Part I. What happened so far? From syllogism to common sense: atourthroughthelogicallandscape. Categorical syllogisms. Categorical propositions (CPs)

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1 What happened so far? From syllogism to common sense: atourthroughthelogicallandscape Categorical syllogisms art 2 art I What happened so far? Mehul Bhatt Oliver Kutz Thomas chneider 17 November 2011 Thomas chneider Categorical syllogisms Thomas chneider Categorical syllogisms What happened so far? What happened so far? Categorical propositions (Cs) What happened so far? Categorical propositions (Cs)... state relations between classes: Eample General form Name All policitians are liars. All is. A Universal affirmation No policitians are liars. No is. E Universal negation ome policitians are liars. ome is. I articular affirmation ome policitians are not liars. ome is not. O articular negation... can be interpreted via Venn diagrammes:... have a quantity, quality and distribution: roposition Quantity Quality Distributes All is. A Universal Affirmative No is. E Universal Negative, ome is. I articular Affirmative ome is not. O articular Negative All is. No is. ome is. ome is not. Distribution: Does the proposition make a statement about all members of or? Thomas chneider Categorical syllogisms What happened so far? Thomas chneider Categorical syllogisms What happened so far?

2 What happened so far? Immediate inferences... via square of opposition: A ubalternation Further inferences: I Contraries Contra dictories Contradictories ubcontraries E ubalternation Conversion (, ; not always successful) Obversion (Change quality, non-) Contraposition ( non-, non-; not always succ.) O What happened so far? Aristotelian versus Boolean interpretation Aristotelian interpretation assumes eistential import: is nonempty A All is. E No is. Boolean interpretation rejects eistential import: in A and E, may be empty A All is. E No is. Thomas chneider Categorical syllogisms What happened so far? Thomas chneider Categorical syllogisms What happened so far? In this part... art II Categorical syllogisms (Cs) 2 tandard-form Cs 3 Venn-diagramme technique for testing Cs 5 The valid Cs 6 ummary and outlook Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 36 Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 37

3 Basic notions Aim: more etended reasoning with Cs 2 tandard-form Cs 3 Venn-diagramme technique for testing Cs 5 The valid Cs 6 ummary and outlook yllogism: Categorical syllogism: deductive argument with 2 premises and 1 conclusion syllogism based on Cs deductive argument of 3 Cs all 3 Cs together contain 3 terms every term occurs in 2 propositions yllogisms are common, clear and easily testable. They are one of the most beautiful and also one of the most important made by the human mind. (GOTTFRIED WILHELM LEIBNIZ, , German philosopher and mathematician, Hannover) Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 38 Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 39 tandard-form Cs Eamples 1 remises and conclusion are standard Cs (A, E, I, O) 2 Cs are arranged in standard order:... 1 is is is... : major term, : minor term Remember: 3 terms altogether, each in 2 propositions! 1, 2, 1, 2 consist of, and a third term: the middle term Major term, minor term, middle term All great scientists are college graduates. ome professional athletes are college graduates. Therefore some professional athletes are great scientists. All artists are egotists. ome artists are paupers. Therefore some paupers are egotists. Major premise contains, M Minor premise contains, M Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 40 Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 41

4 Mood of a C Figure of a C Figure of a C: combination of order of, M, in the premises: Mood of a C is the pattern of types of its three Cs, in the order major premise minor premise conclusions A I I All artists are egotists. ome artists are paupers. Therefore some paupers are egotists. 4 3 = 64 moods Mood AII 4 figures: (1) M M No is M M ome is not M has figure M All is (2) M M (3) M M (4) M M Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 6 9 Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 43 Formal nature of the syllogistic argument There are only 4 64 = 256 possible forms of Cs. Their validity can be ehaustively analysed and established. Only a few will turn out to be valid. Infinitely many (in-)valid syllogistic arguments can be obtained by replacing, M, in a(n in-)valid C with real-world class descriptions. 2 tandard-form Cs 3 Venn-diagramme technique for testing Cs 5 The valid Cs 6 ummary and outlook Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 44 Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 45

5 Testing a form of C for validity Eample... is very simple! 1 Draw three overlapping cycles for,, M: 2 Mark the premises according to their types as earlier. E.g.: AAA-1 All M is. All is M. All is. 3 Try to read off the conclusion without further marking. yllogism type is valid iff reading off was successful. M M What form does this syllogism have? Is it valid? All dogs are mammals. All cats are mammals. Therefore all cats are dogs. Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 46 Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 47 Two cautions Eamples (1) Mark universal before particular premise. aupers All artists are egotists. ome artists are paupers. Therefore some paupers are egotists. Egotists Artists AEE-1 All M is. No is M. No is. Invalid: diagramme does not eclude from. M (2) If a particular premise speaks about two nonempty regions, put the on the boundary of these regions. A G All great scientists are college graduates. ome professional athletes are college graduates. Therefore some professional athletes are great scientists. CG EIO-4 No is M. ome M is. ome is not. Valid: diagramme gives a particular instance of \. M Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 48 Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 49

6 An alternative characterisation of validity of Cs... via rules that focus on the form of the syllogism 2 tandard-form Cs 3 Venn-diagramme technique for testing Cs 5 The valid Cs 6 ummary and outlook Rule 1: Avoid four terms. With 4 terms, it s no syllogism at all Beware of equivocations! (two occurrences of the same word with different meanings) And the Lord spake, saying, First shalt thou take out the Holy in. Then, shalt thou count to three. No more. No less. Three shalt be the number thou shalt count, and the number of the counting shall be three. Four shalt thou not count, neither count thou two, ecepting that thou then proceed to three. Five is right out. Once at the number three, being the third number to be reached, then, lobbest thou thy Holy Hand Grenade of Antioch towards thy foe, who, being naughty in My sight, shall snuff it. (from Monty ython and the Holy Grail, 1975) Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 50 Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 51 Distribute your middle term Watch your distribution Rule 2: Distribute the middle term in at least one premise. (One proposition must refer to all members of M.) Eample: All Russians were revolutionists. All anarchists were revolutionists. Therefore all anarchists were Russians. Fallacy: middle term revolutionists doesn t link, Russians are included in a part of revolutionists Anarchists are included in a part of revolutionists, possibly a different part! Rule 3: Any term distributed in the conclusion must be distributed in the premises. Intuition: if premises speak about some members of a class, we cannot conclude anything about all members of that class. Eample: All dogs are mammals. No cats are dogs. Therefore no cats are mammals. Fallacy: mammals is distributed in the conclusion, but not in the major premise. Fallacy of the undistributed middle Fallacy of illicit process (here: illict process of the major term) Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 52 Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 53

7 Two negative premises are bad Don t turn neg into pos Rule 4: Avoid two negative premises. 2 negative premises 2 class eclusion between, M and between, M Eample: No power to enforce any relation between, Try all nine possibilities in a Venn diagramme! Fallacy of eclusive premises No artists are accountants. ome poets are not accountants. Therefore some poets are not artists. Rule 5: If 1 premise is negative, the conclusion must be neg. Affirmative conclusion ˆ= one of, is (wholly or partly) contained in the other. Can only be inferred if premises assert eistence of M which contains one of, and is contained in the other Eample: Class inclusion only via affirmative propositions No poets are accountants. ome artists are poets. Therefore some artists are accountants. Fallacy of drawing an affirmative conclusion from a neg. premise Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 54 Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 55 Don t be so Aristotelian 2 tandard-form Cs Rule 6: From two universal premises, no particular conclusion may be drawn. Eample: All household pets are domestic animals. No unicorns are domestic animals. Therefore some unicorns are not household pets. 3 Venn-diagramme technique for testing Cs Eistential fallacy (not a fallacy in the Aristotelian interpretation) 5 The valid Cs 6 ummary and outlook Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 56 Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 57

8 The 15 valid forms of syllogisms AAA-1 Barbara AII-3 Datisi EAE-1 Celarent IAI-3 Disamis AII-1 Darii EIO-3 Ferison EIO-1 Ferio OAO-3 Bokardo AEE-2 Camestres AEE-4 Camenes EAE-2 Cesare IAI-4 Dimaris AOO-2 Baroko EIO-4 Fresison EIO-2 Festino 2 tandard-form Cs 3 Venn-diagramme technique for testing Cs 5 The valid Cs 6 ummary and outlook Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 58 Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 59 ummary Literature and outlook Categorical syllogisms... are deductive arguments consisting of 3 Cs require a certain amount of interaction between the terms in their Cs come in 4 figures and 64 moods can be tested for validity using Venn diagrammes or rules/fallacies There are 15 valid forms of syllogisms in Boolean interpretation, 24 in Aristotelian interpretation Contents is taken from Chapters 5, 6 of I. Copi, C. Cohen, K. McMahon: Introduction to Logic, 14th ed., rentice Hall, UUB Magazin 02 E 2115 Link to available copies Chapter 7: transform arguments of everyday speech into syllogistic form, possible difficulties It s almost play time: Try with eamples from ages Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 60 Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 61

9 Literature and outlook Contents is taken from Chapters 5, 6 of I. Copi, C. Cohen, K. McMahon: Introduction to Logic, 14th ed., rentice Hall, UUB Magazin 02 E 2115 Link to available copies Chapter 7: transform arguments of everyday speech into syllogistic form, possible difficulties Thank you. Thomas chneider Categorical syllogisms Categorical syllogisms (Cs) 61

5.1 Standard Form, Mood & Figure

5.1 Standard Form, Mood & Figure 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.