By Eric Bright
Don’t have faith in logic
I am sure the reason you accept the validity of the Law of Noncontradiction (I would call it the law from now on) is not because you have faith in it. It does not even make sense to say that one has faith in the law.
Also, the law is not like physical laws or language laws, or social laws. Certainly we all understand it. But, I have to emphasize on this reality a bit further for my sake.
Laws of physics are called law and they are known to be established facts about the universe. However, they are contingent. There is nothing in the fabric of the universe that necessitates this set of laws over any other conceivable laws. Not a thing. They just happened to be how they are. They very well could have been different and no violation on anything would have happened had they been different. So, the laws of physics are contingent. One way to know it is to imagine universes with different laws and see ig such imaginations ask for assumptions that might be inconceivable to be true. People have done so, and they have discovered that all of “the laws of physics” can be different without talking about anything inconceivable.
In contrast, the laws of logic, are also called law but they are not laws in the same sense that the laws of physics are called so. During the past thousands of years that humans ever lived on the surface of the earth, there has not been even once the slightest shred of hope to even conceive of the three laws of logic to be any different that what they are. Conceiving them to be the opposite of what they are results in inconceivable consequences. Even conceiving them to be the opposite of what they are now is not conceivable. They are necessarily so.
We can only learn about them. We can not even have faith in them for having faith, among many other things needs accepting them in the first place.
The laws of logic cannot be reversed, because to reverse them one has to assume them first.
In the past thousands of years, people have tried to find other logical systems that might be able to do things that the three laws of logic prohibit, like building formal system that allow contradictions for instance, or building logical systems that do not respect the law of identity, and so on. Guess what. They all turned out to be assuming what they were trying to eradicate.
It’s not a trivial thing to witness that the three laws of logic from which everything else can be deducted, cannot be thought about to be any different without knowingly or unknowingly assuming them.
One might say,
The laws of logic?! Hehe… how do we know that there is nothing beyond them? As a cat cannot understand calculus while we do, we might not be able to conceive of laws of logic that are opposite of the ones we have while other, more advanced sentients do.
This reasoning will not cut it. Cats can conceive of things, although a lot simpler than what we can, and it’s inconceivable for what they can conceive to be inconceivable. By the same token, since laws that are opposite to the laws of logic are inconceivable, any other sentient with any other “transcendental” mental faculty would not be able to conceive of it. They are not inconceivable only to us, but they are inconceivable in principle.
For example, if we successfully define numbers 1 and 2 and an operation such as + and the equal sign, then what comes after 1 + 1 would be necessarily 2. It’s not conceivable, even by any other kind of intelligence to define 1, 2, +, =, and then try to operate on them as 1 + 1 and it comes up with a different result. Surely if you define + in a different way, then you will have a similar situation, this time with your new definition. The results would be inevitable by any standards whatsoever.
But, the issue with the laws of logic is even deeper and more fundamental than that. They are not arbitrarily chosen, nor are they arbitrarily defined the same way that we can arbitrarily define a plus sign or a digit such as 1.
But, how did we discovered them? Where did they come from? Aren’t they yet another assumptions accepted by some insane logicians? Couldn’t we define them differently?
No we couldn’t. It is not for the lack of efforts or attempts to define them differently that they are undefinable in different terms than they are now. Almost any smart philosophy major tried it at least once or dreamed of trying it; to come up with different sets of laws of logic; to show that they are bullshit. Even in secret. I did it! It’s a dumb thing to do, I know, but I admit that I did try a few times. And guess what. Everyone, without exception, ends up using them left and right in the process of eliminating them.
It’s not because there are not enough smart people who know a lot of math and logic so to turn it upside down. All of them who attempted to do so were caught using all three of them.
Still, there are a few formal logical systems that try to get rid of one of the three one way or another with very limited results. Very limited result means that they do not work. Those logical systems that have been proposed are not able to do even the simplest things that any other logical system with the laws can do like having a piece of cake.
So, what those systems can do is like this: they assume only two of the three laws. Then they end up having a closed system of interlinked implications with a finite set of members. So finite that one can actually count them in some cases. It’s like, they can prove only a handful number of statements in that closed system and the system becomes locked. Nothing more can be deduced from it anymore. If you know what I am talking about, you would know that these systems are nothing but some thought experiments with no real results. You would know that those systems do not work.
And how did we come to these three laws, and not four or five or ten or any other number? The reason is simple. We, humans, tried to see what statements can conceivably be deduces from what other statements that might be simpler than the ones before them. Then we looked at those simpler statements and tried to see if some of those simpler statement could be deduced from yet simpler statements. If you do it long enough, you start to see a pattern. After a while, you will see a number of statements appearing over and over in every situation. Then you will see two things: (1) that some of these simple statements cannot be broken further down to even simpler one and (2) they also cannot be deduced from any other statement. When you eliminate the duplicated, the redundant, and the compounded ones, you will end up with three. These three statements have the following properties:
(1) They cannot be broken down to simpler statements
(2) Neither of them can be deduced from any of the other two (individually or combined)
(3) If they are eliminated (any of them), no other statement can be made
What would you do when you reach this roadblock? Yes! You would do what any responsible logician would do: To prove that what I just said is not the case.
When logicians tried to prove that the set of statements with the above-mentioned properties are bullshit, along the way the saw themselves coming to an inevitable conclusion: That they are not bullshit. That’s to say, they could only prove that it’s not possible to find any other set of statements that is simpler than this and does not have one or more of the statements involved.
It might not be a news to you, but it might be an interesting observation of other readers. I am sure you have already known all of this. The reason I mentioned it again is to make sure we are on the same page and we are not assuming certain things without knowing what we are assuming.
So far, all I have said was nothing but what you have already asserted: The law of noncontradiction cannot be violated, no matter what. I only elaborated on what you had said, and mentioned the other laws that have the same status in logic. I have never named them though. Here they are: The Law of Identity and the Law of Excluded Middles.
Did anyone try to show that they are superfluous? Oh, yeah baby! They did! They did all they could to show that one of these three laws can be discarded. Not only they couldn’t, but also it was demonstrated that any attempt to do so, inadvertently uses all three of them. So, the case is closed for this.
Here is what we have agreed upon so far, The Law of Noncontradiction, in addition to the other two laws of logic, cannot be discarded, cannot be violated, are not arbitrarily chosen, are not arbitrarily defined, and when/if they are discarded, the results are practically too limited to be even considered as a contender.
There are tens of other very useful logic in the market at the moment. What about them? What about Paraconsistent Logic? What about Fuzzy Logic?
Well, none of them can even come close to such an act as discarding the three laws. Not even close. I studies under Ray Jennings, one of the Canadian founders of the Canadian school of ParaConsistent Logic. It does not reject the law of noncontradiction. It tries to contain the consequences of a contradiction only after when it occurs. That’s all. It cannot even consider rejecting the law of noncontradiction. Now, how about Fuzzy Logic then? It’s a mathematical trick to feed contradictory inputs into a system and calculate them. To do so, it has to do a trick. It has to fuzzify the input, do the calculation, and then defuzzify the results. Why? Because it’s impossible to work with contradictory inputs as they are. So, the system has to turn the information into non-contradictory inputs, operate upon them, and give out an output that can be interpreted by other system that initially gave it the contradictory inputs to start from. It’s only a trick to get around the issue. Once numbers are crunched, the system still assumes that each calculated value is what it is and is not it’s complimentary value. This means that the system still has to work according to the law of noncontradiction (I, as a proud Iranian engineer, studied fuzzy logic to make useful machines; mainly because it’s inventor was an Iranian and I freaking loved it).
Contradictions Really Do Matter
Scientific discoveries and most of our other observations of anything in the universe can allow us form useful, approximate model of the working of the universe. In many cases, these approximations can become very accurate too. They can produce predictions that almost always come true. There are some other scientific predictions that always come true because of the models that we have. Some of our models are incredibly accurate.
However, there is always a margin of error. Science starts with such calculations to predict and consider the ever-present and the omnipresent margins of error. As long as anything relies on observation, it inherits a margin of error. That’s it. There is no way to eliminate this inherent error margin. All observations are contaminated by errors. Even if or when the errors are very small, they are errors still.
So, does that mean that our models are doomed to be laced by errors always and without any exceptions? No. There are some mathematical abstractions that are precise and have no errors in themselves. But, when they are mapped into reality, they do not overlap it as fitly as we wish. I am reminded of Albert Einstein's words when he said,
“As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality.”
And that’s true. Those mathematical models that are certain, usually have very little to do with how the universe works.
So, what? Are we done yet? Did we show that no true knowledge is possible about the universe?
Well, no. We are not done yet and we have not shown such a thing. We only observed that forming positive knowledge of the universe is hard and our models that try to do so have some errors most often than not.
But, still, there are other ways to form knowledge about the universe that are certain. They come in the form of the negations of other models. For example, we know, for sure, that the flat-earth model of our world is false. To know that, one needs to make only one, single, individual observation that can be repeated by everyone and contradict one of the assumptions or consequences of the flat-earth hypothesis. Only one!
This is the whole point of this article so far. Contradictions do matter!
This is how another philosopher would have put it:
“[…] There is no criterion of truth at our disposal, and this fact supports pessimism. But we do possess criteria which, if we are lucky, may allow us to recognize error and falsity. Clarity and distinctness are not criteria of truth, but such things as obscurity or confusion may indicate error. Similarly coherence cannot establish truth, but incoherence and inconsistency do establish falsehood. And, when they are recognized, our own errors provide the dim red lights which help us in groping our way out of the darkness of our cave.”
Karl Popper, 1962, Conjectures and Refutations, New York: Routledge, 1962, Routledge Classics 2002, ISBN: 0415285941
You already know what I am talking about. But you might still wonder what I am trying to do in here.
I am trying to make the point as clear as possible because everything else that will come hinges upon it.
We need to realise that we are not forming a cult in here, nor are we preaching to blindly rely on scientific models with their inherent margins of error. We don't want to remove and idol and replace it with another idol. We don't want to give a special place to science and then deny it from religion. There is no leap of faith in here to be taken. We agree that, “[…] there is a philosophical leap from a good scientific theory to the claim that it is a true description,” so we are not going to torture a believer to submit to this or that interpretations of the quantum theory. We do not try to say, “La, la, la, la, la… I am right, and because of it, you are wrong.”
We don't need any of that to discover falsehood in any theory, be it a scientific theory or a religious one.
What I am trying to convey is to point to the fundamental differences and asymmetry between a verification method of coming at a truth compared to a falsification method of coming to truth.
We know that the flat-earth theory of our world is false. It’s not false today. It has always been false and it will remain false forever. As long as there is only one way to show that the flat-earth theory is false, it is demonstrated to be false. We might not know that “one way” when we are living in an ignorant, primitive, stone-age, ancient village. But that is not the point. There is a way to prove it’s false and today we know that “one way” already. Today, we know a lot more that one way to prove it wrong, but that’s not the point either.
It’s not about observations. It’s about contradictions. If there are two statements that say contradictory things, both of them cannot be true. If such statements belong to a larger set of statements and both can be derived from that larger set of statements, then we got a problem here! Now, not only both of those statements cannot be true, but also we have a certain sign that at least one, and possibly more, statements in the set is/are false.
Yeah! You’re right! Here is when religions’ apologists start to lose track and unconsciously or consciously abandon logic.
Apologists try to excuse the contradictions out of existence. They believe that it is possible. Had they understood what came above, they would have done better. They would have known that it is not possible to excuse any contradiction out of existence. That is not how contradictions work.
We cannot excuse contradictions out of existence.