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Chapter 4

PHI 1101 P - Chapter 4.docx

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University of Ottawa
Iva Apostolova

Deductive Reasoning • 1. The nature of deductive reasoning. • Analytic definitions: ‘Bachelors are unmarried men’: tautological and yet useful. • Deductive reasoning works in much the same way as analytic definitions: You have everything you need to know about the argument contained within the argument. • Deductive arguments do not depend upon facts or new scientific discoveries to work. • Associated with the Greek philosopher Aristotle, Plato’s student; he called deductive arguments ‘syllogisms’(literally meaning ‘inferences’). • When we deduce, we ‘lead’our thoughts into a predetermined path of reasoning, which is what we call logical thinking (see chapter 1). • Deductive arguments are those whose premises, if factually true guarantee the truth of the conclusion. What does it mean? • It doesn`t mean that the premises and the conclusion of a deductive argument have to be always factually true for the argument to work. • It means that, the truth of the conclusion is guaranteed IF the premises were factually true. • In other words, the logical strength of deductive argument does not depend upon the content of the premises but upon the structure/form of the argument. Truth • Truth. The statements which constitute the argument can be factually true or false. • (Arguments are never true or false! Truth and falsehood are properties of the statements, not of the arguments!) • However, ‘truth’has two meanings: 1) truth of the structure and 2) truth of the world (factual truth). • Factual truth is something that deductive arguments don’t care about as much because they are guided by the structure of rational thought, not by facts. Validity • Validity of arguments: a ‘strange’property of arguments! • Validity is meant as a technical term. • In the technical sense validity refers to the structure of the deductive argument not to its content. If the premises support the conclusion, then we have a (technically) valid argument. • The premises, and the conclusion can be true, or they can be false. The falsehood of the premises doesn’t make the argument invalid! • We can have true premises and a true conclusion, false premises and a false conclusion, or false premises and a true conclusion. • However, if we have true premises, we have to get a true conclusion, otherwise the argument is invalid! • Valid arguments can have this structure: • T, T, …  T • T, F, …  T • F, F, …  F • T, F, …  F • F, F, …  T • Invalid arguments always have this structure: T, T, …  F • Ex of valid arguments with false premises and a false conclusions: • Ex 1.All dogs have flippers. All cats are dogs. Therefore, all cats have flippers. • Ex 2.All fish read Russian novels. Goldie is my gold fish. Therefore, Goldie reads Russian novels. • Ex of valid arguments with false premises but true conclusion. • Ex 1. Benny, my dog, is a cat. All cats are mammals. Therefore, Benny is a mammal. • Ex 2. Toronto is the capital of Canada. Canada is in NorthAmerica. Therefore, Toronto is in NorthAmerica. • Ex of valid arguments with true premises and a true conclusion: • Ex 1. Benny is a dog. All dogs are mammals. Therefore, Benny is a mammal. • Ex 2. If abortion is taking of a human life, then it’s murder. Abortion is taking of a human life. Therefore, abortion is murder. • The common link between all of the shown arguments is that they are valid, in the technical sense of the word because the premises, if true, would have made the conclusion true too (that’s what it means to ‘support’the conclusion). • Ex of invalid arguments: • Ex 1. If you don’t wear swim goggles, you can lose your contact lenses. Susan lost her contact lenses. Therefore, Susan must not have worn swim goggles. • Ex 2. If Iva has horns, then she is mortal. Iva is mortal. Therefore, Iva has horns. Soundness • Soundness of arguments: another technical term. • For an argument to be sound it has to have a valid structure, that is, the premises must support the conclusion, and they have to be factually true. • Ex of a sound argument: Ottawa is the capital of Canada. Canada is in NorthAmerica. Therefore, Ottawa is in NorthAmerica. • Sound arguments are rare. Logical Operators II. Logical operators. • Truth-functional statements (statements that are capable of being true or false) include logical operators. • There are five logical operators: • Conjunction: expressed by ‘and’/’but’; it joins together distinct thoughts. • The parts of the conjunction: conjuncts. • It doesn’t matter on which side of the conjunction (‘and’) we place the conjuncts. • Ex: I am a sick man and I am a spiteful man. • Disjunction: expressed by ‘either … or’/ ‘or’; it connects distinct thoughts in a disjoint way. • The parts of the disjunction: disjuncts. • It doesn’t matter on which side of the disjunction (‘or’) we place the disjuncts. • Ex: Either Picasso was this century’s greatest artists or Klimt was. • Implications: expressed by ‘if … then’; connects distinct thoughts in a casual way. • The parts of the implication: antecedent (that which comes immediately after ‘if’and which serves as the cause); consequent (that which comes immediately after ‘then’and serves as the result). • Ex: If I am a sick man, then I am a spiteful man. • Caution! In arguments often ‘if’and ‘then’are omitted. • Caution! You should never swap the places of the antecedent and the consequent! • In English the ‘if’clause often comes at the end of the sentence. • Ex: We’ll go for a picnic if it’s sunny. • Equivalence: expressed by ‘identical to’, ‘equivalent to’, ‘the same as’; connects distinct thoughts through the relation of identity. • Ex: I am myself. • Ex: Your mom is the third person from the left in this old picture. • Negation: expressed by ‘not’or the negative form of any verb or adjective, or adverb (such as ‘dislike’, ‘do not’, ‘unlikely’, etc.). • Ex: This dog does not look like a dog. • Two peculiarities: • ‘Unless’: translated as a negative implication or a disjunction. • Ex: I won’t go out with you unless you shave your beard. • Translate into: ‘I’ll go out with you only if you shave your beard.’Or • ‘Either you shave your beard or I won’t go out with you’. • You can translate most statements that don’t have a clearly stated logical operator into implications (positive or negative). • Example: ‘My dog barks a lot.’ • Translation: ‘If I have a dog, then it barks a lot.’ III. Symbolization. • The goal here is to formalize arguments (turn them into logical formulas) in order to see their structure clearly and avoid any ambiguity of meaning. • The ultimate formalization of deductive arguments is called ‘logical form’. • Sentential variables: variables that replace whole thoughts (sentences of parts of sentences). • We could use standard variables P, Q, R … or we could come up with new variables for each argument. • Ex: ‘I drove my car to the mall today and I picked up my dry cleaning from the dry cleaner.’ • Translation 1: P and Q • Translation 2: C and D (‘C’stands for ‘drove my car’and D stands for ‘picked up my dry cleaning’) • Symbols for the logical operations: • Negation: ~ (tilde) • Conjunction: & (ampersand) • Disjunction: V (vel) • Implication: ᴐ (horseshoe) • Equivalence: ≡ (triple bar) • Ex: This is not my hat. • Logical formula: ~ P • Ex: I am a sick man and I am a spiteful man. • Logical formula: P & Q • Ex: Either picasso was this century’s greatest artists or Klimt was. • Logical for
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