The Laws of Logical Thought

[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.

[Elizabeth]

Thank you for this fun and accessible presentation. Any student of logic would be very lucky to have you as their professor. You're right, we all should be made familiar with these concepts. They are essential for everyone but especially truthseekers.  I'm going to be more mindful now of how these principles can aid my thinking in the future. 

The ancient gods cooked up illogical religions for their human worshippers to follow. Intentionally, unless of course "The Gods Must Be Crazy" and they were logically challenged themselves. You made a good point early on that if too many illogical, inconsistent thoughts were allowed to remain in the membrane then one could go insane? - no, stop thinking - thought locking/blocking or at least the arguments would be full of holes. What perfect soil to plant (whatever the gods wanted planted in there).  Stopping the rational mind is a classic way (NLP) to implant suggestions into the subconscious mind. This is the basis of much soft programming done to the public.  

Have you noticed how many former truthers have gone quiet lately? It's probably due to suffering from major cases of cognitive dissonance. Employing some basic logic would help in clearing up confusion.

[MetaOntosis]

Thank you for this fun and accessible presentation. Any student of logic would be very lucky to have you as their professor. You're right, we all should be made familiar with these concepts. They are essential for everyone but especially truthseekers.  I'm going to be more mindful now of how these principles can aid my thinking in the future. 

You're welcome, and I'm glad to offer my attempt.  Thank you for your kind evaluation of them!  I think it might be worthwhile to have an interlude before going into the third Law of Logical Thought, and go into the various logical operations which are used in these discussions.  I hope it more helpful than confusing. 

[MetaOntosis]

Interlude: Discussion of the Relation of Argument to Implication and other Logical Operations



The first operation which we explored was "implication" which simply says that if the element, statement, entity, what have you, which is to the left of the arrow is true, then whatever follows on the right side of the arrow cannot be false. This is, if the truth value of the implication arrow is to be considered "true".  Otherwise the value of that sentence which says "if theta, then phi", that sentence as a whole is false. This is an interesting way to symbolize the truth functional meaning of phrases like "if A is true, then B is true".  It literally seems to say to us that if we know A is true, then we cannot be wrong about knowing that B is also true, or else what we mean by "if...., then..." is not clear.  It is our confidence that we can proceed from the antecedent to the consequent, which is a perfect parallel to the confidence with which we deduce a conclusion from a set of premises.  It is how we know that, if we know something A, then we cannot be wrong about something else, which is "B".  And that is how knowledge flexes and expands in its vitality, by realizing its full potential, by going beyond the sphere of its mere factuality and "beyond the given".  It is an active, assertive, and forward moving, life affirming mental process.

If I can be sure that the premises are true, in an argument, then I can be sure that the conclusion will be true, if I use a valid deduction.  All valid deductions are built on this principle of implication, which is stated this way:  IF the premises are all true, THEN the conclusion must also be true.
Implication is a very rich idea, even in that it has some "rough edges" which don't capture what we really mean when we ordinarily use the term. For example, if it is false that P, then no matter what Q is, it is true that P -> Q.  That means that if the sun doesn't appear to rise in the east, then Weird Al Yankovic is the King of England, is a true statement!!!  That's  strange indeed, but this is because what implication means is not proven false by this statement, because all implication says is that if the antecedent is true, then it cannot be that the consequent is false.  So the only time implication is "threatened" with being false is when the antecedent is true, but the conclusion turns out to be false.  That's the only clear case when the implication arrow is FALSE.  That is the contrary of the important case of this function, which can be said another way to make it more clear, by using the other operations.

If this statement is made: P -> Q,  then it is only false in the case that P is true, but Q is not true.  That means that in every other case, it is not false, simply because those cases are not defined as what makes it falsifiable.  We only have one clear way to make P -> Q false, and that is to show that its consequent is false even though the antecedent was in fact true.  So if it is the case that "if it is raining, then the streets are wet" is the proposition, then the only way to show that it isn't a true proposition is to find a case where it is raining, and the streets are not wet.  If it is not raining, then the streets not being wet is not a problem for this proposition.  If P is false, then Q being false is not an issue, and the proposition is not proven false, so it still holds true.  It may well be the case, for all we know, that if it were raining, then the streets would be wet.  That's a relevant sort of use of this logical tool. We might wait to find a day when it is raining, and check to see if the streets are wet, and if they aren't, we know that this proposition is not true, because we have an instance when "if it is raining, the streets are not wet", and that is quite different then insisting that "if it is raining, then the streets are wet".  

So, if it is ever the case that the sun does not appear to rise in the east, I can check to see whether the throne of England has a usurper with a penchant for making parodies of top 40 charted pop songs.  If it turns out that it is not the case, then it sure isn't true that if the antecedent is true, then the consequent is also true, and so this proposition is then proven false.  People may rightly ask, should such a proposition be granted the status of serious consideration just because it cannot yet, or perhaps ever be proven wrong.  That is a valid consideration.  But we usually don't come to logical grips with absurd propositions: we usually don't even admit them into consideration.  Think of your reason as being a court: if a case is brought forth that is simply not worth the court's time because it has no merit, then it can be thrown out.  The court is only there to hear serious cases, as when a right has been infringed to a degree that shows demonstrable harm.  Otherwise we must allow nature, and common laws of human communities, to settle "out of court" in the ways that are appropriate in their own domains.  If these were not already reasonable domains for handling "common cases" then can you imagine a world where a court had to exist to hear the facts of every dispute that happens?  We'd all be spending our time in court most of the time, and with little time to create cases outside of it.  Most of our court cases would end up being about things that happened in court, and we'd have to have courts to hear those as well.  Even at the rate that it is now, we already have courts to hear about cases in lower courts, and these are already limited to hearing cases which are "not frivolous".

It is not to say that there is no coherence to the claim of "If absurdity P, then absurdity Q".  It is rather that there is perhaps no sense in our concerning ourselves with it.  What is interesting about implication is that we actually have some rules for making an argument which is not merely a bold statement like that "P -> Q".  That's not an argument.  That's an assertion.  We don't have to take an assertion seriously if the person asserting doesn't want to argue for it.  Why should we believe it? What is the case for this assertion?  What's the argument for it.  Asserting an implication alone is not an argument, since it is simply a statement that may or may not be true!

So, this is the difference now, between implication and argument.  An argument, properly speaking, requires at least two premises in order to be capable of a deduction, which is a translation of meaning between two statements which are premises, and which leads to a resultant statement, a conclusion, which those premises, together, imply.  Those premises must imply that conclusion!  It is not like any random assertion!

So the logical idea of implication is a limited tool in arguments.  It must be part of a collection of statements including at least one more statement in order to "make a point", and be a meaningful part of a valid deduction.  

That means that the general form of an argument, "premises -> conclusion", where we might take premises to be "P", must actually be translated into an implication formula like this: "(P & R) -> Q". It turns out that P and R must have certain well-defined forms in order to validly lead to the conclusion, Q.  That means P must be true, and R must be true, and they must have a certain form, or we cannot guarantee that Q will be true if they are true.  But we know that if they are true, then in those cases, Q will also be true.  And we know in those cases that if Q is found false, then it is because we have erred in our formulation of the premises required to reach it, so that one of them was false, but we also know in those cases that the form of the argument is still valid.  In other words, if those premises were all true, then Q would not have been false!  That's pretty powerful stuff to have handy when hearing arguments!!

So we know that implication is involved in arguments in a specific way.  We know that an argument can be made into an implication, but only with the provision that the premises, as the antecedent, must be just that: premises, plural.  "P" of the "P -> Q" of an argument must include two terms, no less.  And it also requires no more than two in order to make a simple conclusion possible.  Another parallel is that just as in the case when the implication statement "P -> Q" is true, it cannot be the case that Q is false and P is true, or to say it the usual way, it cannot be the case that if this implication is true that P is true, but Q is false.  In the same way, if P is a conjunction of two statements, L and R, symbolized as (L & R), then this conjunction, (L & R), is really one statement, which could be renamed simply "P".  In this case, both L and R must be true for the single statement of their conjunction "P" to be true.  So if only one of these premises, or both, are false, then the statement of their conjunction, P, is also false.  That means that we know that if P is true, that is if L and R both are true, then Q cannot be false; or this implication, which is actually also an argument in this case, is valid, but only if it is true when stated as an implication in this form.

So conjunction, with the "&" symbol, is intimately connected to the idea of "argument", and by means of conjunction operations we can make arguments which imply conclusions such that, if the premises (conjoined into an antecedent) are true, then the conclusion (implied as a consequent of the premises, taken together), cannot be false.  What a powerful set of tools!!

The ancient philosopher Aristotle thoroughly laid out all the possible forms of premises which can be used in an argument which, taken together and both true, lead validly (always and without exception) to a conclusion, and what that conclusion would have to be.  But that is an involved study, and it only involved premises of a certain form, which are called "categorical". Those are pretty powerful, and can easily be diagrammed with Venn diagrams, but they still depend upon the implication operation, and this is more the work of Chrysippus, who worked on "sentential" logic, which is more the sort that I'm going into here.  

Aristotle would have said that a valid syllogism runs, as an example, to say that Universal Affirmative XY and Particular Affirmative X allows us to validly deduce Particular Affirmative Y, as one way of schematizing the form of the argument.  A particular instance of this valid form is the argument "All men are mortal (Universal Affirmative XY), Socrates is a man (Particular Affirmative X), therefore, Socrates is Mortal (Particular Affirmative Y). It is valid because whatever you replace "X and Y" with, you can always conclude that the subject of the particular affirmative has the property Y, because it has the property X, and it was asserted that All subjects that have the property X have the property Y.  That is something we know and sense directly, but Aristotle formalized it into an abstract schema which showed that the argument form itself was valid regardless of the content it carried.  You can know what you want about men and mortality and Socrates, or know nothing at all, but if you put those terms into this argument, and those terms are true, then you can sure that those premises will guarantee that conclusion, that Socrates is mortal.  

That's useful for understanding what is a reasonable argument, even if the facts are not true.  So if I find that Socrates is not mortal, I can reason either that he is not a man, or not all men are mortal, or both! Chyrsippus, if he had his hands on these terms used in this categorical syllogism, would have formed it differently.  He might have said:  M -> D, M, therefore D.  The first premise states that If it is a man, then it is mortal.  The second states that Socrates (or whomever) is a man.  The conclusion, reached in a form we have seen many times already, is that he is mortal.  Most people readily can sense when a valid form of argument is being used, and I've used some in my examples already on good faith that they would be recognized as valid even without remark, though I went into some details about one of them such as Chyrsippus would have used, called modus ponens.  That is of this form:

P -> Q
P

therefore

Q

And that is an argument, written as an implication like this 

[(P -> Q) & P] -> Q

That implication CANNOT BE FALSE if its antecedent is true.  It doesn't matter what terms we put into it, as long as we have both premises as being true, there is no way that the conjunction which is formed from them can't also be true, and so there is no way that the conclusion can be false.  [I'll restate what the conjunction operation means: That if the statements on both sides of it, the statements it conjoins, are true, then it, the conjunction statement as a whole, is true.  If either is false, or both false, then the statement of their conjunction is also false, and only under those conditions is it false.]

If it happens to be true that P -> Q, and if it happens to be true that P, then how can Q not be true? In fact, there is only one thing that might happen other than this.  We might find that P is true, but Q is false, and so we get a false conclusion (because Q is already known to be false), and even though P is true, P -> Q is not true because if P is true, and Q is false (in this case), then P -> Q is explicitly known to be false by the very definition of that operation.

So let's look at some examples of how that can happen.

Let's say by some strange occurrence, the sun does not appear to rise in the east, as per the earlier example I gave, above.  That'd mean that P is true.  But it turns out that the world will not be amused by the spectacle of King Yankovic of England.  That means that one of the premises is false, namely the implication P -> Q.  This makes the conjunction of that premise with the other, P, itself false.  Therefore the conjunction of the premises, we'll call them together "L", is false.  So that means that in the implication representing this argument, L -> Q, is false.  That means that the conclusion being false does not hurt the implication operation in this case, since the antecedent was also false.  But not all of the premises were false, only one.  But that was enough to prevent their implying a truth!  Truth is implied from TRUTH in a valid argument, not from falsehood.  Yet the form of the argument was not injured, just as the form of the implication was not injured, by the falsehood in the antecedent.  That's because we know that if the implication "P -> Q"  were true, then that, conjoined with P, would have to lead to the conclusion Q.  But if Q is not true, then either or both premises would have to be false. Just as the implication L -> Q is not harmed by both L and Q being false, so the form of the argument this implication here represents, [(P -> Q) & P] -> Q, which is valid, is not harmed.  It's validity is preserved, because we can guarantee the truth of Q only in the case that all premises of the valid argument which leads to it is true.

We find that the form of the argument is adequate to produce a true conclusion in every case where the premises are true, and we can be sure in every case that the conclusion, if false, lets us be sure that one or both of the premises are false.  But with a mere implication assertion, by itself, we can't be sure of this.  We know that sometimes implications are true, and sometimes they are false.  But when it comes to a valid argument, the argument form is always valid.  If the premises of that argument were true, the conclusion would follow.  We'd never worry about finding a false conclusion to true premises in that form, as long as we find the conclusion which it predicts (according to the valid form).

Going a bit further, we can draw an interesting implication from all this:

In the cases of a valid argument, when put in the form of an implication. a false conclusion demonstrates one or both premises are false, and true premises demonstrates a true conclusion, without fail, always and in every single case.

Now that is why I was able to draw absurd conclusions from absurd premises in the first example above when I was going on about the Law of Identity.  That's because the forms of those arguments were valid, but the terms were absurd.  But if those absurd premises were true, then we couldn't avoid reaching those absurd conclusions!  Yet, this implies to us that those premises are false, because we know that the conclusions are false.  So therefore, we use a valid argument form to demonstrate a false premise from a false conclusion, and this is to perform another valid argument!

it is called modus tollens.  It is to say that if (P -> Q) is true, then it follows that (~Q -> ~P) must be true! Or as follows:

P -> Q
~Q

therefore

~P

We understand this perfectly simply because we understand that we understand, in this context, that we have assumed P -> Q to be true, and that Q is false, or ~Q is true, and so we know that P must also be false (since P -> Q IS true).  In other words, if P -> Q were shorthand for "an argument", P were shorthand for a conjunction of two premises, and Q were shorthand for a conclusion in that argument, then P -> Q in that case must be true, since it represents a valid argument form. But we'll just say "it's true" in this example.  But we found that the "conclusion" in that argument (now condensed into an implication), Q, is false (premise two in this further argument).  So now we know that the premises cannot all be true, and so the conjunction of them is false, since for a conjunction to be true, both members must be true.  Therefore, the set of premises (their conjunction), P, must be false.

And to reconfirm this in a familiar way, we can show it in this form:

1. [(P -> Q) & P] -> Q        Assumption
2. ~Q                                Assumption
3. Show ~[(P -> Q) & P]
4.   [(P -> Q) & P]              Assumption for Indirect Proof
5.   (P -> Q)                       &E, 4
6.   P                                 &E, 4
7.   Q                                 ->E, 5,6 (modus ponens)
8.   (Q & ~Q)                      &I, 7,2 (But this is a contradiction!)

So we can cross out the show line, and consider that statement in 3. has been shown.  That's because under that statement we assumed the opposite, which is that we assumed that it was false.  That means we assumed ~~[(P -> Q) & P] as statement 4., but I skipped a step and just removed the double negation.  By assuming that opposite statement, we sought to find a contradiction arising directly from it.  We found that, statements 5. & 6. could be derived from that assumption, and that Q could be derived from them, and that this could be conjoined with an already accepted premise in 2., and that this was a contradiction.  But this contradiction means either we dismiss the original premise (which we don't, since we are using it to form an argument and trying to show a conclusion comes from it), or we must reject the other part of the contradiction.  But to reject the other part, we must reject the source of it also, and that means the assumed premise in 4. must be rejected.  That means that its opposite must be true instead.  But that is what we were trying to show.  Quod erat demonstrandum, as they say, or "What was to be demonstrated", or "Q.E.D.".  Thta came right out of the attempt to demonstrate the opposite of what we wanted to show.  That is why that assumption in 4. is called "assumption for indirect proof".  I wanted to show this indirect proof to show the application of the Law of Non-Contradiction in assisting our proofs about arguments.

Now I'll prove a simpler form, by condensing the premise in 1. above into just one term and I'll lengthen the proof by showing all relevant steps:

 1. (P -> Q)      Assumption
 2. ~Q              Assumption
|3. Show ~P    
|4.   ~~P         AIP (Assumption for Indirect Proof)
|5.   P              ~~E, 4  (double negation exploitation on statement 4)
|6.   Q              ->E, 1,5 (modus ponens)
|7.   (Q & ~Q)   &I, 6,2
|8.   ~~~P        !, 7, 4&2 (Contradiction (7) results from later premise (4) and original premise (2) so we negate later one
|9.   ~P            ~~E, 8
Q.E.D.              (Proof completed, as we just showed what was to be shown in 3)

I kept all the "extra steps" because technically those are essential to the proof.  We only leave them out in logic class because the professor and the student both know that is a step and we want to skip it to keep the proofs simpler, "cleaner", and easier for her to grade.  Students usually don't complain about that, but because I was a stickler I kept all these in all of my proofs, except for the Q.E.D. one, as that would have been "over the edge".  I was already in trouble in that class for "political" reasons, and I didn't want to invite disaster any more than I usually did on a daily basis, although in my defense it was usually because someone else invited the situation.  Regardless, we didn't have to spell out the contradiction step, nor the negation of the AIP premise that it implies, nor that first negation in 4. (we'd just skip to 5., as any indirect proof of a negative statement will require assuming the negative of that, and that's always a double negative, leading inevitably to an affirmative).  What we'd usually do is on step 7 we'd just reiterate premise 2 so it is right under statement 6, showing that we had a contradiction, and then draw a line up to the show line and cross it out.  I insisted on using a red line for that, always used blue numbering, put "show" in red, and was highly tempted to make "Q.E.D." the efficient cause of that line to boot.  I say if you are doing something, elaborate, be fussy, enjoy it and savor it.  Do it "your way", as long as it adds a sense of truth and completeness to it all! Aesthetics here seems functional both within the topic and in the pragmatism of seeking to go into the topic with enthusiasm.  Style, which is as one philosopher has said "form existing as content (stressed)", is also a part of our endeavors!  So we learn that we should add art to our science, just we add the cunning of science to our arts.

Still, by keeping all the steps, this whole procedure gives us a sense of the way logic works.  Step by step, like a machine, trying to give us the desired sense of certainty and infallibility.  Look at how each step is clean and simple, and justified on the right side column by some rule or procedure or law which we already understand to be true or valid.  We know that those two premises are supposed to lead to a conclusion.  We think it should lead to the one we want to show in 3.  Unfortunately, we don't have a way to show it directly.  So we show it indirectly by testing the opposite statement as a conclusion.  We find that this produces a contradiction.  This is shown to come straight out of the assumption in 4. We of course won't throw away the original premise, so we throw away this assumed conclusion.  If this assumed conclusion is false, then how can its opposite not be true?  If we get a contradiction from it, then its opposite should be fine and must be consistent with the premises.  

That can be easily tested in any case with a "truth table" or "truth tree" or other decision procedure.  But we can just look at it simply:  If the premises are both true in this argument, then that means P cannot be true, because that would give us a contradiction.  Either it gives us Q, which contradicts premise 2, or it simply falsifies premise one, because if we keep Q as false, but keep (P -> Q) as true (as our assumption dictates), then P cannot be true, because the meaning of that statement being true is that if P were true, then Q could not be false. But we know that Q is false from premise 2!  So we know that P cannot be true, and so it must be false instead.  And with P false along with Q, that conditional statement, that implication, is still true as premise 1, and premise 2 is still true, and also the conclusion, that P is false, is also true.  But with the alternative, that P is true, we end up with premise one being false if  premise 2 is kept as true, or we end up with premise 2 being false if we keep premise 1 as true.  But since our whole proof concerns finding a valid conclusion from those premises if they were true (so they are assumed to be for this purpose), we must not accept a conclusion which contradicts them, and we must accept its opposite if it's consistent with them.  And it is.

These two argument forms, Modus Ponens and its reverse, Modus Tollens, along with the operations here used and the Laws of Logical Thought, are absolutely critical to understanding the flow of logical thought when it is used to create and analyze arguments. Hypothetical reasoning is powerful in a very concrete way is it not?  At least in this context we have a richer understanding of what a logical argument is meant to accomplish, which is that we can demonstrate that if we know something, along with something else, we can know something else we didn't explicitly know but which was implied by the conjunction of knowing those other two things. That is pure certainty, and is a beautiful kind of knowledge which consciousness can attain, but which mere instincts cannot alone discover, although they can often, as Aristotle says, "land many fine blows".  

Instincts on the level of knowledge come in the form of intuitions and hunches, but those are useful mainly because of the fact that if they were converted into a logical form, they would be validated by the fact that in the experience of that person there were experienced, and hence known, or at least reasonably assumed, facts, and that these facts, if tediously articulated into the form of premises would have, if time and circumstances permitted, yielded the deduction of a valid, and hence under the circumstances, a certain conclusion.  It may be the case that we must ride on intuition in order to properly act in many circumstances, and it may be the case that this may be either because it is safer, more efficient, or simply more preferred (perhaps for aesthetic reasons), but in any of these cases it is more practical and livable in that manner.  

But in the way that I view intuition and instinct here, I draw the case for them that logic is not an alien force to them, but the clear, lucid, calm, timeless companion and guarantor of their rightness, which for them is mainly found in their own visceral, qualitative, and spiritual aspects which do not labor with symbolic mechanisms which are properly described in a linear fashion.  Not properly described linearly, but consistent with a logical analysis which would understand their "premises" and their "conclusion" and be able to show that, given the circumstances, and if the mind of the person were "laid out logically", then the intuition/hunch/instinct was a valid process.

Likewise, it is out of the accumulation of much experience and through our good senses that we are able to properly intuit the senses of the ideas we use to create logical systems of thought, and so we find that logic depends upon our "right hemispheric" processes in order to obtain for us the sense of its "rightness" and "trueness" in all that it says and claims.  Also, it tells us something about the processes which inform our intuitions with structural and functional integrity, and we intuit this to be a valid observation that, in a deep way, our intuitions are, when cogent in their own way, are surely not inconsistent with the logical analysis which, if given its own form of the same facts would have deduced the same conclusion, although this wouldn't have helped at the time, and often wouldn't have been possible.

Look at Sherlock Holmes.  All of the deductions which he explains to those who are his awestruck audience are deductions or inductions of logical inference which are only explained in verbal and formal detail when there is time and place for so doing.  When he is going through the processes involved, it is almost certain that he is performing an intuitive shorthand of these processes.  It is just that his intuition has access to a very perceptive fact gathering apparatus, his keen senses, a huge fund of knowledge, a wonderful memory, and an adroit logical intelligence which seems to roll out linear thought with godlike fluidity.  But in the heat of circumstances, and when not formally delineating his processes, he is for the most part operating on intuition, an intuition which is very richly informed by, and cooperative with, his logical-minded consciousness.

So this goes to say that whatever our level of explicit logical knowledge and proficiency, or whatever the lack of that, this is not to say we don't have logical power in our mental processes!  FAR FROM IT. But these Laws of Logical Thought were always operative, and their good sense and scientific precision were offshoots of their already true reality which had always underlain our intuition, which for the most part and in most circumstances that matter, is the preeminent function of human consciousness.  Given that the truth and value of logic is evident to anyone's intuition if they only understand the terms I've explained, albeit in my perhaps rambling and circuitous way, it is then at least clear that we wouldn't want our powers of logic to be abused any more than would we permit our intuitions to be second guessed by others in whom it is not calibrated to protect and promote the interests of life.  Our consciousness is directly tied to our own lives and their vicissitudes, and our intuitions gain their value from being peculiar to our own subjective circumstances, not in being objectively permutative to those of others.  It may also not be possible to clarify to others the significance of what our intuitions inform us about, but there is much about the same circumstances which could be logically explicable to anyone who can get the gist of the main facts and understand certain fundamental implications which are inexorably drawn from them by only a little of the proper effort (logical explanation).  And it may also be that only the person with the correct intuition will hit upon the precise facts and the circumstances of their relevance which will enable them to synthesize the understanding which, if logically explicated, would be appreciable even to someone whose consciousness were under completely different circumstances, and who was without any subjective affinity to the person who was doing the explaining.

How that is practical is in proposing things to others so that they will grasp the significance of our intended actions even though they may not have a motive for readily understanding them at first blush, being that they are not involved materially in the same way, and so do not have the same intuitive comprehension of what they are about to behold.  Likewise it is  useful when explaining ourselves, or events, to third parties not witness to the events, and concerning which those explanations are a key shorthand meant to supplement the lack of experience of those who would hear our story, as is the case in a court of law for jurors listening to testimony, or officials who would hear our circumstances so as to decide what is within their power and purview to do about them.  In those cases then we would prefer a logical, linear, well stacked presentation which did faithful justice to our own understanding, but also to the fact that the other person lacks many of the features of direct involvement which are crucial to a well-formed intuition.  Yet we trust that their specific expertise in hearing certain forms of presented facts and arguments, along with the fund of common sense and breadth of experience which are minimal for holding an office, would enable them to approximate the required intuitions which are better enabled when our logical presentation is sincerely and consistently delivered.

Likewise, in hearing cases by others, we may be enticed by forms of reasoning which sound true, and which boggle our minds because our intuitions say to us that this cannot be a valid line of reasoning or truthful storytelling that we are hearing, but is more like a line of something else... that you wouldn't want to step in.  So we might want for our own sake a clear and clinical procedure for dismissing what we are hearing that does justice to our intuitions without making us merely obtuse, albeit obtuse in all the right ways...     If we can present to our mind, in a clear way that is distinct from the circumstances of the storyteller or casemaker, a well ordered reason for dismissing their claims to our attention, or at least their claims to deserve our assent for some proposed idea about what to think, believe, or do, then we may want to follow our intuitions, of course.  But those intuitions will be stronger and more reliable when we can objectively assess, and analytically construe our reasons for finding the person, as well as their arguments or other rhetoric, to be disagreeable. That would stabilize our consciousness a bit beyond the scope of what intuition alone would allow, and would strengthen our intuitions by discovering the explicit and golden thread of the precise facts, to the precise degree of orientation or disorientation they hold to each other and to the other factors involved, such as in how they are presented, and in what context, which would allow us to create a permanently reliable shorthand for our stand against deception and falsehood which would not require being in exactly those circumstances, under those conditions, in that frame of mind, on that topic, etc, in order to detect and elucidate issues of deception or fallacy or misleading and improper rhetoric which might recur under diverse circumstances, in all kinds of conditions, in many frames of mind which may be fraught by emotion, or on matters perhaps obscured by being only indirect to our experience, or even perhaps because of being in conflict with some of our interests (the human mind is fraught with many biases, for example).

Logic gives an extra and volitionally manipulable instrumentation for taking the sting out of those forms of cognitive turbulence, enabling us to stabilize our mind set with considerations which would otherwise not have occurred to us, or even wouldn't have been intelligible to us, but which nevertheless are intimate to the issue with which we may have to deal, and in which the ethical ramifications may be very great, and the consequences of our decisions upon them no less so. If we can tell apart a meaningful argument from illogical noise just by looking at the form, we already have information we need to judge the case being presented even before knowing what its details are.  It also tells us something about the level of mental competence of the person delivering the case.  It also helps us grasp what he is getting at even without understanding too many details, but simply by hearing the form of what he is saying.  Then we have a tool for refuting an absurdity even without knowing what it was, so imagine if we also knew the detail of the topic, the circumstances, the purpose and intent of the interlocutor, etc.  Logic is a tool not at all overstated in its value in these contexts.  Later analysis of events can be further ramified by this ability.  Research, contemplation, any form of investigation enriched and multiplied in power.  The speed of these processes is accelerated, the quality of their results increased as well.

[MetaOntosis]

Interlude: More on logical operations

We have gone over the following operations so far:

->    Implication Arrow, Conditional Arrow, Implication Operator, or just "Implication" or "Implies"

~     Negation, "Not Operator", or just "Not"

&     Conjunction, "And Operator", or just "And"


We've done rather a lot of work with just these three operators.  We also had some rules for using them in a proof, which are stated in the justification column by showing the operator, then "E" for exploitation, or "I" for introduction, followed by the steps upon which those rules were applied.

We've shown some ways that arguments are constructed in ordinary language, in symbolic form, and the special relationships of these operators with the nature of an argument, especially in the case of implication, since an argument is a special form of that operation.

We've found these handy in exploring the basis of Logical Thought in the first two laws looked at so far, and found those Laws, and the investigations of them, useful in other proofs.  It all hangs together pretty well so far!

There are two more operators commonly used at this level of logic, and those can both be expressed in terms of these operators already given.  One of them is <->, which is biconditional, and means that the statements on both sides of it imply each other.  It stated as "if and only if", or "iff" for short. That is just the same as, given any two terms M and N, to say that (M -> N) & (N -> M) is true.  So it says the exact same thing as that, and that is expressed simply in terms of conditional and conjunction operators.   We won't need to get into that much here.

But the other one is rather more interesting, and is a part of the third Law of Logical Thought, and that is the "or operator", or "inclusive or", or "disjunction", or just "Or":

V

It simply says that a statement it forms with a statement on either side of it is true if and only if (iff) either one, or both of those statements are true.  In other words, it is the like the & operator, but less strict in order to be true. & is true only if both terms it conjoins are true.  V is true if both are true, or if only one is true, and the other false.  Both can be true, or just one, but both cannot be false, if V is true.  If both are false, then the V-statement they compose is also false, and only in that instance.

These operators are sufficient for a well-formed and complete logic.  We like the biconditional, too, because it shortens a statement like (M -> N) & (N -> M) into a simpler form:  (M <-> N).  That's quite handy!  And because this relationship is so important, it really deserves its own symbol.

In proofs, we found that we could introduce a statement with an operator by using a rule called *I, or operation introduction, letting * stand for any operation. We do that by justifying this introduction by this rule itself, and including a reference to any statement numbers in the proof which are necessary to use this rule.

So if in some m) I have X, and in step m+n) I have Y, I can at any step in the proof after that form the statement p) (X & Y) and justifying it with &I, m, m+n.

If we already have that statement and want to get X (or Y), we just "exploit" that statement by using the rule &E, p.

The same process applies for getting rid of an odd number of negations, just use the rule ~~E on the appropriate line.  If we wanted to introduce a double negative for some reason, it could be done by ~~I, referring to the line upon which it was introduced (perhaps to get a statement to look a certain way for formal reasons).

We've seen how to exploit implication, and that is modus ponens.  Introducing such a statement as true, without just assuming it, is not quite as straightforward as the others.  I have 'show' that a conditional is feasible by starting a proof for it, with its own "show line", then under that line I need to assume the antecedent (assumption for conditional proof, or ACP), then I must find, from this and all other assumptions and statements already derived, the consequent is also true. This reminds us of the idea of "entailment" that I used in the proof of the first Law of Logical Thought.  We had "B", and then we needed to find it again within that proof to assert B -> B.  We did do just that, but in a way that was different than this simpler, and indeed less direct way.  In the former case, the implication was derived from an entailment, which itself meant that somewhere prior in the proof, the antecedent of the entailment was part of some other statement that went conjoined therewith directly implied the consequent of the entailment, so that our use of implication as derived from "entailment" in the sense I use it there really suggests this procedure here, if I were to use the method of ACP in the following way:

m)        Show B -> C
m+1)    B                     R, m-n   (We had B earlier in the proof, so we still have it in any subproof. R = "by Reiteration")
m+2)     Show C

According to the rule for ACP, and for proofs in general of these sorts (predicate logic proofs), we can reiterate any line earlier proven or assumed in the proof to any later line, in the main proof or in any subproof.  We can't do the reverse, but can take back into the main proof only that proven (shown) line of a subproof after it has been shown, and only if it was not dependent upon the outcome of another subproof in between it and the main proof.  In that case, we'd have to get those demonstrated also.

If we had B -> C to prove, we'd have to find C from all the statements prior to this, and "ideally" we'd show that assuming B was directly involved in the necessity of then finding C.  But actually, because of the logical nature of the statement, we only need to find that C is deduced from some combination of premises and state that within this subproof so that it follows after we assume B.  That's sufficient.  After all, we could use indirect proof and assume ~C, get a contradiction, and show that within the universe of this total argument that if B is true then C cannot be false.  Some would insist on being more stringent and I can understand why, but in fact that is not necessary here, and that would be a special form of implication and a different form of logic, with a stronger sort of implication really akin to causality!  But in this logic here we are not saying B causes C (nor that if it does that it must cause it).  We are saying that in the universe of assumptions, givens, axioms, derived statements, rules, etc, at our disposal in this proof, that if B is true, then C cannot be false.  To do that, we merely need show that it can be shown true after assuming B, and if no contradiction arises, that is sufficient.  We'll do a little practice proof with the simple Law of Identity (again, modified for being logically relevant in a direct way, this isn't the normal formulation, but phooey, I'm doing it anyway!)

|1)  Show B -> B
|2)  B                        ACP
|3)   |Show B
|4)   |B                     R, 2
|5)  B                        R, 3
|6)  B -> B               CP, 2, 5
Q.E.D.


That is the form of "implication introduction", where "CP" means "conditional proof", and it looks like that no matter how we reach the consequent (as long as we reach the consequent validly based upon all the true statements we have so far). The statements 5 and 6 are technically "not necessary" for most.  5 states what was already stated at 3, but formally takes it out of that sub sub-proof, putting it "with" statement 2 more directly (aesthetic consideration mainly?), and 6 is just saying what we can say in a conditional proof when we assumed the antecedent and ended up validly deducing the consequent, which is what is required.  So we can state it formally and then it is "shown".  And, I like that nifty Q.E.D.!

This qualification, that the consequent be found from all the truthful statements we have found so far, is a lot like saying that it is "entailed" by them as I use the term in the first Law of Logical Thought. It was also in this spirit that I invented that rule.  "Entailment" has other special meanings in logic, but this is not a controversial use I'm making of it here.

So this process basically says:  We want to show a conditional.  We have a "universe" of logical truth to rely upon, and we can expand it or reformulate it any way we like according to the proper logical rules.  If we assume the antecedent is true, will we find that the consequent is also true?  If so, then that is sufficient for us to posit that the conditional itself is true, since it is not contradicted by the logical universe we have so far, and based upon having the antecedent, and including all other entailments which coexist with it, we had the truth of what we wanted to find as the consequent.  And so we have validly shown this conditional statement to be, itself, true.

Obviously, we could disprove that statement by assuming the antecedent, but then finding the consequent as false, and under the same conditions!  

Of course, there is an interesting result if we attempt to do this by using indirect proof, and if by this we reach a contradiction.  That means we have shown that, in this logical universe, and so in the neighborhood of this proof, having the antecedent assumed to be true will end in a result that, if we assume the consequent false for indirect proof, we reach a contradiction, and so the opposite statement, that the consequent is true, must be the case.  That is an even stronger way to suggest that the conditional is acceptable to add to our proof as true, because we have in effect shown it to be true by producing the very definition of it:

Is this conditional (implication) true?
Let the antecedent be assumed true.
Is it possible for the consequent to be false?
Assume it to be false.
Contradiction derived from this assumption.
No, it cannot be false, and so must be true.
Yes, this conditional is, by definition, true because we have found that if the antecedent is true, then the consequent cannot be false.

We have to go through these sorts of steps to satisfy our sense of certainty by ruling out that we have merely somehow spuriously asserted an implication.  We say, basically, if P is assumed true, could we just reach up at random and feel very sure we'd not bring back ~Q in our hand?  If we assume ~Q is brought back and we get a contradiction, but we insist the rest of the statements in the proof so far are true, and we have good justification for that certainty, then we can be absolutely certain that, in this proof, if P is true, Q cannot be false, and hence that is our true conditional statement right there!

So I want to reiterate (no pun intended) here for emphasis that we could just find the consequent also, and this would be enough, because if we can validly find the consequent from all that has come before, then we would only contradict it arbitrarily by assuming the opposite as an indirect way of proving it.  If we can deduce or reiterate the consequent from elsewhere in the proof, or even from within this sub-proof, that is enough. But by showing what is stated by attempting an indirect proof, which is the exact description of the conditional statement which we were seeking, we have shown how we know that when we "just pull down" that consequent from elsewhere in the proof, we have this solid of a proof, because we could have done an indirect proof, then pulled that consequent down, and gotten a contradiction.

But sometimes we cannot directly pull that consequent down into our sub-proof, as with the proof of modus tollens, in the post just before this one.  So it is then sometimes just much more efficient to start with indirect proof of the consequent, just in case starting that is necessary for drawing out the contradiction, but if we tried to find the consequent without doing this first we might not have been able to.

The only thing about the modus tollens proof was that we assumed P -> Q, and already asserted ~Q as another assumed premise.  We wanted to show the following conditional based on that:

~Q -> ~P

So to treat this along the lines of a conditional proof as we have outlined it here, let's do it as follows:


  1) P -> Q                       Assumption (or Premise, or Given)
|2) Show ~Q -> ~P
|3)   ~Q                           ACP
|4)          |Show ~P
|5)          |~~P                AIP
|6)          |P                     ~~E, 5
|7)          |Q                     ->E, 1,6 (modus ponens)
|8)          |(Q & ~Q)         &I, 7,3
|9)          |~~~P              !, 8, 5,3
|10)        |~P                   ~~E, 9
|11) ~P                            R, 4
|12) ~Q -> ~P                 CP, 3, 11
Q.E.D.

In a rather beautiful way, this proof shows formally that we can always validly say the following implication, and from now on use it as a rule of thumb in further proofs as a derived rule:

(P -> Q) -> (~Q -> ~P)

Which is the justification for modus tollens, by showing it in the form of an implication.


So if in any proofs from now on, if I ever have P -> Q and also ~Q, just by justifying it with "MT" using those two lines, I can immediately derive ~P, and without having to go through this (sort of) long but (definitely) pretty proof by which that rule was derived.


The above proof is how it would look in my particular way of drawing up proofs, and to my best understanding this is the most exacting and precise demonstration using the normal rules of predicate logic according to the textbook I prefer, and recommend, which is "Deduction", by Daniel Bonevac, philosophy professor at University of Texas at Austin, where I studied undergraduate philosophy.  I've seen a few logic texts, and this is the best I've ever seen, and still is, hands down.

In reality, there is nothing that can be proven with these "shortcut rules" that can't be proven with just the primitive rules using the basic operations and their introduction and exploitation (or "elimination") rules.  They just make proofs faster, shorter, and more elegant, and save us some time!  Of course, so would leaving out various reiterations, not crossing out show lines in certain ways, not writing Q.E.D., not choosing a certain color for each part of the proof (no need to switch pens), not using a straight edge for making lines and lining up proof statements, justifications, etc.  Sure, but then, for me, they look like an unsatisfying, and rather unpleasant looking mess.  As long as you know what you are doing and are sure about it, you can do it in your head and save time, ink, paper, do it lying in bed in the dark, etc.  To each, his own!

So we've shown how to introduce a conditional statement (implication statement) by means of conditional proof.  That will come in handy in any later proofs for sure.  And of course we covered the uses of all the other rules in various ways in this process and the ones before it, and we've outlined the nomenclature and the methods of employing them in discussions and proofs, and went into at least sufficient detail concerning each operation, and the first two Laws of Logical Thought.

Disjunction, if true, allows either or both terms it connects to be true, but not both to be false. It is inclusive.  But there is also an Exclusive Disjunction, and it is also used in logic, but not normally in this basic context and it is not really needed here. That says (A & ~B) V (~A & B).  Another way to say it is (A V B) & ~(A & B).  Another way is ~(A <-> B) which is also ~[(A -> B) & (B -> A)], and also ~(A -> B) V ~(B -> A).  We don't need to fully understand or test those statements here, but those are all equivalent statements that express the exclusive mode of disjunction, which in ordinary language we use to express that one of two alternatives must be true, but when one of true the other is false.  Inclusive disjunction says that either one will do even if the other is false, and both will also do, but both can't be false, so either one's truth does not exclude the other one's per se, but either one's falsehood demands the other's truth! In contrast, the exclusive disjunction has that both can't be false (as in inclusive), but also that both can't be true.  Therefore, while both forms of disjunction imply that if one statement is false, the other must be true, exclusive disjunction also says that if one is true, the other must be false.

An example of the first case, the inclusive disjunction (or just "disjunction"), we could imagine a circumstance such as would suggest that either of two things might happen, but both might also happen, but for sure at least one will happen.  Take a case where you need a caffeine fix, as I regularly do, and you like tea just as well as coffee.  You are determined to get that fix, and want to drink no less than a pint of beverage.  You might drink some coffee and not tea, or you might drink some tea and not coffee, or you might drink a cup of each, maybe even mixed together.  What seems to make this sort or work is that there is a result that is expected or sought, and either of two modes of reaching it exist, and they don't contradict each other, but the result must take place either way, or both ways.  That works nicely if I want to discuss the fact that I got my caffeine fix already, and we know it had to be one of these two modes by which I got it, but see no reason to rule out that it was by both means.  

In the exclusive mode of disjunction, a result can be reached, and must be reached, by only one of two different modes.  Let's say that there are two doors, and one leads to heaven, and the other to hell, and if you open one you must go through it, and you cannot remain where you are because if you fail to choose a door then the one you looked at last will open automatically at some unknown time, but at a time which is fated to occur.  When a door opens, the agents behind that door guide you into their realm, and you cannot resist even if you wanted to in either case.

You will go to heaven, or you will go to hell, but not both, and not neither.  Then again, this is probably true in real life anyway, even if you are a delusional New Ager!  This sort of choice is an exclusive disjunction, and might be said to be the choice you must make when choosing neither, and choosing both are both impossible.

To leave behind the world of commonplace drugs, let's make the two doors inclusively disjunct!  Let's say that they both can be opened, or just one or the other if you prefer, but regardless, they lead only to one place!  Let's say they are double doors, two doors sharing one portal. And like the other example, you cannot refuse to open at least one of them at some point.  Let's say we don't know whether they open to heaven or hell.  Then all we know for sure is that you are definitely going somewhere very fateful, and your only choice is the possibly superficial one of whether you want to go in through the left door only, the right door only, or if you want to boldly swing both open and embrace your fate in style!

My impression is that New Agers want to believe that the Eternal Destiny of Souls is something like an inclusive disjunct, and that there are two doors which are separate, but they lead to one place on the other side.  They seem to think these are revolving doors also.  "People of the Book" and other cults of specious authority want to convince you that it is an exclusive disjunct, and that they know the exact or best way to ensure that the door you choose is the best one for you.  Some cults say that there are no doors, or that you don't have to go through them, or that nothing is on the other side of either door.  Some cults say that they both lead to a bad place, and you should stay where you are.  Some say they both lead to a good place, and you shouldn't worry which one you choose. Perhaps some would say that one door is worse than the other, but you lose either way. Some would say that both are great, but one is better!  Some say the doors are booby trapped, or stuck, or a trick played on people by others who will swindle their marks with a false alternative, because in fact there is a third door directly behind the person and if they'd only turn around and look, they'd know that this is the door for them, without a doubt...

So we clearly understand, perhaps in spite of this hopefully amusing digression, what inclusive and exclusive disjunction both mean, and from the previous discussion up to this point, we probably had no trouble swimming in these ideas with casual references to logic in ordinary language, and we sense that we have the logical muscle to go back and forth between ordinary language and pure logic without so much trepidation as these doors may have caused! In fact, we may even be better able to face such a situation now because of our endeavors!  After all, is not the essence of these two types of disjunction something rather dynamic, so that perhaps we should have a third form?  I've never heard of it, but it probably exists and I'm sick of doing a google or wiki search on every blazing thing. I'll call it disclusive disjunction, or disclusive or.  That means that either or both doors may lead to hell, but perhaps one in fact leads to heaven.

We can play with these some more, or move on to more lefty left brained stuff.  Okay, till next time, you may do either of these on your own, whether inclusively, exclusively, or disclusively!

[MetaOntosis]

I have to admit that inclusive disjunction is for me the least favorite of logical operations!  I just don't like it!  Then again I don't prefer a universe where evil choices are possible, because someone just might choose them, and cause real harm!  And the more metaphysical the choice and chooser, the more severe and horrific the consequences.  I have to say that this is a concern which is close to the heart of a lot of philosophical thought in metaphysics and ethics, especially where these intersect with theology.  I am working on a logic which works simultaneously as a Gnostic theodicy, but that is something that I may or may not ever get a chance to put forth out for others to examine.  I'm looking for it to ground an ethics which is practicable in the world, but rooted firmly in a metaphysics of morals which is transcendent to it, and to what is evil within it, completely antivalent.  This would be desirable even if, and especially if the entire universe as such were utterly evil, and it would lose none of its sense in more benign contexts, finally being resolved into immediacy of unstated and natural, fully immanent expression in truly Divine Realms, in which we clearly are not (lest we forget... and to our dismay be reminded by some harsh fact which will come as a deserved slap upon our Spiritual Dignity... at least in this way let us not be shocked or confused!).

Inclusive disjunction is the main one we use in standard, classical first-order logic, rather than exclusive or what I called "disclusive" disjunction.  A quick refresh:

(A V B)         Inclusive        ~(~A & ~B), also (A | B) V (A & B)
(A | B)         Exclusive       (A V B) & ~(A & B), also ~(A & B) & ~(~A & ~B), also (A ↧ B) & ~(~A & ~B)
(A ↧ B)         Disclusive      ~(A & B), also (A | B) V (~A & ~B)

And now it seems time to proceed with a discussion of the third Law of Logical Thought.

[Elizabeth]

"My impression is that New Agers want to believe that the Eternal Destiny of Souls is something like an inclusive disjunct, and that there are two doors which are separate, but they lead to one place on the other side.  They seem to think these are revolving doors also.  "People of the Book" and other cults of specious authority want to convince you that it is an exclusive disjunct, and that they know the exact or best way to ensure that the door you choose is the best one for you.  Some cults say that there are no doors, or that you don't have to go through them, or that nothing is on the other side of either door.  Some cults say that they both lead to a bad place, and you should stay where you are.  Some say they both lead to a good place, and you shouldn't worry which one you choose. Perhaps some would say that one door is worse than the other, but you lose either way. Some would say that both are great, but one is better!  Some say the doors are booby trapped, or stuck, or a trick played on people by others who will swindle their marks with a false alternative, because in fact there is a third door directly behind the person and if they'd only turn around and look, they'd know that this is the door for them, without a doubt..." (MetaOntosis post 15).


Thanks for this logic thread and the helpful explanation of the use of the exclusive disjunct  applied to the "fist-pounding" tedious, Jesus is the only way' types who pepper the internet; as well as the equally misguided use of the inclusive disjunct by new agers re: both doors leading to the same place. 


"And the more metaphysical the choice and chooser, the more severe and horrific the consequences.  I have to say that this is a concern which is close to the heart of a lot of philosophical thought in metaphysics and ethics, especially where these intersect with theology.  I am working on a logic which works simultaneously as a Gnostic theodicy, but that is something that I may or may not ever get a chance to put forth out for others to examine"(Meta.post 16). 

I hope you are able to present your Gnostic logic one day soon even if it's just an outline or a few paragraphs. With knowledge comes the responsibility of sharing it. You take your responsibility very seriously and it is helpful and appreciated.

[MetaOntosis]

Example 3) The Law of The Excluded Middle


(A | ~A)


This is to say that every statement which can be made is either true or false.  One cannot assign any truth values other than these, and these are simply either the truth of the statement as stated simply, which is to affirm it, or else to negate it, which is to deny it.  For some odd reason the classical formulation of this Law uses inclusive disjunction, but in fact the whole point of this law is that there is no third value to assign to a single statement, only these two.  Yet these happen to be opposite and by definition cannot both be true of the same statement simultaneously (Law 2).  So, it seems best to state this Law with the operation which is truest to this fact.  And besides, if it really were (A V ~A) then there would be three values, not two.  There would be A, ~A, and (A & ~A).  We know (A & ~A) is not true, and so if it can be only A or ~A, but not both, then it must be (A | ~A).

If there is no way to define a statement other than to say it simply, which is to assert it or affirm it, then that is one truth value. But we know we live in a universe where conditions apply which make some statements contingent upon conditions conjoined to them which may be otherwise than is consistent with these statements' affirmation, and so their truth is contingent upon their conditions also being true.  That is to say, for certain statements P and certain conditions Q, that:

P → Q

If P is true, then those conditions must also be true.  The truth of P depends upon the truth of Q. This implies by its very definition that those conditions may not be true, that Q may be false.  If this were not the case, then all statements would be unconditionally true!  No conditions would be inconsistent with any statement's truth, and so inconsistency would be impossible.  That is clearly not the case in the world in which we exist.  We have direct and empirical evidence that things seem to be as they seem, but that they also seem to be otherwise from different perspectives.  Even if things always are what they are, and nothing ever changed except how they seem, that would be a change which means that seemings were different from each other, and so therefore that there would have to be at least two which were "not" each other.  If one, then not the other, for that perception.  Likewise, if the other, then not the one.

"Not-ness" is this property such that two conditions may apply so that it is, under certain conditions, one of them, and under any other conditions, the other one of them, but not both of them.  That's the "not-ness". This dichotomy.  It is not merely that a multiplicity exists, but that they have aspects which do not overlap. Yes, a red apple and a green apple have something in common in that both are apples, but they do not have in common which color that they are. Truth, like being, may be ultimate in nature, but it admits of modification in that what is true of one thing may be untrue of another, or they are really the same thing. Each thing has a property which is, with respect to the other, its exclusive property, a truth unique to it compared to the other, which is therefore a truly different property.

These things may also be truly the same in other respects.  Both exist or can exist, for example.  But if both exist at the same time, they must exist in different respects, as different entities.  If they exist as the same entity, then only at different times (i.e. child or adult), or in different aspects of it whether or not simultaneously (arm and hand, open hand and closed hand, wet open hand or dry open hand, etc).

For our perception, it is down to a distinction between seeming and being, but for beings "as such" it is down to changes in time and space throughout their existences, as well as alterations in their very essence.  Either it is this, or it is that (not this), either it is now, or it is then (not now), either it is here, or it is there (not here), or in short, either it is, or it isn't.  But since if it is, it is, but if it isn't, it is really "something, but something else", and we may have a name for that something else, or we may not. If not, it is still something else, but not this.  If this, then it is not something else, so not that.  If it exists actually, we say it exists per se, if it only might exist or only potentially exists, but doesn't actually, we say that it exists potentially.  But either way, it either exists actually, or not.

Meaningless coexistence is not possible. Either we are essentially restating the same thing, or we are stating something else, with a different meaning, however "slight" in difference.  Things don't exist trivially, but in ways which are distinguishable from one another, or else if one thing then in distinct ways from itself at different times and/or (inclusive or) in different parts, and perhaps in different ways even in those aspects.

But these two are the manner in which all things exist, and are the manner in which all statements can be "true", either simply, or alternatively.  If simply, then not alternatively.  If alternatively, then not simply.  And since a threshold has been crossed somewhere in being so that these two are distinct, and so they are two and not one, then we can speak of these distinctions only as distinct, and name them distinctly. Let the simplest name be A, so that the simplest signifier attached to it as an alternate is ', so A' is the alternative.  We use ~ for "not", and we mean by "not" whatever the minimal necessary distinction is so that in a universe formerly, or just seemingly unified, we must refer to it now as being in two states, such as "former" and "latter", or "this" and "that", or "self" and "other", or "same" and "different".  If the changes are superficial, such as many perceptual qualities are superficial with respect to a being's essence, then these may be on a continuum marked by distinctions in kind which shade into each other, and which perhaps alternate over a common space (or alternate over time in a given space).  Blue shades through green into Yellow, which shades through orange into Red, which shades through purple into Blue again.

But at least two of these are not the same.  Blue is not Yellow.  Yellow is not Blue.  More generally, This Color is not That Color.  Kinds of qualia also are not the same: extension is not color is not form is not pleasure is not pressure etc.  One amount or intensity is not another.  Where these things merge, they merge, but not where they do not.

The minimal understanding might be that everything that is, simply in that it is, is not something else.  That's called "haecceity".  Even two apples which are otherwise perfectly and exactly the same, are not the same, in that they are two, not one.  Even if the universe existed forever and were infinite in dimensions, then it would be "not" the contrary of these, which is nothing which doesn't exist forever and has no dimensions.  These would both not be something which existed for a time and which had some dimensions.  Those are two ways things can be "not" something else, either as a contrary or as a gradational alternative (always a mixture).  Green is not Blue, and not Yellow, but is some Blue, and some Yellow.  Blue is not Red, and not Yellow, but is not only not them, but also what it, itself, is.  That is an essence.  A True Essence is not merely "not something else", but has "actual, positive content".

Truly, identity cannot be duplicated without also being distinguished at least in the number of its resulting duplicate instances!  Each instance is not any other instance, although with respect to what is invariant about them all, they may be interchangeable so that, with respect to those features, if you were to "look away" then if they were interchanged some unknown number of times then you, upon looking back, wouldn't know whether they each were in their original places or not.  But even then, the understanding that they are distinct in that regard is preserved, and this is underlain by the fact that their places are still distinct, and so are they each that are in each distinct place, so that if they were two in number, and were interchanged an odd number of times, then you'd know either one is in the other's place, but if an even number of times, then that they remain in their own places, and those are distinct situations.

And then let us say they were not merely apples, but minds!  Then though they were in all ways distinct, touching one would give it a feeling not felt by the other at all, and vice versa.  And even if you touched the other in the exact same way, and even if you interchanged their places, their feelings would be just as distinct as they are in themselves, and as each felt its own feeling uniquely.  And these things are identical in form, but their substances are distinct.  They are "like" each other in their form, but are not each other in their substances.  One form, but two substances.  This is more true if their forms differ in any way.  Just as this kind is not that kind, and this aspect of a thing not that aspect, so this specimen is not that specimen.  "This" is always "not that".

The general statement of this situation with regard to all things that can be stated, and so of all statements, is that "this is not that", and as to truth value that "True is not false", and "False is not true".  When correspondence between a statement and the thing about which this statement is made is exact, then there is truth; and when not, there is falsehood. The distinction is infinitely fine, but absolutely sufficient, or it is no distinction at all. Gross distinctions are made up of many fine ones, or of one gross one, or several intermediate ones, but each one is sufficient to demarcate a true versus false distinction.

And since what we mean by "distinct" is that these alternatives are not the same, then even if they refer to "the same thing", they cannot refer to it in the same part, or not at the same time.  Or, even if the same part of the same thing, then not at the same time.  Or even if so, then it is in some other way "not" the same.  Otherwise there are no distinct alternatives!  But if there ever are any distinct alternatives, then there are at least two in total, and one is "not" the other.  Even if there are more, then any one is "not" the rest, however many.  And as to whether it is that one or not, the nature of "not" isn't concerned with which of the others it isn't, nor in how many other alternatives there are.  It is "not any of them". It is not each of them the same way it is not any of them and not all of them, just as it is in the same way that it is itself simply in all cases.

So before going further into this let's sum up by saying of anything that is that can be stated truly, that this is possible only if it is stated exactly "as it is". And if it is stated any other way then it is false, however slight the difference between what is stated and what actually is the case. And that anything that is that can be stated truly is the proper subject matter of logical statements, and so of logical thought. And so whatever can be stated logically and thought logically, it is stated exactly as it is in fact, and so is "true", or it is not, and so is "false" (not true), and never both at the same time and/or same respect, and never neither if it has any meaning and reality. 

[MetaOntosis]

As an aid to decoding the sense of some of these preceding statements I'll here post some "truth tables" which show the "truth functional" properties of the five major operators used in this logic.  They are tableaux of decision procedures which demonstrate the mechanical precision and certainty of reaching a result for any given statement such that if we know the truth values of the component statements then we know the truth values of the ultimate operators of those statements.

The ultimate operator of a single statement is the singulary operator of negation, or "not".

   

There are several binary operators, which operate on exactly two statements.  The first here is the conditional operator, or "implication", or "if, then".

   

The second binary operator is conjunction, or "and".

   

The third binary operator is inclusive disjunction, or "or".

   

And the last of the five major operators, the fourth binary operator, is biconditional, or "if and only if" (iff).

   

As can be seen in each of these truth tables, the left column is the input column giving the truth values possible for all of the variables given, in all permutations.  Then these are translated into the statements in the columns at their right as interpretations of those statements' truth values.  These interpretations of the statements' variables gives the operator its needed input so as to reach its own truth value.  These results as given are the definitions of these operations.  The truth values are color coded to help train the eye to view them distinctly as having a "product truth value", given as a result of the inputs of the statements over which they operate.

[MetaOntosis]

It should be noted that in the not operator truth table there is a clear repetition of the transformation from one truth value to its reversal, then back again, with each addition of that operator to the original statement.  The input value can be either truth or falsehood, and the resulting negation is indicated as the color-coded value to the left of the statement.  That becomes the new complex statement, and the last, or ultimate operator is the leftmost, which is the operator upon which the next operator will operate.

The various operations are clearly defined by these truth tables.

IMPORTANT
We'll now go into some further truth tables which show equivalencies between statements given the same truth value inputs.  This is where the color coding might really help out.  Note that the ultimate truth values of these statements are the same, thought the color corresponds not to ultimacy, but order from innermost to outermost operator in a complex statement.  Note that the simple statement is at left, and the equivalent, sometimes more complex statements are to the right.  Be aware that the color coding "orange" is the first operator of the simpler statements, and of the complex statements, the "innermost".  Blue is the next operator, and green is the third operator where a statement is that complex.  It is not that the colors correspond between statements, but that they reflect the order of operators from the first and innermost to the last, ultimate, outermost and final operator of any statement. Always compare the outermost operators of any equivalent statements.  Therefore, if the simple statement is compared with the complex, then look to compare the orange of the first with the blue, or green of the latter.  I also will keep the third and outermost operator the same color as that of the outermost operator of the statements it operates together, just so as to show they are connected, though I technically should have made the central operator "outermost" and so green in some cases.  But you understand it was to simplify it for the eye.  The logic is the same.  Check the meanings of the statements' truth values anyway, and don't take my word for it.  Even if they were all the same color, a trained eye can just look at them and tell, or even if each one were a different color.  Here I tried to make it easier, however, with some consistent, if also adaptable and context sensitive color coding.

First we have a simpler table giving the equivalence between biconditional and conjoined conditionals in both directions with respect to A and B.  In this one I made the simpler statement's operator blue to match the conjunction at right more clearly.  The rest are as stated above in the note.

   

Note how the values for the two conditional statements are interpreted through the conjunction, and only by this is a final set of values reached, and that these values are identical with the biconditional operator's.



Next we have inclusive disjunction, compared to equivalent statements which are more complex ways of saying the same thing.

   

It is clear that these statements are equivalent by sheer examination of the outputs being validly derived from the inputs.  It is more difficult to visualize their equivalence, but it is possible to examine "Venn diagrams" of each of them for a satisfying assist in that regard.  I recommend looking each operation up on wikipedia.  Warning, the symbols may vary.



Next there is exclusive disjunction, which is in fact an expression of the third Law of Logical Thought.

   

This is the way this truth table looks with two distinct variables.  It will look interesting when viewed as an expression of the third Law, as it will then be a "tautology".


Finally here is "disclusive" disjunction, which I call it because it is the inverse of inclusive disjunction.

   

And those are most of the operations we'll ever have to concern ourselves with in the normal predicate logic which we can use to construct and examine a huge number of arguments.  We can modify them in the future and keep these as a foundation, and add more operations which increase the power of this logic.  But for the purposes of this thread, we'll stick with these.