Example 1) The Law of Identity (Modified)
A -> A
This says plainly "If A, then A". This is the implication arrow (
->), and it is the arrow which functions to mean that if what is to the left of it is true, then what is to the right cannot be false. If what is to the left is true, and what is to the right is false, then this arrow is, in this case,
itself false (doesn't operate truthfully in this case, except in saying that its operation is false, which is itself "true" about it).
If an argument were in the form of an implication, then the conclusion would go to the right of the arrow, because the conclusion is what is implied from the premises. The conclusion "follows" from them because, as already defined, if the premises are all true, then the conclusion cannot be false, and must be true. The premises would all go to the left of the same arrow of implication.
The implication arrow says the same exact thing as an argument, except in the ideal form of an implication statement, we only present two statements, one which is to the left of the arrow (the antecedent), and one to the right of the arrow (the consequent). It is a "binary" operator. In the case of the question of what an argument is in respect to an implication statement, it is a statement of the form
P -> Q such that
P represents
the conjoined totality of all the premise statements (usually two), stated together as "P" in this form, it means that
if they are all true then "P",
their conjunction, is true. In this case
Q,
their conclusion, is the consequent of this implication, and cannot be false
if the implication is true, as that requires that "if the antecedent is true (in this case all the premises of the argument), then the consequent (in this case the conclusion) cannot be false (or must be true)". Arguments whose forms are such that when they are converted into this implication we know the implication is true, are valid arguments. This though they may have false statements
in them, because, for example, if happens to be the case that the argument form is valid (and so this construction of the argument as an implication statement
is true), then if the consequent (here, our "conclusion") is
false, then so also must be the antecedent (here the statement that "all the premises are true"), for it
cannot be true, as then we would have an implication with a false consequent and a true antecedent, which is not possible if the implication is
itself true.
Look at it freshly
(set of premises) -> (conclusion). If the premises are
all true, then the conclusion is true. So, if we want this guarantee of truth such that we can say this: that
if there are certain things that we know then therefore, and with no doubt at all, we can be certain that there are other things we
also know, then this idea of implication serves that purpose.
Now look again at the Law of Identity as formulated here:
A -> A
If we want A to imply A, then when we know that A is true, we cannot then say that A is false, because that would mean that A cannot be certain after it is certain, but that means that nothing can ever be implied, no matter what. That would mean that if we knew something, it wouldn't ever be useful for guaranteeing we knew anything else which it would imply. But we know that would be absurd, because we would be unable to support that. I want to point out here that I am taking the Law of Identity into a modified form, because that Law simply states that
A = A. I don't take this to be logical
per se, and I would say that is truly a "Law of Thought" as such. But what is interesting about this rule is simply that without it we cannot even coherently communicate. What is
logically interesting about it is that if we have asserted "A" as true, then we can be sure that it is not false. In other words "If A is true, then A is true." The
truth of A is what is
logically crucial
about its identity, which is already thought to generally be preserved in all iterations of it. A hybrid way to speak of these two ways of talking about identity would go something like this:
Let it be true that:
= = "corresponds identically with"
B = The meaning of A'
C = The meaning of A''
A = The form of a term called "A" or designated "A"
A' = One iteration of A
A" = Another iteration of A different in time or space or in location within the same argument, as A', but in form identical to A' so that as to form (A = A' = A")
Then we know the following:
[(B = A)
& (C = A)
] -> (B
= C)
Or "If the meaning of one iteration of a term corresponds identically to the form of that term, and the meaning of another iteration of the same term as the first iteration
also corresponds to its form, then we must take it that the meaning of the first iteration of a term corresponds identically with the meaning of the second iteration of that same term. But this doesn't really look the same as "A = A", although this is what that formula is really saying in the first Law of Thought.
But I wish to construe the significance of what this identity implies about a term and focus it upon something interesting about that term, namely that it's truth is preserved along with its identity, across any series of manipulations so that, if its truth is once accepted in a "logical space" or "universe" then that truth must always be accepted as a valid consequence. So this isn't just
any implication, this is an implication about the identity and preservation of
truth itself. It is also a great way to introduce perhaps the most important logical operator,
implication (or "conditional"). So let's explore the significance of this modified form of the law of identity. We can ask ourselves, how is it evident to us that this implication is always true, no matter what form of content is placed within "A". What is the reason for this?
The reason can be shown by letting us say that the thesis "A" is true, but not let us immediately imply it from itself, so that we can't say it again in an infinitely small time afterwards (or beforehand), and if we allowed that then maybe that would make sense. After all, it is conceivable that "A" was true one planck second ago, and now false, or else now true, and one plank second later will be false. But if we say it is true now, then, is it not true, now? The intuition we had in saying it is true now immediately and absolutely prevents us from saying, in that moment and with that thought, anything else. "It is true that A" our mind says, "so, it is true, that A". Imagine if you could think that thought in an exact moment. All you are doing is duplicating the same content. Theoretically, you could repeat the stated truth of A an infinite number of times in one instant, or else repeat it infinitely on a time line perpendicular to the dimension in which you are saying it here, and in that moment when these two timelines intersect, you'd be saying "A", and in the timeline perpendicular to here, you'd be saying one LOOOOOONG "A", and that is consistent with saying it "once" in this instant, here on this horizontal timeline.
Saying "A is true" is just consistent with saying that A is true, unless something else prevents it. But in one single instance, what could prevent you from repeating it as many times as you want, as long as you "had the time", which is to say an infinite amount of time transverse to normal time and intersecting it in one instant, and if you had the energy, which is to say an infinite amount of mental energy for filling up that time with the contemplation of the truth of A?
So since an implication operator works "in an instant", and for that instant, it operates "infinitely"
within that instant, and
preserves truth within it. And since our example gives us a good intuition of how that might work, it seems clear that it is nothing but sensible to say that, at least for this one instance, for this one notion, call it "A", that (A -> A) cannot be anything but true. "If A,"
: "then A." (That's what you look like when you infinitely process the truth of A in one instant). Now, we have a strong sense of the nature of not only the law of identity as a meaningful axiom, but we see it is intimately connected to the nature of logical thought by way of the idea of "implication".
I'm not telling you that you must hold this axiom to be true, nor that you must agree with it, or with me for that matter, in any way. I'm just saying... well, try to show me a reason to think that it isn't, and that will be awesome to behold! But you mustn't use this axiom in that process, nor can you draw implications...
Because if you do, that will backfire.
That would be like trying to imply that something is false from the assumption that it is true... and implication means exactly
not that. In logic, in fact, it means
anything but that. You can get away with implying what is true from what is true, what is false from what is false, or what is true from what is false! But you can't, by the very meaning of what we mean by "implication", imply what is false from what is true! Mainly, however, the value of implication is to draw true conclusions from true premises. Therefore, if you wanted to draw a conclusion that you claim is true, that the law of identity is false, you'd couldn't include in your premises that the law of identity is true! Logic is instantaneous in this regard.
This is something that people out there in the world are very slippery about. They like to use the law of identity when it is convenient for them, but drop it when it is not. But the problem with this approach is that it is not valid. It means that you are dealing with a sloppy-headed person, who can't think straight, or at least doesn't talk straight, and so is a 'fibber'. If you were at the bar next to the capital building which was called "The Cloak and Dagger" at 8:31 pm and 22 seconds, sitting at the counter, then you were doing precisely that, precisely there, precisely then. This was true. How can it be false as an implication? So if they lie, they can say they were there, then, and in that way, and then the next second they can say that they weren't, but we know that is impossible to be the case that they were, and then they weren't! People can mix up how the talk about the truth, but the truth doesn't mix itself up with them to suit their fantasies...
But with logic, fibbers are notoriously sloppy, because logic always preserves truth, and when we want to say something is true, then for those conditions when that happened, we can never say it is false, EVER. Not unless when we first said it was true, it really wasn't in the first place, and then we must eventually admit that it was
false. And in that case
this is the truth, and will never change!
Liars find this a scary universe, this logical universe, so they usually go into politics or acting (pretending), or other forms of fraud. Both lie and are falsifications of what is real which each must pretend that they are doing something which appears real, but in fact is just a well-contrived appearance that deceives. The first pretends to be taking care of the good of everyone and in good faith (they do not and are not), and the second pretends to be something in order to convey the images of a story, and really they "act like" and only in that way do they "act". The
real action is what is done in the world by real actors, and this all without undue regard for appearances (although appearances must also be taken into account sometimes. it is just that in "theater" that is in essence
all of what is important to regard. Politics, and much of culture and human society is such a contrived falsehood meant to look true but only appearing to be so, that in fact stage and film acting are actually imitations
of real life in a more exact way than it would at first appear! "Real life" is a lot of pretending, with a lot of malintent behind it, so that the deceptions which are produced by the pretending enable the malintent behind that facade to reach its goal. It is
organized crime. Then, within that organized crime of fraudulent pretenses, culture itself becomes a reification of it, framing into a "larger than life status" the fantasy that what goes on in the first place was genuine when it wasn't! Art in this context
doubles the intensity of the delusion which was already presented in the world at large. This is what Shakespeare was saying in fact. But this is a bit of a digression, although a pertinent one.
As to attempting to imply what is false from what is true, you can try it, but the problem is that you can't have the law of identity in your premises
and not have it in your conclusion. It is not possible to imply the falsehood of something if, in the premises used, it is true. This is because it is instantaneously true, over the whole domain of the formula of implication, because logic is instantaneous (and eternal), and so a term isn't true on the left side of the implication arrow and then not true after it gives rise to a conclusion on the other side! It isn't like a hen that lays an egg and disappears afterward! The truth is immutable on both sides of the arrow, and what is true on one side cannot become untrue on the other. We cannot say "B -> ~B" unless B is actually false! But down below I promise to demonstrate even this, mechanically, using the rules we have, to show that not only is the Law of Identity (or, ID) consistent, but it is consistent in showing how the contrary of itself is
not consistent, and I'll also show that the contrary of the Law of Identity must be inconsistent with itself in an attempt to establish itself! Which is to say it cannot be established, but must be arbitrarily asserted, but since when was an arbitrary assertion ever a demonstration?
Therefore, to logically demonstrate the untruth (how else can untruth be demonstrated except logically?) of the logical Law of Identity, you cannot employ it in your premises. And if you want to demonstrate something to be true, not just whenever, but for all cases, then you must demonstrate that it is illogical to deny it in any case. Since if something is always true then it is implied in all cases, all the time. That means also that I should be able to pick
any case, and find that would be illogical to deny a law in that case, or how can it truly be a
law?
So how, pray tell, can you ever
demonstrate the falsehood of the Law of Identity? Here's a way I think it would have to go:
Let A be the statement "the law of identity is false", which is our desired conclusion (it no longer capitalized, as it is stripped of its Glory)
If we know the set of premises called "B" is true, upon which we would imply A, then this implies "A", which is to say, (B -> A)
Therefore A
QED
But hold on, if we know that by the premises, "B", being true, and that this is the basis of knowing "A" to be true, "A" meaning that the law of identity is false, then B must first be found true for this to work. And to say that the law is false, this must be found to be true in just one case. In that case the truth of B must show that "A" is true, which is that the Law of Identity, or "ID", is false, etc.
But we have to get B already true if we demonstrate that A is true (ID is false) based
on B, so we can't just start with (B -> A) to get A, because that would be like saying "There is a condition B, such that
if B is true, then A is true, therefore A is true. That is just as if to say "There is something, rain, such that IF it is raining, then the streets are wet, therefore the streets are wet". But that is not valid. This statement right now just says that rain exists in general, and that if it is raining, then the streets are wet. It doesn't say that it is raining! So how can we conclude that the streets are wet? Because it is a hypothetical statement, the implication (B -> A) requires an
instantiation of the antecedent, B, so as to enable the implication of the consequent, A to come into effect as a conclusion of B
actually being true (not just saying 'if'). But for A to be a conclusion of B in a hypothetical implication, it must be taken out of that hypothetical arrangement by the truth of B alone being asserted. Then the force of truth of the fact of B itself being true will enable what is hypothetically true in the implication, that A is true if B is true, to become a fact. If we add that B is asserted to be true, so the truth of the antecedent is now provided (so now "B" is not only the "if" part of a hypothetical statement), then therefore the general statement of conditionality, that on the condition of B being true, A must also be true, is instantiated or able to be shown valid in one instance. That leads to the logical conclusion, which was "A" in our original implication, but only by means of the new premises which comprise the set of statements that are conjoined together into one statement: the hypothetical (B -> A), plus an existential fact, B. Thus leading to a conclusion which is the consequent of the hypothetical statement, again the consequent being what "follows" the arrow of implication, now follows yet a further implication, which is of that hypothetical arrow being activated by an existing fact. That fact, in this case, is that B is true.
So we can't say (B -> A)
-> A, as this is not an argument. It is a statement saying what is not true in our interpretation. That just says that if it is true that, if B is true, then A is true, then A is true. We have to to assume, or find that it is the case, that B is true in order to complete this into a valid argument.
So we have to say:
(B -> A) Our Hypothesis for demonstration of A
B Our fact we need to see if this hypothesis is true
Now, we need to have "A" result. So this is to say,
(B -> A)
B
therefore
A
As can be seen, now "B" has been expanded to include two premises, one of which is the very fact of B, and the other is the implication that if the fact of B is true, then A would follow. If both of these are true, then A follows. So to keep these terms distinct from the "set of premises which imply A" by using a different term, this is now to be called "Bx" instead of just B, so that Bx = [(B -> A) & (B)], and Bx -> A means [(B -> A) & (B)] -> A.
Let's allow this for the moment. Let's allow it to be true that this was demonstrated. Let it be true that there is a hypothetical that is itself true, that there is some statement "B" which if true then requires "A" be true, and then let's just assume that B, in fact,
is true, enabling that conclusion, A. Let's see what happens.
It was found that in order for A to be true, and in order for (B -> A) to lead to that by way of demonstration, that it was necessary that we also had to have B itself be true. So (B -> A) and (B) are the premises of the implication, and A is the conclusion. That looks like this:
[(B -> A) & (B)] -> A
That is, the conjunction of the true premises implies the true conclusion, A.
But that means that whenever and however this is true, that the following is also true:
Because A is the the negation of the law of identity, and the law of identity states that for some statement L representing all possible statements, (L -> L). so if we take any statement instead of the general schema statement "L", say we take the "B" of this example, then we have A = ~(B -> B), because it is not possibly true now to say (B -> B), since that requires the law of identity to be true, in order for this to be true. So here, A = ~(L -> L) = ~(B -> B). In other words, if any true statement does not guarantee itself as a true statement, does not imply itself, then no other true statement guarantees itself as a as a true statement. If it ever happens to be true as a fact that a statement is true, then it just happened to be true for some other reason, or for no reason, but never on the basis that it, itself, was true! This is the negation of the Law of Identity, and so this is what we must now say if the Law of Identity is proven false. Now it is "the law of identity", negated...
And since we negated the Law, it is a lesser law that we have created to replace it which says no premise can guarantee itself as a conclusion, because this sort of law is not guaranteed, and we cannot pick and choose when we want to make up a law "just this once". If generally, A = ~(L -> L), then specifically, for any B, A = ~(B -> B). As a further note, to round out the idea here, since A is the negation of the law of identity, then in this construction here the Law of Identity is ~A, which is generally (L -> L), and for the specific case of B, if ~A, then (B -> B).
So since we chose B out of convenience to our proof for a specific instance to use for instantiating A, we use it also to proceed in our investigation with a new formula as follows:
A -> ~(B -> B)
But we know two things now:
[(B -> A) & (B)] -> A
A -> ~(B -> B)
We believe that there is some sort of thing B that, if it is true that B in any given instance, then A is true, so that if we ever find out B, then we would get A. We also know that if we ever had A, then we'd not be able to imply B from B, because A means exactly that. This will be key to finding out why the Law of Identity is immune to contradiction. So for that purpose we also need to make sure we agree on an interesting point.
So we know that A = A, right? I'm not saying that A implies A here, but just that we mean "A", and just "A", whenever and wherever we say "A", and that this meaning doesn't change. That is the original and logically uninteresting "Law of Identity". We also already know what it means since it was discussed above. But let me reiterate the discussion to show how it will help us make a clear demonstration in what follows.
The fact that A = A means that we can, appropriately, say that it is the same A that is the consequent in the first formula, which is then used as an antecedent in the second formula in the two formulas shown above. But this just says that whenever we deny that the law of identity is true, we are always denying that the law of identity is true, wherever we say it. In other words, whenever we use the term "A", we always mean what we mean when we use that term elsewhere. That means that the "A" terms in these formulas are interchangeable, and it will be shown, can bridge their formulas.
Now when we say that the law of identity is false, we aren't just denying that law here and there, but we mean to say that
we can guarantee that it
cannot be guaranteed that the law of identity
is true. Otherwise the best we have "demonstrated" is that we cannot guarantee that the law of identity is not true in most instances except one, which means we cannot guarantee even one instance of it being found false outside of one instance where it should have applied. But to falsify a law, we simply need to show that it is broken once under conditions in which it supposedly held true. So in fact, if we falsify a law in any instance, we falsify its power as a law, and so we falsify its application to
any instance. If what the law described turned out to be true in any instance, it was for perhaps some reason, but not because of the law which stated that it had to be the case in every instance. What is described then is something that is true for some additional reason, and not because it itself must be true, simply. That is what is claimed to be true by the Law of Identity, and if it is true in any instance, the Law declares that it
had to be true, and could not have been false, and this is therefore not limited to any given instance, but takes any instance and applies to it, just as long as the same implication form is in place.
Above it was already claimed that there were conditions in which this could be guaranteed, namely that if B, then A. B, and so A. That was our demonstration (or we have none). So let it have been demonstrated. That means it is now guaranteed not to be true, which means we cannot ever guarantee, not even in one instance, that a premise guarantees itself as a conclusion. So we know we have the same "A" in both sentences. It means the same thing when it results from an instantiated conditional which is the antecedent of a conditional statement, just as it does mean that same thing when it is the antecedent of another conditional statement. We can close that gap then, because we already know that the "A" said twice is the same "A", but said in two places for two reasons.
Since the consequent of the first formula is the guarantee that "A" whenever its antecedents are all true (conjoined together, and all true, as though a set of premises), and that A is "~ID", which is ~(L -> L), and since we know that A is thus guaranteed because it is implied by a true antecedent, then we know that whatever guarantees A also guarantees whatever A guarantees elsewhere, because this is just to keep the law of implication, which it was asserted for the purposes of demonstration could be kept without the law of identity anyway, and which being a logical operator, always acts the same way in all statements. Implications always imply.
In other words, we said we could assert an implication from some antecedent, and that means that if we choose, we can simultaneously use that consequent as an antecedent to imply other consequents with no impedance. There is nothing to prevent us. This much common sense must be maintained to demonstrate anything at all. We must mean what we mean by "A", and always mean that, or we cannot use it in a demonstration to deduce "A" from premises, or imply "A" from antecedents in a conditional formula, or even insist that the meaning of "A" is true, whatever that meaning would happen to be. In this case, it is ~ID, or the falsehood of the law of identity, in logic. We say that "A = A", which is to say the use of the term must be consistent in our demonstrations or no demonstration is possible, because our terms otherwise don't mean anything consistently enough to enable a demonstration. If they do hold with the same meaning whenever they are stated, then they must, and they can't drop that requirement ever, or we have the possibility that the demonstration was false because the terms were not consistent, which is the opposite of a guarantee that a truth is demonstrated. But this guarantee is required for an implication to be how we defined it, which is necessary for a demonstration in this case. So for the purposes of any demonstration, terms must be self-same in meaning, such that for any statement "A", and any other statement "A", A = A, in both form and meaning.
I know it is convoluted. That is why I decided to minimize this thread to the Law of Identity for now. It started that I would do an example of each one and then do an exegesis on their metaphysical and moral correlations, but I had to pare it down to just one example for now.
So that means if it is the same A in both cases, then we can just splice them into one iteration, and let the consequent in the one sentence serve as the antecedent in the other, because it does do that, by the fact that "A = A", which is the Law of Identity for thought in general, but which I modified for use here in a logical sense.
So that fully justifies us to say the following:
[(B -> A) & (B)] -> A -> ~(B -> B)
Just as whatever implies A, implies that
same "A" whenever that A also implies anything else, so this formula, which implies A, implies the same A which also implies this other formula here.
And that means that just as this formula implies A, it implies, through A, anything that A also implies, since these implications all hold simultaneously!!! (why wouldn't they, since we are talking about truth as such, and laws of truth at that?).
So this means that we can actually say that, in addition to what we have just said, that the
premises of A imply,
through A, the
conclusions of A. We can actually reduce A to being a mere bridge for another implication which we can state, and therefore take it out of that implication and still be saying a true formula! We don't mean that we don't imply A anymore, in fact we insist that we always do in this situation. We simply say that we can make an additional statement, a further consequence from this one, namely that:
[(B -> A) & (B)] -> ~(B -> B)
Now remember how we added B separately from our taking (B -> A) to be provisionally true for our demonstration? So really we derived the conjunction seen in the premise here (called an "antecedent" in the first part of an implication formula, but here I have played loose and used "premise" and "antecedent" interchangeably, and also "consequent" and "conclusion", because in this case the antecedent is a set of statements which
are premises in an argument, and so a complex statement). We didn't assume these premises all at once as a conjunction forming one antecedent, but we started by assuming the formula that lets B imply A, which is (B -> A), and then by the Grace of the God of Demonstrations, pretending we found B on a lark, just so we could ensure A, and did all this just so we could see what would happen if it were ever the case that A were demonstrated. Then we found a case where it makes sense to say something because we found A to be true, namely that formula ~(B -> B), which here is an instance of denying ID more generally as ~(L -> L). That is where we are. But since we synthesized the premise here out of granting B, that just means that B is true as well as that formula (which we assumed for the sake of argument, to test some further implications).
Well, that means B is true.
So we can say "B", simply, is true (apart from its conjunction with the other premise).
(NOTE: This is a reiteration, NOT an implication!)
And just as we can leave out A in the above case, we can leave out that other premise in the antecedent and say, given all the other conditions we just proved and which are still held true, that those,
plus B, gives us ~(B -> B). That other premise alone doesn't yield ~(B -> B), and so neither does B alone. We're not saying that. But we can say that it is true that
in the context of that other premise, that B itself does imply ~(B -> B). Let's agree that we can say it is true, therefore, that B in a premise has resulted in ~(B -> B) in a conclusion, along with other premises in the process. That means, from B (not only B, but from B in at least part), we arrived at ~(B -> B). That means we can say there is an "entailment" from B in this context, namely that ~(B -> B) is a consequence of B's being true. Let us say entailments are implications which are nested in a context that the entailments themselves may leave unstated, but that these entailments, when in their own valid context of unstated (or muted) implications, are also valid implications. That's because implications are always implications, even when one implication depends on a context of others which are left unstated temporarily, or even indefinitely. After all, perhaps many implications left unstated actually imply B, and so even though we didn't start our demonstration with those, they still allowed us to draw implications by means of B. Let the symbol for entailment be
~>.
Let's say that if m ~> n then, in some context left unstated but held true, m -> n.
That is to say
(m ~> n) -> (m -> n).
So that means that B, known to be true for our demonstration, entails ~(B -> B), and so it also implies it, because entailment is a subset of implication (as I use it here), and so we can say, finally:
Because
B ~> ~(B -> B)
Therefore
B -> ~(B -> B)
That seems quite valid to say, in the context of our other statements in the demonstration (not by itself arbitrarily, or "always", just always in this logical neighborhood of explicit statements, but this will be sufficient for our purposes).
So we know how entailment works to produce special forms of implication which are also just as strong as the implications which support them, at least in the context of the argument that made their construction validly argued (or else in which they were assumed already).
Let us take these two techniques, the transitive property of semantic identity, and the entailment modality of implication, and construct an interesting proof that the demonstration of the falsehood of the law of identity results in implying the Law of Identity!
The implication was discovered valid that:
If (B -> A)
and (B), then
[(B -> A)
& (B)
]
This is to say that in the context of these arguments including (B -> A) as an active premise, asserting B will imply [(B -> A) & (B)]. Therefore B ~> [(B -> A) & (B)]. It was already shown that this would imply B when we demonstrated the property of entailment which allowed us to show that B -> ~(B -> B). Therefore the following logical sequence can be delivered:
B (already assumed)
B ~> [(B -> A) & (B)] (entailed by the fact that B is part of what implies the introduction of this conjunction)
B -> [(B -> A) & (B)] (implied by property of entailment)
[(B -> A) & (B)] -> B (our rule of conjunction exploitation, the converse of conjunction introduction, which we already used to get B out to demonstrate that B ~> ~(B -> B), above)
All this gives us, through the transitive property of semantic identity:
B -> [(B -> A) & (B)] -> B
And by simplification as before:
B -> B
But this is an instantiation of the Law of Identity, which has now been resurrected into Glory, through the self-abnegation of its being contradicted by an inferior assertion, which is proven false by this result. This sets the stage for the second law, which is the Law of Non-Contradiction, in the next example. Interestingly, if (B -> B) is true, then, by using as a substitutive term for B the statement itself as a whole, (B -> B), and by employing the transitive property of semantic identity (the law of identity for thought generally), the following can be shown to be true:
(B -> B) -> (B -> B),
But because
~(B -> B) is of the contrary nature, it cannot lead to such consistency, before or now, i.e., we could never have demonstrated the following:
~(B -> B) -> ~(B -> B)
And that is shown by substituting its own formula into its own terms, which instead gives us the opposite of this:
~[~(B -> B) -> ~(B -> B)]
The Law of Identity validates itself, and is also implied by its own contradiction, but the contradiction of the Law cannot validate
its self , but can and does imply what it at first contradicted.
And now it is handy to utilize our understanding of the nature of valid argumentation.
So now, by demonstrating A to be impossible, and so always false, we know that a premise is false, namely Bx, and so we know that either or both (B -> A) and (B) is/are false. The first could be false because while something called "B" may be true, it would be impossible to infer A from it, since A is shown to be impossible. On the other hand, "B" could be something impossible, but if it were, then the strange statement "(B -> A)", normally ridiculous, would be made possibly useful in showing A, and so in this way possibly true and assumable for purposes of demonstration. But the results of this demonstration are not kind for any attempt upon the Dignity of the Law of Identity!
It has been shown how it is true, also, to say that the Law of Identity is a True Defender of the very rule of implication, perhaps the fundamental cornerstone of Logic, and this because it is the most fundamental implication that is possible to state! We may imply a lot of fanciful things, and pur
pose something based on a lot of fanciful reasons, but surely, at least it is true that, if we assume anything, then it ought to imply itself consequently! After that we must be more careful with our discipline about what we assume in our premises (antecedents), and what other steps we take on the way to our ultimate implication, which will involve many "sub-implications" along the way. Such a Noble Law of (Logical) Thought indeed~
Many fun examples of those implications can be created, by the way!
So by proving this Law of Identity through a convoluted, yet interesting mechanical process, we've discovered the fundamental significance of certain properties and rules, and their distinctions, which we take for granted in some of our normal thinking processes, but which when elucidated in the context of fundamental Laws of Logic, demonstrate for us a parallel between Logical Consistency and the Virtue of Honesty, and also demonstrates the Immutability and Immortality of the Glory of Truth, vis-a-vis the impossibility of any attempt to suppress it by way of a vain and arrogant contradiction, which is proven always to be parasitical upon it, and hypocritical in relying upon, and implying what it claims to have usurped.
I hope it wasn't too dense, and if there were any errors of any kind, please feel free to point them out, because I am always trying to improve, and this was meant to improve the reader, not mislead him.
NEXT: I will endeavor to demonstrate the metaphysical and moral significance of the Law of the Excluded Middle.
(Enchanting music fades in and out with their galloping)
"Yaaayyyy" 
"Aye that... 
principles of logic onto the dominions governed by the FAITH...the final frontier.