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Syllogisms Fun with Deductive Reasoning What is a syllogism? » A syllogism is a deductive argument comprising three categorical propositions: a major premise, a minor premise and the conclusion. Categorical propositions have four standard forms: » » » » A = All S are P E = No S are P I = Some S are P O = Some S are not P In the mood » The mood of a syllogism is defined by which of the forms appear and where. So, for example: All M are P Some S are M Therefore, All S are P has the mood: AIA. Syllogism overview » A categorical syllogism contains only three categorical terms: a major term, minor term and middle term. » The major term appears as the predicate in the conclusion, and only once in the major premise (i.e., the first premise). » The minor term appears as the subject in the conclusion, and only once in the minor premise (i.e,. the second premise). » The middle term appears once in the major premise, once in the minor premise, and once in the conclusion. Distribution of terms » A term is said to be distributed when all members of the class denoted by the term are affected by a proposition. – All S are P » S is distributed; P is not distributed » Example “All cows are mammals” tells us something about cows but nothing about mammals Syllogism Examples Correct Syllogism: • Major Premise: All mammals are warm-blooded animals. • Minor Premise: No lizards are warm-blooded animals. • Conclusion: Therefore, no lizards are mammals. Correct Syllogism: • Major Premise: All humans are mortal. • Minor Premise: All Greeks are human. • Conclusion: Therefore, all Greeks are mortal. Descartes’ Syllogism (correct) • Major Premise: Existence has to be true if one is thinking. • Minor Premise: I am thinking. • Conclusion: I think, therefore, I am. Syllogisms can be • Valid or Invalid (reasoning in incorrect order) AND • True or False (reasoning from a faulty major premise) • If a syllogism is both true and valid then it is said to be sound Examples of Faulty Syllogisms FALSE Syllogism (not TRUE -- false major premise) – Major Premise: Blondes have more fun – Minor Premise: Mary is blonde; Jane is brunette – Conclusion: Mary has more fun than Jane. INVALID Syllogism (not VALID – order of reasoning is incorrect): – Major Premise: All dogs eat meat – Minor Premise: Bob (a human) eats meat – Conclusion: Bob is a dog. Corrections Syllogism One: The first faulty syllogism proceeds from a FALSE major premise and therefore can be thrown out entirely. Syllogism Two: – Major Premise: All dogs eat meat – Minor Premise: Rover is a dog. – Conclusion: Therefore, Rover eats meat. Valid or invalid? True or False? Example One: • Major Premise: When it snows the streets get wet. • Minor Premise: The streets are getting wet. • Conclusion: Therefore, it is snowing. Example Two: • Major Premise: If you buy a Ferrari, you will instantly be popular. • Minor Premise: Ed just bought a Ferrari. • Conclusion: Ed will achieve instant popularity. Example Three: • Major Premise: When the battery is dead, the car will not start. • Minor Premise: The car will not start. • Conclusion: Therefore, the battery is dead. Corrections: Valid and True Example One: • Major Premise: When it snows, the streets get wet. • Minor Premise: It is snowing. • Conclusion: Therefore, the streets are getting wet. Example Two: • Example Two proceeds from the beginning from a FALSE major premise (Ferraris give instant popularity) and therefore can be thrown out entirely. Example Three: • Major Premise: When the battery is dead, the car will not start. • Minor Premise: The battery is dead. • Conclusion: Therefore, the car will not start. Types of valid syllogisms • • • • Modus Ponens (Affirming the antecedent) Modus Tollens (Denying the consequent) Hypothetical Syllogism (Chain argument) Disjunctive Syllogism Modus Ponens 1. If A then B 2. A 3. Therefore, B Examples: • If it’s spring, then the birds are chirping • It’s spring. • The birds are chirping. • • • If a world government doesn’t evolve soon, then wars will continue to occur A world government isn’t going to evolve soon. Wars will continue to occur Modus Tollens 1. If A then B 2. Not B 3. Not A Example: 1. If it’s spring then the birds are chirping 2. The birds aren’t chirping 3. Therefore, it isn’t spring. Hypothetical Syllogism 1. If A then B 2. If B then C 3. If A then C Example: 1. If we successfully develop nuclear fusion power, then power will become plentiful and cheap. 2. If power becomes cheap and plentiful, then the economy will flourish. 3. If we successfully develop nuclear fusion power, then the economy will flourish. Disjunctive Syllogism 1. A or B 2. Not A 3. B Example: 1. Either Romney won in 2012 or Obama did. 2. Romney didn’t win. 3. Obama did win. Valid or invalid? » Download Socrative app or go to Socrative.com » Room 769815 Syllogism no-no’s » Syllogisms need to follow 6 rules in order to be valid. If they violate one of these rules then that syllogism commits a formal fallacy and is invalid Rule 1 » There needs to be three categorical terms and those terms cannot vary in how they are used » A fallacy of equivocation occurs when a term is used in a different way within the course of an argument. So, for example: – The priest told me I should have faith. – I have faith that my son will do well in school this year. – Therefore, the priest should be happy with me. » “faith” is being used in two different ways in this argument Rule 2 » The middle term of a valid syllogism is distributed in at least one of the premises. The fallacy of the undistributed middle occurs when this doesn't happen. For instance, the middle term (furry animals) in this syllogism – All dogs are furry animals – Some cats are furry animals – Therefore,some dogs are cats isn't distributed, and the argument is clearly fallacious Rule 3 » If a term is distributed in the conclusion it must be distributed in at least one of the premises – All Protestants are Christians – No Catholics are Protestants – Therefore, no Catholics are Christians doesn't work, because the term "Christians" is distributed in the conclusion, but not in the (major) premise. Rule 3 » The fallacy of illicit major occurs (as above) when the major term is distributed in the conclusion, but not in the (major) premise. » » The fallacy of illicit minor occurs when the minor term is distributed in the conclusion, but not in the (minor) premise Rule 4 » A valid syllogism can't have two negative premises » The fallacy of exclusive premises occurs when a syllogism has two premises that are negative. » A negative premise is either an "E" statement ("No S are P") or an "O" statement ("Some S are not P"), and if you've got two of them in your premises, your syllogism isn't valid. Rule 5 » The conclusion of a syllogism must be negative, if either premise is negative » The fallacy of drawing an affirmative conclusion from a negative premise occurs if this rule is violated. Similarly, if a conclusion is negative, then one of the premises must be negative (which rule, if broken, constitutes the fallacy of drawing a negative conclusion from an affirmative premise). Rule 6 » No particular conclusion can be drawn from two universal premises » This is arguably the most counterintuitive of the rules for validity. An existential fallacy occurs whenever a particular conclusion appears with two universal premises (for example, All M are P, All S are M, Therefore, some S are P).