1 Philosophy 105: Critical Reasoning Modesto Junior College Instructor: Categorical Logic 1.1 Venn Diagrams for Simple Propositions 1.2 Venn Diagrams for Syllogisms 1.3 Conversion, Obversion and Contraposition 1.4 Reducing the Number of Terms
2 1.1 Venn Diagrams for Simple Propositions Notes Venn Diagrams are ways of pictorially representing the informational content of categorical propositions. The basic conventions or such diagrams are as follows: 1. Each term is represented by a circle. (When two terms are involved, the diagram contains two overlapping circles, one for each term.) 2. The area inside the circle for a term represents the extension of that term, and the area outside the circle represents everything not in the extension of that term. 3. The areas of overlap and non-overlap of the circles represents things in an obvious way, given convention (2). Thus, for example, the football-shaped region of overlap between the two circles represents things that are in both the extensions of the two terms involved. 4. Shading of an area means that the class of things represented by that are is empty; placing an X in an are indicates that the correlated class is not empty that is, that there is at least one thing in it. 5. If a given area is neither shaded out nor filled in with an X, nothing is said about the corresponding class. A-Claim: E-Claim: I-Claim: O-Claim:
3 1.1 Venn Diagrams for Simple Propositions Exercises Directions: Draw Venn diagrams for the following propositions. 1. No life decisions are happenings based solely on logic. 2. All electric motors are machines that depend on magnetism. 3. Some political campaigns are mere attempts to discredit opponents. 4. Some rock music lovers are not fans of Madonna. 5. All redistricting plans are sources of controversy. 6. No tax audits are pleasant experiences for cheaters. 7. Some housing developments are complexes that exclude children. 8. Some cruise ships are not steam-driven vessels.
4 1.2 Venn Diagrams for Syllogisms Notes Venn diagrams provide a convenient technique for testing the validity of categorical syllogisms. M 1 8 5 2 4 3 6 7 S P Area 1 is non-s and non-p and M. Area 2 is S and non-p and M. Area 3 is S and P and M. Area 4 is non-s and P and M. Area 5 is S and non-p and non-m. Area 6 is S and P and non-m. Area 7 is non-s and P and non-m. Area 8 is non-s and non-p and non-m. To use a Venn Diagram to test for the validity of a categorical syllogism, follow this procedure. 1. Enter the information of the premises into the diagram. (Do not enter the conclusion.) 2. Look at the resulting diagram to see whether it implies the information of the conclusion. If it does, the syllogism is valid. If it does not, the syllogism is invalid. Several pointers will facilitate using this procedure: 1. Diagram universal premises before particular premises. 2. When diagramming a premise, concentrate only on the two circles representing the terms of the premise. The third circle can largely be ignored. 3. When placing a plain X or a circled X in the diagram, note that it always goes into an area that is split by the third (largely ignored) circle. This third circle divides that are into two parts. If neither of these pats has been shaded, the X is placed on the line from the third circle that divides the two parts. If one of these parts has been shaded, the X is placed in the unshaded part. (NOTE: In a diagram for a categorical syllogism, it will never happen that both parts are shaded.)
5 1.2 Venn Diagrams for Syllogisms Exercises 1. All corporations that overcharge their customers are unethical businesses. Some unethical businesses are investor-owned utilities. Therefore, some investor-owned utilities are corporations that overcharge their customers. 2. No AIDS victims are people who pose an immediate threat to the lives of others. Some kindergarten children are AIDS victims. Therefore, some kindergarten children are not people who pose an immediate threat to the lives of others. 3. No individuals truly concerned with the plight of suffering humanity are people motivated primarily by self-interest. All television evangelists are people motivated primarily by self-interest. Therefore, some television evangelists are not individuals truly concerned with the plight of su!ering humanity. 4. All high-fat diets are diets high in cholesterol. Some diets high in cholesterol are not healthy food programs. Therefore, some healthy food programs are not high-fat diets. 5. No engineering majors are candidates for nightly hookups. No candidates for nightly hookups are deeply emotional individuals. Therefore, no deeply emotional individuals are engineering majors. 6. All impulse buyers are consumers with credit cards. All shopaholics are impulse buyers. Therefore, all shopaholics are consumers with credit cards. 7. No pediatricians are individuals who jeopardize the health of children. All faith healers are individuals who jeopardize the health of children. Therefore, no faith healers are pediatricians. 8. Some individuals prone to violence are not men who treat others humanely. Some police o"cers are individuals prone to violence. Therefore, some police o"cers are not men who treat others humanely. 9. Some ATM locations are places criminals lurk. All places criminals lurk are places to avoid at night. Therefore, some places to avoid at night are ATM locations. 10. No corporations that defraud the government are organizations the government should deal with. Some defense contractors are not organizations the government should deal with. Therefore, some defense contractors are not corporations that defraud the government. 11. All circular triangles are plane figures. All circular triangles are three-sided figures. Therefore, some three-sided figures are plane figures.
6 12. All supernovas are objects that emit massive amounts of energy. All quasars are objects that emit massive amounts of energy. Therefore, all quasars are supernovas. 13. No people who profit from the illegality of their activities are people who want their activities legalized. All drug dealers are people who profit from the illegality of their activities. Therefore, no drug dealers are people who want their activities legalized. 14. Some individuals who risk heart disease are people who will die young. Some smokers are individuals who risk heart disease. Therefore, some smokers are people who will die young. 15. Some communications satellites are rocket-launched failures. All communications satellites are devices with antennas. Therefore, some devices with antennas are rocket-launched failures. 16. All currently living dinosaurs are giant reptiles. All giant reptiles are ectothermic animals. Therefore, some ectothermic animals are currently living dinosaurs. 17. All survivalists are people who enjoy simulated war games. No people who enjoy simulated war games are soldiers who have tasted the agony of real war. Therefore, all soldiers who have tasted the agony of real war are survivalists. 18. No spurned lovers are Valentine s Day fanatics. Some moonstruck romantics are Valentine s Day fanatics. Therefore, some moonstruck romantics are not spurned lovers. 19. No theocracies are regimes open to change. All theocracies are governments that rule by force. Therefore, some governments that rule by force are not regimes open to change. 20. Some snowflakes are not uniform solids. All snowflakes are six-pointed crystals. Therefore, some six-pointed crystals are not uniform solids.
7 1.3 Conversion, Obversion and Contraposition Notes General Notes: We are adding a new truth-value for propositions in Section 1.3. It is called undetermined. Each move below can only be performed on the statements indicated within the explanation. When a move is applied to a statement according to the rules below, it is called a legal move. When you make a legal move, statements keep/retain their originally truth values. Thus, if a statement is true and you make a legal move, the statement stays true. When a statement is false and you make a legal move, the statement stays false. When a move is applied to a statement and violates one of the rules below, it is called an illegal/illicit move and the truthvalue for the resulting statement will be undetermined. When you make a legal move, the beginning statement and the statement generated after making the move are said to be logically equivalent (i.e., this indicates both statements mean the same thing).
8 Conversion: switch subject and predicate Review the diagrams to be sure you understand the relation being expressed. What is it? Conversion is a move that allows us to switch subjects and predicates for E & I statements only. When is it a legal move? Conversion can only be used on E & I statements. How do I convert E & I statements? Switch the subject and predicate terms. Examples: To convert the statement: No cats are fish, we switch the subject and predicate terms to arrive @ No fish are cats. Since these two statements mean the same thing, we state that they are logically equivalent. Contraposition: two steps Review the diagrams to be sure you understand the relation being expressed. What is it? Contraposition is a move that allows us to switch the subject and predicate terms for A & O statements while also changing each term to its complement. When is it a legal move? Contraposition can be used on A & O statements only. How do I contrapose statements? Contraposition is a two-step process Switch subject and predicate terms. Change both the subject and predicate term to its complement (i.e., opposite). Examples: To contrapose the statement: All cats are fish, we first switch the subject & predicate terms to the A statement yielding "All fish are cats. Next we change each term to its complement and the final statement is All nonfish are non-cats. You do not have to show each step, just the original and final statements. Since these two statements mean the same thing, we state that they are logically equivalent. Obversion: two steps Review the diagrams to be sure you understand the relation being expressed. What is it? Obversion is a move that allows us to switch the quality and predicate terms for all four statements: A, E, I & O. When is it a legal move? Obversion can be used on A, E I & O statements. How do I obvert statements? Obversion is a two-step process Change the quality of the statement (i.e., move horizontally across the square from the current statement letter) and, Change the predicate term to its complement (i.e., opposite). Examples: To obvert the statement: No cats are fish, we first move horizontally to the A statement yielding "All cats are. Next we change the predicate term to its complement and the final statement is All cats are non-fish.. You do not have to show each step, just the original and final statements. Since these two statements mean the same thing, we state that they are logically equivalent.
9 Testing arguments for validity. Three Steps: Symbolize the argument. Determine which of the three new moves: conversion, contraposition or obversion has taken place between the premise and conclusion of the argument. If the move is a legal move, then the argument is valid. If the move is illegal/illicit, the argument is invalid. Note: it is critical that you memorize legal versus illegal moves (or learn which statements are not logically equivalent) to do well on the subsequent test on this material.
10 1.3 Conversion, Obversion and Contraposition Exercises I. Exercises 1 through 6 provide a statement, its truth-value in parentheses, and an operation to be performed on that statement. Supply the new statement and the truth-value of the new statement. Exercises 7 through 12 provide a statement, its truth-value in parentheses, and a new statement. Determine how the new statement was derived from the given statement and supply the truthvalue of the new statement. Truth Given statement Operation New statement Value 1. No A are non-b. (T) conv. 2. Some A are B. (T) contrap. 3. All A are non-b. (F) obv. 4. All non-a are B. (F) contrap. 5. Some non-a are not B. (T) conv. 6. Some non-a are non-b. (T) obv. 7. No non-a are non-b. (F) No B are A. 8. Some A are not non-b. (T) Some A are B. 9. All A are non-b. (F) All non-b are A. 10. No non-a are B. (F) All non-a are non-b. 11. Some non-a are not B. (T) Some non-b are not A. 12. Some A are non-b. (F) Some non-b are A.
11 II. Perform the operations of conversion, obversion, and contraposition as indicated. 1. Convert the following propositions and state whether the converse is logically equivalent or not logically equivalent to the given proposition. a. All hurricanes are storms intensified by global warming. b. No sex-change operations are completely successful procedures. c. Some murals by Diego Rivera are works that celebrate the revolutionary spirit. d. Some forms of carbon are not substances with a crystalline structure. 2. Obvert the following propositions and state whether the obverse is logically equivalent or not logically equivalent to the given proposition. a. All radically egalitarian societies are societies that do not preserve individual liberties. b. No cult leaders are people who fail to brainwash their followers. c. Some college football coaches are people who do not slip money to their players. d. Some budgetary cutbacks are not actions fair to the poor. 3. Contrapose the following propositions and state whether the contrapositive is logically equivalent or not logically equivalent to the given proposition. a. All physicians whose licenses have been revoked are physicians ineligible to practice. b. No unprosecuted migrants are migrants granted asylum. c. Some politicians who do not defend Social Security are politicians who do not want to increase taxes. d. Some opponents of gay marriage are not opponents of civil unions.
12 1.4 Reducing the Number of Terms Notes Many two-premise arguments contain complemented terms, so that the number of terms in the entire argument exceeds three. Thus, these arguments are not categorical syllogisms in the strict sense. Nevertheless, through use of the operations of conversion, obversion and contraposition, such arguments can often be replaced by logically equivalent arguments that are categorical syllogisms in the strict sense. The important things to remember in employing these operations are that conversion may be applied only to E and I propositions and that contraposition may be applied only to A and O propositions. Example: All non-p are non-m. Contra. All M are P. All non-m are non-s. Contra. All S are M. -------------------------- No S are non-p. Obv. All S are P. Applying contraposition to both premises and obversion to the conclusion, we may replace this argument with the logically equivalent argument All M are P. All S are M. -------------- All S are P. Venn diagram the syllogism to determine if it is Valid or Invalid.
13 1.5 Reducing the Number of Terms Exercises Rewrite the following arguments using letters to represent the terms, reduce the number of terms, and put the arguments into standard form. Then test the new forms with Venn diagrams or by means of the five rules for syllogisms to determine the validity or invalidity of the original arguments. 1. Some intelligible statements are true statements, because all unintelligible statements are meaningless statements and some false statements are meaningful statements. 2. Some people who do not regret their crimes are convicted murderers, so some convicted murderers are people insusceptible of being reformed, since all people susceptible of being reformed are people who regret their crimes. 3. All Peace Corps volunteers are people who have witnessed poverty and desolation, and all people insensitive to human need are people who have failed to witness poverty and desolation. Thus, all Peace Corps volunteers are people sensitive to human need. 4. Some unintentional killings are not punishable o!enses, inasmuch as all cases of self-defense are unpunishable o!enses, and some intentional killings are cases of self-defense. 5. All aircraft that disintegrate in flight are unsafe planes. Therefore, no poorly maintained aircraft are safe planes, because all wellmaintained aircraft are aircraft that remain intact in flight. 6. No objects that sink in water are chunks of ice, and no objects that float in water are things at least as dense as water. Accordingly, all chunks of ice are things less dense than water. 7. Some proposed flights to Mars are inexpensive ventures, because all unmanned space missions are inexpensive ventures, and some proposed flights to Mars are not manned space missions. 8. All schools driven by careerism are institutions that do not emphasize liberal arts. It follows that some universities are not institutions that emphasize liberal arts, for some schools that are not driven by careerism are universities. 9. No cases of AIDS are infections easily curable by drugs, since all diseases that infect the brain are infections not easily curable by drugs, and all diseases that do not infect the brain are cases other than AIDS. 10. Some foreign emissaries are people without diplomatic immunity, so some people invulnerable to arrest and prosecution are foreign emissaries, because no people with diplomatic immunity are people vulnerable to arrest and prosecution.
14 1.5 Reducing the Number of Terms Exercises Part II 1. Some metals are rare and costly substances, but no welder s materials are nonmetals; hence some welder s materials are rare and costly substances. 2. Some Asian nations were nonbelligerents, because all belligerents were allies either of Germany or Britain, and some Asian nations were not al-lies of either Germany or Britain. 3. Some nondrinkers are athletes, because no drinkers are persons in perfect physical condition, and some people in perfect physical condition are not nonathletes. 4. All things inflammable are unsafe things, so all things that are safe are nonexplosives, because all explosives are flammable things. 5. All worldly goods are changeable things, for no worldly goods are things immaterial, and no material things are unchangeable things. 6. All those who are neither members nor guests of members are those who are excluded; therefore no nonconformists are either members or guests of members, for all those who are included are conformists. 7. All mortals are imperfect beings, and no humans are immortals, whence it follows that all perfect beings are nonhumans. 8. All things present are nonirritants; therefore no irritants are invisible objects, because all visible objects are absent things. 9. All useful things are objects no more than six feet long, because all difficult things to store are useless things, and no objects over six feet long are easy things to store.