Conundrums of Conditionals in Contraposition
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Section 2.1: Proof Techniques
Section 2.1: Proof Techniques January 25, 2021 Abstract Sometimes we see patterns in nature and wonder if they hold in general: in such situations we are demonstrating the appli- cation of inductive reasoning to propose a conjecture, which may become a theorem which we attempt to prove via deduc- tive reasoning. From our work in Chapter 1, we conceive of a theorem as an argument of the form P → Q, whose validity we seek to demonstrate. Example: A student was doing a proof and suddenly specu- lated “Couldn’t we just say (A → (B → C)) ∧ B → (A → C)?” Can she? It’s a theorem – either we prove it, or we provide a counterexample. This section outlines a variety of proof techniques, including direct proofs, proofs by contraposition, proofs by contradiction, proofs by exhaustion, and proofs by dumb luck or genius! You have already seen each of these in Chapter 1 (with the exception of “dumb luck or genius”, perhaps). 1 Theorems and Informal Proofs The theorem-forming process is one in which we • make observations about nature, about a system under study, etc.; • discover patterns which appear to hold in general; • state the rule; and then • attempt to prove it (or disprove it). This process is formalized in the following definitions: • inductive reasoning - drawing a conclusion based on experi- ence, which one might state as a conjecture or theorem; but al- mostalwaysas If(hypotheses)then(conclusion). • deductive reasoning - application of a logic system to investi- gate a proposed conclusion based on hypotheses (hence proving, disproving, or, failing either, holding in limbo the conclusion). -
'The Denial of Bivalence Is Absurd'1
On ‘The Denial of Bivalence is Absurd’1 Francis Jeffry Pelletier Robert J. Stainton University of Alberta Carleton University Edmonton, Alberta, Canada Ottawa, Ontario, Canada [email protected] [email protected] Abstract: Timothy Williamson, in various places, has put forward an argument that is supposed to show that denying bivalence is absurd. This paper is an examination of the logical force of this argument, which is found wanting. I. Introduction Let us being with a word about what our topic is not. There is a familiar kind of argument for an epistemic view of vagueness in which one claims that denying bivalence introduces logical puzzles and complications that are not easily overcome. One then points out that, by ‘going epistemic’, one can preserve bivalence – and thus evade the complications. James Cargile presented an early version of this kind of argument [Cargile 1969], and Tim Williamson seemingly makes a similar point in his paper ‘Vagueness and Ignorance’ [Williamson 1992] when he says that ‘classical logic and semantics are vastly superior to…alternatives in simplicity, power, past success, and integration with theories in other domains’, and contends that this provides some grounds for not treating vagueness in this way.2 Obviously an argument of this kind invites a rejoinder about the puzzles and complications that the epistemic view introduces. Here are two quick examples. First, postulating, as the epistemicist does, linguistic facts no speaker of the language could possibly know, and which have no causal link to actual or possible speech behavior, is accompanied by a litany of disadvantages – as the reader can imagine. -
Three Ways of Being Non-Material
Three Ways of Being Non-Material Vincenzo Crupi, Andrea Iacona May 2019 This paper presents a novel unified account of three distinct non-material inter- pretations of `if then': the suppositional interpretation, the evidential interpre- tation, and the strict interpretation. We will spell out and compare these three interpretations within a single formal framework which rests on fairly uncontro- versial assumptions, in that it requires nothing but propositional logic and the probability calculus. As we will show, each of the three intrerpretations exhibits specific logical features that deserve separate consideration. In particular, the evidential interpretation as we understand it | a precise and well defined ver- sion of it which has never been explored before | significantly differs both from the suppositional interpretation and from the strict interpretation. 1 Preliminaries Although it is widely taken for granted that indicative conditionals as they are used in ordinary language do not behave as material conditionals, there is little agreement on the nature and the extent of such deviation. Different theories tend to privilege different intuitions about conditionals, and there is no obvious answer to the question of which of them is the correct theory. In this paper, we will compare three interpretations of `if then': the suppositional interpretation, the evidential interpretation, and the strict interpretation. These interpretations may be regarded either as three distinct meanings that ordinary speakers attach to `if then', or as three ways of explicating a single indeterminate meaning by replacing it with a precise and well defined counterpart. Here is a rough and informal characterization of the three interpretations. According to the suppositional interpretation, a conditional is acceptable when its consequent is credible enough given its antecedent. -
The Incorrect Usage of Propositional Logic in Game Theory
The Incorrect Usage of Propositional Logic in Game Theory: The Case of Disproving Oneself Holger I. MEINHARDT ∗ August 13, 2018 Recently, we had to realize that more and more game theoretical articles have been pub- lished in peer-reviewed journals with severe logical deficiencies. In particular, we observed that the indirect proof was not applied correctly. These authors confuse between statements of propositional logic. They apply an indirect proof while assuming a prerequisite in order to get a contradiction. For instance, to find out that “if A then B” is valid, they suppose that the assumptions “A and not B” are valid to derive a contradiction in order to deduce “if A then B”. Hence, they want to establish the equivalent proposition “A∧ not B implies A ∧ notA” to conclude that “if A then B”is valid. In fact, they prove that a truth implies a falsehood, which is a wrong statement. As a consequence, “if A then B” is invalid, disproving their own results. We present and discuss some selected cases from the literature with severe logical flaws, invalidating the articles. Keywords: Transferable Utility Game, Solution Concepts, Axiomatization, Propositional Logic, Material Implication, Circular Reasoning (circulus in probando), Indirect Proof, Proof by Contradiction, Proof by Contraposition, Cooperative Oligopoly Games 2010 Mathematics Subject Classifications: 03B05, 91A12, 91B24 JEL Classifications: C71 arXiv:1509.05883v1 [cs.GT] 19 Sep 2015 ∗Holger I. Meinhardt, Institute of Operations Research, Karlsruhe Institute of Technology (KIT), Englerstr. 11, Building: 11.40, D-76128 Karlsruhe. E-mail: [email protected] The Incorrect Usage of Propositional Logic in Game Theory 1 INTRODUCTION During the last decades, game theory has encountered a great success while becoming the major analysis tool for studying conflicts and cooperation among rational decision makers. -
Immediate Inference
Immediate Inference Dr Desh Raj Sirswal, Assistant Professor (Philosophy) P.G. Govt. College for Girls, Sector-11, Chandigarh http://drsirswal.webs.com . Inference Inference is the act or process of deriving a conclusion based solely on what one already knows. Inference has two types: Deductive Inference and Inductive Inference. They are deductive, when we move from the general to the particular and inductive where the conclusion is wider in extent than the premises. Immediate Inference Deductive inference may be further classified as (i) Immediate Inference (ii) Mediate Inference. In immediate inference there is one and only one premise and from this sole premise conclusion is drawn. Immediate inference has two types mentioned below: Square of Opposition Eduction Here we will know about Eduction in details. Eduction The second form of Immediate Inference is Eduction. It has three types – Conversion, Obversion and Contraposition. These are not part of the square of opposition. They involve certain changes in their subject and predicate terms. The main concern is to converse logical equivalence. Details are given below: Conversion An inference formed by interchanging the subject and predicate terms of a categorical proposition. Not all conversions are valid. Conversion grounds an immediate inference for both E and I propositions That is, the converse of any E or I proposition is true if and only if the original proposition was true. Thus, in each of the pairs noted as examples either both propositions are true or both are false. Steps for Conversion Reversing the subject and the predicate terms in the premise. Valid Conversions Convertend Converse A: All S is P. -
On the Analysis of Indirect Proofs: Contradiction and Contraposition
On the analysis of indirect proofs: Contradiction and contraposition Nicolas Jourdan & Oleksiy Yevdokimov University of Southern Queensland [email protected] [email protected] roof by contradiction is a very powerful mathematical technique. Indeed, Premarkable results such as the fundamental theorem of arithmetic can be proved by contradiction (e.g., Grossman, 2009, p. 248). This method of proof is also one of the oldest types of proof early Greek mathematicians developed. More than two millennia ago two of the most famous results in mathematics: The irrationality of 2 (Heath, 1921, p. 155), and the infinitude of prime numbers (Euclid, Elements IX, 20) were obtained through reasoning by contradiction. This method of proof was so well established in Greek mathematics that many of Euclid’s theorems and most of Archimedes’ important results were proved by contradiction. In the 17th century, proofs by contradiction became the focus of attention of philosophers and mathematicians, and the status and dignity of this method of proof came into question (Mancosu, 1991, p. 15). The debate centred around the traditional Aristotelian position that only causality could be the base of true scientific knowledge. In particular, according to this traditional position, a mathematical proof must proceed from the cause to the effect. Starting from false assumptions a proof by contradiction could not reveal the cause. As Mancosu (1996, p. 26) writes: “There was thus a consensus on the part of these scholars that proofs by contradiction were inferior to direct Australian Senior Mathematics Journal vol. 30 no. 1 proofs on account of their lack of causality.” More recently, in the 20th century, the intuitionists went further and came to regard proof by contradiction as an invalid method of reasoning. -
Proof by Contrapositive July 12, 2012
Proof by Contrapositive July 12, 2012 So far we've practiced some different techniques for writing proofs. We started with direct proofs, and then we moved on to proofs by contradiction and mathematical induction. The method of contradiction is an example of an indirect proof: one tries to skirt around the problem and find a clever argument that produces a logical contradiction. This is not the only way to perform an indirect proof - there is another technique called proof by contrapositive. Suppose that we are asked to prove a conditional statement, or a statement of the form \If A, then B." We know that we can try to prove it directly, which is always the more enlightening and preferred method. If a direct proof fails (or is too hard), we can try a contradiction proof, where we assume :B and A, and we arrive at some sort of fallacy. It's also possible to try a proof by contrapositive, which rests on the fact that a statement of the form \If A, then B." (A =) B) is logically equivalent to \If :B, then :A." (:B =):A) The second statement is called the contrapositive of the first. Instead of proving that A implies B, you prove directly that :B implies :A. Proof by contrapositive: To prove a statement of the form \If A, then B," do the following: 1. Form the contrapositive. In particular, negate A and B. 2. Prove directly that :B implies :A. There is one small caveat here. Since proof by contrapositive involves negating certain logical statements, one has to be careful. -
Working with Logic
Working with Logic Jordan Paschke September 2, 2015 One of the hardest things to learn when you begin to write proofs is how to keep track of everything that is going on in a theorem. Using a proof by contradiction, or proving the contrapositive of a proposition, is sometimes the easiest way to move forward. However negating many different hypotheses all at once can be tricky. This worksheet is designed to help you adjust between thinking in “English” and thinking in “math.” It should hopefully serve as a reference sheet where you can quickly look up how the various logical operators and symbols interact with each other. Logical Connectives Let us consider the two propositions: P =“Itisraining.” Q =“Iamindoors.” The following are called logical connectives,andtheyarethemostcommonwaysof combining two or more propositions (or only one, in the case of negation) into a new one. Negation (NOT):ThelogicalnegationofP would read as • ( P )=“Itisnotraining.” ¬ Conjunction (AND):ThelogicalconjunctionofP and Q would be read as • (P Q) = “It is raining and I am indoors.” ∧ Disjunction (OR):ThelogicaldisjunctionofP and Q would be read as • (P Q)=“ItisrainingorIamindoors.” ∨ Implication (IF...THEN):ThelogicalimplicationofP and Q would be read as • (P = Q)=“Ifitisraining,thenIamindoors.” ⇒ 1 Biconditional (IF AND ONLY IF):ThelogicalbiconditionalofP and Q would • be read as (P Q)=“ItisrainingifandonlyifIamindoors.” ⇐⇒ Along with the implication (P = Q), there are three other related conditional state- ments worth mentioning: ⇒ Converse:Thelogicalconverseof(P = Q)wouldbereadas • ⇒ (Q = P )=“IfIamindoors,thenitisraining.” ⇒ Inverse:Thelogicalinverseof(P = Q)wouldbereadas • ⇒ ( P = Q)=“Ifitisnotraining,thenIamnotindoors.” ¬ ⇒¬ Contrapositive:Thelogicalcontrapositionof(P = Q)wouldbereadas • ⇒ ( Q = P )=“IfIamnotindoors,thenIitisnotraining.” ¬ ⇒¬ It is worth mentioning that the implication (P = Q)andthecontrapositive( Q = P ) are logically equivalent. -
4.6 Indirect Argument: Contradiction and Contraposition
4.6 Indirect Argument: Contradiction and Contraposition “Reductio ad absurdum is one of a mathematician’s finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 “Reductio ad absurdum is one of a mathematician’s4.6 Indirect finestArgument: weapons. Contradiction It is a far andfiner Contraposition gambit than any chess gambit: a1 chess/ 7 player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 false." A Logical Defense “Suppose I did commit the crime. Then, at the time of the crime, I would have to be at the scene of the crime. In fact, at the time of the crime, I was in SM242, far from the scene of the crime, as the midshipmen will testify. This contradicts the assumption that I committed the crime, since it is impossible to be in two places at once. Hence the assumption is “Reductio ad absurdum is one of a mathematician’s4.6 Indirect finestArgument: weapons. Contradiction It is a far andfiner Contraposition gambit than any chess gambit: a2 chess/ 7 player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 A Logical Defense “Suppose I did commit the crime. Then, at the time of the crime, I would have to be at the scene of the crime. In fact, at the time of the crime, I was in SM242, far from the scene of the crime, as the midshipmen will testify. -
Critical Thinking
Critical Thinking Mark Storey Bellevue College Copyright (c) 2013 Mark Storey Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is found at http://www.gnu.org/copyleft/fdl.txt. 1 Contents Part 1 Chapter 1: Thinking Critically about the Logic of Arguments .. 3 Chapter 2: Deduction and Induction ………… ………………. 10 Chapter 3: Evaluating Deductive Arguments ……………...…. 16 Chapter 4: Evaluating Inductive Arguments …………..……… 24 Chapter 5: Deductive Soundness and Inductive Cogency ….…. 29 Chapter 6: The Counterexample Method ……………………... 33 Part 2 Chapter 7: Fallacies ………………….………….……………. 43 Chapter 8: Arguments from Analogy ………………………… 75 Part 3 Chapter 9: Categorical Patterns….…….………….…………… 86 Chapter 10: Propositional Patterns……..….…………...……… 116 Part 4 Chapter 11: Causal Arguments....……..………….………....…. 143 Chapter 12: Hypotheses.….………………………………….… 159 Chapter 13: Definitions and Analyses...…………………...…... 179 Chapter 14: Probability………………………………….………199 2 Chapter 1: Thinking Critically about the Logic of Arguments Logic and critical thinking together make up the systematic study of reasoning, and reasoning is what we do when we draw a conclusion on the basis of other claims. In other words, reasoning is used when you infer one claim on the basis of another. For example, if you see a great deal of snow falling from the sky outside your bedroom window one morning, you can reasonably conclude that it’s probably cold outside. Or, if you see a man smiling broadly, you can reasonably conclude that he is at least somewhat happy. -
Chapter 8: the Logic of Conditionals
Chapter 8: The Logic of Conditionals § 8.1 Informal methods of proof Conditional elimination This method of proof is also known by its Latin name, modus ponens (literally, “method of affirming”—roughly, having affirmed the antecedent of a conditional, you may affirm the consequent). From P and P → Q , you may infer Q. Biconditional elimination This is sometimes called “modus ponens for the biconditional.” From P and P ↔ Q , you may infer Q. From P and Q ↔ P , you may infer Q. Some handy equivalences Contraposition P → Q ⇔ ¬Q → ¬P The “conditional – disjunction” equivalence P → Q ⇔ ¬P ∨ Q The “negated conditional” equivalence ¬(P → Q) ⇔ P ∧ ¬Q The “biconditional – conjunction” equivalence P ↔ Q ⇔ (P → Q) ∧ (Q → P) The “biconditional – disjunction” equivalence P ↔ Q ⇔ (P ∧ Q) ∨ (¬P ∧ ¬Q) In some systems of deduction for the propositional portion of FOL, these equivalences are used as rules. In the system F, and in Fitch, these are not going to be rules. In fact, we will be using Fitch to prove these equivalences. Still, it is useful to be aware of them. Note that the equivalence symbol ⇔ is not a connective of FOL, but a symbol we use in the “metalanguage” (in this case, English, the language in which we talk about FOL), and the statements of equivalence are not FOL sentences. For a useful chart of tautological equivalences, see the Supplementary Exercises page on the course web site. Look under the listings for Chapter 7. The method of conditional proof This very important method of proof is a way of establishing conditional sentences. In using this method, we make a provisional assumption, P, and deduce some consequences of it. -
Modal Logic NL for Common Language
Modal logic NL for common language William Heartspring Abstract: Despite initial appearance, paradoxes in classical logic, when comprehension is unrestricted, do not go away even if the law of excluded middle is dropped, unless the law of noncontradiction is eliminated as well, which makes logic much less powerful. Is there an alternative way to preserve unrestricted comprehension of common language, while retaining power of classical logic? The answer is yes, when provability modal logic is utilized. Modal logic NL is constructed for this purpose. Unless a paradox is provable, usual rules of classical logic follow. The main point for modal logic NL is to tune the law of excluded middle so that we allow for φ and its negation :φ to be both false in case a paradox provably arises. Curry's paradox is resolved dierently from other paradoxes but is also resolved in modal logic NL. The changes allow for unrestricted comprehension and naïve set theory, and allow us to justify use of common language in formal sense. Contents 1 Expressiveness of Language, No-no paradox1 2 Renements to classical logic2 2.1 Role of : in Rened Not-1' 6 3 Paradoxes of classical logic examined in NL6 3.1 Liar, Russell's, Barber paradox7 3.2 Curry's paradox, Then-4 7 3.3 No-no paradox8 3.4 Knower and Fitch's Paradox9 4 Conclusion 10 1 Expressiveness of Language, No-no paradox Ideally, logic underlies both language and mathematics. However, dierent paradoxes in logic, such as the liar paradox, Russell's paradox and Fitch's paradox[1], suggest expres- siveness of common language must be restricted.