## Saturday, 28 January 2017

### Deductions & Syllogism

The chapter of Deductions or Syllogism deals with finding various conclusions from a given set of statements or premises. A combination of two or more of the following types of premises are given which are then combined to draw conclusions –

1. All A’s are B’s – which can alternately be written as:
Some A’s not being B is not a possibility | Only B’s are A

2. No A is B – which can alternately be written as:
Some A’s being B is not a possibility | A cannot be B

3. Some A’s are B’s - which can alternately be written as:
Many/Most/Few A’s are B’s | No A being B is not a possibility

4. Some A’s are not B’s - which can alternately be written as:
All A’s are not B’s | Many/Most/Few A’s are not B’s | All A’s being B is not a possibility

Term & its Distribution:

Terms: There are always exactly two terms in a premise. In the above examples A and B are the terms between which a relation is being formed with the help of a premise. For ease of communication we shall refer to the first term (“A” in all the above examples) to be present in the subject of the premise and the second one (“B” in all the above examples) to be in the predicate of the premise.

Distribution of a Term: Whenever the whole of a term is being referred to, the term is said to be distributed. For instance, in Eg 1 the whole of “A” is being described as “B”; hence the term A is said to be distributed (while “B” isn’t). Again in Eg 2 whole of “A” & “B” are described as separate from each other, so both the terms have been distributed. In Eg 3 only a portion of “A” is described as common with a portion of “B”, hence none of the terms are distributed. If read carefully, Eg 4 describes a portion of “A” as being separate from all of “B” which means only B is distributed in this premise.

Classification of Premises:

Universal & Particular: Whenever the term in the subject of the premise is taken as a whole and described together, such premise is called Universal Premise [Eg 1 & 2 in the above cases where we come to know that the whole of “A” is “B” & is not “B” respectively] – alternately we can say, whenever the term in the subject of the premise is distributed, such premise is said to Universal. On the other hand, if only a portion of the term in the subject of the premise is described then such a premise is called a Particular Premise [Eg 3 & 4 where the word “All” and “No” is replaced by “Some”]

Positive & Negative: If a premise is conveying something to be common between the two terms then it is said to be a Positive Premise [Eg 1 & 3 which informs All and Some A’s are B’s respectively]. On the other hand if the premise states something uncommon between the two terms then it is said to be a Negative Premise [Eg 2 & 4 where All and Some A’s have been stated to be separate from B].

Types of Conclusion:

Conclusion Sub-set: A Particular premise can be extracted from any Universal premise between the same two terms. Also some premises can be reversed by interchanging the subject and predicate.

Eg 3 is implied from Eg 1 – If “All A’s are B’s” is given then it implies that if some portion of A are selected; they shall also be B.

Eg 4 implied from Eg 2 – If “No A’s are B’s” is given then it implies that if some portion of A are selected; they shall not be B.

Following is a comprehensive list of Conclusions which are sub sets of respective given premises:

 PREMISE CONCLUSION (Sub-Set) All A's are B's Some A's are B's Some B's are A's No A is B No B is A Some A's are not B's Some B's are not A's Some A's are B's Some B's are A's

Deductions: When two premises with one term in common, are combined together to draw a conclusion between first and third terms, such a conclusion is called Deduction.

Eg:       i) All A’s are B’s
ii) Some A’s are C’s

From i) & ii) the following Deduction can be derived –
Some B’s are C’s

In the above example, A is the term common in both premises – also called Middle Term; while B and C are the first and third terms. Any relation between A & B or A & C will be considered to be a Conclusion Sub-set and only a relation between B & C will be termed as Deduction.

Rules for Deduction:

1. The Middle term must be distributed at least once in the two premises; else there shall be no deduction

2. If both the premises are Particular; there will be no deduction

3. If both the premises are Negative; there will be no deduction

4. If one of the premise is Particular, the deduction, if any, will be Particular

5. If one of the premise is Negative, the deduction, if any, will be Negative

6. If a term is distributed in the deduction, it must have been distributed in the Premise

Application of Rules:

In order to derive Deductions the above mentioned rules can be applied. The following examples will help in understanding the approach –

#1. Rivals are not friends | Friends cannot be foes

Here both statements are in the form of Universal Negative hence there can be no Deduction (i.e. relation between first and third term, Rivals and Foes is not clear) – refer Rule 3

#2. Some vegetables are fruits | Some fruits are not grain

Here both premises are Particular hence there can be no Deduction (i.e. relation between first and third term, Vegetables and Grain is not clear) – refer Rule 2

#3. Patriotism is a feeling | Courage is a feeling

Here both statements are in the form of Universal Positive, however the middle term has not been distributed, so there can be no Deduction (i.e. relation between first and third term, Patriotism and Courage is not clear) – refer Rule 1

#4. All planets are vehicles | Vehicles move in orbits

Here both premises are in the form of Universal Positive and the middle term has been distributed. In the premises, the term Planets has been distributed while Orbits has not been so. The Deduction hence, can be a Universal Positive with Planets distributed in it.
i.e. All planets move in orbits – refer Rule 6

#5. All sportsmen are athletes | Some joggers are not athletes

Here first statement is Universal Positive and the second one is Particular Negative. Also the middle term, Athletes has been distributed once in the premise. From Rule 4 & 5 the Deduction will be Particular Negative (which has predicate distributed), hence Sportsmen will be the predicate (as among Sportsmen and Joggers only the former has been distributed in the premise).
i.e. Some joggers are not sportsmen – refer Rule 6

#6. Some possibilities are not happening | All happenings are destiny

Here, just like #5 we have a combination of Universal Positive and Particular Negative premises. With the middle term “happening(s)” distributed; from Rule 4 & 5 the Deduction must be Particular Negative (with the predicate being distributed). However, if we look closely, neither of the first and third terms “possibilities” and “destiny” have been distributed in the premises; which make them ineligible to be the predicate of a Particular Negative statement (refer Rule 6). Hence there is no Deduction.

Contradictory Premises

There are times when no particular relation can be established between two terms and any option given will be wrong individually; however if the given options cover all possibilities then invariably the answer becomes either of the two possibilities (eg #7).

#7. Some patients are not sick | Some sick are under illusion

Here as both premises are particular, no conclusive relations can be established between the terms “patient” and “under illusion” (refer Rule 2). However if the following options are given,

i) All patients are under illusion
ii) Some patients are not under illusion
iii) No patient is under illusion
iv) Either i) or ii)

In the above case though both i) and ii) do not follow, one has to chose option iv) as there are no other possibilities beyond i) and ii).

How to identify contradictory premises

If a given premise is Universal, its opposite or contradictory premise shall be Particular and vice versa.

Also, if a premise is Positive, its opposite or contradictory premise shall be Negative and vice versa.
It is also worth noting that a Particular Positive and a Particular Negative premise between two same terms can both be true at the same time and at least one of them must be true at any point of time.
Following is a list of pair of premises in which one of the two shall always be held true:

 PREMISE (i) PREMISE (ii) Relation Case 1 All A's are B's Some A's are not B's Contradictory Case 2 No A is B Some A's are B's Contradictory Case 3 Some A's are B's Some A's are not B's Parallel existence possible

Possibility Questions

With the above discussion we can now find definite conclusions, but sometimes the question only asks to verify possibility. Note the following to understand how to solve possibility based questions -

i) In case there is no definite conclusion between two terms at all (i.e. cases where middle term is not distributed / both premises are negative or particular etc.) then all relations (i.e. all four types of relation discussed right at the beginning) between the two terms are possible.

Hence in such cases all conclusions don’t hold true, but all conclusions are possible.

ii) In case there is a definite conclusion between two terms then all relations are possible, except the ones contradicting the definite conclusions.

#8. Some countrymen are traitors | Some traitors are alive

Here, as both premises are particular there cannot be any conclusive relation between first and third terms (countrymen and alive). Hence none of the following relations are correct –

All countrymen are alive; No countrymen are alive; Some countrymen are alive; Some countrymen are not alive.

However all of the above 4 are possible. For example if it is given that “No countrymen are alive is a possibility”, we shall take it as correct.

#9. All questions are answerable | All answerable are debatable

Here the primary conclusion between first and third term shall be – All questions are debatable
and from this conclusion we can also derive Some questions are debatable

Hence here all relations are possible except the contradictory statements of the definite conclusions deduced above.

All debatable being questions is a possibility – Correct

Some debatable not being questions is a possibility – Correct

No question is debatable – Incorrect (contradictory to the definite conclusion)

Some questions are not debatable - Incorrect (contradictory to the definite conclusion)

#### 13 comments:

1. Sir...it was a very good explanation... But i have one doubt . Restatement follows or not ??

e.g. Statement :All A Are B
Conclusion: all A Are B

1. Thanks for the comment..
Restatement certainly follows, however we should search for an options which are not only logically correct but also requires some amount of deduction from our end. In the absence of such an option we can go ahead with the restatement.
There are also certain questions where they specifically mention - which of the following can be derived from "all the statements" given below:
Eg- 1)All A are B 2)All B are C 3)All C are D
Then we only accept conclusions which are relations between A and D as only those would require all three premises to be combined together.

2. Very nice article Sir

Can a definite conclusion be a possibility ?

3. Thanks.
Definite is certainly a subset of Possibility. Hence yes.

4. Sir if statement is some a are b then all a are b is a possibility is true or false..

5. If "some A are B" is given then "All A are B" and "All B are A" both are possible. Hence, answer to your question is "True"

6. Thnks.very useful article. But i have doubt in only a few & only few pls explain me.

1. Sure. You can reach out any time for the doubts.

7. Thank you sir.
Can you also expalin only and alone type statements.
ex:
---------------------------
statemets
all 50 are 40.
only 40 are 80
alone 90 are 40.
Conclusions:
1.all 40 are 50 is a possibility.
2.all 50 are 90 is a possibility.
3.all 80 are 90 is a possibility.
-----------------------------------
thanks in advance sir.

1. Thanks for the comment. If you scroll above you would find that All A's are B's is same as Only B's are A's.

So Only 40 are 80 means All 80 are 40.

Also, Alone 90 are 40 means All 40 are 90.

Hence all 3 given conclusions are correct (in fact 2 & 3 are definitely true)

2. Thank you so much sir!You are the best.

8. What is the meaning of only some A are B

1. Although I have never seen "only" and "some" being used together; I would treat the above statement same as "Some A are B" because even though we are using "only some" before describing A we are giving a definite status to partial set of A and not all of them.