the logic behind basic mathematical operation

Sorry!! But this is not what am I seeking.
While doing addition we follow a certain process... such as we do a carryover. I want to know why such process is followed.

Logic (set theory) is itself a branch of mathematics and Bertrand Russell failed to establish it as the universal mathematical substrate.

I suggest you redirect your focus to the word "explanation", and answers to this regarding mathematics include " a reflection of bodily interactions with the world" (see Lakoff and Nunez). For example, one well known concept of "addition" is moving to the right along a number line.

Irrespective of how fundamental operations (like addition( are "explained", one these are established, other operations like multiplication and division can be defined in terms of them. (NB Square rooting also involves the concept of "inverse mapping" which in terms of the body implies " a return journey")

If you are merely asking about calculation techniques, "carrying" is related to "place value" involving the practice of writing digits in columns relating to successive powers of counting base from right to left.

Thnx... Can i get an explanation... Is that possible. Let take n example... 356+344... Can we begin from here

Write 356+344 as 3H+5T+6U plus 3H+4T +4U
(H=hundreds T=tens U=units)

Adding units 6U+4U =1T(to carry) +0U
Adding tens 5T +4T + 1T =1H(to carry) +oT
Adding hundreds 3H+3H+1H =7H
Final answer 7H+0T+0U =700

First question... Why the addition has to begin from the right or from the unit place

Because addition from the right can affect values on the left, but addition on the left can never affect values on the right. If you start on the left, you will have to rework your answer every time a sum to the right produces exceeds ten.

Forget about the process... Say you don't know how to add then which logical process you will follow and why?

Maybe this guy needs an abacus !

Are you perhaps suggesting something like using a "look up table" to find number bonds that make ten ? There seems to be no answer to your question unless you do that, or convert everything to base two and refer to the "logic" of a binary adder.

The logic we generally used for the basic mathematical operations are to know each and every formulas of a mathematics and all basics formulas should understood by a person who is solving a maths. BODMAS concept is very important to break many question and get answers of different question.