>>12239558it allows for a rigorous definition of implication that is logical. So with using boolean operators, for "¬P?Q", that is the implication, to be true, we study the cases for P.
If P is false then Q may be whatever the fuck it wants, that means that the result of the implication (Q) can be false, because as we said P is false (therefore ¬P is true, then the whole expression would be true according to "true?false is true")
Now again, when P is false ( ¬P is true), Q might also be true, and the whole expression would still be true, for a practical example: -1=1 => 1=1
You see that P (the first statement) is clearly false, still, by squaring both sides we get a valid result. With booleans we can say: "true?true is true"
Now when P is true, and this is where the boolean expression comes in handy, Q MUST BE TRUE FOR THE WHOLE EXPRESSION TO BE TRUE, and this constitutes a very important concept which is: THE TRUE DOES NOT IMPLY THE FALSE
To explain this, we just study boolean cases. When P is true (¬P is false), then for the whole expression to be true Q must be true, it's clear with the expression: false?true is true
Otherwise, if Q is false, the whole expression is false
To remind you again, this means that THE TRUE DOES NOT IMPLY THE FALSE
Note: Q doesn't "necessarily" have to be deduced from P, for example "1+1=2 => N is an infinite set" is valid and true, with both propositions considered already true. I know this might seem weird because the proposition are not "clearly" related, still, if they're considered ALREADY true then the expression is valid. But usually, we start with a true proposition, then we imply step by step (a chain of implications, true => true => ... => true, and remember THE TRUE DOES NOT IMPLY THE FALSE) to get a true result at the end. So this definition is the most rigorous, and mostly based on logic rather than intuition.
