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.
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”]
Types of Conclusion:
Conclusion
Subset: 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 (SubSet)

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
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
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
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)
Sir...it was a very good explanation... But i have one doubt . Restatement follows or not ??
ReplyDeletee.g. Statement :All A Are B
Conclusion: all A Are B
Thanks for the comment..
DeleteRestatement 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.
Very nice article Sir
ReplyDeleteCan a definite conclusion be a possibility ?
Thanks.
ReplyDeleteDefinite is certainly a subset of Possibility. Hence yes.
Sir if statement is some a are b then all a are b is a possibility is true or false..
ReplyDeleteIf "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"
ReplyDeleteThnks.very useful article. But i have doubt in only a few & only few pls explain me.
ReplyDeleteSure. You can reach out any time for the doubts.
DeleteThank you sir.
ReplyDeleteCan 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.
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.
DeleteSo 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)
Thank you so much sir!You are the best.
DeleteWhat is the meaning of only some A are B
ReplyDeleteAlthough 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.
Delete