This second law, the Law of Noncontradiction, and the Third law both follow from the Law of Identity. They are correlates and consequences of the first law.
Aristotle gives us the Second Law in three different forms.
……the same attribute cannot at the same time belong and not belong to the same subject in the same respect………Metaphysics Book IV, 3 Translated W.D. Ross
This is his ontological argument. It’s an argument based on how things “be.” It’s about existence. The subject cannot have an attribute AND not have the attribute at the same time and in the same way. This is the important beginning of all syllogisms. You remember those from geometry. Here’s one:
All men are mortals.
Socrates is a man.
Socrates is a mortal.
For this to hang together the attributes “mortal” and “man” must belong to the subjects in the same way at the same time. That’s self evident, but it took a genius to recognize it and put it in words for us the first time.
Aristotle states this law in a second way.
This, then, is the most certain of all principles, since it answers to the definition given above. For it is impossible for any one to believe the same thing to be and not be…………Metaphysics Book IV, 3, Translated W.D. Ross
This is where cognitive dissonance lives. That distress is evidence of the truth of this psychological claim that Aristotle is making. It’s possible to EVALUATE contrary truth or falsehood of two different subjects, but it is psychologically impossible for a mind to hold two contrary beliefs simultaneously. When presented with evidence contrary to a closely held belief, people can experience extreme distress. That distress often leads to people equivocating, moving the goal posts, ignoring, destroying friendships, or even screaming in an attempt to retain the first, wrongly held belief. When we are at our best we hear the arguments and change our beliefs based on the evidence.
If you are going to operate on our Socrates demonstration above, you can’t believe that Socrates is both a man AND not a man. Luckily, it’s impossible to actually believe both. (This is just one reason why it is important to actually define what a man is. Equivocating on man-ness changes everything.)
Here’s his third run at this law.
….the most indisputable of all beliefs is that contradictory statements are not at the same time true………..Metaphysics Book IV, 6, Translated W.D. Ross
This is strictly a logical argument and relies on the law itself. I think it’s a tautology, but I’m ok with that. Things are strange down here on bedrock. I also believe that this is “the most indisputable of all beliefs…”
Here’s a little fun from Uncle Ari.
………..if all are alike both right and wrong, one who believes this can neither speak nor say anything intelligible; for he says at the same time both “yes” and “no.” And if he makes no judgement but thinks and does not think, indifferently, what difference will there be between him and plants? (LULZ) — Thus, then, it is in the highest degree evident that neither any one of those who maintain this view nor any on else is really in this position. For why does a man walk to Megara and not stay at home thinking he ought to walk? Why does he not walk early some morning into a well or over a precipice , if one happens to be in his way? Why do we observe him guarding against this, evidently not thinking that falling in is alike good and not good? Evidently he judges one thing to be better and another worse. And if this is so, he must judge one thing to be man and another to be not-man, one thing to be sweet and another to be not-sweet. For he does not aim at and judge all things alike, when, thinking it desirable to drink water or to see a man, he proceeds to aim at these things; yet he ought, if the same thing were alike man and not-man. But, as was said there is no one who does not obviously avoid some things and not others. Therefore, as it seems, all men make unqualified judgments, if not about all things, still about what is better and worse. And if this is not knowledge but opinion, they should be all the more anxious about the truth, as a sick man should be more anxious about his health than one who is healthy; for he who has opinions is, in comparison with the man who knows, not in a healthy state as far as the truth is concerned.Metaphysics, Book IV, 4, Translated by W.D. Ross
Judging and knowing truth and/or falsehood when we see it is the difference between safe travels and falling down a well. Judge well, and don’t let the bastards lie to you.