Three Statement Syllogisms With Clear Explanation

PROPOSITION:A proposition is a sentence that makes a statement and gives a relation between two or more terms.The parts of proposition are given below.

i) Subject : A subject is the part of the proposition about which something is being said.

ii) Predicate : Predicate is the part of the proposition denoting that which is affirmed or denied about the subject.

eg : In the proposition All novels are songs, something is being said about novels. So novels is the subject. Songs is the predicate here because it affirmed about the subject.

Syllogism Rules:

Rule 1: Statement starting will keyword ‘ALL’

Ex. – Statement : All dogs are cats

          Conclusion : Some cats are dogs

Rule 2: Statement starting will keyword ‘SOME’

Ex. – Statement : Some dogs are cats

          Conclusion : Some cats are dogs

Rule 3: Statement starting will keyword ‘NO’

Ex. – Statement : No dog is a cat

         Conclusion : No cat is a dog

Rule 4: Statement starting will keyword ‘SOME NOT’

Ex. – Statement : Some dogs are not cats

Read Below table Carefully:

statement	statement	conclusion
All	All	All
All	No	No
All	Some	No Conclusion
Some	All	Some
Some	Some	No Conclusion
Some	No	Some Not
No	No	No Conclusion
No	All	Some not reversed
No	Some	Some not reversed

Note: You can cancel out common terms in two statements given, then on the remaining terms apply the syllogisms rules and solve...

                                        THREE - STATEMENT SYLLOGISM

This type of syllogism problems consist of 3 statements which are followed by 4 or moreconclusions.A typical three - statement syllogism problem is given below.

Directions : Below are given three statements followed by several conclusions based on them. Examine the conclusions and decide whether they logically follow from the given statements.You have to take the given statements as true even if they appear to be at variance with commonly known facts.

Statements : A) All bags are hats.

                       B) Some pins are bags.

                       C) No hats are needles.

Conclusions : I) Some pins are hats.

                       II) No needles are bags.

                       III) Some pins are needles.

                     IV) Some pins are not needles.

1) Only I and II follow

2) Only I and IV follow

3) I, II and IV follow

4) Either III or IV, and I follow

5) Either III or IV and I and II follow.

Before solving this example, let us see the steps in solving a three-statement syllogism

problems.

Step I:

i) Consider a given conclusion.

ii) Note the subject and predicate of this given conclusion.

iii) Now find which of the two given statements has this subject and predicate.

iv) a) If there is a common term between the two statements chosen in the previous part, then consider only these two statements.

b) If there is no common term between the two statements chosen in the previous part, then we should consider all the three statements.

Step II:

i) If two statements are relevant for a given conclusion, align them.

ii) If three statements are relevant, write them as a chain. That is, align them in such a way that the predicate of the first sentence and subject of the second are the same, and the predicate of the second sentence and the subject of the third sentence are the same.

iii) Now arrive at the conclusion using the table.

iv) Now compare the given conclusion with the conclusion drawn using the tables. If they match, the given conclusion is true. If they do not match, it is false.

Step III:

i) If a given statement has already been marked as a valid conclusion after step II, then leave it. Otherwise check if it is an immediate inference of any of the three given statements of the conclusion derived.

ii) Search for complementary pair :

a) Check if any two given conclusions have the same subject and the same predicate.

b) If (a) is satisfied, then check whether any of them has been marked as a valid conclusion after step II or as an immediate inference.

c) If none of them has been marked as a valid conclusion, then they will form a

complementary pair if they are an All - Some Not or Some – Some Not or Some - No.

d) If they do make a complementary pair, then mark the choice "either of the two follows".If a conclusion is marked as a valid conclusion after step II, then it is not necessary to perform

step III (i). Again if a given conclusion has already been accepted in step III (i), then it is not necessary to perform step III (ii).

The learner should understand these steps clearly. Now follow the solution to the example which is already given. Here we have to check the validity of each and every conclusions one by one.

Conclusion I : Here the subject is pin and the predicate is hat. So let us consider (A) and (B) as our relevant statements because they have a common term 'bags'.The second step is to align the sentences.

The aligned pair is,

Some pins are bags.

All bags are hats.

Some + All = I. So we arrive at the conclusion,

'Some pins are hats'. So conclusion I is valid.

Conclusion II : Here the subject is 'needles' and the predicate is 'bags'. Statement C contains the subject 'Needles'. But 'bags' appears in both A and B. We should select A because there is a common term between A and C. This is an aligned pair and so we arrive at the conclusion No bags are needles which implies No needles are bags. Hence conclusion II is valid.

Conclusion III : Here the subject is 'pins' and the 'predicate' is needles. These words appear in statements (B) and (C) respectively which have no term in common. So all the three statements should be taken as relevant. Now align the statements as Step II (ii) So we get,

Some pins are bags

All bags are hats.

No hats are needles.

Some + All + No = (Some + All) + No= Some + NO = Some Not.

So the conclusion is 'Some pins are not needles', which is conclusion IV. So conclusion IV is valid.Since conclusion III is not valid in step II, let us perform step III (i). The conclusion, Some pins are not needles is not an immediate inference of any of the three given statements. So the next step is to check the existence of a complementary pair in the given conclusions.We see that conclusion III and conclusion IV form a omplementary pair of Some- Somenot type.So the choice "either III or IV follows" could be selected. But we find that conclusion IV is valid from the previous step. So conclusion III is not valid. Hence for this given example, the third choice which is 'I, II and IV follow' is true…

Some Examples With Explanation:Example-1:

Statements: a. All lions are ducks.

                          b. No duck is a horse.

                          c. All horses are fruits.

Conclusions: I. No lion is a horse.

                           II. Some fruits are horses.

                           III. Some ducks are lions.

                            IV. Some lions are horses.

1) All follows

2) Only either I or II and both III and IV follow

3) Only either I or IV and both II and III follow

4) Only either I or IV and II follow

5) None of theseAns 5; 

Only I, II and III follow. Statement (a) +Statement (b) gives conclusion I. [ All+No=No]. Hence, conclusion I follows but conclusion IV does not follow. Conclusion II follows from conversion of statement (c).Similarly, conclusion III follows from conversion of statement (a).

Example-2:

Statements: Some goats are hammers.

                         All hammers are diamonds.

                          No diamond is green.

Conclusions: I. No goat is green.

                           II. Some diamonds are hammers.

                           III. Some goats are diamonds.

                            IV. Some greens are hammers.

1) Only I and IV follow

2) Only II and IV follow

3) Only II and III follow

4) Only either II or III and I follow

5) None of these

Ans:3;

1st statement + 2nd statement gives: Some goats are diamonds. Hence III follows.III + last statement gives: Somegoats are not green. Hence I does not follow. Conversion of second statement gives II. IV does not follow because of the last two statements.

Example-3:

Statements: Some flowers are rods.

                          Some rods are doors.

                          Some doors are houses.

Conclusions:I. Some houses are flowers.

                          II. Some doors are flowers.

                          III. Some flowers are doors.

                          IV. No house is flower.

1) Only I and IV follow

2) Only II and III follow

3) Only either I or II follows

4) Only either I or IV follows

5) None of these

Ans:4;

As all the statements are I-type, hence no conclusion follows from their combinations.

But I and IV make a complementary pair, hence either I or IV follows.