1. Logic HUM 200 Syllogisms in Ordinary Language 1
2. Objectives 2 When you complete this lesson, you will be able to: Identify the three ways an argument in ordinary language deviates from standard form Reduce the number of terms in a syllogism to three terms Translate categorical propositions into standard form Use a parameter to conduct uniform translation Identify three types of enthymemes Construct a sorites to test the validity of an argument Identify disjunctive and hypothetical syllogisms Describe three methods of responding to a dilemma
3. Syllogistic Arguments 3 Any argument that is a standard-form categorical syllogism, or can be reformulated as a standard-form categorical syllogism Reduction to standard form results in a standard-form translation
4. Syllogistic Arguments, continued 4 First deviation Order of the premises and conclusion not the same as a standard-form argument Second deviation Premises appear to have more than three terms Third deviation Component propositions may not be standard-form propositions
5. Reducing the Number of Terms to Three 5 Eliminate synonyms No wealthy persons are vagrants. All lawyers are rich people. Therefore no attorneys are tramps. Six terms can be reduced to three No wealthy persons are vagrants. All lawyers are wealthy persons. Therefore no lawyers are vagrants.
6. Reducing the Number of Terms to Three, continued 6 Eliminate class complements All mammals are warm-blooded animals. No lizards are warm-blooded mammals. Therefore all lizards are nonmammals. Use immediate inferences All mammals are warm-blooded animals. No lizards are warm-blooded mammals. Therefore no lizards are mammals.
12. Translating Categorical Propositions into Standard Form 12 Singular propositions Asserts that a specific individual belongs to a particular class Unit class One-member class whose only member is that object itself “All S is P” Issues Existential import (some is complicated) Fallacy of the undistributed middle
13. Translating Categorical Propositions into Standard Form 13 Consider the following argument: All mammals are warm-blooded animals. No snakes are warm-blooded animals. Therefore, All snakes are nonmammals. If we applied our general rules for syllogisms to the above argument, we would judge it to be invalid because (1) it contains four terms, and (2) it has an affirmative conclusion drawn from a negative premise. We can, however, modify it slightly without changing the substance of the argument and see that it is perfectly valid. Consider this change: All mammals are warm-blooded animals. No snakes are warm-blooded animals. Therefore, No snakes are mammals. We have reduced the number of terms to three by simply obverting the conclusion: "All snakes are nonmammals" becomes "No snakes are mammals." These two propositions are equivalent. The syllogism is now in standard-form and is known to be valid.
14. Translating Categorical Propositions into Standard Form, continued 14 Categorical propositions that have adjectives or adjectival phrases as predicates Some flowers are beautiful Replace the adjective with a term designating the class of all objects that possess that attribute Some flowers are beauties
15. Translating Categorical Propositions into Standard Form, continued 15 Many categorical propositions contain adjectives or adverbs as predicates instead of terms denoting a class of objects. For example, consider the following categoricals: Some animals are mean. No automobiles are available for lease. All our students are handsome. Mary is always late. The predicates in the above propositions convey attributes of the subject. Some animals are "mean." No automobiles are "available for lease." All our students are "handsome." May is "always late." Every attribute, however, determines a class, a group of things possessing that attribute. We can always change the proposition to indicate a class of objects to which the attribute applies. While there are other ways of expressing these propositions, the following examples should suffice so you get the idea. The above propositions could be put into standard form: Some animals are "things (or objects) that are mean." [Here we have a class of objects, those that are mean] No automobiles are "things (or objects) available for lease." All our students are "handsome persons" (or objects). Mary is "a person who is always late."
16. Translating Categorical Propositions into Standard Form, continued 16 Categorical propositions whose main verbs are other than the standard form “to be” All people seek recognition Create a class and use the standard form “to be” All people are seekers of recognition
17. Translating Categorical Propositions into Standard Form, continued 17 The standard copula for categorical propositions used in syllogisms is a form of the verb "to be" (such as is, was, are, etc.). Consider the following propositions: All children desire attention. Some people drink lemonade. These propositions are easily translated into standard form by regarding all of the proposition except the subject term and the quantifier as naming a class-defining attribute, and replace it by a standard copula and a term designating the class determined by that class-defining attribute. The above propositions would then become: All children are desirers of attention. Some people are drinkers of lemonade. “Desirers of attention" has now become a class of people (or objects), those that "desire attention." The standard copula "are" is inserted. "Drinkers of lemonade" is now a class, those people that "drink lemonade." The standard copula "are" is again inserted.
18. Translating Categorical Propositions into Standard Form, continued 18 Standard-form ingredients are all present, but not arranged in standard-form order Racehorses are all thoroughbreds Decide which term is the subject term and then rearrange the words to reflect a standard-form categorical proposition All racehorses are thoroughbreds
19. Translating Categorical Propositions into Standard Form, continued 19 Categorical propositions whose quantities are indicated by words other than “all,” “no,” and “some” “every” or “any” are translated to “all” “a” or “an” may mean “all” or “some,” depending on context “the” may refer to a particular individual or all members of the class “not every” and “not any” should be translated with care
20. Translating Categorical Propositions into Standard Form, continued 20 Exclusive propositions Assert that the predicate applies only to the subject named Only citizens can vote Reversing the subject and predicate, and replace the “only” with “all” All those who can vote are citizens
21. Translating Categorical Propositions into Standard Form, continued 21 Categorical propositions that contain no words at all to indicate quantity Examine the context “Dogs are carnivores” becomes “All dogs are carnivores” “Children are present” becomes “Some children are beings who are present”
22. Translating Categorical Propositions into Standard Form, continued 22 Propositions that do not resemble standard-form categorical propositions, but can be translated Nothing is both round and square No round objects are square objects
23. Translating Categorical Propositions into Standard Form, continued 23 Exceptive propositions Makes two assertions: that all members of some class, except for members of one of its subclasses, are members of some other class All but employees are eligible All non-employees are eligible No employees are eligible Translate into an explicit conjunction of two standard-form categoricals All non-employees are eligible persons, and no employees are eligible persons
25. Uniform Translation 25 Parameter An auxiliary symbol that aids in reformulating an assertion into standard form The poor always you have with you Use “times” as the parameter (temporal) All times are times when you have the poor with you
26. Uniform Translation, continued 26 Consider reducing by using a parameter Soiled paper plates are scattered only where careless people have picnicked. There are soiled paper plates scattered about here. Therefore careless people must have been picnicking here. Use “places” as the parameter All places where soiled paper plates are scattered are places where careless people have picnicked. This place is a place where soiled paper plates are scattered. Therefore this place is a place where careless people have picnicked.
28. Enthymemes 28 An argument that contains an unstated proposition Jones is a native-born American. Therefore Jones is a citizen. Missing a premise that is thought to be understood All native-born Americans are citizens First-order enthymeme The proposition that is taken for granted is the major premise
29. Enthymemes, continued 29 Second-order enthymemes Proposition taken for granted is the minor premise All students are opposed to the new regulations. Therefore all sophomores are opposed to the new regulations. Missing minor premise All sophomores are students.
30. Enthymemes, continued 30 Third-order enthymeme Proposition taken for granted is the conclusion No true Christian is vain, but some churchgoers are vain Infer the conclusion Some churchgoers are not true Christians
33. Sorites 33 Sometimes a single categorical proposition will not suffice for drawing a desired conclusion from a group of premises. The evidence for a conclusion consists of more than two propositions. The inference is not a syllogism in such cases but a series of syllogisms. Consider the following argument: All dictatorships are undemocratic. All undemocratic governments are unstable. All unstable governments are cruel. All cruel governments are objects of hate. Therefore, All dictatorships are objects of hate. The inference (stated in the conclusion) may be tested by means of the syllogistic rules. The argument is a chain of syllogisms in which the conclusion of one becomes a premise of another. In the above syllogism, however, the conclusions of all the syllogisms except the last remain unexpressed. A sorite is a chain of syllogisms in which the conclusion of one is a premise in another, in which all the conclusions except the last one are unexpressed, and in which the premises are so arranged that any two successive ones contain a common term.
34. Sorites, continued 34 Sorites appear in two distinct types: the Aristotelian and the Goclenian. It is the arrangement of the propositions within the sorite which determines what type it is. In the Aristotelian, the first premise contains the subject of the conclusion and the common term of two successive propositions appears first as a predicate and next as a subject. Here is an example of the Aristotelian sorite, using letters to indicate its special arrangement: A = B : Aristotle is a man.B = C : All men are mammals.C = D : All mammals are living beings.D = E : All living beings are substances.--------A = E : Therefore, Aristotle is a substance.
35. Sorites, continued 35 In a Gocleniansorite the arrangement is different. The first premise contains the predicate of the conclusion and the common term of two successive propositions appears first as subject and next as predicate. Here is an example of the Gocleniansorite, again using letters to indicate its special arrangement: D = E : One who has no peace of mind is miserable.C = D : One who lacks much has no peace of mind.B = C : One who has many desires lacks much.A = B : One who has many vices has many desires.---------A = E : Therefore, One who has many vices is miserable.
40. Disjunctive and Hypothetical Syllogisms 40 Disjunctive proposition Contains two component propositions Either she was driven by stupidity or arrogance. Disjuncts She was driven by stupidity She was driven by arrogance
41. Disjunctive and Hypothetical Syllogisms, continued 41 Disjunctive syllogism Disjunction in one premise Denial or contradictory of one of its two disjuncts in other premise Validly infer that the other disjunct is true Either Mr. Smith is the brakeman’s next door neighbor or Mr. Robinson is the brakeman’s next door neighbor. Mr. Robinson is not the brakeman’s next door neighbor. Therefore Mr. Smith is the brakeman’s next door neighbor.
42. Disjunctive and Hypothetical Syllogisms, continued 42 Hypothetical proposition If the first native is a politician, then the first native lies Contains two propositions Antecedent follows the “if” Consequent follows the “then”
43. Disjunctive and Hypothetical Syllogisms, continued 43 Hypothetical syllogism Contains at least one conditional proposition as a premise Pure hypothetical syllogism All premises are conditional (if p then l)If the first native is a politician, then he lies. (if l then denies p) If he lies, then he denies being a politician. (therefore, if p then denies p) Therefore if the first native is a politician, then he denies being a politician.
44. Disjunctive and Hypothetical Syllogisms, continued 44 Mixed hypothetical syllogism One premise is conditional, the other is not Modus Ponens (VALID) Categorical premise affirms the antecedent of the conditional premise, and the conclusion affirms its consequent If the second native told the truth, then only one native is a politician. The second native told the truth. Therefore only one native is a politician.
45. Disjunctive and Hypothetical Syllogisms, continued 45 Fallacy of affirming the consequent Categorical premise affirms the consequent of the conditional premise rather than the antecedent If Bacon wrote Hamlet, then Bacon was a great writer. Bacon was a great writer. Therefore Bacon wrote Hamlet. (Any great writer could have written Hamlet)
46. Disjunctive and Hypothetical Syllogisms, continued 46 Mixed hypothetical syllogism Modus tollens (VALID) Categorical premise denies the consequent of the conditional premise and the conclusion denies its antecedent If the one-eyed prisoner saw two red hats, then he could tell the color of the hat on his own head. The one-eyed prisoner could not tell the color of the hat on his own head. Therefore the one-eyed prisoner did not see two red hats.
47. Disjunctive and Hypothetical Syllogisms, continued 47 Fallacy of denying the antecedent Categorical premise denies the antecedent of the conditional premise, rather than the consequent If Carl embezzled the college funds, then Carl is guilty of a felony. Carl did not embezzle the college funds. Therefore Carl is not guilty of a felony. (Carl could have committed another felony)
49. The Dilemma 49 Claims that a choice must be made between two alternatives, both of which are usually bad Simple dilemma (positive – from text) Conclusion is a single categorical proposition If the blest in heaven have no desires, they will be perfectly content; so they will be also if their desires are fully gratified; but either they have no desires, or have them fully gratified; therefore they will be perfectly content.
50. The Dilemma, continued 50 Complex dilemma Conclusion is a disjunction Every time we talked to higher level managers, they kept saying they didn’t know anything about the problems below them…Either the group at the top didn’t know, in which case they should have known, or they did know, in which case they were lying to us. One is said to be caught on “the horns” of the dilemma There are three solutions….
51. The Dilemma, continued 51 First, escaping between the horns Reject the disjunctive premise If students are fond of learning, they need no stimulus, and if they dislike learning, no stimulus will be of any avail. But any student is either fond of learning or dislikes it. Therefore a stimulus is either needless or of no avail. Introduce a third type of student: one who is indifferent to learning
52. The Dilemma, continued 52 Second, grasp the dilemma by the horns Reject the premise that is a conjunction If students are fond of learning, they need no stimulus Even students who are fond of learning may sometimes need stimulus (grades)
53. The Dilemma, continued 53 Third, rebut the dilemma by means of a counterdilemma Dilemma to not enter politics If you say what is just, men will hate you; and if you say what is unjust, the gods will hate you; but you must say either one or the other; therefore you will be hated. Counterdilemma If I say what is just, the gods will love me; and if I say what is unjust, men will love me. I must say either one or the other. Therefore I shall be loved!
54. Summary 54 Deviations from standard-form arguments Translating to three terms Reduction to standard form Parameters Enthymemes Sorites Disjunctive and hypothetical syllogisms The dilemma
