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Venn Diagrams (For those who don’t like to sing) Barbara Celarent Remix • Barbara, Celarent, Darii, Ferio are in the prime. • Baroco, Camenes, and Cesare, Ferison: the second line. • Bocardo, Datisi, and Dimaris, Festino—they’re all in the third. • Camestres, Disamis, and Fresison are fourth, lest you sound absurd. Constructing Venn Diagrams • A note on this PowerPoint presentation – Our text begins, prudently, by describing how to construct Venn diagrams for • A single proposition (one step and one circle) – The text then presents how to diagram an argument (hopefully valid) that draws an immediate inference from • One categorical proposition to a categorical conclusion (two steps and two circles) – This discussion, however, will explain the process involving standard • Three-step syllogisms of three propositions (two premises and a conclusion) and three circles. Constructing Venn Diagrams • 3 intersecting circles representing the 3 classes designated by the 3 terms in the 3 propositions, following these conventions: – Upper left circle = subject term in conclusion – Upper right circle = predicate in conclusion – Lower middle circle = middle term in premises – Circumscribed by a square to indicate the universe of everything else not included in one of the three circles. • Must be drawn so some areas overlap and others don’t. This configuration allows the efficient illustration – of both classes and complements – that an object in any class may or may not belong to another class – of eight possible relationships (three classes taken two at a time, or 23 relationships) Basic practices • What does shading indicate? – Emptiness • What does an ‘x’ indicate? – That some (one or more) element/s exist/s there. • What does a blank area indicate? – There may or may not be anything there. Nothing definitive is known about the area in question. • What is a complement? – Everything within the universe of discourse (the set of all classes being discussed) but outside of a particular class, e.g., not-x Using Venn Diagrams (This is really complex, Brother Murray. Can’t we just sing, instead?) 1. Translate the everyday language into categorical propositions. 2. Arrange the propositions into standard syllogistic form. • If necessary, use the three processes of categorical equivalence to unify non-matching (complementary) terms in the simplest possible configuration. 3. Using shading (for A & E propositions) and ‘x’s (for I & O propositions), record the information given in the premises of the syllogism (as demonstrated in an earlier lesson), indicating which areas are empty and which are not. a.Shade areas indicated by universal propositions before indicating the possible placement of particular propositions. b.If an I or O proposition calls for the indeterminate placement of an ‘x,’ place it on the line separating the areas in question. 4. Inspect the diagram to determine if the conclusion necessarily follows. • If it does, the argument is valid. • If it does not, the argument is invalid. FYI • Different diagrams for the same, logically equivalent syllogism may appear different, depending on how the – Propositions are rendered (as A,E,I,O) – Classes are identified (as complementary or not) – Circles are arranged & labeled (class-circle assignment) • BUT the validity outcome will be the same assuming – The syllogisms are logically equivalent, and – The diagrams have been manipulated properly. • These facts have two implications: – Two diagrams for any one of the 15 valid argument forms will have the same validity outcome, regardless of substance. – The diagram of an invalid argument can suggest substitution instances for purposes of counterexample refutation. Review ① Change the everyday argument into a standard-form syllogism. ② Draw and label the basic Venn diagram (including the universe outside of the three classes) according to convention. ③ Shade areas indicating universal premises before demonstrating particular premises. ④ Place an x in the appropriate area for particular premises. • If there is no compelling reason to place the x in a particular area, place it on the line straddling any areas under consideration. ⑤ Check to see whether the content of the conclusion has been diagrammed as a necessary consequence of diagramming the premises (i.e., you don’t have to diagram the conclusion: it will diagram itself, or not. • • If the conclusion appears necessarily, the syllogism is valid, If it has not, the syllogism is invalid. Now, let’s have some FUN! YEAH, Baby! Venn diagramming for the obsessive compulsive