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Branden Fitelson Philosophy 57 Lecture ' 1 Branden Fitelson $ ' 2 Philosophy 57 Lecture $ Chapter 5: Categorical Syllogisms & Venn Diagrams I • Four Examples (see next two slides for final diagrams & answers): Philosophy 57 — Day 17 (1) If a person is a republican, then that person is not a democrat. Therefore, all big spenders are not Republicans, since all big spenders are democrats. • Mid-Term returned today (briefly discuss) (2) Some latchkey children are not kids who can stay out of trouble, for some – 87–100 (A); 80–86 (B); 74–79 (C); 60–73 (D); < 60 (F) youngsters prone to boredom are latchkey children, and no kids who can stay out of trouble are youngsters prone to boredom. • WebVenn Demonstration (A very useful online tool!) (3) Some individuals prone to violence are are not men who treat others • Back to Chapter 5 humanely. Some police officers are individuals prone to violence. Therefore, – I will be skipping sections 5.3, 5.6, and 5.7 some police officers are not men who treat others humanely. – Today: sections 5.2–5.3 (4) No M are P. All M are S . Therefore, Some S are not P. – Time Permitting: section 5.4 (reducing # of terms) & SJSU Philosophy Branden Fitelson ' % & Chapter 5 Philosophy 57 Lecture 04/03/04 3 & SJSU Philosophy ∗ ∗ ∗ ∗ Step 1: Step 2: Step 3: Step 4: SJSU Philosophy Branden Fitelson ' $ Translate from English passage into three categorical claims. Symbolize, and place categorical syllogism in standard form. Draw Venn Diagram for premises of Categorical Syllogism. Is conclusion information contained in the premise diagram? % Chapter 5 Philosophy 57 Lecture 04/03/04 4 % & Chapter 5 04/03/04 SJSU Philosophy $ % Chapter 5 04/03/04 Branden Fitelson ' 5 Philosophy 57 Lecture Branden Fitelson $ ' 6 Philosophy 57 Lecture $ Chapter 5: Categorical Syllogisms & Venn Diagrams II • Four More Examples (see next two slides for diagrams & answers): (5) A few communications satellites are rocket-launched failures. The only communications satellites are devices with antennas. Therefore, there are rocket-launched failures which are devices with antennas. (6) No C are O. Some D are not O. Therefore, Some D are not C. (7) Not all snowflakes are uniform solids. Only six-pointed crystals are snowflakes. Therefore, some six-pointed crystals are not uniform solids. (8) No P are M. Some M are not S . Therefore, Some S are not P. & SJSU Philosophy Branden Fitelson ' % & Chapter 5 Philosophy 57 Lecture 04/03/04 7 % SJSU Philosophy Branden Fitelson ' $ Chapter 5 04/03/04 8 Philosophy 57 Lecture $ Chapter 5: Categorical Syllogisms & Venn Diagrams III • 3-Circle Venn Diagram Rules and Tips 1. Marks (shading, or placing an “X”) are entered only for the premises. No marks are made for the conclusion. 2. If the argument contains one universal premise, then this premise should be entered first in the diagram. If there are two universal premises, either one can be done first. 3. When entering the information contained in a premise, one should concentrate on the circles corresponding to the two terms in the statement. While the third circle cannot be ignored altogether, it should be given only minimal attention. 4. When inspecting a completed diagram to see whether it supports a particular conclusion, one should remember that particular statements assert two things: “Some S are P” means “At least one S exists and that S is a P. 5. When shading a region, one must be careful to shade all of the area in question. 6. The region in which an “X” goes is initially always divided up into two parts. If one of these parts has been shaded, then the “X” goes in the other part of the region. & SJSU Philosophy % & Chapter 5 04/03/04 SJSU Philosophy % Chapter 5 04/03/04 Branden Fitelson 9 Philosophy 57 Lecture ' Branden Fitelson $ ' 7. If neither of the two parts in a region is shaded, then the ‘X’ goes on the line separating the two parts of the region. Right 10 Philosophy 57 Lecture $ Chapter 5: Section 5.4 — Reducing the # of Terms • Consider the following English syllogism: Wrong All photographers are non-writers. Some editors are writers. Therefore, some non-photographers are not non-editors. • In symbolic form, this can be written as follows: All P are non-W. Some E are W. Therefore, some non-P are not non-E. • This syllogism is not in standard form, since it has 6 terms, but it can be fixed. • We need to transform the three statements in such a way that: (1) we do not change their meaning, and (2) we end-up with 3 terms in the syllogism. • In fact, we can transform this syllogism into standard form in such a way that we use only the three terms P, W, and E. But, how? Recall one of our tables. % & & SJSU Philosophy 04/03/04 Chapter 5 Branden Fitelson 11 Philosophy 57 Lecture ' Chapter 5: Section 5.4 — Reducing the # of Terms, Cont’d Proposition Converse Obverse Contrapositive (A) All A are B. All B are A. (,) No A are non-B. (=) All non-B are non-A. (=) (E) No A are B. No B are A. (=) All A are non-B. (=) No non-B are non-A. (,) (I) Some A are B. Some B are A. (=) Some A are not non-B. (=) Some non-B are non-A. (,) (O) Some A are not B. Some B are not A. (,) Some A are non-B. (=) Some non-B are not non-A. (=) • And, recall the syllogism we are trying to convert into standard form: All P are non-W. Some E are W. ∴ Some non-P are not non-E. • We need to transform the syllogism in such a way that: (1) we preserve the meaning of each claim, and (2) the syllogism only contains 3 terms. • We also need the result to be in standard form (may require re-arranging premises). It’s OK to have complements (“non-X”s) in the resulting syllogism. • There may be multiple correct transformations! Here, there are at least two. & SJSU Philosophy Chapter 5 04/03/04 SJSU Philosophy Branden Fitelson ' $ Chapter 5 % 04/03/04 12 Philosophy 57 Lecture $ Chapter 5: Section 5.4 — Reducing the # of Terms, Cont’d • Here are two correct transformations into standard form: All P are non-W. No P are W. 7→ Some E are W. Some E are W. ∴ Some non-P are not non-E. ∴ Some E are not P. or Some W are not non-E. 7→ All W are non-P. ∴ Some non-P are not non-E. • In the 2nd case, we had to re-arrange the premises in order to get the syllogism into standard form. It’s OK that we have “non-P” and “non-E” in the result. • Let’s do Venn Diagrams to check and see if these transformations are valid categorical syllogisms. They had better yield the same result! • NOTE: No special technique is involved for doing 3-circle Venn Diagrams for syllogisms which contain complements (“non-X”s). % & SJSU Philosophy Chapter 5 04/03/04 % Branden Fitelson ' Philosophy 57 Lecture 13 & SJSU Philosophy $ % Chapter 5 04/03/04