syllogisms

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1. If A then B : 2 conclusions possible: A implies B
~B implies ~A
Eg:- If it rains I will take an umbrella: Then:
It rained => I took an umbrella
I did not take an umbrella => It did not rain.

2. Only if A then B : 2 conclusions possible: B implies A
~A implies ~B
Eg:- Only if it rains I will take an umbrella: Then:
I took an umbrella => It rained
It did not rain => I did not take an umbrella.

3 Unless A, B : 2 conclusions possible: ~A implies B
~B implies A
Unless it rains I won't take an umbrella: Then:
It did not rain => I did not take an umbrella
I took an umbrella => It rained.

Syllogisms (All, No, Some, Some - Not)
It's not possible to explain this in detail here..I'll try my best

Basically there are four types of sentences (category of quantifier) here:
Universal Affirmative : All A's are B's
Universal Negative : No A is a B
Particular Affirmative : Some A's are B's
Particular Negative : Some A's are not B's

Here in each sentence A is the subject and B is predicate.

There is something called distribution ie in each sentence A or B is said to be distributed if it has information about all the entities of the category to which A or B belongs. For eg:- All cats are animals - This statement has information about all the entities belonging to the category 'cat'. But it doesn't have info regarding all the entities belonging to the category 'animals'. So subject(cats) is distributed here and predicate (animals) is not distributed.
If you find it difficult to fully understand the concept of distribution (which will mostly be the case) you can simply remember the general rule:

Category of quantifier: Subject Predicate

Universal Affirmative D ND
Universal Negative D D
Particular Affirmative ND ND
Particular Negative ND D

# D - Distributed ; ND - Not distributed

The 2 statements (which are usually given in the question) are called premises. And the statement which logically follows these 2 statements (which is usually asked for) is the conclusion.

There are 7 rules for deductions:
1. Every deduction should contain three and only three distinct terms.
2. The middle term must be distributed at least once in the two premises.
3. If one premise is negative, the conclusion, if any, must be negative.
4. If one premise is particular, the conclusion, if any, must be particular.
5. If both the premises are negative, no conclusion can be drawn.
6. If both the premises are particular, no conclusion can be drawn.
7. If a term is not distributed in the premises, then that term cannot be distributed in the conclusion.

For Eg:-
No cat is a dog.
All donkeys are animals.
No conclusion possible because there are 4 terms (cat, dog, donkeys, animals). There can be only 3 distinct terms.

No cat is a dog.
No dog is a donkey.
No conclusion possible because both the premises are negative. Both the premises cannot be negative.

Some people are vegetarians.
Some vegetarians are healthy.
No conclusion possible because both the premises are particular. Both the premises cannot be particular.


All humans have two legs.
Some humans are men.
So this implies Some men have two legs.
Here humans is the middle term (that which appears in both the premises). Middle term will not appear in the conclusion. So once you find out that the two premises do not violate any of the conditions and a conclusion is possible, you can say that the conclusion will have 'two legs' and 'men'. Now in, All humans have two legs - 'humans' is distributed and 'two legs' is not distributed (in universal affirmative subject is distributed and predicate is
not distributed). In, Some humans are men - both 'humans' and 'men' are not distributed. So in the conclusion the subject and predicate should retain their distribution from the premises (D or ND) ie we know that the conclusion should have 'two legs' and 'men' and we also know that 'two legs' should not be distributed in the conclusion (as is the case with 'two legs' in the premise) and 'men' should not be distributed in the conclusion (as is the case with 'men' in the premise)
So in the conclusion we have two legs - not distributed ; men - not distributed. So both subject and predicate are not distributed. So it is particular affirmative (Some). Hence the required conclusion is Some men have two legs.

Hope I have confused you enough. Just try out a few syllogism questions from the mocks so far.

Ref: http://www.pagalguy.com/forum/cat-and-related-discussion/34711-pg-underdogs-team-udt-08-a-43.html#post1259928