12-27-2014, 07:22 PM
(This post was last modified: 12-27-2014, 09:49 PM by MetaOntosis.)
Example 2) The Law of Non-Contradiction
~(A & ~A)
Some thoughts before we proceed
We should first discuss the logical operations we've already begun using, now that we've displayed them in context. The "not" operator, which is here represented with a tilde, ~, simply negates the thesis which follows it. It means, "to negate", to declare that it is "not true that (thesis)". If we declare that a thesis is not true, then that means we simply have a case where the truth is outside that thesis, because there is no truth in it.
What happens to the thesis? Nothing. It is just an idea, anyway. It sits in the repository of ideas and does nothing except carry the "not" operator in that context. Perhaps in some other context it will be freed of the not operator, but not in that precise context. For example, of the thesis is "My alien friends are Divine Messengers", then it is saying all that with the implied preface that "it is true that" my alien.. etc. So when we negate this, we are in effect saying that "it is not true that"... etc. Why it is not true is another matter. We simply declared the whole sentence false.
Aristotle goes into this in his writings where he declares that logic operates upon basic sentence grammar, insofar as we use it ordinarily. A sentence has a subject and a predicate. If "it is true", then "it" is the subject and "true" is the predicate. The verb "is" connects the two, and is technically considered part of the predicate phrase. That being the case, the logical operator "true", which is not written, is, metaphysically, a statement about the being of the subject: "It is" it seems to declare. If it turns out we want to say more than something barely "is", then we say "how it is", and we add further qualifications to that basic existential statement. In other words, we add to the predicate.
But everything that follows the copular verb either is or is not, just as the subject either is or is not. Specifically, since the predicate is conjoined onto the subject as a "member" of its own being, we say that the predicate train either is or is not true of the predicate. Even existence itself, the is verb, is really a predicate (in logic for sure it is). Either "is" is true of the subject, or it is not true. There are two is's. The is which is true of the subject itself in reality, and the is which connects the statement itself as a subject, as a whole, to its predicates. The predicates available to any statement in logic of this ordinary sort are "true" and "false".
Look at the sentence, "It is true". Now if it (the whole sentence) is true, then "it is true", is true! The whole sentence has been turned into a subject of our consideration, and we have added a predicate phrase to describe its qualities. Clearly the sentence simply "is", so we leave that unremarked in this extended analysis. But the interest for us is whether or not the sentence is true, which is to say does it describe reality in the way it purports to do. If it is true, then we can, in the context of our discussion, just say the sentence without further qualification. Saying a sentence like this is assumed to be saying something meaningful about reality, and so is assumed to be a declaration of the truth, as qualified by this sentence as spoken.
So we say "My alien friends are Divine Messengers". The subjects is "My alien friends" and the predicate is "are divine messengers". In our logical analysis we'll just "fuse" the two is's into the phrase as stated, so that if it is true, we just let it stand. "Are", in this case, stands for the relation between the subject and predicate, but it also stands for the truth of that connection (in our analysis, and as used in practice since people say these things as if they mean them), and so we don't need to say "This sentence is true, namely that..." etc. We let it stand if it is true, unmodified.
If it is not true, then we modify the subject predicate relation. We add "not" to the copula. "My alien friends are not divine messengers". So we have a sense of the not operator right there. It disjoins the subject and predicate as related in that sentence. As to our justification for doing that, it requires further analysis of the subject and the qualifications it has both for being as such and for being as stated, as "Divine Messengers". If he has no friends, then has no alien friends, then he has no alien friends who are divine messengers. Maybe that's why this sentence is false. Or, if he has friends, maybe they are people who tell him what to say, or inspire him with EM stimulation to his brain, are human and not alien (although perhaps they work with aliens, who are their friends). Maybe they are Divine Messengers, maybe not, but they are not alien, and that matters here, as someone is either confused or deceived or lying. Or maybe he has alien friends, but they aren't divine at all.
In any event, this is "sentential" logic, and we are modifying the copula or other verb, with our "not" operator. That is sufficient for our uses of logic here. There are other ways of going about logical analysis, but they are not introductory, and we are just looking at the Laws of Logical Thought and the rules sufficient to introduce and explicate them.
So the "not" operator is well-understood by now. Let's look at it further. What do we mean by "not" when it is applied to a sentence which declares already that something is "not" true? We are saying that "it is not true that it is not true" and then whatever the content is. How does "not" operate upon itself? Well it must operate upon the same verb that is already operated upon by the not which has come before it, but it is already modified. So it would be redundant to modify it to no effect, as if saying "not" more than once emphasized it or something. That'd just be stuttering. If you said "that not not true", I'd think you meant that is true! Why? Because the "not" operator makes the truth of a sentence false. If a sentence "is true" then it makes it false by declaring it "is not true". If it is not true, then it is false. Those are our primary values in logic, are they not? If we mean by "false" or "not true" that the truth is completely outside the space in which it is asserted that something is true, then that means all truth is outside that space, and the falsehood of that assertion is what is in that space. If we mean that this is not the case, then we must reverse this situation, and put the truth of that statement inside its sphere of expression, and put the falsehood back out. Its truth in, false out, or false in, truth out. There is only one way it goes in each case, and it only goes these two ways (as the "not" operator is concerned). When we look at truth and falsity as the values of an expression, and if we operate on them with the "not" operator, then it takes the truth of the assertion that is stated an reverses its inclusion into this assertion, or else it reverses this reversal, and reverses the exclusion of it out. What it negates, it reverses in this way. And when it negates itself, it reverses that reversal which would have held if it hadn't reversed itself. And something that reverses its own reversal of something... it goes back to the beginning, before it reversed anything at all.
That means if we started with truth (a), then reversed it, we have case (b). If we started with case (b), then reversed that, we'd be back to case (a).
When someone wanted to deny our propositions, did they not say that our propositions were false? That is "not true"? When they offered their own, did they not have to say that they were true? And when we denied their propositions did we not declare them "false" and "not true" with the same meaning? If not, what other meaning? So in this case we are clear about what "not" does. What about when we say that our own sentence is "not false"? That is to say "not not true"? That means it is true! It doesn't mean something else. Whatever else someone might try to mean by "not", what we mean is clear, and it is clear to most people who have anything to say, and that is that "not" reverses the value of truth in a sentence. If we say it is true, then not true means it is false. If we say that it is false, then not false means that it is true, whatever the sentence.
We see that to deny a denial is to affirm the truth that was first denied. A = ~~A. These are the same. Also ~A = ~~~A, but that is rather awkward and not useful in most cases. Let's keep it down to three nots or less. Don't be naughty by making your nots too many, and too knotty.
So we have "not" pretty well down, even when it is a knotty proposition full of nots. What of the "&" operator? Well, it is simply what says about any two theses that they are both true. It doesn't mean they are joined together in any other way than that both of them are true, and neither of them are false. If we say that that sentence is true, then we are moving from the predicates of the theses in in it and into the the truth value of the logical operator. That's what happens when we move to any binary operator. We move away from then sentences and operate on the connection, the logical connection, which holds them together. If the rule of that logical connection, which is a logical operator, is satisfied by the sentences over which it operates, then that rule holds true in this case. If it is not the case, then it is not true in this case, and is false. The rule doesn't change, but the truth values of the sentences over which it operates sometimes do change. In cases where that change obeys the rule, then the truth value of the rule over those sentences remains the same. When those sentences' truth values do not match the requirements of the operator's rules which must be satisfied to make its own value true, then its value becomes false in those cases. The value of the truth operator depends, in each case, upon the truth value of the sentences over which it operates. "&" is a binary operator, but "~" is a singulary operator, operating on only one sentence. When a binary operator unites two sentences, it becomes the primary factor in their logical unity, making them one new logical sentence, upon which we may operate with a singulary operator, or else combine into other sentences by means of another binary operator, and so on. We use singulary because "unary" doesn't work, aesthetically.
Operators, like the sentences over which they operate, can be true or false. If they are true, then they are not false, if they are false, then they are not true.
So for the & operator, called "and", it is actually simple. It operates on two and only two sentences. The operator is true if, and only if, both sentences are true. If one or both are false, then the and operator, in this instance, is false. That is to say, it doesn't operate truly, but falsely here. That is to say that ~& is true. That's our thesis in the Law of Non-Contradiction, shown again here:
~(A & ~A)
The parentheses are used to group the sentences which have a main operator, and that operator is that upon which the operator immediately outside the parentheses operates. I made them both bold and red to indicate this. So this Law asserts that it is false that both A and ~A are true. It is always false, no matter what thesis you put in for A. Think about it carefully. If A is false, then that means ~A is true. If ~A is false, then that means that A is true. If A is true, that means ~A is false. If ~A is true, then that means A is false. All this makes good sense since we understand "not" so well. That is, that we understand it so well. We do, unless we've been naughty and not understood it. We must rectify that by being not naughty, and undoing the knots in our understanding of "not", so that we not not understand "not".
It is clear, however, all knots aside, that a proposition cannot be both true and false in the same way, at the same time! It can be true that it is true and false in different ways at the same time: If it is true that it is true, then it is false that it is not true. If it is false that it is true, then it is true that it is false. It can be true in the same way, but at different times: It can be true now, and false later, or false now, and true later (but only if it is not a Law). In that way, but only in that way, can a thesis, statement, sentences, fact, idea, and so on, be both true and false. It cannot be in both ways simultaneously, however. That is, not at the same time and in the same respect. So we can have the following:
(A & ~~A)
But we cannot have:
(A & ~A)
Because these formula are instantaneous, and without further qualification, eternal, and because the same exact meaning must be found for every instance of a term in such a context, then the first sentence is just fine, as it is simply saying "A is true, and A is true". The second sentences is saying "A is true, and A is not true". They both imply that they are saying this "at the same time, and in the same respect".
So we have instead the contrary of the second of the formula's above. We have this, our Logical Law of Non-Contradiction:
~(A & ~A)
We see clearly how this rule is true, and we can see pretty clearly right away how it isn't false. And we can see, by our understanding of ~ and & that it must be true by the definitions of the operations, and even a schoolchild gets this. So how do we get a clearer sense that it cannot be false? How do we demonstrate this so that it plays out like a confession in a mechanical way? We once again show the incredible feat that a True Law, and only a True Law, can accomplish. It will be demonstrated not only by its own declaration, which is so obvious it seems almost comical to declare it, it seems almost like a King stepping off his throne, and declaring himself King when it was already obvious and unchallenged by the entire realm of un-beheaded people. (Is that why they behead people who offend a king? So he can emphasize just how "uncrowned" they are, especially in contrast to himself?). Not only can this happen, and it does, but the opposition will declare it as well!! If it takes the crown and tries to put it on its own head, it cannot help but realize its awful deed, put the crown back where it belongs, and behead itself!!
On into the logical meat
Let B stand for (A & ~A)
Let A stand for (D & ~D)
Let D stand for (B & ~B)
(everything looks bad so far...)
B -> (A & ~A)
B
(A & ~A)
(D & ~D) & ~(D & ~D)
(D & ~D)
(B & ~B) & ~(B & ~B)
(B & ~B)
~B
B -> ~B
B
~(A & ~A)
So, by the rules already explored in the example concerning the Law of Identity, we've shown that contradicting the Law of Non-Contradiction literally contradicts itself, and allows us to firmly and plainly deduce the Law of Non-Contradiction. The Law of Non-Contradiction already implies itself because of the Law of Identity, so we find that the Law of Non-Contradiction is implied by itself (fittingly) and by its contrary, its nemesis, its opposite, the negation of the Law of Non-Contradiction (call it the law of contradiction). If both sides imply one side and not the other, then between the two of them, that one is the correct and true side of a controversy.
Granted that it allows us to imply this as a falsehood, and so we have implied what is false from what is initially stated as true. We have broken the Law of Identity, since (B -> ~B) says what an implication specifically does not say. But that means that this implication is not true. If it is not true, then we cannot imply the Law of Non-Contradiction from Contradicting the Law of Non-Contradiction, but also we know that if it were true, then we would have. But since we know that it is not true, then we know that its contrary, the Law of Non-Contradiction, must be true. Surely enough, this is exactly what is implied by the Law of Non-Contradiction as well. So if we keep the mechanics of implication, then we not only imply the Law of Non-Contradiction either way we proceed, whether we take it to be true or false as a premise, but we also prove that the contrary is false, because it leads to an implication of its own falsehood, which is patently impossible to imply, but even if we made it possible, then it implies itself false, and its contrary true!! What a whammy. That's like an episode of the three stooges in logical terms, with some Cosmo Kramer the Logician thrown in for good measure.
The questions arise, did I cheat? Did I put in that D = (B & ~B) unfairly? I think not, because I separated it a full two implications from B. Sometimes things do go in a circle. For example, the Law of Identity: (A -> A) -> [(A -> A) -> (A -> A)], and this can go on forever without contradiction. And we could instantiate all those implications by just declaring "A". We could let "B" stand for (A -> A) and then say A -> B, and then instantiate with A, and that brings us verily to B. Then we could say for (A -> A) -> (A -> A) we substitute the symbol "C" and so on. In each case we have no risk of a contradiction. But is that perhaps because we don't use "not" in this anywhere?
So let's look at the example given above:
I said "Let D stand for (B & ~B).
It was not ever stated that (B & ~B) was true verbatim when we said B -> (A & ~A), but only because, out of noble charity, we arbitrarily chose B in order to give a "nick name" to the thesis of contradiction. That is sensible, since if we said "A" for the nick name, it would look confusing. But, technically, there is no reason why we cannot. After all, we can say A -> A, so why can't we say A stands for something with "A" in it? We could say A = (A & ~A), and that is really what they are saying. Their position implies that any position, and its opposite, are both true. This includes their own position that, any position and its opposite are true, with its own opposite (our position, in fact) are both true! They must be saying this or it is no "law" at all. Their law is that our our Law is not truly a Law. They need show only one instance where it isn't, and then it will not be a Law anymore, since it is not guaranteed in all cases. Their law is pretty weak, since in merely needs be true "in at least one case, ~(A & ~A) must be false!" Wow... not as ambitious as ours. Ours must be true in every single case in existence, ever, and always, no matter what ifs, ands, buts, or nots. So we aimed to show simply that one case is really not true, the contrary of our Law itself. We went right for the jugular. But instead of doing so immediately, we gave it a fighting chance by way of letting it have a few intermediaries help shield it from direct assault. That's just how weak the opposing thesis is...
Because the form of their claim is, abstractly, is like this:
(@ & ~@)
then it really doesn't matter what we substitute for those terms. It only matters that they be substituted for both terms in the same way, with the same meaning, and that all the other symbols are applied in the same way. That goes for any thesis, including their own so:
(@ & ~@) & ~(@ & ~@) should be just as good as the formula for which it is an instance. Really, one is just as good as the other, as they are logically equivalent. Both say something, and its opposite, are both true, and one can stand for the other in any instance.
But instead of just jumping straight to that, I was courteous, and tried to separate the thesis from self-reference, and "filter" it through a go-between, to see if it would help. It didn't. Does it matter what I labeled "D"? No, because it wouldn't have mattered what I labeled the initial thesis when I labeled that "B". I could have labeled that D, and then found something else, say Z, to be (D & ~D). Again, D, in and of itself, implies (D & ~D) according to their rules. I simply made it mechanically more formal to reach it by putting it through a few gears. I'm not saying a more clever set up couldn't have been arranged at this level of simplicity, and it would be intriguing to see (anyone?). I just think that this suffices for the demonstration that it is easy to find a case where the thesis contradicts itself, and that is because the thesis does, by its definition, contradict itself.
In any event, it is clear that this could be no other way, since we cannot imply what is false from what is true, and if (A & ~A) is true, then we can immediately imply (A -> ~A), which is itself an impossibility, because the antecedent of a proposition cannot be true and then imply the consequent that is false, and that is literally what this formula does. It does this because it says that A is true in the antecedent (and requires this for an implication), then in the consequent it denies that premise, which we know is false because A is true, not ~A. It is impossible to imply this, yet this is just what the contradictory thesis enables us to do. That means it violates the law of identity as well, because it declares that the opposite of that Law must also be true alongside it, and we already saw what happened to that pretender's usurping head. Again, as stated above, if it can't imply this because the conclusion is false, then the conclusion must be true, and that is the opposite of the premise, and that premise therefore, the premise "A" (their usurping un-law) is false. And this is one of those cases where it is meaningful to imply what is true, from what is false. From bad rebellions come great boons!
A few concrete examples and ethical considerations
Aside from these logical considerations, what could we put into these schematics to give a sense of how this could be true "in the flesh"? We could use some creative and fun ones!
Let's say that someone who is asserting this proposition comes along, and we need to show them a case which would convince them that they are absurd. Let's borrow a page from Aristotle, who made quite a big deal out of this in his day, and say that the fact of the person who asserts this, that he is asserting it, is "A".
If he is asserting it, then he is asserting that he has made a rule saying that it is also true that he is asserting it and not asserting it. But that means when he is talking, he is not talking. And when he is not talking, we are not listening....
If he deplores that we are rude, we just explain that, when we are rude, we really are not rude.
If he claims that these maneuvers prove his point, then we explain that when we prove his point, we really do not prove it. So when we prove his point for him, we leave it unproven. Therefore, his point is unproven.
Of course, if he is truly an ass, he'll say that this proves his point. But... Doesn't that really just prove our point? That he is an ass? If he says that he is not an ass, then we say that this proves that he is, because if he is not, then he also is, an ass.
Again, we are not being rude, because if we are being rude, then we are also not being rude. We are being honest. He says we aren't, but that means we are.
Etc.
It is just one way it could happen.
Let's say it is less adversarial. Imagine taking a contradictorialist on a date! Every place you go is great, because it isn't, and every thing you do is fun, because it isn't. Everything you say is funny, because it isn't. And every time she says no, she is implying she means yes.
Well, it is going to be fun and not fun at the same time, perhaps. It depends on how you interpret the formula. We mean at the same time, and in the same respect. That means it must be the same subject that receives the predicate, and its contrary, and both in the same respect. It can't be fun for one person, and not the other, and satisfy this formula. It can't be fun now, and in this way, but not then, and in another way, and really be a "contradiction". That is just a "change", and that is something that happens because it isn't a contradiction, but a complete alteration, or growth, or diminution, or movement, or generation, or destruction of the predicate itself or some of its attributes. It must be both fun and not fun, at the same time, in the exact same way, and for the exact same person if it is to be called a true contradiction. We assert that this is impossible, and therefore in logical thought, we needn't ever entertain it. It is therefore a Law of Logical Thought, and it is ironclad in its guarantee, always and forever.
As Avicenna meanly said, let us beat them until they declare that one cannot be beaten, and not beaten, at the same time. I would add, just as a precaution, that if they say they are being beaten here, but not there, then beat them everywhere at once, and stop only when they admit that they cannot be beaten everywhere at once, and not beaten everywhere at once. If they claim that it is at this time, but not another time, beat them incessantly, everywhere at once.. you get the picture.
The best way this rule is demonstrated, in fact, is by finding someone who understands it and notice the smooth flow of communication, and the progress of logical reasoning that is possible. Notice that when contradictions are found, both are perplexed and wish to resolve them. Notice that people who don't respect this rule, really aren't worth reasoning with, and probably not worth talking with anyway.
After all, if you say no, they may think you mean yes... and if they are evil enough, they may pretend that they respect your wishes, or will honor their agreements with you, or even that they understand what you are saying. They may be very good actors. After all, actors who no nothing about physics or theology play these characters rather convincingly all the time, at least within the frame or upon the stage in which they are "in their element". Criminals and cons, and demons, are always looking for a way to get a victim (also called a "mark") into the element in which the crime has optimal chance of success.
Even these buffoons cannot truly contradict, but they let you believe your assumptions about them even though those assumptions are false... namely that they are how they appear...
But they are not, and they will contradict this when the time comes to get what they want from you. Until then, they just nod their head and smile... But when finally they prove that they are not what they seem, at least we have the logical process which explains to us this very fact, for if they were what they seem, they'd not contradict it with actions which are not befitting, and so we have that much more caution from the beginning, and if we are tricked we have that much more reflex and integrity to act in our own defense, without "self-contradiction".
It seems only evil beings truly seek to benefit from the principles of asserted contradictions, which are really reversals of truth which are not, themselves, true. But they want to share a space with someone who has integrity, and consistency, just long enough to get their goods, so they pretend to be consistent with their appearance, and make their appearance consistent with a decent person, for as long as possible in order to get the most gains, and minimize their risks. If we let them also take our very own Logical Thought from us, then we are doomed at the highest levels of our minds, and all else is like taking candy from a baby.
Everyone in this cafe has some experience with what I'm talking about here.
~(A & ~A)
Some thoughts before we proceed
We should first discuss the logical operations we've already begun using, now that we've displayed them in context. The "not" operator, which is here represented with a tilde, ~, simply negates the thesis which follows it. It means, "to negate", to declare that it is "not true that (thesis)". If we declare that a thesis is not true, then that means we simply have a case where the truth is outside that thesis, because there is no truth in it.
What happens to the thesis? Nothing. It is just an idea, anyway. It sits in the repository of ideas and does nothing except carry the "not" operator in that context. Perhaps in some other context it will be freed of the not operator, but not in that precise context. For example, of the thesis is "My alien friends are Divine Messengers", then it is saying all that with the implied preface that "it is true that" my alien.. etc. So when we negate this, we are in effect saying that "it is not true that"... etc. Why it is not true is another matter. We simply declared the whole sentence false.
Aristotle goes into this in his writings where he declares that logic operates upon basic sentence grammar, insofar as we use it ordinarily. A sentence has a subject and a predicate. If "it is true", then "it" is the subject and "true" is the predicate. The verb "is" connects the two, and is technically considered part of the predicate phrase. That being the case, the logical operator "true", which is not written, is, metaphysically, a statement about the being of the subject: "It is" it seems to declare. If it turns out we want to say more than something barely "is", then we say "how it is", and we add further qualifications to that basic existential statement. In other words, we add to the predicate.
But everything that follows the copular verb either is or is not, just as the subject either is or is not. Specifically, since the predicate is conjoined onto the subject as a "member" of its own being, we say that the predicate train either is or is not true of the predicate. Even existence itself, the is verb, is really a predicate (in logic for sure it is). Either "is" is true of the subject, or it is not true. There are two is's. The is which is true of the subject itself in reality, and the is which connects the statement itself as a subject, as a whole, to its predicates. The predicates available to any statement in logic of this ordinary sort are "true" and "false".
Look at the sentence, "It is true". Now if it (the whole sentence) is true, then "it is true", is true! The whole sentence has been turned into a subject of our consideration, and we have added a predicate phrase to describe its qualities. Clearly the sentence simply "is", so we leave that unremarked in this extended analysis. But the interest for us is whether or not the sentence is true, which is to say does it describe reality in the way it purports to do. If it is true, then we can, in the context of our discussion, just say the sentence without further qualification. Saying a sentence like this is assumed to be saying something meaningful about reality, and so is assumed to be a declaration of the truth, as qualified by this sentence as spoken.
So we say "My alien friends are Divine Messengers". The subjects is "My alien friends" and the predicate is "are divine messengers". In our logical analysis we'll just "fuse" the two is's into the phrase as stated, so that if it is true, we just let it stand. "Are", in this case, stands for the relation between the subject and predicate, but it also stands for the truth of that connection (in our analysis, and as used in practice since people say these things as if they mean them), and so we don't need to say "This sentence is true, namely that..." etc. We let it stand if it is true, unmodified.
If it is not true, then we modify the subject predicate relation. We add "not" to the copula. "My alien friends are not divine messengers". So we have a sense of the not operator right there. It disjoins the subject and predicate as related in that sentence. As to our justification for doing that, it requires further analysis of the subject and the qualifications it has both for being as such and for being as stated, as "Divine Messengers". If he has no friends, then has no alien friends, then he has no alien friends who are divine messengers. Maybe that's why this sentence is false. Or, if he has friends, maybe they are people who tell him what to say, or inspire him with EM stimulation to his brain, are human and not alien (although perhaps they work with aliens, who are their friends). Maybe they are Divine Messengers, maybe not, but they are not alien, and that matters here, as someone is either confused or deceived or lying. Or maybe he has alien friends, but they aren't divine at all.
In any event, this is "sentential" logic, and we are modifying the copula or other verb, with our "not" operator. That is sufficient for our uses of logic here. There are other ways of going about logical analysis, but they are not introductory, and we are just looking at the Laws of Logical Thought and the rules sufficient to introduce and explicate them.
So the "not" operator is well-understood by now. Let's look at it further. What do we mean by "not" when it is applied to a sentence which declares already that something is "not" true? We are saying that "it is not true that it is not true" and then whatever the content is. How does "not" operate upon itself? Well it must operate upon the same verb that is already operated upon by the not which has come before it, but it is already modified. So it would be redundant to modify it to no effect, as if saying "not" more than once emphasized it or something. That'd just be stuttering. If you said "that not not true", I'd think you meant that is true! Why? Because the "not" operator makes the truth of a sentence false. If a sentence "is true" then it makes it false by declaring it "is not true". If it is not true, then it is false. Those are our primary values in logic, are they not? If we mean by "false" or "not true" that the truth is completely outside the space in which it is asserted that something is true, then that means all truth is outside that space, and the falsehood of that assertion is what is in that space. If we mean that this is not the case, then we must reverse this situation, and put the truth of that statement inside its sphere of expression, and put the falsehood back out. Its truth in, false out, or false in, truth out. There is only one way it goes in each case, and it only goes these two ways (as the "not" operator is concerned). When we look at truth and falsity as the values of an expression, and if we operate on them with the "not" operator, then it takes the truth of the assertion that is stated an reverses its inclusion into this assertion, or else it reverses this reversal, and reverses the exclusion of it out. What it negates, it reverses in this way. And when it negates itself, it reverses that reversal which would have held if it hadn't reversed itself. And something that reverses its own reversal of something... it goes back to the beginning, before it reversed anything at all.
That means if we started with truth (a), then reversed it, we have case (b). If we started with case (b), then reversed that, we'd be back to case (a).
When someone wanted to deny our propositions, did they not say that our propositions were false? That is "not true"? When they offered their own, did they not have to say that they were true? And when we denied their propositions did we not declare them "false" and "not true" with the same meaning? If not, what other meaning? So in this case we are clear about what "not" does. What about when we say that our own sentence is "not false"? That is to say "not not true"? That means it is true! It doesn't mean something else. Whatever else someone might try to mean by "not", what we mean is clear, and it is clear to most people who have anything to say, and that is that "not" reverses the value of truth in a sentence. If we say it is true, then not true means it is false. If we say that it is false, then not false means that it is true, whatever the sentence.
We see that to deny a denial is to affirm the truth that was first denied. A = ~~A. These are the same. Also ~A = ~~~A, but that is rather awkward and not useful in most cases. Let's keep it down to three nots or less. Don't be naughty by making your nots too many, and too knotty.
So we have "not" pretty well down, even when it is a knotty proposition full of nots. What of the "&" operator? Well, it is simply what says about any two theses that they are both true. It doesn't mean they are joined together in any other way than that both of them are true, and neither of them are false. If we say that that sentence is true, then we are moving from the predicates of the theses in in it and into the the truth value of the logical operator. That's what happens when we move to any binary operator. We move away from then sentences and operate on the connection, the logical connection, which holds them together. If the rule of that logical connection, which is a logical operator, is satisfied by the sentences over which it operates, then that rule holds true in this case. If it is not the case, then it is not true in this case, and is false. The rule doesn't change, but the truth values of the sentences over which it operates sometimes do change. In cases where that change obeys the rule, then the truth value of the rule over those sentences remains the same. When those sentences' truth values do not match the requirements of the operator's rules which must be satisfied to make its own value true, then its value becomes false in those cases. The value of the truth operator depends, in each case, upon the truth value of the sentences over which it operates. "&" is a binary operator, but "~" is a singulary operator, operating on only one sentence. When a binary operator unites two sentences, it becomes the primary factor in their logical unity, making them one new logical sentence, upon which we may operate with a singulary operator, or else combine into other sentences by means of another binary operator, and so on. We use singulary because "unary" doesn't work, aesthetically.
Operators, like the sentences over which they operate, can be true or false. If they are true, then they are not false, if they are false, then they are not true.
So for the & operator, called "and", it is actually simple. It operates on two and only two sentences. The operator is true if, and only if, both sentences are true. If one or both are false, then the and operator, in this instance, is false. That is to say, it doesn't operate truly, but falsely here. That is to say that ~& is true. That's our thesis in the Law of Non-Contradiction, shown again here:
~(A & ~A)
The parentheses are used to group the sentences which have a main operator, and that operator is that upon which the operator immediately outside the parentheses operates. I made them both bold and red to indicate this. So this Law asserts that it is false that both A and ~A are true. It is always false, no matter what thesis you put in for A. Think about it carefully. If A is false, then that means ~A is true. If ~A is false, then that means that A is true. If A is true, that means ~A is false. If ~A is true, then that means A is false. All this makes good sense since we understand "not" so well. That is, that we understand it so well. We do, unless we've been naughty and not understood it. We must rectify that by being not naughty, and undoing the knots in our understanding of "not", so that we not not understand "not".
It is clear, however, all knots aside, that a proposition cannot be both true and false in the same way, at the same time! It can be true that it is true and false in different ways at the same time: If it is true that it is true, then it is false that it is not true. If it is false that it is true, then it is true that it is false. It can be true in the same way, but at different times: It can be true now, and false later, or false now, and true later (but only if it is not a Law). In that way, but only in that way, can a thesis, statement, sentences, fact, idea, and so on, be both true and false. It cannot be in both ways simultaneously, however. That is, not at the same time and in the same respect. So we can have the following:
(A & ~~A)
But we cannot have:
(A & ~A)
Because these formula are instantaneous, and without further qualification, eternal, and because the same exact meaning must be found for every instance of a term in such a context, then the first sentence is just fine, as it is simply saying "A is true, and A is true". The second sentences is saying "A is true, and A is not true". They both imply that they are saying this "at the same time, and in the same respect".
So we have instead the contrary of the second of the formula's above. We have this, our Logical Law of Non-Contradiction:
~(A & ~A)
We see clearly how this rule is true, and we can see pretty clearly right away how it isn't false. And we can see, by our understanding of ~ and & that it must be true by the definitions of the operations, and even a schoolchild gets this. So how do we get a clearer sense that it cannot be false? How do we demonstrate this so that it plays out like a confession in a mechanical way? We once again show the incredible feat that a True Law, and only a True Law, can accomplish. It will be demonstrated not only by its own declaration, which is so obvious it seems almost comical to declare it, it seems almost like a King stepping off his throne, and declaring himself King when it was already obvious and unchallenged by the entire realm of un-beheaded people. (Is that why they behead people who offend a king? So he can emphasize just how "uncrowned" they are, especially in contrast to himself?). Not only can this happen, and it does, but the opposition will declare it as well!! If it takes the crown and tries to put it on its own head, it cannot help but realize its awful deed, put the crown back where it belongs, and behead itself!!
On into the logical meat
Let B stand for (A & ~A)
Let A stand for (D & ~D)
Let D stand for (B & ~B)
(everything looks bad so far...)
B -> (A & ~A)
B
(A & ~A)
(D & ~D) & ~(D & ~D)
(D & ~D)
(B & ~B) & ~(B & ~B)
(B & ~B)
~B
B -> ~B
B
~(A & ~A)
So, by the rules already explored in the example concerning the Law of Identity, we've shown that contradicting the Law of Non-Contradiction literally contradicts itself, and allows us to firmly and plainly deduce the Law of Non-Contradiction. The Law of Non-Contradiction already implies itself because of the Law of Identity, so we find that the Law of Non-Contradiction is implied by itself (fittingly) and by its contrary, its nemesis, its opposite, the negation of the Law of Non-Contradiction (call it the law of contradiction). If both sides imply one side and not the other, then between the two of them, that one is the correct and true side of a controversy.
Granted that it allows us to imply this as a falsehood, and so we have implied what is false from what is initially stated as true. We have broken the Law of Identity, since (B -> ~B) says what an implication specifically does not say. But that means that this implication is not true. If it is not true, then we cannot imply the Law of Non-Contradiction from Contradicting the Law of Non-Contradiction, but also we know that if it were true, then we would have. But since we know that it is not true, then we know that its contrary, the Law of Non-Contradiction, must be true. Surely enough, this is exactly what is implied by the Law of Non-Contradiction as well. So if we keep the mechanics of implication, then we not only imply the Law of Non-Contradiction either way we proceed, whether we take it to be true or false as a premise, but we also prove that the contrary is false, because it leads to an implication of its own falsehood, which is patently impossible to imply, but even if we made it possible, then it implies itself false, and its contrary true!! What a whammy. That's like an episode of the three stooges in logical terms, with some Cosmo Kramer the Logician thrown in for good measure.
The questions arise, did I cheat? Did I put in that D = (B & ~B) unfairly? I think not, because I separated it a full two implications from B. Sometimes things do go in a circle. For example, the Law of Identity: (A -> A) -> [(A -> A) -> (A -> A)], and this can go on forever without contradiction. And we could instantiate all those implications by just declaring "A". We could let "B" stand for (A -> A) and then say A -> B, and then instantiate with A, and that brings us verily to B. Then we could say for (A -> A) -> (A -> A) we substitute the symbol "C" and so on. In each case we have no risk of a contradiction. But is that perhaps because we don't use "not" in this anywhere?
So let's look at the example given above:
I said "Let D stand for (B & ~B).
It was not ever stated that (B & ~B) was true verbatim when we said B -> (A & ~A), but only because, out of noble charity, we arbitrarily chose B in order to give a "nick name" to the thesis of contradiction. That is sensible, since if we said "A" for the nick name, it would look confusing. But, technically, there is no reason why we cannot. After all, we can say A -> A, so why can't we say A stands for something with "A" in it? We could say A = (A & ~A), and that is really what they are saying. Their position implies that any position, and its opposite, are both true. This includes their own position that, any position and its opposite are true, with its own opposite (our position, in fact) are both true! They must be saying this or it is no "law" at all. Their law is that our our Law is not truly a Law. They need show only one instance where it isn't, and then it will not be a Law anymore, since it is not guaranteed in all cases. Their law is pretty weak, since in merely needs be true "in at least one case, ~(A & ~A) must be false!" Wow... not as ambitious as ours. Ours must be true in every single case in existence, ever, and always, no matter what ifs, ands, buts, or nots. So we aimed to show simply that one case is really not true, the contrary of our Law itself. We went right for the jugular. But instead of doing so immediately, we gave it a fighting chance by way of letting it have a few intermediaries help shield it from direct assault. That's just how weak the opposing thesis is...
Because the form of their claim is, abstractly, is like this:
(@ & ~@)
then it really doesn't matter what we substitute for those terms. It only matters that they be substituted for both terms in the same way, with the same meaning, and that all the other symbols are applied in the same way. That goes for any thesis, including their own so:
(@ & ~@) & ~(@ & ~@) should be just as good as the formula for which it is an instance. Really, one is just as good as the other, as they are logically equivalent. Both say something, and its opposite, are both true, and one can stand for the other in any instance.
But instead of just jumping straight to that, I was courteous, and tried to separate the thesis from self-reference, and "filter" it through a go-between, to see if it would help. It didn't. Does it matter what I labeled "D"? No, because it wouldn't have mattered what I labeled the initial thesis when I labeled that "B". I could have labeled that D, and then found something else, say Z, to be (D & ~D). Again, D, in and of itself, implies (D & ~D) according to their rules. I simply made it mechanically more formal to reach it by putting it through a few gears. I'm not saying a more clever set up couldn't have been arranged at this level of simplicity, and it would be intriguing to see (anyone?). I just think that this suffices for the demonstration that it is easy to find a case where the thesis contradicts itself, and that is because the thesis does, by its definition, contradict itself.
In any event, it is clear that this could be no other way, since we cannot imply what is false from what is true, and if (A & ~A) is true, then we can immediately imply (A -> ~A), which is itself an impossibility, because the antecedent of a proposition cannot be true and then imply the consequent that is false, and that is literally what this formula does. It does this because it says that A is true in the antecedent (and requires this for an implication), then in the consequent it denies that premise, which we know is false because A is true, not ~A. It is impossible to imply this, yet this is just what the contradictory thesis enables us to do. That means it violates the law of identity as well, because it declares that the opposite of that Law must also be true alongside it, and we already saw what happened to that pretender's usurping head. Again, as stated above, if it can't imply this because the conclusion is false, then the conclusion must be true, and that is the opposite of the premise, and that premise therefore, the premise "A" (their usurping un-law) is false. And this is one of those cases where it is meaningful to imply what is true, from what is false. From bad rebellions come great boons!
A few concrete examples and ethical considerations
Aside from these logical considerations, what could we put into these schematics to give a sense of how this could be true "in the flesh"? We could use some creative and fun ones!
Let's say that someone who is asserting this proposition comes along, and we need to show them a case which would convince them that they are absurd. Let's borrow a page from Aristotle, who made quite a big deal out of this in his day, and say that the fact of the person who asserts this, that he is asserting it, is "A".
If he is asserting it, then he is asserting that he has made a rule saying that it is also true that he is asserting it and not asserting it. But that means when he is talking, he is not talking. And when he is not talking, we are not listening....
If he deplores that we are rude, we just explain that, when we are rude, we really are not rude.
If he claims that these maneuvers prove his point, then we explain that when we prove his point, we really do not prove it. So when we prove his point for him, we leave it unproven. Therefore, his point is unproven.
Of course, if he is truly an ass, he'll say that this proves his point. But... Doesn't that really just prove our point? That he is an ass? If he says that he is not an ass, then we say that this proves that he is, because if he is not, then he also is, an ass.
Again, we are not being rude, because if we are being rude, then we are also not being rude. We are being honest. He says we aren't, but that means we are.
Etc.
It is just one way it could happen.
Let's say it is less adversarial. Imagine taking a contradictorialist on a date! Every place you go is great, because it isn't, and every thing you do is fun, because it isn't. Everything you say is funny, because it isn't. And every time she says no, she is implying she means yes.
Well, it is going to be fun and not fun at the same time, perhaps. It depends on how you interpret the formula. We mean at the same time, and in the same respect. That means it must be the same subject that receives the predicate, and its contrary, and both in the same respect. It can't be fun for one person, and not the other, and satisfy this formula. It can't be fun now, and in this way, but not then, and in another way, and really be a "contradiction". That is just a "change", and that is something that happens because it isn't a contradiction, but a complete alteration, or growth, or diminution, or movement, or generation, or destruction of the predicate itself or some of its attributes. It must be both fun and not fun, at the same time, in the exact same way, and for the exact same person if it is to be called a true contradiction. We assert that this is impossible, and therefore in logical thought, we needn't ever entertain it. It is therefore a Law of Logical Thought, and it is ironclad in its guarantee, always and forever.
As Avicenna meanly said, let us beat them until they declare that one cannot be beaten, and not beaten, at the same time. I would add, just as a precaution, that if they say they are being beaten here, but not there, then beat them everywhere at once, and stop only when they admit that they cannot be beaten everywhere at once, and not beaten everywhere at once. If they claim that it is at this time, but not another time, beat them incessantly, everywhere at once.. you get the picture.
The best way this rule is demonstrated, in fact, is by finding someone who understands it and notice the smooth flow of communication, and the progress of logical reasoning that is possible. Notice that when contradictions are found, both are perplexed and wish to resolve them. Notice that people who don't respect this rule, really aren't worth reasoning with, and probably not worth talking with anyway.
After all, if you say no, they may think you mean yes... and if they are evil enough, they may pretend that they respect your wishes, or will honor their agreements with you, or even that they understand what you are saying. They may be very good actors. After all, actors who no nothing about physics or theology play these characters rather convincingly all the time, at least within the frame or upon the stage in which they are "in their element". Criminals and cons, and demons, are always looking for a way to get a victim (also called a "mark") into the element in which the crime has optimal chance of success.
Even these buffoons cannot truly contradict, but they let you believe your assumptions about them even though those assumptions are false... namely that they are how they appear...
But they are not, and they will contradict this when the time comes to get what they want from you. Until then, they just nod their head and smile... But when finally they prove that they are not what they seem, at least we have the logical process which explains to us this very fact, for if they were what they seem, they'd not contradict it with actions which are not befitting, and so we have that much more caution from the beginning, and if we are tricked we have that much more reflex and integrity to act in our own defense, without "self-contradiction".
It seems only evil beings truly seek to benefit from the principles of asserted contradictions, which are really reversals of truth which are not, themselves, true. But they want to share a space with someone who has integrity, and consistency, just long enough to get their goods, so they pretend to be consistent with their appearance, and make their appearance consistent with a decent person, for as long as possible in order to get the most gains, and minimize their risks. If we let them also take our very own Logical Thought from us, then we are doomed at the highest levels of our minds, and all else is like taking candy from a baby.
Everyone in this cafe has some experience with what I'm talking about here.
~ ++ Hanc Defendemus ++ ~