From syllogism to common sense: a tour through the logical landscape. Categorical syllogisms
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1 From syllogism to common sense: a tour through the logical landscape Categorical syllogisms Part 2 Mehul Bhatt Oliver Kutz Thomas Schneider 17 November 2011 Thomas Schneider Categorical syllogisms
2 What happened so far? Part I What happened so far? Thomas Schneider Categorical syllogisms What happened so far?
3 What happened so far? Categorical propositions (CPs)... state relations between classes: Example General form Name All policitians are liars. All S is P. A Universal affirmation No policitians are liars. No S is P. E Universal negation Some policitians are liars. Some S is P. I Particular affirmation Some policitians are not liars. Some S is not P. O Particular negation... can be interpreted via Venn diagrammes: S All S is P. No S is P. Some S is P. Some S is not P. P S P S P S P x x x x Thomas Schneider Categorical syllogisms What happened so far?
4 What happened so far? Categorical propositions (CPs)... have a quantity, quality and distribution: Proposition Quantity Quality Distributes All S is P. A Universal Affirmative S No S is P. E Universal Negative S, P Some S is P. I Particular Affirmative Some S is not P. O Particular Negative P Distribution: Does the proposition make a statement about all members of S or P? Thomas Schneider Categorical syllogisms What happened so far?
5 What happened so far? Immediate inferences... via square of opposition: A Contraries E Subalternation Contra dictories Contradictories Subalternation I Subcontraries O Further inferences: Conversion (S P, P S; not always successful) Obversion (Change quality, P non-p) Contraposition (S non-p, P non-s; not always succ.) Thomas Schneider Categorical syllogisms What happened so far?
6 What happened so far? Aristotelian versus Boolean interpretation Aristotelian interpretation assumes existential import: S is nonempty S A All S is P. E No S is P. x P S x P Boolean interpretation rejects existential import: in A and E, S may be empty A All S is P. E No S is P. S P S P Thomas Schneider Categorical syllogisms What happened so far?
7 Part II Categorical syllogisms (CSs) Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 36
8 In this part... 2 Standard-form CSs 3 Venn-diagramme technique for testing CSs 4 Rules and fallacies 5 The valid CSs 6 Summary and outlook Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 37
9 And now... 2 Standard-form CSs 3 Venn-diagramme technique for testing CSs 4 Rules and fallacies 5 The valid CSs 6 Summary and outlook Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 38
10 Basic notions Aim: more extended reasoning with CPs Syllogism: deductive argument with 2 premises and 1 conclusion Categorical syllogism: syllogism based on CPs deductive argument of 3 CPs all 3 CPs together contain 3 terms every term occurs in 2 propositions Syllogisms 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 Schneider Categorical syllogisms Categorical syllogisms (CSs) 39
11 Standard-form CSs 1 Premises and conclusion are standard CPs (A, E, I, O) 2 CPs are arranged in standard order:... S 1 is... P 1... S 2 is... P 2... S is... P P: major term, S: minor term Remember: 3 terms altogether, each in 2 propositions! S 1, S 2, P 1, P 2 consist of P, S and a third term: the middle term Major premise contains P, M Minor premise contains S, M Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 40
12 Examples Major term, minor term, middle term All great scientists are college graduates. Some professional athletes are college graduates. Therefore some professional athletes are great scientists. All artists are egotists. Some artists are paupers. Therefore some paupers are egotists. Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 41
13 Mood of a CS Mood of a CS is the pattern of types of its three CPs, in the order major premise minor premise conclusions A I I All artists are egotists. Some artists are paupers. Therefore some paupers are egotists. 4 3 = 64 moods Mood AII Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 6 9
14 Figure of a CS Figure of a CS: combination of order of S, M, P in the premises: No P is M P M Some S is not M has figure S M All S is P S P 4 figures: (1) M P S M S P (2) P M S M S P (3) M P M S S P (4) P M M S S P Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 43
15 Formal nature of the syllogistic argument There are only 4 64 = 256 possible forms of CSs. Their validity can be exhaustively analysed and established. Only a few will turn out to be valid. Infinitely many (in-)valid syllogistic arguments can be obtained by replacing S, M, P in a(n in-)valid CS with real-world class descriptions. Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 44
16 And now... 2 Standard-form CSs 3 Venn-diagramme technique for testing CSs 4 Rules and fallacies 5 The valid CSs 6 Summary and outlook Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 45
17 Testing a form of CS for validity... is very simple! S P 1 Draw three overlapping cycles for S, P, M: 2 Mark the premises according to their types as earlier. M E.g.: AAA-1 All M is P. All S is M. All S is P. S P M 3 Try to read off the conclusion without further marking. Syllogism type is valid iff reading off was successful. Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 46
18 Example What form does this syllogism have? Is it valid? All dogs are mammals. All cats are mammals. Therefore all cats are dogs. Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 47
19 Two cautions (1) Mark universal before particular premise. Paupers All artists are egotists. Some artists are paupers. Therefore some paupers are egotists. x Egotists Artists (2) If a particular premise speaks about two nonempty regions, put the x on the boundary of these regions. PA GS All great scientists are college graduates. Some professional athletes are college graduates. Therefore some professional athletes are great scientists. x CG Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 48
20 Examples AEE-1 All M is P. No S is M. No S is P. S P Invalid: diagramme does not exclude S from P. M EIO-4 No P is M. Some M is S. Some S is not P. S x P Valid: diagramme gives a particular instance of S \ P. M Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 49
21 And now... 2 Standard-form CSs 3 Venn-diagramme technique for testing CSs 4 Rules and fallacies 5 The valid CSs 6 Summary and outlook Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 50
22 An alternative characterisation of validity of CSs... via rules that focus on the form of the syllogism 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 Pin. 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, excepting 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 Python and the Holy Grail, 1975) Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 51
23 Distribute your middle term Rule 2: Distribute the middle term in at least one premise. (One proposition must refer to all members of M.) Example: All Russians were revolutionists. All anarchists were revolutionists. Therefore all anarchists were Russians. Fallacy: middle term revolutionists doesn t link S, P Russians are included in a part of revolutionists Anarchists are included in a part of revolutionists, possibly a different part! Fallacy of the undistributed middle Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 52
24 Watch your distribution 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. Example: 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 illicit process (here: illict process of the major term) Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 53
25 Two negative premises are bad Rule 4: Avoid two negative premises. 2 negative premises 2 class exclusion between S, M and between P, M No power to enforce any relation between S, P Try all nine possibilities in a Venn diagramme! Example: No artists are accountants. Some poets are not accountants. Therefore some poets are not artists. Fallacy of exclusive premises Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 54
26 Don t turn neg into pos Rule 5: If 1 premise is negative, the conclusion must be neg. Affirmative conclusion ˆ= one of S, P is (wholly or partly) contained in the other. Can only be inferred if premises assert existence of M which contains one of S, P and is contained in the other Example: Class inclusion only via affirmative propositions No poets are accountants. Some artists are poets. Therefore some artists are accountants. Fallacy of drawing an affirmative conclusion from a neg. premise Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 55
27 Don t be so Aristotelian Rule 6: From two universal premises, no particular conclusion may be drawn. Example: All household pets are domestic animals. No unicorns are domestic animals. Therefore some unicorns are not household pets. Existential fallacy (not a fallacy in the Aristotelian interpretation) Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 56
28 And now... 2 Standard-form CSs 3 Venn-diagramme technique for testing CSs 4 Rules and fallacies 5 The valid CSs 6 Summary and outlook Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 57
29 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 Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 58
30 And now... 2 Standard-form CSs 3 Venn-diagramme technique for testing CSs 4 Rules and fallacies 5 The valid CSs 6 Summary and outlook Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 59
31 Summary Categorical syllogisms... are deductive arguments consisting of 3 CPs require a certain amount of interaction between the terms in their CPs 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 It s almost play time: Try with examples from Pages Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 60
32 Literature and outlook Contents is taken from Chapters 5, 6 of I. Copi, C. Cohen, K. McMahon: Introduction to Logic, 14th ed., Prentice Hall, SUUB Magazin 02 E 2115 Link to available copies Chapter 7: transform arguments of everyday speech into syllogistic form, possible difficulties Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 61
33 Literature and outlook Contents is taken from Chapters 5, 6 of I. Copi, C. Cohen, K. McMahon: Introduction to Logic, 14th ed., Prentice Hall, SUUB Magazin 02 E 2115 Link to available copies Chapter 7: transform arguments of everyday speech into syllogistic form, possible difficulties Thank you. Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 61
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