Part One
Logic is like math. Here is a basic mathematical operation:
Addend: 1
Addend: 2
Sum: 3
A person who performs this operation is using math—they're being mathematical.
Here is a basic logical operation:
Premise: All humans are mortal.
Premise: Socrates is human.
Conclusion: Socrates is mortal.
A person who performs this operation is using logic—they're being logical.
Part Two
As in math, it is possible to do it wrong. Here is an inaccurate mathematical operation:
Addend: 1
Addend: 2
Sum: 54
A person who performs this operation is not being mathematical. I don't know what they're doing, but it's not math. No: not even if they say "When I say 1, what I actually mean is 52."
Here is an inaccurate logical operation:
Premise: All humans are mortal.
Premise: Socrates is human.
Conclusion: Socrates is a three-winged bat.
A person who performs this operation is not being logical. I don't know what they're doing, but it's not logic. No: not even if they say "When I say 'mortal', what I actually mean is 'three-winged bat'."
Part Three
Anyone who uses math is using the same math as anyone else who uses math. "New math" is not that new. You can't add 1 and 2 and get 54, or 46, or even 8. One plus two is always three—if you're using math.
Anyone who uses logic is using the same logic as anyone else who uses logic. You can't add "all humans are mortal" to "Socrates is human" and get "Socrates is a three-winged bat," or "Socrates has hair," or even "Socrates is a twit who likes asking questions to make people look stupid." "All humans are mortal" plus "Socrates is human" always means "Socrates is mortal"—if you're using logic.
No one gets "his own math." No one gets "his own logic."
Part Four
Also, accurate information is essential. If you're trying to figure out how much space you need for your new bookcase, you have to put the right numbers into your mathematical equation and do the math properly—if you're trying to figure out whether Socrates is mortal, you have to put the right premises into your logical equation and do the logic properly.
It is perfectly possible to do the math right and get the answer wrong. Whoops, your bookcase is longer than you thought.
It is perfectly possible to do the logic right and get the answer wrong. Whoops, Socrates is a computer program.
Logic, like math, is a process. You get out what you put in.
And now I'm going to go mend some watches.
[Edit: Whoops, I put the last sentence of part three in part four. Also, I guess I'm getting ready to go mend some watches. Not actually mending them yet, just cleaning my teeth and getting my shoes on.]