5 - Categorical syllogism.ppt

1、1Syllogism A syllogism is a two-premise deductive argument.Or,a categorical syllogism is an argument in which both the premises and the conclusion are categorical propositions.2 There are 3 propositions.There are 3 terms.Each term is used twice.None of the term is used twice in the same proposition.

2、3 All soldiers are patriots.No traitors are patriots.Therefore,no traitors are soldiers.4Major term Major term is the predicate of the conclusion.Soldiers5Minor term Minor term is the subject of the conclusion.Traitors6Middle term Middle term is the link between the two premises.It does not occur in

3、 the conclusion.Patriots7Major and minor premises The major premise is the premise that contains the major term:“All soldiers are patriots”.The minor premise is the premise that contains the minor term:“No traitors are patriots”.8Standard form The major premise is listed first,the minor premise seco

4、nd,and the conclusion last.Major premise Minor premise Conclusion9Mood The mood is the order of the type of propositions in a categorical syllogism.In the previous example,the mood is AEE.A is the type of the major premise,E for minor premise,the final E for the conclusion.10Figure The figure of a c

5、ategorical proposition is determined by the location of the two occurrences of the middle term in the premises.Therefore,there are 4 possible figures.11Figure 1Figure 2Figure 3Figure 4M PS MS PP MS MS PM PM SS PP MM SS P12Positions of the 4 Sets of Middle Terms Remember the W-shaped figure13 Since t

6、here are four kinds of categorical propositions and three categorical propositions in each syllogism,we can have 64(4x4x4)possible moods.And since we have 4 possible figures,there are a total of 256 different forms(mood-figure)of categorical syllogism.14 If you can write out the form of a particular

7、 categorical syllogism,then you can check whether it is unconditionally valid or not.If we further require certain terms in the syllogism to refer to actually existing things,then we are concerned with conditional validity.15Unconditionally ValidFigure 1Figure 2Figure 3Figure 4AAAEAEIAIAEEEAEAEEAIII

8、AIAIIEIOOAOEIOEIOAOOEIO16 Dont worry that you have to remember all the forms.You are simply required to understand the underlying principle.Just by knowing the mood-figure of a syllogism,we can check whether the syllogism is unconditionally valid or not.People like Aristotle have checked all 256 for

9、ms and recorded the results.So,in a way,you can refer to their results as if you are checking a periodic table.17 For beginners like you,you have to know how to determine the validity of a syllogism without the help of such reference.You can use Venn diagrams to judge case by case.18 This is an exam

10、ple of an unconditionally valid form:No painters are sculptors.Some sculptors are artists.Therefore,some artists are not painters.19Identify the major termNo painters are sculptors.Some sculptors are artists.Therefore,some artists are not painters.20Identify the minor termNo painters are sculptors.S

11、ome sculptors are artists.Therefore,some artists are not painters.21Identify middle term and figureNo painters are sculptors.Some sculptors are artists.Therefore,some artists are not painters.Make sure that the first premise contains the major term.By locating the middle term,we know the figure is 4

12、.22Determine the moodNo painters are sculptors.(E)Some sculptors are artists.(I)Therefore,some artists are not painters.(O)23 Since we have got the figure and the mood of the syllogism,we can label the syllogism as EIO-4.By crosschecking the result provided,we immediately see that the syllogism is u

13、nconditionally valid.24Venn Diagrams Again Venn diagrams are useful for testing the validity of categorical syllogisms.As mentioned,we are only dealing with three terms and that each proposition contains two terms,we are only required to draw 3 circles.In practice,we never draw more than 3 circles e

14、ven for problems that involve more than three terms(see the later section on sorites).25 Each specific area of the three circles gives us certain information of the membership relation between the three types of things mentioned.By using the learnt techniques to label each area one by one,we can see

15、 whether the syllogism is valid or not.26Procedures Draw three circles:the first two is like the ones we have drawn before,with the right-hand side one labeling the major term,the left-hand side one labeling the minor term.The third circle is put on top of the bottom two,labeling the middle term.27P

16、lacing of the circles Middle term Minor term Major term28 Shading or placing an X are done in the same way as before,with certain minor details to note.We draw two separate Venn diagrams for the premises and the conclusion.And see whether the diagram of the premises covers the diagram of the conclus

17、ion.If this is so,the argument is valid.Otherwise,it is invalid.29 Diagram the universal premise first if one premise is universal and the other particular.If both premises are universal or both are particular,it does not matter which one you draw first.When entering the information,always concentra

18、te on the two circles corresponding to the two terms in the proposition.30 Shading can be done in all possible ways,but make sure that the entire area is shaded.If the same area is to be shaded twice by two propositions,the two shadings should be done in lines roughly perpendicular to each other.31

19、If possible,X goes to the unshaded part(The shaded part is the area in which no objects can occur).If both parts are not shaded,X is put on the line separating the two parts.That is why if shading is done before putting a cross,unnecessary shifting of the cross is avoided.32 X should never dangle ou

20、tside the diagram.X should never be placed on the intersection of two lines.33Trial No P are M.All S are M.No S are P.34 All M are P.No S are M.No S are P.35 All M are P.Some S are not M.Some S are not P.36 Some M are not P.Some S are M.Some S are not P.37Check for Traditional standpoint One further

21、 step is required in case the syllogism is invalid in the Boolean standpoint.We need to ask whether the type of thing mentioned by certain terms in the syllogism actually exists.This should be done only when the syllogism is judged unconditionally invalid.If an originally unconditionally invalid syl

22、logism is judged valid after the consideration the Traditional standpoint,then we label the syllogism as conditionally valid.38Conditionally validFigure 1Figure 2Figure 3Figure 4AAIAEOAAIAEOEAOEAOEAOEAOAAI39ExamplesNo fighter pilots are tank commanders.All fighter pilots are courageous individuals.T

23、herefore,some courageous individuals are not tank commanders.No F are T.All F are C.Some C are not T.40All reptiles are scaly animals.All currently living tyrannosaurs are reptiles.Therefore,some currently living tyrannosaurs are scaly animals.All R are S.All C are R.Some C are S.41 From the two exa

24、mples,we see that practically speaking,we need to consider the Traditional standpoint only if 3 of the 4 areas of a circle has been shaded(each circle is divided into 4 areas).Also,we need to consider the Traditional standpoint only if the 2 premises are universal and the conclusion is particular.42

25、Reducing the number of terms We can only deal with three terms in a syllogism.However,some arguments are syllogistic and yet seem to contain more than three terms.The reason is that some of these terms form a pair,such as clever and non-clever(stupid).43Reducing the number of terms We can thus perfo

26、rm the three transformation procedures(conversion,contraposition,and obversion)to reduce the number of terms to exactly three.It does not matter which one of a pair is to be retained.Remember that obversion applies to all propositions,contraposition to A and O types,and conversion to E and I types.4

27、4ExamplesAll P are non-W.(obversion)Some E are W.Some non-P are not non-E.(contraposition)No P are W.Some E are W.Some E are not P.45Examples Some intelligible statements are true statements,because all unintelligible statements are meaningless statements and some false statements are meaningful sta

28、tements.46Examples All aircrafts that disintegrate in flight are unsafe planes.Therefore,no poorly maintained aircraft are safe planes,because all well-maintained aircraft are aircraft that remain intact in flight.47Enthymemes It is an argument that is expressible as a categorical syllogism but that

29、 is missing a premise or a conclusion.It is common in daily conversation.We need to supply the implicit proposition.48Example Venus completes its orbit in less time than the Earth,because Venus is closer to the sun.49 Any planet closer to the sun completes its orbit in less time than the Earth.50All

30、 planets closer to the sun are planets that complete their orbit in less time than the Earth.All planets identical to Venus are planets closer to the sun.All planets identical to Venus are planets that complete their orbit in less time than the Earth.51Can you provide the missing propositions that c

31、an turn the following into two syllogisms?I will not be the patient of those doctors who smoke.If they care so little about their own health,they would not care about mine.Hint:One of the missing premise is“All doctors who smoke are doctors who do not care about their own health.”52Sorites A sorites

32、 is a chain of categorical syllogisms in which the intermediate conclusions have been left out.In fact,we can see it as composing of two or more sets of categorical syllogisms.Consequently,in order to see whether a sorities is valid by means of Venn diagrams,we will draw a series of three-circles Ve

33、nn diagrams.53Evaluating sorites Put it in standard form as follows:1.Put the premise contains either the subject or the predicate of the conclusion as the first premise.2.Arrange the other premises so that each premise shares a single term with its consecutive premise.3.Each term must occur twice.R

34、educe the number of terms when necessary.54 Check the validity:1.Using Venn Diagrams,deduce the sub-conclusion from the first and second premises.2.Then treat the sub-conclusion as a premise and deduce another sub-conclusion with the third premise.3.Perform the same step until we have the final prem

35、ise,the last sub-conclusion and the conclusion of the whole argument.In drawing the Venn diagram this time,we test for the validity of the whole argument.55ExampleNo B are C.Some E are A.All A are B.All D are C.Some E are not D.56Rearranged into standard form1.All D are C.2.No B are C.No B are D3.All A are B.No A are D4.Some E are A.Some E are not D.57 Unless the examiner asks you to draw the accompanying Venn diagrams for each step,you are only required to present the standard form as a solution to the sorites problem.