Reasoning SYLLOGISM RULES FOR DERIVING CONCLUSIONS 1. The Conclusion does not contain the Middle Term (M). Premises : All spoons are plates. Some spoons are cups. Invalid Conclusion : All spoons are cups. 2. If both Premises are negative, no Conclusion follows Premises : No black is white. No white is green. Invalid Conclusion : No black is green. 3. If both the Premises are Particular, no Conclusion follows. Some sticks are elephants. Some elephants are lions. Invalid Conclusion : Some sticks are lions. Ex. 2. Premises : Some books are photos. Some photos are not pens. Invalid Conclusion : Some books are not pens, 4. From a Particular Major Premise and a negative Minor Premise, no Conclusion can be drawn. Premises : Some drums are boxes. Some cups are not drums. Invalid Conclusion : Some cups are not boxes. 5. In a Categorical argument, there must be a Middle Term. If there is no Middle Term, no Conclusion can be derived. Premises : All ships are girls. All trees are men. The given Premises have no Middle Term, hence" no Conclusion follows, 6. The Middle Term must be distributed at least once in Premises. Otherwise, no Conclusion can be derived. Premises : All ships are rivers. Some rivers are ponds. Conclusion : Some ships are ponds. Explanation : Here, the. Middle Term "rivers" is not' distributed in any of the Premises and therefore, the Conclusion is invalid. For the Middle Term to be distributed in a Premise (a) The Middle Term (M) must be the Subject if the Premise is an A-Proposition. (b) The Middle Term (M) must be the Subject or Predicate if Premise is an E-Proposition. (c) The Middle term (M) must be the Predicate if Premise is an O-Proposition. Ex. Premises : All pens are inks. Some inks are black. Conclusions : I. All inks are pens. II. Some pens are black. In the Premises, the Middle Term is 'ink'. Clearly, it is not distributed in the first Premise, which is an A-Proposition as it does not form its Subject. Again, it is not distributed in the second Premise, which is an I-Proposition. Since the Middle Term is not distributed even once in the Premises, so no Conclusion follows. 7. If Major Term is distributed in the Inference it must be distributed in the Major Premise. Otherwise, the Conclusion cannot be valid. Premises : All books are kites No book is a desk. Conclusion : No desk is kite Explanation : The Inference is an E-Proposition, so both the terms are distributed. Major Premise is A- Proposition, so only the Subject (Books) is distributed. Obviously, the Major Term is distributed in the Inference but not in the Premise. So, the Inference is invalid. 8. If the Minor Term is distributed in the Inference it must be distributed in the Minor Premise. Otherwise, the Inference cannot be valid. Premises : No flower is fruit. All fruits are seeds Conclusion : No seed is flower. Explanation : Here, the term "seed" which is Minor Term is distributed in the Inference but not in the Minor Premise, so the Inference is invalid. Valid Conclusion : Some seeds are not flowers. 9. If one Premise is negative, the Conclusion must be negative. Some books are photos. No photos is coloured. Valid Conclusion : Some book" ^re not coloured. Ex. 2. Premises : All birds are black. No black is white Valid Conclusion : No bird is white. 10. If one Premise is Particular, the conclusion must be Particular Some cups are plates. All plates are spoons. Valid Conclusion : Some cups are spoons. Ex. 2. Premises : Some chairs are tables. No table is desk. Valid Conclusion : Some chairs are not desks.
11. If both the Premises are Affirmative, the Conclusion be Affirmative. All boxes are chairs. All chairs are tables. Valid Conclusion : All boxes are tables Ex. 2. Premises : Some dogs are cats. All cats are lions. Valid Conclusion : Some dogs are lions. 12. If Major Premise is Affirmative the Conclusion must be Particular. Premises : Some dogs are cats All cats are lions. Valid Conclusion : Some dogs are lions. 13. If the Conclusion is Universal, both the Premises must be Universal. All boys are players. All players are tall. Valid Conclusion : All boys are tall. Ex. 2. Premises : All boys are players. No player is tall. Valid Conclusion : No boy is tall. 14. No Conclusion follows if both the Premises are Particular. Ex. Premises : Some horses are lions. Some lions are cats. Invalid Conclusion : I. Some horses are cats. 15. No Conclusion follows if both the Premises are Negative. Ex. Premises : No book is journal. No journal is magazine. Invalid Conclusion : No book is magazine. 16. No Conclusion follows if the Major Premise is Particular and the Minor Premise is Negative. Ex. Premises : Some girls are dancers. No singer is girl. Conclusions : I. No dancer is singer. II. Some dancers are singers. Here, the first Premise containing the Middle Term 'girls' as the Subject is the Major Premise and the second Premise containing the Middle Term 'girls' as the Predicate is the Minor Premise. Since, the Major Premise is Particular and the Minor Premise is Negative, so no Conclusion follows. 17. If the Middle Term is distributed twice, the Conclusion cannot be universal. Ex. Premises : All coolers are fans. No heater is cooler. Conclusions : I. No heater is fan. II. Some heaters are fans. Here, the first Premise is an A-Proposition and so the Middle term 'coolers' forming the Subject is distributed. The second Premise is an E-Proposition and so the Middle Term 'Cooler' forming the Predicate is distributed. Since, the Middle Term is distributed twice, so the Conclusion cannot be Universal. ANALYTICAL METHOD FOR DERIVING CONCLUSIONS Analytical Method for deriving valid Conclusion! (rather say determining the validity of a Categorical Argument) consists of following simple steps : STEP I : ALIGNMENT OF THE PREMISES : It is well known that if there is no Middle Term, no Conclusion follows. Therefore, in the given Premises; there should be a common term. By alignment ofthq Premises we mean that the two Premises should bi written in such a way that the common term (M)i the Predicate of the first Premise and it is the Subjea of the second Premise. If the given Premises are not already aligned, can align them by : (i) changing the order of the Premises and/or (ii) converting one of the Premises. Now consider the following examples : (1) No photo is coloured. (2) Some books are photos. Explanation : We can align these Premises by changing the order of the Premises. Thus, (1) Some books are photos (2) No photo is coloured. Ex. 2. Premises : (1) All plates are spoons. (2) Some cups are plates. Explanation : We can align the Premises by chai ing the order of the Premises. Thus, (11 Some cups are plates. (2) All plates are spoons. Ex. 3. Premises : (1) All books are pens (2) All tables are pens. Explanation : We can align the Premises by verting the second Premise. Thus, (1) All books are pens (2) Some pens are tables. Ex. 4. Premises : (1) Some girls are beautiful, (2) Some Indians are beautiful. Explanation : We can align the Premises by verting the second Premise. Thus, Ml Some ffirls are beautiful (2) Some beautiful are Indians.
We can also align the Premises by converting the first Premise and changing the order of the Premises. Thus. (1) Some Indians are beautiful fi) some oeautitui are gins. Conversion becomes necessary when the Middle Term (M) is either Subject in both the Premises or the Predicate in both the Premises. In such a case we have to convert one of the Premises. Here, obvious question arises: Which of the two Premises be chosen for conversion? The answer to this question lies in the following rule of the Conversion : ORDER OF CONVERSION If in the given pair of Premises, one is I-Proposition and another E-Proposition, then the I-Proposition should be converted. STEP II : INFERENCE CHART After the alignment, refer to the following chart to draw valid Conclusion : ' Note : (i) We see that we get valid Conclusions only from six combinations. (ii) The chart gives correct results if and only if the two given Premises have been properly aligned. FORMAT OF THE CONCLUSION While deriving Conclusion by referring to the above chart, it is important to note the format of the Conclusion. I. The Subject of the Conclusion should be the Subject of the first Premise and the Predicate of the Conclusion should be the Predicate of the second Premise (after alignment). F'or example : Premises : Conclusion : No girl is singer. Explanation : The given Premises are already aligned. From the chart : E-type of Conclusion. Thus, according to the format of the conclusion. No girl is singer. II. In the case of Oj-type Conclusion, the Subject of the Conclusion is the Predicate of the second Premise and the Predicate of the Conclusion is the Subject of the first Premise. For example : (lj No book is pencil. (2) Some pencils are copies. Explanation : The given Premises are already aligned. From the chart type ot Conclusion. Conclusion according to the format STEP III: TO CHECK IMMEDIATE INFERENCE : After determining the validity of given Conclusions, check for any immediate inferences Conversion and/ or Implication. For example : Premises : (1) All pens are buses. (2) All buses are cars. Conclusions : I. All pens are cars. II. Some buses are pens. III. Some cars are buses. IV. Some cars are pens. V. Some pens are buses. Explanation : The given Premises are already aligned. From the chart : A-type of Conclusion Conclusion according to the format All pens are cars Conclusion I Conclusion II Conversion 01 iirst Fremise Conclusion II! Conversion oi second Premise Conclusion IV Conversion oi Conclusion I Conclusion V Implication of first Premise Therefore, all the Conclusions are valid. STEP IV : EVALUATE COMPLEMENTARY PAIR : This Step is applicable to only those Conclusions which do not follow from Steps II and III. If one of the Conclusions has already been found to be valid in Step II or III, this Step is not required, if there are only two Conclusions. A pair of contradictory statements, i.e., a pair of statements such that if one is true, the other is false and when no definite Conclusion can be drawn, using
the rules of Syllogism, eitehr of them is bound to follow, is called a Complementary Pair. Two Propositions make a Complementary Pair, if: (i) both of them have the same Subject and the same Predicate and (ii) there are an I-O type pair or an A-O type pair or an I-E type pair. For example : Remember that in a Complementary Pair at least one of the Conclusions is always true and therefore if you find a Complementary pair, then mark the choice 'either I or II follows" to be true. TYPICAL EXAMPLES SET 1 Directions (1-15) : In each question below are given two statements followed by two conclusions numbered I and II. You have to take the two statements to be true even if they seem to be at variance with commonly known facts then decide which of the given conclusions logically follows from the two given statements disregarding commonly known facts. Give answer (1) if only conclusion I follows Give answer (2) if only conclusion II follows. Give answer (3) if either I or II follows Give answer (4) if neither I nor II follows and Give answer (5) if both I and II follow Ex.1. Statements : Some chairs are tables. All furniture are tables Conclusions : I. Some stools are tables II. Some furniture are chairs. Ex.2. Statements : All games are outdoor. All outdoor are indoor. Conclusions : I. All games are indoor. II. Some indoor are outdoor. Ex.3. Statements : Some books are photos. No photo is coloured. Conclusions : I. No book is coloured. II. Some photos are coloured. Ex.4. Statements : All birds are black. No black is white Conclusions : I. No bird is white. f II. Some birds are not black. Ex.5. Statements : Some cups are plates. All plates are spoons. Conclusions : I.«All spoons are plates. II. Some spoons are cups. Ex.6. Statements : No machine is green. Conclusions : I. No green is oily. No machine is oily. II. All machines are oily. Ex.7. Statements : Some projects are research. Conclusions : I. Some research are data. Some projects are data. II. No project is data. Ex.8. Statements : All drums are flutes. Some flutes are long. Conclusions: 1. Some drums are long. II. No drum is long Ex.9. Statements : All Ts are Us. No U is Z. Conclusions : I. No T is Z. II. Some Ts are Zs. Ex.10. Statements : Some pins are clips. All clips are darts. Conclusions : I..Some darts are pins. II. Some pins are darts. Ex.11. Statements : All toys are dolls. Conclusions : I. All dolls are sweet. All dolls are toys. II. All toys are sweet. Ex. 12. Statements : No paper is pencil Conclusions.: I. Some papers are clips. No clip is pencil. II. Some pencils are papers. Ex.13. Statements : All stones are watches. All watches are balls. Conclusions : I. All balls are stones. Ex.14. Statements : Conclusions: I. II. Some watches are stone; Some pockets are bags. Some bags are empty. Some pockets are empty II. No pocket is empty. Ex.15. Statements : All keys are locks. Conclusions : I. Some locks are doors. Some doors are keys. II. No door is key. ANSWERS WITH EXPLANATION 1.(4) We can align the Premises by changing the order of the Premises and converting the first Pre; mises. Thus All furniture are tables. Some tables are chairs.
No Conclusion. Now look for any Conversion and Implication There is no such Conclusion. 2.(5) The given Premises are already aligned. All games are outdoor. No Conclusion. The conclusions form Complementary Pair, therefore either of them is true. 9.(1) The given Premises are already aligned., AllTs are Us. All outdoor are indoor. type Conclusion "All games are indoor." This is Conclusion I. Conclusion II is the Conversion of the second Premise. Therefore, both the Conclusions follow. 3.(4) The given Premises are already aligned. Some books are photos. No photo is coloured. type of Conclusion Conclusion according to the format: "Some books are not coloured.".(1) The given Premises are already aligned. All birds are black. No black is white. A + E type of Conclusion "No bird is white." This is Conclusion I. 5.(2) The given Premises are already aligned. Some cups are plates. All plates are spoons- type of Conclusion Conclusion according to the format: "Some cups are spoons." Conclusion II is the Conversion of our derived Conclusion. Therefore, Conclusion II follows. 6.(4) Since both the Premises are negative, no conclusions follows from them. 7.(3) Since both the Premises are Particular, no conclusion follows from them. But the Conclusions form Complementary Pair. Therefore, either Conclusion I or II follows. 8.(3) The given Premises are already aligned. All drums are flutes. borne tlutes are long. No U is Z. E-lype of Conclusion. "No. T is Z" This is Conclusion I. 10.(5) The given Premises are already aligned. Some pins are clips. All clips are darts. We know that.type of Conclusion "Some pins are darts." This is Conclusion II. Conclusion I is the Conversion of our derived Conclusion. 11.(2) The given Premises are already aligned. All toys are dolls. All dolls are sweet. "All toys are sweet." This is Conclusion II. 12.(4) We can align the Premises by changing the order of the Premises and converting the second Premise. Thus, Some clips are papers. No paper is pencil. "Some clips are not pencils." 13.(2) The given Premises are already aligned. All stones are watches. "All stones are balls." Conclusion II is the Conversion of the first Premise.
14.(3) Since both the Premises are Particular, no Conclusion follows. But the Conclusions form Complementary Pair, therefore either of them is true. 15.(3) The given Premises are already aligned. All keys are locks. bome locks are doors, No Conclusion. Both the Conclusions iorrn Complementary Pair, therefore, either of them is true. SET-2 [ Directions (16-20): In each question below are given three statements followed by four conclusions - I, II, III and IV. You have to take the given statements to be true even if they seem to be at variance with commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follow (s) from the given statements disregarding commonly known facts. Ex.16. Statements : Some trees are roads. All roads are stones. Conclusions : I. All trees are roads. II. Some stones are trees. III.No stone is tree. (1) Only either II or III follows (2) Only either I or II follows (3) Only either I or111 follows (4) Only II follows Ex.17. Statements : Some pencils are houses. All houses are buses. Conclusions : I. Some pencils are buses. II. Some buses are pencils. III. All buses are pencils. (1) All follow (2) None follows (3) Only II follows (4) Only I and II follow Ex.18. Statements : Some trees are flowers. Some flowers are pencils. Some pencils are tables. Conclusions : I. Some tables are flowers. II. Some pencils are trees. III. Some tables are trees. IV. Some trees are pencils. (1) All follow (2) Nonevfollows (3) Only I and III follow (4) Only II and IV follow Ex.19. Statements : All rods are bricks. Some bricks are ropes. All rooes are doors. Conclusions : I. Some rods are doors. II. Some doors are bricks. III. Some rods are not doors IV.All doors are ropes. (1) Only I and II follow (2) Only I, II and III follow (3) Only either I or III ad II follow (4) Only either I or III and IV follow Ex.20. Statements : Some books axe pens. Some pens are watches. All watches are radios. Conclusions : I. Some radios are watches. II. Some radios are pens. III. Some watches are books. IV.Some books are watches. (1) A11 follow (2) Only I and III follow (3) Only II and IV follow (4) Only I and IV follow Directions (21-25) : In each question below there are two statements followed by four conclusions numbered I, II, III, and rv. You have to take the two given statements to be true even if they seem to be at variance with commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follows from the given statements, disregarding commonly known facts. Ex.21. Statements : All boys are girls. All girls are monkeys. Conclusions : I. Some monkeys are girls. II. All girls are boys. III. Some boys are not monkeys. IV.Some monkeys are boys. (1) All follow (2) Only I and III (3) Only I and IV (4) Only I Ex.22. Statements : Some tables are rats. Some rats are balls. Conclusions : I. Some rats are tables. II. Some balls are rats. III. Some balls are tables. IV. Some rats are both tables as well as balls. (1) All follow (2) Only I (3) Only I and III (4) Only II and IV Ex.23. Statements : Some apes are bears. Some bears are children
Conclusions : I. Some bears are apes. II. Some apes are children. III. Some children are apes. IV.Some children are bears. (1) Only I and III follow (2) Only I and IV follow (3) Only IV follows (4) Only I follows (5) None follows Ex.24. Statements : Some paints are boxes. No box is rain. Conclusions : I. Some paints are rain II. Some boxes are rain III. Some paints are not rain IV.Some boxes are paints. (1) Only III (2) Only IV (3) Either III or IV (4) Only III and IV Ex.25. Statements : All cars are trains. All trains are, planes. Conclusions : I. All cars are planes. (1)All follow (2) None follows (3) Only I and II follow (4) Only II, III and IV follow (5) Only I, II and III follow II Some trains are cars. III.Some planes are trains. IV.Some planes are cars. ANSWERS WITH EXPLANATION 16. (4) First Premise is Particular Affirmative (I-type). Second Premise is Universal Affirmative (A-type). Both the Premises are already aligned. Some trees are roads. All roads are stones type of Conclusion. Thus, our derived Conclusion would be : "Some trees are stones." Conclusion II is the Conversion of our derived Conclusion. 17. (4) First Premise is Particular Affirmative (I-type). Second Premise is Universal Affirmative (A-type). Both the Premises are already aligned. Some pencils arehouses All houses are buses. type of Conclusion. Thus, our derived Conclusion would be : "Some pencils are buses". This is the Conclusion I. Conclusion II is the Conversion of the derived Conclusion. Therefore, only Conclusions I and II follow. 18. (2) All the three Premises are Particular Affirmative (I - type). Therefore, no conclusion can be derived from these Premises. Now look for any Conversion and/or Implication: There is no such Conclusion. 19. (3) First premise is Universal Affirmative (A-type). Second Premise is Particular Affirmative (I-type). Third Premise is Universal Affirmative (A-type). Some bricks are rorjes. All ropes are doors. -type of Conclusion Conclusion: Some bricks are doors. It is the Conversion of Conclusion II. Conclusions I and III form Complementary Pair. Therefore, either Conclusion I or III follows. Therefore, our required answer is option (3). 20. (5) First and second Premises are Particular Affirmative, I-type). Third Premise is Universal Affirmative (A-type). Conclusion I is the conversion of the third premise. Second and third premises are relevant for the Conclusion II, < Thus, Some pens are watches. All watches are radios.
of Conclusion Conclusion: Some pens are radios. Conclusion II is the Conversion of this Conclusion. Therefore, only Conclusions 1 and II follow. Venn-Diagram We know that if both the Premises are Particular, no Conclusion follows from them. Now look for any Conversion or Implication. Conclusion I is the Converse of the first Premise. Conclusion IV is the Converse of the second Premise. Therefore, only Conclusions I and IV follow. Possible Venn-diagrams 21. (3) Both the Premises are Universal Affirmative and they are already aligned. All boys are qirls. All girls are monkeys. type of Conclusion Thus, our derived Conclusion would be : "All boys are monkeys." Conclusion IV is the Converse of our derived conclusion. Conclusion I is the Converse of the second Premise. Therefore, only Conclusions I and IV follow. Possible Venn-diagrams 24. (4) First Premise is Particular Affirmative (I-type). Second Premise is Universal Negative (E-type). Both the Premises are already aligned. Some oaints are boxes. NO DOX is rain. We know that 22. (5) Both the Premises are Particular Affirmative and they are already aligned. We know that if both the Premises are Particular, no conclusions follows from them. Now look for any Conversion or Implication. Conclusion I is the Converse of the first Premise. Conclusion II is the Converse Of the second Premise. Therefore, only Conclusions I and II follow. 23. (2) Both the premises are Particular Affirmative and they are already aligned. Thus,