I have a few questions about the law of non-contradiction that have kind of been bothering me. I would appreciate some help on how to prove this law to be true and impossible to violate:

1. I’ve heard that you can only prove the law if you use the law, thus nothing really gets proven. Is there a rebuttal for this?

2. Why does something either have to be true or false. Why can’t something be true and false at the same time?

I think that something can be both true and false at the same time. Take for example the sentence:
This sentence is false.

1. The law exists independently of our proofs. We “discover” it; we didn’t invent it. There will never be a square-circle, for example.

2. The two terms are mutually exclusive.

Yes, I think there is a rebuttal for that. First, let’s look at the two halves of their claim: “you can only prove the law if you use the law” – “thus nothing really gets proven.” The first part sounds true enough because we can’t even Think without using the law. But the second part doesn’t seem to follow from the first. Why can’t you use the law to prove the law? You can prove that other things work by using them, cars for example. Suppose someone came along and said, “You used the car to prove that it works, thus nothing really got proven.” Well I think using the car is a pretty Good proof that it works, and using the law of non-contradiction is a pretty good proof that It works. The real problem for skeptics is that if they try Not to use the law, That is what doesn’t work. You can’t think properly unless you believe that the conclusions you come up with are true and not false.

1. Why does something either have to be true or false. Why can’t something be true and false at the same time?

Because of something called the Principle of Explosion. Wikipedia explains:

[quote=Wikipedia][C]onsider two contradictory statements – “All lemons are yellow” and “Not all lemons are yellow”, and suppose (for the sake of argument) that both are simultaneously true. If that is the case, anything can be proven, e.g. “Santa Claus exists”, by using the following argument:

1. We know that “All lemons are yellow” as it is defined to be true.
2. Therefore the statement that (“All lemons are yellow" OR "Santa Claus exists”) must also be true, since the first part is true.
3. However, if “Not all lemons are yellow” (and this is also defined to be true), Santa Claus must exist – otherwise statement 2 would be false. It has thus been “proven” that Santa Claus exists. The same could be applied to any assertion, including the statement “Santa Claus does not exist”. source
[/quote]

I hope that makes sense. If the law of non-contradiction was false, the result would be that everything is both true and false. “Truth” would then be a meaningless concept. “There is no truth” would be true, which is self-contradictory, because that itself is a truth-claim. Moreover, thinking would be useless and maybe even impossible. Our own experience that thinking works thus may suggest that truth exists, and that the law of non-contradiction is true.

[As a complete aside, I think it is ironic and funny that the law of Non-contradiction is Also known as the law of Contradiction. :shrug:]]

Not really. Some things are true and some things are false. But there are things that are both true and false at the same time.

It can’t be both true and false at the same time. Using the example of the square-circle - the truth of the matter is that you have either a square OR a circle, never both. With any other statement, the principle still applies.

True for your examples. But not true for all statements. The example given in #2 shows that a statement can be both true and false at the same time. If the sentence is true it is false and if it is false it is true.

AND you have just broken the third principle of the LAW, namely that “opposite assertions cannot be both true at the same time”.
Aristotle the fellow that first postulated the LAW assumes that any assertion involves predicating one thing of another. Your example does not. It is self referential violating the third principle.

Unfortunately your example is a prime example of a paradox and does not, either prove or disprove the Law of Contradiction.

By the way the guy who came up with that example was “Eubulides of Miletu” 4th century BCE while the LAW of Non-Contradiction was postulated by Aristotle who died in 232 BCE came almost 200 years after Eubulides.

It is obvious that Aristotle was aware of Eubulides and saw fit to enumerate a principle so that one was NOT tempted to use such a ruse to thwart his LAW

Spatial and concrete thinking helps one avoid the pitfalls of assuming everything in life is binary and denying the existence of paradox.

Everything in God is paradoxical to put it mildly.

Not much in real life is binary. Logic is essential, people mostly don’t know how to apply it, it gets a bad name then it gets applied even worse.

What does it mean when you call something a paradox? Doesn’t it mean that it has broken the law of non-contradiction. If something has broken the law of non-contradiction, then that means that the law of non-contradiction is not a universal law because it does not apply in all cases.
A sentence can be both true and false at the same time.
Take the example illustrated above:

**This sentence is false **

If the sentence is true, then the sentence is false.
OTOH
If the sentence is false, then the sentence is true.

Actually it is quite useless that you keep repeating your “example”. We got it, it is a Paradox. Paradoxes cannot be used within the Law of Non Contradiction.

Aristotle, the guy who proposed the Law of Non-Contradiction, enumerated 3 principles for the Law.
You are breaking the third one.
Namely that the effect (true/false) is directly affecting the proposition. But for any proposition to be valid Aristotle predicates that “any assertion involves predicating one thing of another”.

Now you can: either understand how the law is to be applied to logic AND follow the 3 principles enumerated by Aristotle or continue in your false assertion that your example somehow applies to this law. Which It clearly does NOT.

Why not? Is it because they prove that the Law of non-Contradiction is not a universal law which applies to all statements?

The principle of explosion ought to be rejected. Given how many beliefs we all have, it is very likely true that at least two of them together form a contradiction (either directly or inferentially) - it is unlikely that the set of beliefs that any of us in fact holds is totally consistent - we do try to be as consistent as we can be, but it is impossible for mere humans to be perfectly consistent, for there often exist contradictions which are not immediately apparent. So, the conjunction of all our beliefs very likely constitutes or implies a contradiction. By the principle of explosion, from such a contradiction, any belief logically follows. Hence, it is very likely that we are logically justified in believing anything whatsoever. Clearly, this is absurd - where did we go wrong? I think we went wrong in accepting the principle of explosion.

The purpose of logic should be to model human thought, and improve it through formalisation and regularisation. But if the model, the formalisation, yields conclusions that contradict common sense, then should we reject common sense, or should we go looking for an alternative model which corresponds with it better?

Classical logic is based on material implication, for which it is true that “from a falsehood anything follows” (ex falso quodlibet). The principle of explosion, “from a contradiction anything follows” (ex contradictione quodlibet), is just a special case of this more general principle of material implication. As an example of an application of that principle, consider the following inference: “If John F. Kennedy was President of the Soviet Union, then dinosaurs live on the moon”. Now, the antecedent is false, therefore according to material implication, the conditional is true, no matter how absurd the consequent - indeed, despite the fact that the consequent has nothing to do with the antecedent, and that even were the antecedent true, that would give us no reason to believe the consequent. So, if material implication violates common sense, which should yield: common sense, or material implication? I say, keep common sense, and get rid of material implication. But since classical logic is based on material implication, that means we need to abandon classical logic and turn to non-classical logics such as relevance logics (which are a type of paraconsistent logic). And in relevance logics and paraconsistent logics, the principle of explosion is rejected.

Simon

You have a distorted view of an “Universal Law”.

The Law of Gravity IS a Universal Law, however there are distinct exception to it.
For any given Law, there are principles that inform us, how to properly apply said Law.
Nothing inconsistent with that.

Physicists have spent the last 40 years, trying to come up with a unified Law, that can explain both the Macro and the Micro Cosmos. They have yet to attain their goal.
It is called the Holy Grail of Physics. :rolleyes:

Why it is important you may ask! Well quite simply the Laws that Apply to the Micro Cosmos (Quantum Mechanics, String Theory, etc.). Fail miserably if you try to apply them to the Macro Cosmos.
On the other hand the Theory of Relativity works remarkably well at the Macro Cosmos level yet fail miserably if applied to the Micro Cosmos.

That does not mean we throw them away. Not at all.
We simply specify principles of applicability for those Laws.

In other words “Universal” does not quite mean what you have in mind.
For the Law of Gravity, it means that this Law applies everywhere in the Universe. However there are particular conditions where it breaks down.
In those cases, we outline the principles of the Law’s applicability. It is still Universal because we know where it works and where it does not.

I hope this helps you a little in understanding this issue.

I thought that relativistic quantum mechanics worked remarkably well in chemistry when studying elements such as the lanthanides and actinides. I thought that relativistic quantum mechanics had many successful applications in high energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics.
I was surprised to learn from you that the theory of relativity fails miserably if applied to the micro cosmos as described by quantum mechanics because the following references do not seem to agree with that:
M.Reiher, A.Wolf (2009). Relativistic Quantum Chemistry
P. Strange (1998). Relativistic Quantum Mechanics: With Applications in Condensed Matter and Atomic Physics
W. Greiner (2000). Relativistic Quantum Mechanics. Wave Equations
A. Wachter (2011). “Relativistic quantum mechanics”
H. Pilkuhn (2005). Relativistic Quantum Mechanics
P.A.M Dirac (1932). “Relativistic Quantum Mechanics”
J.P. Antoine (2004). “Relativistic Quantum Mechanics”.
Relativistic quantum mechanics of particles with direct interactions
by F. Coester and W. N. Polyzou
Phys. Rev. D 26, 1348 – Published 15 September 1982
C. Fronsdal, L.E. Lundberg (1970). “Relativistic Quantum Mechanics of Two Interacting Particles”. Phys. Rev. D 1. arXiv:physics/9703008
T. Ohlsson (2011). Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory
I.J.R. Aitchison, A.J.G. Hey (2002). Gauge Theories in Particle Physics: From Relativistic Quantum Mechanics to QED
I. P. Grant (2007). Relativistic Quantum theory of atoms and molecules
Can you tell us where the above books have gone wrong in applying the theory of relativity to the micro cosmos as described by quantum mechanics?

In JerryZ’s defence, I think when he said “relativity” he meant “general relativity”. There is a real open issue in how to get general relativity and quantum theory to work together. You are right that getting special relativity and quantum theory to work together is a problem that was solved decades ago.

Simon

Yes. Special relativity and quantum mechanics are compatible, and I don’t see any contradictions between the two. In fact, quantum field theory,the framework used to understand non-gravitational forces, unites the two. Although there are problems with developing a consistent theory of quantum gravity, my understanding is that general relativity theory works with questions such as low-energy scattering of photons and gravitons, but breaks down when you get to the Planck scale, which is the problem of "nonrenormalizability."But doesn’t this just mean that quantum field theory is not sufficient for understanding gravity?

I didn’t know that the laws from string theory fail miserably if you try to apply them to the Macro Cosmos. I thought that string theory was a quantum mechanical theory that includes general relativity as a low-energy limit. More generally, i thought that string theory was developed as a theoretical framework which could provide a unification of general relativity with the principles of quantum mechanics.

I cannot see any problem with ANY of the books you quoted above, my question to you is: CAN YOU see any problem with them?

OK, I will tell you. You are comparing apples to oranges.

The Theory of Relativity was formulated by Albert Einstein. It is comprised of 2 parts: “General Relativity” and “Special Relativity”.

“Quantum Mechanics” can ONLY be applied to Micro Cosmos (Atoms, Sub Atomic Particles), you CANNOT apply it to Galaxies, Solar Systems, etc. (Macro Cosmos).
At the Macro scales the General Relativity Theory works very well.

An attempt to merge the 2 theories into a single Unified Theory HAS NOT been accomplished yet.
The 2 theories do not get along with each other. BUT each works nicely within their own domains.
An attempt to merge the 2 theories has had some success in very limited cases by using the “Special Relativity Theory” ONLY together with “Quantum Mechanics” which has created it’s own particular Theory called “Relativistic Quantum Mechanics”.

However I hope that you understand that “Relativistic Quantum Mechanics” is neither The “Theory of Relativity” (General Relativity) nor “Quantum Mechanics”. It is a completely distinct theory with it’s own set of rules and principles.

Merging “General Relativity” AND “Quantum Mechanics” HAS NOT BEEN ACCOMPLISHED. It is called the “Holy Grail of Physics”

Which going back to the Law of Non Contradiction has it’s own set of rules that you HAVE to follow if you pretend to use it. Like any Law in the Universe.

I don’t see what this has to do with the “law” of non-contradiction. AFAIK, there are no contradictions between quantum mechanics and special relativity. Further I thought that for questions like the low-energy scattering of photons and gravitons, there was no problem coupling the Standard Model to general relativity. Although quantum field theory is not sufficient for understanding gravity, it does not mean that there is a contradiction between general relativity and quantum mechanics. So this discussion of relativity and quantum mechanics seems to me to be off the topic of a discussion about the “law” of non-contradiction.

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