Syllogisms Questions and Answers

Answer: A

The answer is Only conclusion I and conclusion II are correct.

Conclusion I: the bottles that are cups are, like every cup, knives, so some bottles are knives. Conclusion II: since every cup is a knife and cups exist, some knives are cups; this is the converse of 'All cups are knives' restricted to 'some'.

Conclusion III does not follow: only some bottles are known to be cups, and nothing says every cup is a bottle. Conclusion IV reverses 'All cups are knives'; there may be knives that are not cups. So any option containing III or IV is wrong.

Answer: B

The valid inferences are Only II and III.

From 'all heroes are winners' and 'all winners are lucky', every hero is lucky. So, assuming heroes exist as the usual reading of such statements does, some lucky people are heroes (II), and some winners are heroes (III), the converse of statement 1 limited to 'some'.

I does not follow: the lucky people may include many who are not heroes, so 'all lucky people are heroes' reverses the inclusion. Every option containing I is therefore wrong, and 'Only III' leaves out II, which follows just as directly.

Answer: C

The valid inference is If (i) and (ii) are true, then (iii) is true.

If some roses are red and all red flowers fade quickly, then those red roses fade quickly, so some roses fade quickly.

The other options try to infer something from a false premise, which does not work. If (ii) is false, some red flowers may still fade quickly, and other roses may too, so (iii) could be either true or false; neither "then (iii) is false" nor "then (iii) is true" follows. Likewise, if (i) and (ii) are both false, nothing is settled about whether some roses fade quickly.

Answer: A

Only conclusion III follows.

All teachers are professors and no professor is male, so no teacher can be male: conclusion III is certain.

About engineers we know only that some males are engineers. Those male engineers cannot be professors, but other engineers, who are not male, may or may not be professors. So 'No engineer is a professor' (I) and 'Some engineers are professors' (II) are both possible but neither is certain. (They are complementary, so exactly one of them is true, but the statements do not tell us which.) That rules out every option containing I or II.

Answer: D

Only conclusions I and III are correct.

'Some soaps are clean' converts directly to 'some clean objects are soaps', which is conclusion I. Those clean soaps are wet, because all clean objects are wet, so at least some wet objects are soaps: conclusion III.

Conclusion II, 'no clean object is a soap', contradicts statement 1. Conclusion IV, 'all wet objects are soaps', does not follow: the statements allow wet objects that are neither clean nor soaps. Since I and III both follow, 'only I' is incomplete, and the 'either ... or' options pair a valid conclusion with an invalid one.

Answer: D

The correct choice is Conclusion I and III follow.

From 'All bottles are glasses' and 'All glasses are jugs', every bottle is a glass and every glass is a jug, so all bottles are jugs: conclusion I follows. 'All bottles are glasses' also means that those glasses which are bottles exist, so 'Some glasses are bottles' follows by conversion: conclusion III follows.

Conclusion II, 'All jugs are bottles', reverses a universal statement. There may be jugs that are not glasses, and glasses that are not bottles, so it does not follow. That rules out every option containing II.

Answer: A

The key gives Only one, counting statement 3 as correct.

Statements 1 and 2 reverse a universal statement. 'All cats are animals' does not mean 'all animals are cats', and 'all dogs are mammals' does not mean 'all mammals are dogs'. Both conclusions are false.

Statement 3 ends in a true conclusion: all parrots are birds. Strictly, 'some birds are parrots' does not by itself prove that every parrot is a bird; the key treats the statement as correct because its conclusion is true, while the conclusions of 1 and 2 are plainly false. On that reading exactly one statement is correct, which rules out 'Only two', 'All three' and 'None'.

Answer: C

Both 1 and 2 are correct.

Statement 1: all jasmines are flowers and some flowers are white. The white flowers may or may not include jasmines, so 'some jasmines may be white' is a possibility that the premises allow. A definite 'some jasmines are white' would not follow, but 'may be' does.

Statement 2 is a valid chain: all rats are mammals and all mammals are animals, so all rats are animals.

So neither statement oversteps its premises, and the answer is both.

Answer: B

Only two pairs are correctly matched.

Each pair gives a statement and a conclusion drawn by converting it.

• Pair 1: 'All roses are flowers' converts by limitation to 'Some flowers are roses'. Valid.
• Pair 2: 'Some students are athletes' converts only to 'Some athletes are students'; it says nothing about all athletes. Invalid.
• Pair 3: a universal negative converts simply: 'No cats are dogs' means 'No dogs are cats'. Valid.

So pairs 1 and 3 match and pair 2 does not.

Answer: C

The valid sets are 1 and 3.

Set 1: all Ts are squares and all squares are rectangular, so all Ts are rectangular. A valid chain.

Set 3: idiots are bumblers and bumblers fumble, so idiots fumble. Also a valid chain.

Set 2 is invalid: the fat things that are elongated need not be among the elongated things that are huge, so two 'some' premises prove nothing. Set 4 is invalid: water is good for health, but that does not make water a 'health food', and even if it were, 'health foods are rare' would not make water rare. So '4 only' and '1, 2, 3 & 4' are wrong, and '3 only' misses set 1.