Truth Tables for Negation, Conjunction, and Disjunction

Total Page：16

File Type：pdf, Size：1020Kb

3.2 Truth Tables for Negation, Conjunction, and Disjunction Copyright © 2005 Pearson Education, Inc. Truth Table A truth table is used to determine when a compound statement is true or false. Copyright © 2005 Pearson Education, Inc. Slide 3-2 Conjunction Truth Table Click on speaker for audio The symbol ^ is read as “and” p q pq∧ Case 1 T T T Case 2 T F F Case 3 F T F Case 4 F F F The conjunction is true only when both p and q are true. Copyright © 2005 Pearson Education, Inc. Slide 3-3 Disjunction Click on speaker for audio The symbol V is read as “or” p q p ∨ q Case 1 T T T Case 2 T F T Case 3 F T T Case 4 F F F The disjunction is true when either p is true,qis true, or both p and q are true. Copyright © 2005 Pearson Education, Inc. Slide 3-4 Making a truth table Let’s construct a truth table for p v ~q. This is read as “p or not q”. Step 1: Make a table with different possibilities for p and q .There are 4 different possibilities. p q Case 1 T T Case 2 T F Case 3 F T Case 4 F F Copyright © 2005 Pearson Education, Inc. Slide 3-5 Making a truth table (cont’d) Click on speaker for audio Step 2: Now, make a column for ~q (“not” q) since we want to ultimately find p v ~q p q ~q Case 1 T T F Case 2 T F T Case 3 F T F Case 4 F F T Copyright © 2005 Pearson Education, Inc. Slide 3-6 Making a truth table (cont’d) Step 3: Next, make a column for p v ~q. Use the first and third columns to decide the truth values for p v ~q The truth table is now finished. p q ~q p v~q T T F T T F T T F T F F F F T T Copyright © 2005 Pearson Education, Inc. Slide 3-7 Check it out Use the truth table above to decide the truth value of p V ~q if p is false and q is true. (Answer: p V ~q is false) Copyright © 2005 Pearson Education, Inc. Slide 3-8 Next steps Read Examples 1-6 Work problems 5-13,odds; 43-50,all; 51-54,all from p. 115 in text Do online homework for Sec . 3.2 Copyright © 2005 Pearson Education, Inc. Slide 3-9 3.3 Truth Tables for the Conditional and Biconditional Copyright © 2005 Pearson Education, Inc. Conditional Click on speaker for audio p -> q is read as “if p then q” p q p → q Case 1 T T T Case 2 T F F Case 3 F T T Case 4 F F T The conditional statement p → q is true in every case except when p is a true statement and q is a false statement. Copyright © 2005 Pearson Education, Inc. Slide 3-11 Biconditional Click on speaker for audio The biconditional statement, pq ↔ means that pq → and qp → , or, symbolically ( pq→∧→) ( qp). Take these 2 columns to get column 7 p q (p → q) ∧ (q → p) case 1 T T T T T T T T T case 2 T F T F F F F T T case 3 F T F T T F T F F case 4 F F F T F T F T F order of steps 1 3 2 7 4 6 5 Copyright © 2005 Pearson Education, Inc. Slide 3-12 Next steps Read Examples 1 and 5 only Work problems 7-13, odds; 71-76, all from p. 125 in text Do online homework for Sec . 3.3 Do online quiz for 3.2 and 3.3 Copyright © 2005 Pearson Education, Inc. Slide 3-13.

Copy Link

Recommended publications

Dialetheists' Lies About the Liar
PRINCIPIA 22(1): 59–85 (2018) doi: 10.5007/1808-1711.2018v22n1p59 Published by NEL — Epistemology and Logic Research Group, Federal University of Santa Catarina (UFSC), Brazil. DIALETHEISTS’LIES ABOUT THE LIAR JONAS R. BECKER ARENHART Departamento de Filosofia, Universidade Federal de Santa Catarina, BRAZIL [email protected] EDERSON SAFRA MELO Departamento de Filosofia, Universidade Federal do Maranhão, BRAZIL [email protected] Abstract. Liar-like paradoxes are typically arguments that, by using very intuitive resources of natural language, end up in contradiction. Consistent solutions to those paradoxes usually have difficulties either because they restrict the expressive power of the language, orelse because they fall prey to extended versions of the paradox. Dialetheists, like Graham Priest, propose that we should take the Liar at face value and accept the contradictory conclusion as true. A logical treatment of such contradictions is also put forward, with the Logic of Para- dox (LP), which should account for the manifestations of the Liar. In this paper we shall argue that such a formal approach, as advanced by Priest, is unsatisfactory. In order to make contradictions acceptable, Priest has to distinguish between two kinds of contradictions, in- ternal and external, corresponding, respectively, to the conclusions of the simple and of the extended Liar. Given that, we argue that while the natural interpretation of LP was intended to account for true and false sentences, dealing with internal contradictions, it lacks the re- sources to tame external contradictions. Also, the negation sign of LP is unable to represent internal contradictions adequately, precisely because of its allowance of sentences that may be true and false.
[Show full text]
Chapter 18 Negation
Chapter 18 Negation Jong-Bok Kim Kyung Hee University, Seoul Each language has a way to express (sentential) negation that reverses the truth value of a certain sentence, but employs language-particular expressions and gram- matical strategies. There are four main types of negatives in expressing sentential negation: the adverbial negative, the morphological negative, the negative auxil- iary verb, and the preverbal negative. This chapter discusses HPSG analyses for these four strategies in marking sentential negation. 1 Modes of expressing negation There are four main types of negative markers in expressing negation in lan- guages: the morphological negative, the negative auxiliary verb, the adverbial negative, and the clitic-like preverbal negative (see Dahl 1979; Payne 1985; Zanut- tini 2001; Dryer 2005).1 Each of these types is illustrated in the following: (1) a. Ali elmalar-i ser-me-di-;. (Turkish) Ali apples-ACC like-NEG-PST-3SG ‘Ali didn’t like apples.’ b. sensayng-nim-i o-ci anh-usi-ess-ta. (Korean) teacher-HON-NOM come-CONN NEG-HON-PST-DECL ‘The teacher didn’t come.’ c. Dominique (n’) écrivait pas de lettre. (French) Dominique NEG wrote NEG of letter ‘Dominique did not write a letter.’ 1The term negator or negative marker is a cover term for any linguistic expression functioning as sentential negation. Jong-Bok Kim. 2021. Negation. In Stefan Müller, Anne Abeillé, Robert D. Borsley & Jean- Pierre Koenig (eds.), Head-Driven Phrase Structure Grammar: The handbook. Prepublished version. Berlin: Language Science Press. [Pre- liminary page numbering] Jong-Bok Kim d. Gianni non legge articoli di sintassi. (Italian) Gianni NEG reads articles of syntax ‘Gianni doesn’t read syntax articles.’ As shown in (1a), languages like Turkish have typical examples of morphological negatives where negation is expressed by an inflectional category realized on the verb by affixation.
[Show full text]
A Symmetric Lambda-Calculus Corresponding to the Negation
A Symmetric Lambda-Calculus Corresponding to the Negation-Free Bilateral Natural Deduction Tatsuya Abe Software Technology and Artificial Intelligence Research Laboratory, Chiba Institute of Technology, 2-17-1 Tsudanuma, Narashino, Chiba, 275-0016, Japan [email protected] Daisuke Kimura Department of Information Science, Toho University, 2-2-1 Miyama, Funabashi, Chiba, 274-8510, Japan [email protected] Abstract Filinski constructed a symmetric lambda-calculus consisting of expressions and continuations which are symmetric, and functions which have duality. In his calculus, functions can be encoded to expressions and continuations using primitive operators. That is, the duality of functions is not derived in the calculus but adopted as a principle of the calculus. In this paper, we propose a simple symmetric lambda-calculus corresponding to the negation-free natural deduction based bilateralism in proof-theoretic semantics. In our calculus, continuation types are represented as not negations of formulae but formulae with negative polarity. Function types are represented as the implication and but-not connectives in intuitionistic and paraconsistent logics, respectively. Our calculus is not only simple but also powerful as it includes a call-value calculus corresponding to the call-by-value dual calculus invented by Wadler. We show that mutual transformations between expressions and continuations are definable in our calculus to justify the duality of functions. We also show that every typable function has dual types. Thus, the duality of function is derived from bilateralism. 2012 ACM Subject Classification Theory of computation → Logic; Theory of computation → Type theory Keywords and phrases symmetric lambda-calculus, formulae-as-types, duality, bilateralism, natural deduction, proof-theoretic semantics, but-not connective, continuation, call-by-value Acknowledgements The author thanks Yosuke Fukuda, Tasuku Hiraishi, Kentaro Kikuchi, and Takeshi Tsukada for the fruitful discussions, which clarified contributions of the present paper.
[Show full text]
Journal of Linguistics Negation, 'Presupposition'
Journal of Linguistics http://journals.cambridge.org/LIN Additional services for Journal of Linguistics: Email alerts: Click here Subscriptions: Click here Commercial reprints: Click here Terms of use : Click here Negation, ‘presupposition’ and the semantics/pragmatics distinction ROBYN CARSTON Journal of Linguistics / Volume 34 / Issue 02 / September 1998, pp 309 350 DOI: null, Published online: 08 September 2000 Link to this article: http://journals.cambridge.org/abstract_S0022226798007063 How to cite this article: ROBYN CARSTON (1998). Negation, ‘presupposition’ and the semantics/pragmatics distinction. Journal of Linguistics, 34, pp 309350 Request Permissions : Click here Downloaded from http://journals.cambridge.org/LIN, IP address: 144.82.107.34 on 12 Oct 2012 J. Linguistics (), –. Printed in the United Kingdom # Cambridge University Press Negation, ‘presupposition’ and the semantics/pragmatics distinction1 ROBYN CARSTON Department of Phonetics and Linguistics, University College London (Received February ; revised April ) A cognitive pragmatic approach is taken to some long-standing problem cases of negation, the so-called presupposition denial cases. It is argued that a full account of the processes and levels of representation involved in their interpretation typically requires the sequential pragmatic derivation of two different propositions expressed. The first is one in which the presupposition is preserved and, following the rejection of this, the second involves the echoic (metalinguistic) use of material falling in the scope of the negation. The semantic base for these processes is the standard anti- presuppositionalist wide-scope negation. A different view, developed by Burton- Roberts (a, b), takes presupposition to be a semantic relation encoded in natural language and so argues for a negation operator that does not cancel presuppositions.
[Show full text]
Expressing Negation
Expressing Negation William A. Ladusaw University of California, Santa Cruz Introduction* My focus in this paper is the syntax-semantics interface for the interpretation of negation in languages which show negative concord, as illustrated in the sentences in (1)-(4). (1) Nobody said nothing to nobody. [NS English] ‘Nobody said anything to anyone.’ (2) Maria didn’t say nothing to nobody. [NS English] ‘Maria didn’t say anything to anyone.’ (3) Mario non ha parlato di niente con nessuno. [Italian] ‘Mario hasn’t spoken with anyone about anything.’ (4) No m’ha telefonat ningú. [Catalan] ‘Nobody has telephoned me.’ Negative concord (NC) is the indication at multiple points in a clause of the fact that the clause is to be interpreted as semantically negated. In a widely spoken and even more widely understood nonstandard dialect of English, sentences (1) and (2) are interpreted as synonymous with those given as glosses, which are also well-formed in the dialect. The examples in (3) from Italian and (4) from Catalan illustrate the same phenomenon. The occurrence in these sentences of two or three different words, any one of which when correctly positioned would be sufficient to negate a clause, does not guarantee that their interpretation involves two or three independent expressions of negation. These clauses express only one negation, which is, on one view, simply redundantly indicated in two or three different places; each of the italicized terms in these sentences might be seen as having an equal claim to the function of expressing negation. However closer inspection indicates that this is not the correct view.
[Show full text]
Propositional Logic (PDF)
Mathematics for Computer Science Proving Validity 6.042J/18.062J Instead of truth tables, The Logic of can try to prove valid formulas symbolically using Propositions axioms and deduction rules Albert R Meyer February 14, 2014 propositional logic.1 Albert R Meyer February 14, 2014 propositional logic.2 Proving Validity Algebra for Equivalence The text describes a for example, bunch of algebraic rules to the distributive law prove that propositional P AND (Q OR R) ≡ formulas are equivalent (P AND Q) OR (P AND R) Albert R Meyer February 14, 2014 propositional logic.3 Albert R Meyer February 14, 2014 propositional logic.4 1 Algebra for Equivalence Algebra for Equivalence for example, The set of rules for ≡ in DeMorgan’s law the text are complete: ≡ NOT(P AND Q) ≡ if two formulas are , these rules can prove it. NOT(P) OR NOT(Q) Albert R Meyer February 14, 2014 propositional logic.5 Albert R Meyer February 14, 2014 propositional logic.6 A Proof System A Proof System Another approach is to Lukasiewicz’ proof system is a start with some valid particularly elegant example of this idea. formulas (axioms) and deduce more valid formulas using proof rules Albert R Meyer February 14, 2014 propositional logic.7 Albert R Meyer February 14, 2014 propositional logic.8 2 A Proof System Lukasiewicz’ Proof System Lukasiewicz’ proof system is a Axioms: particularly elegant example of 1) (¬P → P) → P this idea. It covers formulas 2) P → (¬P → Q) whose only logical operators are 3) (P → Q) → ((Q → R) → (P → R)) IMPLIES (→) and NOT. The only rule: modus ponens Albert R Meyer February 14, 2014 propositional logic.9 Albert R Meyer February 14, 2014 propositional logic.10 Lukasiewicz’ Proof System Lukasiewicz’ Proof System Prove formulas by starting with Prove formulas by starting with axioms and repeatedly applying axioms and repeatedly applying the inference rule.
[Show full text]
Sets, Propositional Logic, Predicates, and Quantifiers
COMP 182 Algorithmic Thinking Sets, Propositional Logic, Luay Nakhleh Computer Science Predicates, and Quantifiers Rice University !1 Reading Material ❖ Chapter 1, Sections 1, 4, 5 ❖ Chapter 2, Sections 1, 2 !2 ❖ Mathematics is about statements that are either true or false. ❖ Such statements are called propositions. ❖ We use logic to describe them, and proof techniques to prove whether they are true or false. !3 Propositions ❖ 5>7 ❖ The square root of 2 is irrational. ❖ A graph is bipartite if and only if it doesn’t have a cycle of odd length. ❖ For n>1, the sum of the numbers 1,2,3,…,n is n2. !4 Propositions? ❖ E=mc2 ❖ The sun rises from the East every day. ❖ All species on Earth evolved from a common ancestor. ❖ God does not exist. ❖ Everyone eventually dies. !5 ❖ And some of you might already be wondering: “If I wanted to study mathematics, I would have majored in Math. I came here to study computer science.” !6 ❖ Computer Science is mathematics, but we almost exclusively focus on aspects of mathematics that relate to computation (that can be implemented in software and/or hardware). !7 ❖Logic is the language of computer science and, mathematics is the computer scientist’s most essential toolbox. !8 Examples of “CS-relevant” Math ❖ Algorithm A correctly solves problem P. ❖ Algorithm A has a worst-case running time of O(n3). ❖ Problem P has no solution. ❖ Using comparison between two elements as the basic operation, we cannot sort a list of n elements in less than O(n log n) time. ❖ Problem A is NP-Complete.
[Show full text]
From Fuzzy Sets to Crisp Truth Tables1 Charles C. Ragin
From Fuzzy Sets to Crisp Truth Tables1 Charles C. Ragin Department of Sociology University of Arizona Tucson, AZ 85721 USA [email protected] Version: April 2005 1. Overview One limitation of the truth table approach is that it is designed for causal conditions are simple presence/absence dichotomies (i.e., Boolean or "crisp" sets). Many of the causal conditions that interest social scientists, however, vary by level or degree. For example, while it is clear that some countries are democracies and some are not, there are many in-between cases. These countries are not fully in the set of democracies, nor are they fully excluded from this set. Fortunately, there is a well-developed mathematical system for addressing partial membership in sets, fuzzy-set theory. Section 2 of this paper provides a brief introduction to the fuzzy-set approach, building on Ragin (2000). Fuzzy sets are especially powerful because they allow researchers to calibrate partial membership in sets using values in the interval between 0 (nonmembership) and 1 (full membership) without abandoning core set theoretic principles, for example, the subset relation. Ragin (2000) demonstrates that the subset relation is central to the analysis of multiple conjunctural causation, where several different combinations of conditions are sufficient for the same outcome. While fuzzy sets solve the problem of trying to force-fit cases into one of two categories (membership versus nonmembership in a set), they are not well suited for conventional truth table analysis. With fuzzy sets, there is no simple way to sort cases according to the combinations of causal conditions they display because each case's array of membership scores may be unique.
[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]
Logic, Proofs
CHAPTER 1 Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: “Paris is in France” (true), “London is in Denmark” (false), “2 < 4” (true), “4 = 7 (false)”. However the following are not propositions: “what is your name?” (this is a question), “do your homework” (this is a command), “this sentence is false” (neither true nor false), “x is an even number” (it depends on what x represents), “Socrates” (it is not even a sentence). The truth or falsehood of a proposition is called its truth value. 1.1.1. Connectives, Truth Tables. Connectives are used for making compound propositions. The main ones are the following (p and q represent given propositions): Name Represented Meaning Negation p “not p” Conjunction p¬ q “p and q” Disjunction p ∧ q “p or q (or both)” Exclusive Or p ∨ q “either p or q, but not both” Implication p ⊕ q “if p then q” Biconditional p → q “p if and only if q” ↔ The truth value of a compound proposition depends only on the value of its components. Writing F for “false” and T for “true”, we can summarize the meaning of the connectives in the following way: 6 1.1. PROPOSITIONS 7 p q p p q p q p q p q p q T T ¬F T∧ T∨ ⊕F →T ↔T T F F F T T F F F T T F T T T F F F T F F F T T Note that represents a non-exclusive or, i.e., p q is true when any of p, q is true∨ and also when both are true.
[Show full text]
Logic, Sets, and Proofs David A
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical statements with capital letters A; B. Logical statements be combined to form new logical statements as follows: Name Notation Conjunction A and B Disjunction A or B Negation not A :A Implication A implies B if A, then B A ) B Equivalence A if and only if B A , B Here are some examples of conjunction, disjunction and negation: x > 1 and x < 3: This is true when x is in the open interval (1; 3). x > 1 or x < 3: This is true for all real numbers x. :(x > 1): This is the same as x ≤ 1. Here are two logical statements that are true: x > 4 ) x > 2. x2 = 1 , (x = 1 or x = −1). Note that \x = 1 or x = −1" is usually written x = ±1. Converses, Contrapositives, and Tautologies. We begin with converses and contrapositives: • The converse of \A implies B" is \B implies A". • The contrapositive of \A implies B" is \:B implies :A" Thus the statement \x > 4 ) x > 2" has: • Converse: x > 2 ) x > 4. • Contrapositive: x ≤ 2 ) x ≤ 4. 1 Some logical statements are guaranteed to always be true. These are tautologies. Here are two tautologies that involve converses and contrapositives: • (A if and only if B) , ((A implies B) and (B implies A)). In other words, A and B are equivalent exactly when both A ) B and its converse are true.
[Show full text]
RELATIONAL PARAMETRICITY and CONTROL 1. Introduction the Λ
Logical Methods in Computer Science Vol. 2 (3:3) 2006, pp. 1–22 Submitted Dec. 16, 2005 www.lmcs-online.org Published Jul. 27, 2006 RELATIONAL PARAMETRICITY AND CONTROL MASAHITO HASEGAWA Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 Japan, and PRESTO, Japan Science and Technology Agency e-mail address: [email protected] Abstract. We study the equational theory of Parigot’s second-order λµ-calculus in con- nection with a call-by-name continuation-passing style (CPS) translation into a fragment of the second-order λ-calculus. It is observed that the relational parametricity on the tar- get calculus induces a natural notion of equivalence on the λµ-terms. On the other hand, the unconstrained relational parametricity on the λµ-calculus turns out to be inconsis- tent. Following these facts, we propose to formulate the relational parametricity on the λµ-calculus in a constrained way, which might be called “focal parametricity”. Dedicated to Prof. Gordon Plotkin on the occasion of his sixtieth birthday 1. Introduction The λµ-calculus, introduced by Parigot [26], has been one of the representative term calculi for classical natural deduction, and widely studied from various aspects. Although it still is an active research subject, it can be said that we have some reasonable un- derstanding of the ﬁrst-order propositional λµ-calculus: we have good reduction theories, well-established CPS semantics and the corresponding operational semantics, and also some canonical equational theories enjoying semantic completeness [16, 24, 25, 36, 39]. The last point cannot be overlooked, as such complete axiomatizations provide deep understand- ing of equivalences between proofs and also of the semantic structure behind the syntactic presentation.
[Show full text]