The meaning of T (or True) is true and the meaning of F (or False) is false. The meaning of any other wff is its truth table, where in the absence of parentheses, we define the hierarchy of evaluation to be ¬, ∧, ∨, →, and we assume ∧, ∨, → are left associative. Examples. ¬ A ∧ B means (¬ A) ∧ B A ∨ B ∧ C means A ∨ (B ∧ C) A ∧ B → C means (A ∧ B) → C A → B → C means (A → B) → C. Three Classes A Tautology is a wff for which all truth table values are T. A Contradiction is a wff for which all truth table values are F.
    [Show full text]
  • B1.1 Logic Lecture Notes.Pdf
    Oxford, MT2017 B1.1 Logic Jonathan Pila Slides by J. Koenigsmann with some small additions; further reference see: D. Goldrei, “Propositional and Predicate Calculus: A Model of Argument”, Springer. Introduction 1. What is mathematical logic about? • provide a uniform, unambiguous language for mathematics • make precise what a proof is • explain and guarantee exactness, rigor and certainty in mathematics • establish the foundations of mathematics B1 (Foundations) = B1.1 (Logic) + B1.2 (Set theory) N.B.: Course does not teach you to think logically, but it explores what it means to think logically Lecture 1 - 1/6 2. Historical motivation • 19th cent.: need for conceptual foundation in analysis: what is the correct notion of infinity, infinitesimal, limit, ... • attempts to formalize mathematics: - Frege’s Begriffsschrift - Cantor’s naive set theory: a set is any collection of objects • led to Russell’s paradox: consider the set R := {S set | S 6∈ S} R ∈ R ⇒ R 6∈ R contradiction R 6∈ R ⇒ R ∈ R contradiction ❀ fundamental crisis in the foundations of mathematics Lecture 1 - 2/6 3. Hilbert’s Program 1. find a uniform (formal) language for all mathematics 2. find a complete system of inference rules/ deduction rules 3. find a complete system of mathematical axioms 4. prove that the system 1.+2.+3. is consistent, i.e. does not lead to contradictions ⋆ complete: every mathematical sentence can be proved or disproved using 2. and 3. ⋆ 1., 2. and 3. should be finitary/effective/computable/algorithmic so, e.g., in 3. you can’t take as axioms the system of all true sentences in mathematics ⋆ idea: any piece of information is of finte length Lecture 1 - 3/6 4.
    [Show full text]