Logic

Logic to most is all about critical thinking, rational step by step deduction and reasonable arguments. While for most practical purposes this approach does fit the bill ;at times we are encountered with simple expressions or statements that have profound meaning or insights.

Let take for example a simple statement

A=B is same as B=A ........ (1)

If the above A,B were mathematical equations or expressions then statement (1) is an absolute truth.

For example:

(a+b)^2 = (b+a)^2

Or using a more famous equation,

F = m×a is the same as m×a = F ( in classical mechanics)

Proving A=B will automatically implies proving B=A

Now, let us consider a statement Apple= Fruit and Fruit = Apple

The above statements are not same because Apple is a subset of the class of objects called fruits. So, while all apples are fruits ; all fruits (Oranges, Watermelons, Bananas etc) are not apples. In fact speaking technically using the '=' sign above is not correct and is misleading.

I have found in my conversations with several professionals, professors and researchers that the above distinction is overlooked which results in erroneous conclusions.

The above is only a very simple demonstration of the vast topic called Logic. While I don't claim to be an authority in this area I try to keep my fundamentals clear and ensure that while communicating we don't end up speaking across one another.

Another example of such logical statement deduction is using them in conjunction.

For example,

If a statement U is correct then on using another unknown statement V with it if we arrive at another correct statement W then the unknown statement V is also correct.

Here is an illustration for it when the statement is an equation

Let us consider x,y,z as three distinct positive integers

(x+y) ^2 =x^2+y^2 +2×x×y ...... (2 )...... statement U ; is always true we can interchange the variables and still it will be true.

Let another unknown equation in this case be z= x+y....... (3) .....statement V . Does this hold correct for some of the distinct positive integers we chose ?

Let us use (2) & (3) or U and V in conjunction we get

z^2 = x^2 +(z-x)^2 +2×z×x

Solving , we get 0 =0 ....... (4)...... statement W which is a correct equation or expression and hence our initial unknown z=x+y is correct for some distinct positive integers.

Another, way of looking at this is analysing the statements

All engineers are intellectuals ( unknown)

All engineers are educated in science. ( given to be true)

On using the above in conjunction we get All intellectuals are educated in science. Which is not necessarily true as some may belong to Arts background as well.

Many mathematical conjectures based on my experiences can be simply resolved if the above approach is taken. It also lays the foundation of constructive and collaborative arguments which add value to the subject matter being discussed. Laying the above upfront ensures that correct deductive reasoning gets employed. While I am not aware of all the logic in the world ; I can reasonable assume people are not as well. Hence these ground rules make it simple to understand the statements, expressions, equations etc whatever they might be and their meanings.