Truth Tables, Tautologies, and Logical Equivalences Mathematicians Normally Use a Two-Valued Logic: Every Statement Is Either True Or False
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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. -
Chrysippus's Dog As a Case Study in Non-Linguistic Cognition
Chrysippus’s Dog as a Case Study in Non-Linguistic Cognition Michael Rescorla Abstract: I critique an ancient argument for the possibility of non-linguistic deductive inference. The argument, attributed to Chrysippus, describes a dog whose behavior supposedly reflects disjunctive syllogistic reasoning. Drawing on contemporary robotics, I urge that we can equally well explain the dog’s behavior by citing probabilistic reasoning over cognitive maps. I then critique various experimentally-based arguments from scientific psychology that echo Chrysippus’s anecdotal presentation. §1. Language and thought Do non-linguistic creatures think? Debate over this question tends to calcify into two extreme doctrines. The first, espoused by Descartes, regards language as necessary for cognition. Modern proponents include Brandom (1994, pp. 145-157), Davidson (1984, pp. 155-170), McDowell (1996), and Sellars (1963, pp. 177-189). Cartesians may grant that ascribing cognitive activity to non-linguistic creatures is instrumentally useful, but they regard such ascriptions as strictly speaking false. The second extreme doctrine, espoused by Gassendi, Hume, and Locke, maintains that linguistic and non-linguistic cognition are fundamentally the same. Modern proponents include Fodor (2003), Peacocke (1997), Stalnaker (1984), and many others. Proponents may grant that non- linguistic creatures entertain a narrower range of thoughts than us, but they deny any principled difference in kind.1 2 An intermediate position holds that non-linguistic creatures display cognitive activity of a fundamentally different kind than human thought. Hobbes and Leibniz favored this intermediate position. Modern advocates include Bermudez (2003), Carruthers (2002, 2004), Dummett (1993, pp. 147-149), Malcolm (1972), and Putnam (1992, pp. 28-30). -
Chapter 5: Methods of Proof for Boolean Logic
Chapter 5: Methods of Proof for Boolean Logic § 5.1 Valid inference steps Conjunction elimination Sometimes called simplification. From a conjunction, infer any of the conjuncts. • From P ∧ Q, infer P (or infer Q). Conjunction introduction Sometimes called conjunction. From a pair of sentences, infer their conjunction. • From P and Q, infer P ∧ Q. § 5.2 Proof by cases This is another valid inference step (it will form the rule of disjunction elimination in our formal deductive system and in Fitch), but it is also a powerful proof strategy. In a proof by cases, one begins with a disjunction (as a premise, or as an intermediate conclusion already proved). One then shows that a certain consequence may be deduced from each of the disjuncts taken separately. One concludes that that same sentence is a consequence of the entire disjunction. • From P ∨ Q, and from the fact that S follows from P and S also follows from Q, infer S. The general proof strategy looks like this: if you have a disjunction, then you know that at least one of the disjuncts is true—you just don’t know which one. So you consider the individual “cases” (i.e., disjuncts), one at a time. You assume the first disjunct, and then derive your conclusion from it. You repeat this process for each disjunct. So it doesn’t matter which disjunct is true—you get the same conclusion in any case. Hence you may infer that it follows from the entire disjunction. In practice, this method of proof requires the use of “subproofs”—we will take these up in the next chapter when we look at formal proofs. -
In Defence of Constructive Empiricism: Metaphysics Versus Science
To appear in Journal for General Philosophy of Science (2004) In Defence of Constructive Empiricism: Metaphysics versus Science F.A. Muller Institute for the History and Philosophy of Science and Mathematics Utrecht University, P.O. Box 80.000 3508 TA Utrecht, The Netherlands E-mail: [email protected] August 2003 Summary Over the past years, in books and journals (this journal included), N. Maxwell launched a ferocious attack on B.C. van Fraassen's view of science called Con- structive Empiricism (CE). This attack has been totally ignored. Must we con- clude from this silence that no defence is possible against the attack and that a fortiori Maxwell has buried CE once and for all, or is the attack too obviously flawed as not to merit exposure? We believe that neither is the case and hope that a careful dissection of Maxwell's reasoning will make this clear. This dis- section includes an analysis of Maxwell's `aberrance-argument' (omnipresent in his many writings) for the contentious claim that science implicitly and per- manently accepts a substantial, metaphysical thesis about the universe. This claim generally has been ignored too, for more than a quarter of a century. Our con- clusions will be that, first, Maxwell's attacks on CE can be beaten off; secondly, his `aberrance-arguments' do not establish what Maxwell believes they estab- lish; but, thirdly, we can draw a number of valuable lessons from these attacks about the nature of science and of the libertarian nature of CE. Table of Contents on other side −! Contents 1 Exordium: What is Maxwell's Argument? 1 2 Does Science Implicitly Accept Metaphysics? 3 2.1 Aberrant Theories . -
CSE Yet, Please Do Well! Logical Connectives
administrivia Course web: http://www.cs.washington.edu/311 Office hours: 12 office hours each week Me/James: MW 10:30-11:30/2:30-3:30pm or by appointment TA Section: Start next week Call me: Shayan Don’t: Actually call me. Homework #1: Will be posted today, due next Friday by midnight (Oct 9th) Gradescope! (stay tuned) Extra credit: Not required to get a 4.0. Counts separately. In total, may raise grade by ~0.1 Don’t be shy (raise your hand in the back)! Do space out your participation. If you are not CSE yet, please do well! logical connectives p q p q p p T T T T F T F F F T F T F NOT F F F AND p q p q p q p q T T T T T F T F T T F T F T T F T T F F F F F F OR XOR 푝 → 푞 • “If p, then q” is a promise: p q p q F F T • Whenever p is true, then q is true F T T • Ask “has the promise been broken” T F F T T T If it’s raining, then I have my umbrella. related implications • Implication: p q • Converse: q p • Contrapositive: q p • Inverse: p q How do these relate to each other? How to see this? 푝 ↔ 푞 • p iff q • p is equivalent to q • p implies q and q implies p p q p q Let’s think about fruits A fruit is an apple only if it is either red or green and a fruit is not red and green. -
Conditional Statement/Implication Converse and Contrapositive
Conditional Statement/Implication CSCI 1900 Discrete Structures •"ifp then q" • Denoted p ⇒ q – p is called the antecedent or hypothesis Conditional Statements – q is called the consequent or conclusion Reading: Kolman, Section 2.2 • Example: – p: I am hungry q: I will eat – p: It is snowing q: 3+5 = 8 CSCI 1900 – Discrete Structures Conditional Statements – Page 1 CSCI 1900 – Discrete Structures Conditional Statements – Page 2 Conditional Statement/Implication Truth Table Representing Implication (continued) • In English, we would assume a cause- • If viewed as a logic operation, p ⇒ q can only be and-effect relationship, i.e., the fact that p evaluated as false if p is true and q is false is true would force q to be true. • This does not say that p causes q • If “it is snowing,” then “3+5=8” is • Truth table meaningless in this regard since p has no p q p ⇒ q effect at all on q T T T • At this point it may be easiest to view the T F F operator “⇒” as a logic operationsimilar to F T T AND or OR (conjunction or disjunction). F F T CSCI 1900 – Discrete Structures Conditional Statements – Page 3 CSCI 1900 – Discrete Structures Conditional Statements – Page 4 Examples where p ⇒ q is viewed Converse and contrapositive as a logic operation •Ifp is false, then any q supports p ⇒ q is • The converse of p ⇒ q is the implication true. that q ⇒ p – False ⇒ True = True • The contrapositive of p ⇒ q is the –False⇒ False = True implication that ~q ⇒ ~p • If “2+2=5” then “I am the king of England” is true CSCI 1900 – Discrete Structures Conditional Statements – Page 5 CSCI 1900 – Discrete Structures Conditional Statements – Page 6 1 Converse and Contrapositive Equivalence or biconditional Example Example: What is the converse and •Ifp and q are statements, the compound contrapositive of p: "it is raining" and q: I statement p if and only if q is called an get wet? equivalence or biconditional – Implication: If it is raining, then I get wet. -
7.1 Rules of Implication I
Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms) to derive either a conclusion or a series of intermediate conclusions that link the premises of an argument with the stated conclusion. The First Four Rules of Inference: ◦ Modus Ponens (MP): p q p q ◦ Modus Tollens (MT): p q ~q ~p ◦ Pure Hypothetical Syllogism (HS): p q q r p r ◦ Disjunctive Syllogism (DS): p v q ~p q Common strategies for constructing a proof involving the first four rules: ◦ Always begin by attempting to find the conclusion in the premises. If the conclusion is not present in its entirely in the premises, look at the main operator of the conclusion. This will provide a clue as to how the conclusion should be derived. ◦ If the conclusion contains a letter that appears in the consequent of a conditional statement in the premises, consider obtaining that letter via modus ponens. ◦ If the conclusion contains a negated letter and that letter appears in the antecedent of a conditional statement in the premises, consider obtaining the negated letter via modus tollens. ◦ If the conclusion is a conditional statement, consider obtaining it via pure hypothetical syllogism. ◦ If the conclusion contains a letter that appears in a disjunctive statement in the premises, consider obtaining that letter via disjunctive syllogism. Four Additional Rules of Inference: ◦ Constructive Dilemma (CD): (p q) • (r s) p v r q v s ◦ Simplification (Simp): p • q p ◦ Conjunction (Conj): p q p • q ◦ Addition (Add): p p v q Common Misapplications Common strategies involving the additional rules of inference: ◦ If the conclusion contains a letter that appears in a conjunctive statement in the premises, consider obtaining that letter via simplification. -
Two Sources of Explosion
Two sources of explosion Eric Kao Computer Science Department Stanford University Stanford, CA 94305 United States of America Abstract. In pursuit of enhancing the deductive power of Direct Logic while avoiding explosiveness, Hewitt has proposed including the law of excluded middle and proof by self-refutation. In this paper, I show that the inclusion of either one of these inference patterns causes paracon- sistent logics such as Hewitt's Direct Logic and Besnard and Hunter's quasi-classical logic to become explosive. 1 Introduction A central goal of a paraconsistent logic is to avoid explosiveness { the inference of any arbitrary sentence β from an inconsistent premise set fp; :pg (ex falso quodlibet). Hewitt [2] Direct Logic and Besnard and Hunter's quasi-classical logic (QC) [1, 5, 4] both seek to preserve the deductive power of classical logic \as much as pos- sible" while still avoiding explosiveness. Their work fits into the ongoing research program of identifying some \reasonable" and \maximal" subsets of classically valid rules and axioms that do not lead to explosiveness. To this end, it is natural to consider which classically sound deductive rules and axioms one can introduce into a paraconsistent logic without causing explo- siveness. Hewitt [3] proposed including the law of excluded middle and the proof by self-refutation rule (a very special case of proof by contradiction) but did not show whether the resulting logic would be explosive. In this paper, I show that for quasi-classical logic and its variant, the addition of either the law of excluded middle or the proof by self-refutation rule in fact leads to explosiveness. -
Boolean Satisfiability Solvers: Techniques and Extensions
Boolean Satisfiability Solvers: Techniques and Extensions Georg WEISSENBACHER a and Sharad MALIK a a Princeton University Abstract. Contemporary satisfiability solvers are the corner-stone of many suc- cessful applications in domains such as automated verification and artificial intelli- gence. The impressive advances of SAT solvers, achieved by clever engineering and sophisticated algorithms, enable us to tackle Boolean Satisfiability (SAT) problem instances with millions of variables – which was previously conceived as a hope- less problem. We provide an introduction to contemporary SAT-solving algorithms, covering the fundamental techniques that made this revolution possible. Further, we present a number of extensions of the SAT problem, such as the enumeration of all satisfying assignments (ALL-SAT) and determining the maximum number of clauses that can be satisfied by an assignment (MAX-SAT). We demonstrate how SAT solvers can be leveraged to solve these problems. We conclude the chapter with an overview of applications of SAT solvers and their extensions in automated verification. Keywords. Satisfiability solving, Propositional logic, Automated decision procedures 1. Introduction Boolean Satisfibility (SAT) is the problem of checking if a propositional logic formula can ever evaluate to true. This problem has long enjoyed a special status in computer science. On the theoretical side, it was the first problem to be classified as being NP- complete. NP-complete problems are notorious for being hard to solve; in particular, in the worst case, the computation time of any known solution for a problem in this class increases exponentially with the size of the problem instance. On the practical side, SAT manifests itself in several important application domains such as the design and verification of hardware and software systems, as well as applications in artificial intelligence. -
Logical Connectives Good Problems: March 25, 2008
Logical Connectives Good Problems: March 25, 2008 Mathematics has its own language. As with any language, effective communication depends on logically connecting components. Even the simplest “real” mathematical problems require at least a small amount of reasoning, so it is very important that you develop a feeling for formal (mathematical) logic. Consider, for example, the two sentences “There are 10 people waiting for the bus” and “The bus is late.” What, if anything, is the logical connection between these two sentences? Does one logically imply the other? Similarly, the two mathematical statements “r2 + r − 2 = 0” and “r = 1 or r = −2” need to be connected, otherwise they are merely two random statements that convey no useful information. Warning: when mathematicians talk about implication, it means that one thing must be true as a consequence of another; not that it can be true, or might be true sometimes. Words and symbols that tie statements together logically are called logical connectives. They allow you to communicate the reasoning that has led you to your conclusion. Possibly the most important of these is implication — the idea that the next statement is a logical consequence of the previous one. This concept can be conveyed by the use of words such as: therefore, hence, and so, thus, since, if . then . , this implies, etc. In the middle of mathematical calculations, we can represent these by the implication symbol (⇒). For example 3 − x x + 7y2 = 3 ⇒ y = ± ; (1) r 7 x ∈ (0, ∞) ⇒ cos(x) ∈ [−1, 1]. (2) Converse Note that “statement A ⇒ statement B” does not necessarily mean that the logical converse — “statement B ⇒ statement A” — is also true. -
Contradiction Or Non-Contradiction? Hegel’S Dialectic Between Brandom and Priest
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Padua@research CONTRADICTION OR NON-CONTRADICTION? HEGEL’S DIALECTIC BETWEEN BRANDOM AND PRIEST by Michela Bordignon Abstract. The aim of the paper is to analyse Brandom’s account of Hegel’s conception of determinate negation and the role this structure plays in the dialectical process with respect to the problem of contradiction. After having shown both the merits and the limits of Brandom’s account, I will refer to Priest’s dialetheistic approach to contradiction as an alternative contemporary perspective from which it is possible to capture essential features of Hegel’s notion of contradiction, and I will test the equation of Hegel’s dialectic with Priest dialetheism. 1. Introduction According to Horstmann, «Hegel thinks of his new logic as being in part incompatible with traditional logic»1. The strongest expression of this new conception of logic is the first thesis of the work Hegel wrote in 1801 in order to earn his teaching habilitation: «contradictio est regula veri, non contradictio falsi»2. Hegel seems to claim that contradictions are true. The Hegelian thesis of the truth of contradiction is highly problematic. This is shown by Popper’s critique based on the principle of ex falso quodlibet: «if a theory contains a contradiction, then it entails everything, and therefore, indeed, nothing […]. A theory which involves a contradiction is therefore entirely useless I thank Graham Wetherall for kindly correcting a previous English translation of this paper and for his suggestions and helpful remarks. Of course, all remaining errors are mine. -
Deduction (I) Tautologies, Contradictions And
D (I) T, & L L October , Tautologies, contradictions and contingencies Consider the truth table of the following formula: p (p ∨ p) () If you look at the final column, you will notice that the truth value of the whole formula depends on the way a truth value is assigned to p: the whole formula is true if p is true and false if p is false. Contrast the truth table of (p ∨ p) in () with the truth table of (p ∨ ¬p) below: p ¬p (p ∨ ¬p) () If you look at the final column, you will notice that the truth value of the whole formula does not depend on the way a truth value is assigned to p. The formula is always true because of the meaning of the connectives. Finally, consider the truth table table of (p ∧ ¬p): p ¬p (p ∧ ¬p) () This time the formula is always false no matter what truth value p has. Tautology A statement is called a tautology if the final column in its truth table contains only ’s. Contradiction A statement is called a contradiction if the final column in its truth table contains only ’s. Contingency A statement is called a contingency or contingent if the final column in its truth table contains both ’s and ’s. Let’s consider some examples from the book. Can you figure out which of the following sentences are tautologies, which are contradictions and which contingencies? Hint: the answer is the same for all the formulas with a single row. () a. (p ∨ ¬p), (p → p), (p → (q → p)), ¬(p ∧ ¬p) b.