5.1 Standard Form, Mood & Figure

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1 Unit Five Categorical Syllogisms Page 1 Unit Five Categorical Syllogisms A Categorical Syllogism is a deductive argument consisting of two premises and a conclusion in which three terms each appear twice. 5.1 Standard Form, Mood & Figure Key Concept Syllogism is the basic deductive argument consisting of two premises and a conclusion. All M are P. All S are M. Therefore, All S are P. (AAA-1) Key Terms Major Term Mood Minor Term Figure 3. Sixteen Examples AAA AAE AAI AAO AEA AEE AEI AEO AIA AIE AII AIO AOA AOE AOI AOO B. Figure refers to the location of the middle term in the premises. Four figures multiplied by the 64 moods results in a total of 256 categorical syllogisms. C. Medieval Logicians shunned the fourth figure.. 1. It is possible to render a valid syllogism in the fourth figure. 2. The controversy arises from the fact that a Figure Four syllogism does not follow the natural deductive process that begins at the most universal and ends at the least universal. 3. This is true to a lesser extent in figures two and three. Figure one is the perfect syllogism form. Outline Syllogism Construction A. All Categorical Syllogisms in standard form follow this pattern. 1. The Major Term is the predicate term. 2. The Minor Term is the subject term. B. The terms appearing in the major premise are the predicate term and the middle term. C. The terms appearing in the minor premise are the subject term and the middle term. Form of the Syllogism A. Mood is a consequence of the quality and quantity of the propositions making up the syllogism. 1. The letter short hand applied to each of the three propositions designates mood. 2. There are 64 combinations that exhaust all possible moods. Constructing the Syllogism A. We construct a categorical syllogism by reading the mood and figure. B. Suppose we wanted to construct EIO We identify the mood of the major premise. Universal Negative.

2 Unit Five Categorical Syllogisms Page 2 2. We identify the mood of the minor premise. Particular Affirmative 3. We identify the mood of the conclusion and mark the subject and predicate. Particular negative 4. We mark the middle terms. 5. We mark the subject and predicate terms in their respective premises. C. Aristotle s Conditionally Valid Forms F 1 F 2 F 3 F 4 Condition AAI EAO AEO EAO AEO Subject exists AAI EAO EAO Middle Term exists AAI Predicate exists D. Validity of an AAI-1 categorical syllogism. 1. Valid All mammals are animals All dogs are mammals Some dogs are animals 2. Invalid All mammals are animals All unicorns are mammals Some unicorns are animals Validity A. A categorical syllogism can be valid, conditionally valid, or invalid. 1. A categorical syllogism is valid when its conclusion cannot be false if its premises are true. All dudus are occos. All ivies are dudus. Therefore all ivies are occos. All mammals are animals. All dogs are mammals. Therefore, all dogs are animals. 2. A categorical syllogism is conditionally valid according to Aristotle, when its conclusion cannot be false if its premises are true providing that a critical term denotes an existing thing. Boole denies conditional validity. All dudus are occos. All ivies are dudus. Therefore some ivies are occos (Valid if dudus exist). All mammals are animals. All dogs are mammals. Therefore, some dogs are animals. (Valid if dogs exist). 3. A categorical syllogism is invalid when its conclusion can be false even if its premises are true. All cats are animals. All dogs are animals. Therefore, All dogs are cats. B. Unconditionally Valid Forms Figure 1 Figure 2 Figure 3 Figure 4 AAA EAE IAI AEE EAE AEE AII IAI AII EIO OAO EIO EIO AOO EIO The Name Game A. Medieval logicians developed an ingenious list of names to make the reduction of 2 nd and 3 rd figure syllogisms to 1 st figure relatively easy. B. The vowel positions in each of the names correspond to the mood of the categorical syllogism. 1. Some of the consonants indicate the operation that advances the reduction. The consonant always refers to the vowel that precedes it. 2. S means a simple conversion. All S are P converse simply to All P are S. P means accidental conversion. All S are P converts accidentally to Some P are S. 3. M means to change or transpose the premises. The minor premise and major premise change places. 4. C means to reduce by contradiction. Valid Syllogism Figure 1 AAA-1 EAE-1 AII-1 EIO-1 Figure 2 EAE-2 AEE-2 EIO-2 AOO-2 Figure-3 AAI-3 EAO-3 IAI-3 Medieval Code Name BARBARA CELARENT DARII FERIO CESARE CAMESTRES FESTINO BAROCO DARAPTI FELAPTON DISAMIS

3 Unit Five Categorical Syllogisms Page 3 AII-3 OAO-3 EIO-3 DATISI BOCARDO FERISON Example: Reducing Disamis to Darii Disamis (Figure 3) Darii (Figure 1) Some B are C All B are A All B are A Some C are B Some A are C Some C are A 1. Major Premise I is converted simply (DIS). Some B are C converts simply to Some C are B. 2. Minor Premise A changes places with the major premise. (AM) Makes the syllogism: All B are A Some C are B Some A are B 3. The Conclusion is converted simply. (IS) Final rendering: All B are A Some C are B Some C are A C. Because this technique was used in the Middle Ages, there is not Figure Four. Distribution A. The syllogism s validity rests on the capacity to predicate one of many. B. When a term is predicated, distribution may occur. Distribution follows the pattern that we studied earlier in the course. Here s a review. C. When applied to the pattern of the various categorical syllogisms, we get the following conclusion. 1. The term in the subject column of a universal affirmative (A) distributes. 2. The term in the subject and predicate columns of a universal negative (E) distributes. 3. Neither of the terms in particular affirmative (I) distributes. 4. The term in the predicate column of a particular negative (O) distributes. Statement Subject Column Predicate Column A All S are P S - E No S are P S P I Some S are - - O Some S are not P - P Statement Subject Column Predicate Column Major Premise A, E E, O Minor Premise A, E E, O Conclusion A, E E, O Textbook Exercise 5.1 I 1-5 Exercise 5.1 II 1-5 Exercise 5.1 III 1-5 Exercise 5.1 IV 1-5

4 Unit Five Categorical Syllogisms Page Venn Diagrams Key Concept The Venn diagram offers a visual way to determine if a Categorical Syllogism is or is not valid. Outline Venn Diagrams A. The Venn diagram for Categorical Syllogisms adds another circle for the Middle Term. B. The goal of the diagram is to determine the validity of a Categorical Syllogism. C. As the map below indicates, there are seven distinct areas of the Venn. 1. The potential membership in each area is fixed and a product of the structure of the diagram. 2. This is why the Venn can reveal if an argument is or is not valid. Primary Venn Location A. Only Premises are graphed onto the Venn diagram. 1. We graph a universal by shading or lining the area of the circle that is empty. 2. We graph a particular by placing an X in the Venn to identify where it is located. B. There are four universal affirmative possibilities requiring four Venn diagrams to illustrate. 1. All S are M 2. All M are S 3. All P are M 4. All M are P

5 Unit Five Categorical Syllogisms Page 5 E. There are four particular negative possibilities requiring four Venn diagrams to illustrate. 1. Some S are not M 2. Some M are not S 3. Some P are not M 4. Some M are not P C. There are four universal negative possibilities requiring two Venn diagrams to illustrate. 1. No S are M 2. No M are S 3. No P are M 4. No M are P D. There are four particular affirmative possibilities requiring two Venn diagrams to illustrate. 1. Some S are M 2. Some M are S 3. Some P are M 4. Some M are P Venn for Unconditionally Valid Arguments A. The Process 1. Diagram only the Premises. 2. Diagram the Universals first by closing off any area that the proposition indicates is empty. 3. Diagram the Particulars by indicating with an X any area that is occupied. 4. If the conclusion appears then the Categorical Syllogism is valid. If it doesn t appear then the categorical syllogism is invalid. B. Categorical Syllogism AAA-1 All M are P All S are M All S are P

6 Unit Five Categorical Syllogisms Page 6 1. We begin with the Major Premise All M are P. a. We ignore the S circle and darken out the part of the M circle that has to be empty if all of the Ms are Ps. 3. Examine the S circle to determine which areas of it are occupies and which are empty. Is the only part of the S circle that is occupied also in the P circle? If the answer is yes, it s valid. Otherwise it s invalid. B. The particular prime place refers to the primary location of the X when its final location cannot be determined. b. The Venn now looks like this 2. Next Venn the Minor Premise All S are M. a. We ignore the P circle and darken out the part of the S circle that has to be empty if all of the Ss are Ms. 1. This occurs because the proposition relates only two of the terms to each other leaving the relationship to the third term undetermined. 2. This is why the universal must be venned before the particular. b. The Venn now looks like this.

7 Unit Five Categorical Syllogisms Page 7 C. Categorical Syllogism EIO-1 No M are P Some S are M Some S are not P 3. If there is an X somewhere in the S circle and not in the P circle then the Venn is illustrating Some S are not P and its valid. Otherwise, it is invalid. Venn for Conditionally Valid Arguments A. A Venn can be used to prove a Conditionally Valid statement by following the pattern we used in the last unit. No M are P All M are S Some S are not P 1. We begin with the Universal premise No M are P. 1. We begin with the Major Premise 2. We then diagram the Particular premise. We know it has moved off of it prime location because one of the areas in which it might have been is empty, so it can t be in that area.

8 Unit Five Categorical Syllogisms Page 8 2. Next we Venn the Minor Premise 5.3 Rules & Fallacies Key Concept We can quickly determine the validity of a categorical syllogism by examining against a criterion made up a few rule. Key Terms Illicit Major Undistributed Middle Existential Fallacy Illicit Minor Exclusive Fallacy Extension Fallacy 3. Finally, we add the circled X which represents the assumption that area in question is not empty. a. If the Middle Term denotes a thing that exists, the syllogism is valid b. If not, the syllogism commits the Existential Fallacy. Outline Which Rules? A. Various renderings of a criterion for evaluating the validity of Categorical Syllogism have been employed over the centuries. B. Criteria for Validity 1. The middle term must be distributed at least once. (Undistributed Middle) Correct Example AAA-1 All M are P All S are M All S are P Incorrect Example AAA- 2 All P are M All S are M All S are P 2. Any term distributed in the conclusion must be distributed in the premises. (Illicit Major, Illicit Minor) Textbook Exercise 5.2 I 1-10 Exercise 5.2 II 1-10 Exercise Lessons Correct Example AAA-1 All M are P All S are M All S are P Incorrect Example AAA- 3 All M are P All M are S All S are P 3. From two negative premises, no valid conclusion follows. (Exclusive Fallacy) Example EEE-3 No M are P No M are S No S are P

9 4. If both premises are universal, the conclusion can only be valid if the requirements for conditional validity are met. (Existential Fallacy) All M are P All S are M Some S are P (For Aristotle Invalid if the subject is not real (Unicorns for example) (For Boole, always invalid) 5. The conclusion always follows from the weakest part. (Extension Fallacy) Unit Five Categorical Syllogisms Page 9 Any valid Categorical syllogism in which the premises are Universal Affirmatives will have a Universal Affirmative conclusion. Correct: AAA Incorrect: AEA Any valid Categorical syllogism in which at least one of the premises is Negative will have a negative conclusion. Correct: AEE Incorrect: AEA Any valid Categorical syllogism in which at least one of the premises is Particular will have a Particular conclusion. Correct: AII (If the Existential Fallacy is avoided) Incorrect: AIA Any valid Categorical syllogism in which at least one of the premises is Particular and at least one premise Negative will have a Particular Negative conclusion. Correct: EIO Incorrect: EIA Syllogistic Fallacies A. Syllogistic fallacies are formal fallacies. 1. The presence of a formal fallacy makes the argument invalid because the truth-value of the conclusion cannot be predicted. 2. The conclusion may in fact be true, but not because the syllogism inferred it. All dogs are mammals. All beagles are mammals. Therefore, all beagles are dogs. B. List of syllogistic fallacies 1. Undistributed Middle occurs when the middle term is not distributed in at least one of the premises. 2. Illicit Major occurs when the predicate term is distributed in the conclusion, but not distributed in the major premise. 3. Illicit Minor occurs when the subject term is distributed in the conclusion, but not distributed in the minor premise. 4. Exclusive Fallacy occurs when both premises are negative. 5. Existential Fallacy occurs when the existence of a particular is inferred from a universal. 6. Extension Fallacy occurs when the quantity or quality of the conclusion are over extended because it doesn t follow from the weakest part.

10 Unit Five Categorical Syllogisms Page Reducing the Number of Terms 1. Frame the Syllogism. E No are A All are E No S are P 2. Set the figure (Figure 2). E No P are M A All S are M E No S are P 3. Determine distribution. E No P are M A All S are M E No S are P Textbook Exercise 5.3 I 1-10 Exercise 5.3 II 1-10 Apply the rules. Key Concept When categorical syllogisms occur in ordinary language, they are seldom expressed in standard form. In this lesson we will learn ways to reduce terms. Key Terms Main Term Derivative Term Outline When Can We Reduce A. Arguments that contain three main terms and a number of derivate terms can be reduced. 1. Main Term refers to the rendering of a word without qualification. Runner, speaker, & player 2. Derivative Term refers to a rendering of a word with qualification. Non-Runner, Non- Speaker, & Non-Player. B. To determine if reducing terms is an applicable technique, identify the main terms and their derivative. 1. There can be no more than three main terms after the derivatives are reduced. 2. If there are more than three main terms, this reduction method won t work. How to Reduce 1. Begin by putting the statements into the form of a categorical proposition. No players are non-speakers. Some coaches are speakers. Some non-players are not non-coaches 2. Translate the syllogism into symbols. All P are non-m Some S are M Some non-p are not non-s 3. Using obversion, conversions, and contraposition. (obvert) (contra pose) All P are non-m Some S are M Some non-p are not non-s 4. This reduces the number of terms to No P are M Some S are M Some S are not P

11 Unit Five Categorical Syllogisms Page Now evaluate the argument as you would any other categorical syllogism. Textbook Exercise Ordinary Language Arguments Key Concept When categorical syllogisms occur in ordinary language, they are seldom expressed in standard form. In this lesson we will learn ways to translate these into standard form. Outline Translations Techniques A. Rules for translations offer guidance in the process. 1. Rules do not exist for every situation. 2. The key is to understand the meaning of the statement and carry that meaning through the translation process. B. Rules for Translations 1. Terms without Nouns (Some roses are red. Some roses are red flowers.) 2. Nonstandard Verbs (Some students will become educated. Some college students are persons who will become educated.) 3. Singular Propositions (John went home. All person identical to John are persons who went home.) 4. Adverbs and Pronouns (Whoever works hard will succeed. All persons who work hard are persons who succeed.) 5. Unexpressed Qualifiers (There are lions in the zoo. Some lions are animals in the zoo.) 6. Nonstandard Qualifier (All newborns are unable to talk. No newborns are persons who are able to talk.) 7. Conditional Statement (If it s a mouse, then it s a mammal. All mice are mammals.) 8. Exclusive Proposition (None but the brave deserve the fair. All persons who deserve the fair are brave persons.) 9. The Only (Only teachers will be hired. All the persons who will be hired are teachers.) 10. Exceptive Proposition (Everyone accept for students are invited. All non-students are persons who are invited, and no students are invited persons.) Adjusting for Syllogisms 1. We begin with an argument. Autistic children are occasionally helped by aversion therapy, but sometimes aversion therapy can be inhumane. I guess inhumane treatment can sometimes help people. 1. Employing some of the translation techniques, we rephrase the statement.

12 Unit Five Categorical Syllogisms Page 12 Some autistic children are sometimes helped by aversion therapy sessions. Some aversion therapy sessions are inhumane. Therefore, some autistic children are sometimes helped by aversion therapy sessions. 2. Replace the terms with symbols. Some A are S Some S are I Some A are I 3. Adjust into standard form Some S are I Some A are S Some A are I (III-3) 4. Proceed as with any other categorical syllogism. Textbook Exercise Enthymemes Key Concept An Enthymeme (en-thi-mēm) is an argument that is expressible as a categorical syllogism, but is missing a premise or a conclusion. Key Terms Enthymeme Outline Conversion Process A. The Missing Conclusion People who have planned well for retirement will be financially secure in their golden years and Maynard has planned well for his retirement. 1. Determine what is missing: Conclusion 2. Add the conclusion so as to be consistent with terms in use. People who have planned well for retirement will be financially secure in their golden years and Maynard has planned well for his retirement. Maynard will be financially secure in their golden years. 3. Rewrite in syllogistic form. People who planned well for retirement are people who will be financially secure in their golden years. All people like Maynard are people who planned well for retirement. Therefore, all people like Maynard are people who will be financially secure in their golden years. 4. Replace the terms with symbols. All R are G All M are R All M are R (AAA-1) 5. No further adjustment need. Proceed as with any other categorical syllogism. B. The Missing Premise Maynard has planned well for his retirement. Maynard will be financially secure in their golden years. 1. Determine what is missing: Premise 2. Add the conclusion so as to be consistent with terms in use. People who have planned well for retirement will be financially secure in their golden years. Maynard has planned well for his retirement.

13 Unit Five Categorical Syllogisms Page 13 Maynard will be financially secure in their golden years. 3. Rewrite in syllogistic form. People who planned well for retirement are people who will be financially secure in their golden years. All people like Maynard are people who planned well for retirement. Therefore, all people like Maynard are people who will be financially secure in their golden years. 4. Replace the terms with symbols. All R are G All M are R All M are R (AAA-1) 5. No further adjustment need. Proceed as with any other categorical syllogism. Textbook Exercise Sorites Key Concept A Sorites is a chain of categorical syllogism in which intermediate conclusions have been left out. There are two processes for testing the validity of a Sorites. Key Terms Sorites Enthymeme Technique Outline Sorites Basics A. The criterion in evaluating sorites is the notion that the chain is only as strong as its weakest link. B. A standard form sorites is one in which each of the component propositions is in standard form. 1. Each term appears twice. 2. The predicate term of the conclusion is in the first premise (Major Premise). 3. Each succeeding premise has a term in common with the preceding premise. Enthymeme Technique A. The Enthymeme Technique supplies the missing parts to the sorites in order test for validity. The syllogism is valid if a conclusion can be logically inferred from one step to the next through to the final conclusion. B. Example All of the divas are constantly seeking attention. No humble souls are constantly seeking attention. All the happy people are humble souls. Some logic student are happy people. Some logic students are not divas. C. Execution 1. Argument is in blue and the intermediate conclusions are in green. 2. In a valid syllogism, each intermediate conclusion can be validly inferred and the final conclusions match. 3. In an invalid conclusion, either each intermediate conclusion cannot be validly inferred or the final conclusions don t match.

14 Unit Five Categorical Syllogisms Page 14 Testing Validity with Rules A. The rules for testing validity can be adapted to test for the validity of sorites. B. Rules Reviewed 1. The middle term must be distributed at least once. (Undistributed Middle) 2. Any term distributed in the conclusion must be distributed in the premises. (Illicit Major, Illicit Minor) 3. From two negative premises, no valid conclusion follows. (Exclusive Fallacy) 4. If both premises are universal, the conclusion can only be valid if the requirements for conditional validity are met. (Existential Fallacy) 5. The conclusion always follows from the weakest part. (Extension Fallacy) C. Execution (Example) All of the divas are constantly seeking attention. No humble souls are constantly seeking attention. No humble souls are divas All the happy people are humble souls. No happy people are divas. Some logic students are happy people. Some logic students are not divas. Textbook Exercise 5.7 I 1-5 Exercise 5.7 II 6-10 Exercise 5.7 III 1-5 Exercise Lessons