Are the fundamental laws of logic true?

[trenton]

I have been exploring the most foundational laws in logic. I have been trying to stress test them to see if they are true. This includes

1. The law of identity 

2. The law of non-contradiction (also known as the law of contradiction)

3. The law of the excluded middle

If you would like to try to refute them you can. The law of non-contradiction in particular seems to be the closest to refuted, though this seems debatable.

Firstly, the law of identity is the most stable one in my opinion. A=A. If it is the case that existence is, then existence is. I haven't come up with a clear counter example.

Secondly, that law of non-contradiction seems to hold in most cases. The closest thing to a test would be "this statement is false." Some argue that it is genuinely both true and false at the same time. Dialetheism is something I would like to check for those who claim contradiction is not an error. The counterargument is that "this statement is false" doesn't give any truth value at all and is completely vacuous. But then would that make this neither true nor false? That is still a third category of truth.

The way I think about contradiction is that there are different types of contradictions that are allowed. Some contradictions resolve by disambiguating. Some by setting a reference point. Some are a difference in perspective. Some may be the result of limited human concepts for explaining reality, and this would be semantic or linguistic contradictions. The contradictions I treat as genuine problems would be self defeating performative contradictions that fail their own test. 

For example, "all statements are meaningless unless they can be empirically verified." Logical positivism has an actually damaging contradiction because it fails by its own standard. This is the same as claiming "nothing is true." The problem is that if that statement is true, then there is at least one truth. It the statement is false, then the truth is that the statement is false. A contradiction is not necessarily an error, but there are specific types of contradictions which are actually problematic for a logical system. Some contradictions are self defeating within a logical system. Beyond this domain, I'm not sure I would say that no contradiction is ever allowed. That can be debated.

The law of the excluded middle appears to hold when formalized in a narrow and specific way. That is that a proposition is either true or false by virtue of the structure of a proposition which thus serves as a truth claim. It seems the liar paradox would be the closest attempt at a counter example.

Self reference is the main source of paradox in these cases. I haven't seen any attempt that isn't self referencing.

Do you think any of the laws of logic are true? If one is false, then which one and why?

[Mixcoatl]

I think the third one is non true in quantum physics. That's why new kinds of logic had to be developed. 

I agree that the first one is the most stable law: It's a squared triangle possible? 

By the way, some people say that God has absolute power. I don't know. I think God can't break the most fundamental laws of logic. A squared triangle is absolutely impossible.

[trenton]

1 hour ago, Mixcoatl said:

I think the third one is non true in quantum physics. That's why new kinds of logic had to be developed. 

I agree that the first one is the most stable law: It's a squared triangle possible? 

By the way, some people say that God has absolute power. I don't know. I think God can't break the most fundamental laws of logic. A squared triangle is absolutely impossible.

By a squared triangle, I assume you mean a three sided shape with four sides. A counter example would be a shape that is both a square and a triangle by having three and four sides. It is a fairly stable example.

I think once you pin down limitations, it creates a situation in which one finite thing cannot be another finite thing. Therefore do long as you have defined parameters, there is by necessity limitations in which the defined thing logically can't be other than what it is given the limitations imposed by the definition. 

From that standpoint a true counterexample involving a squared triangle would probably be non duality. If you and I are one, then a square and a triangle are one. You may or may not accept this, but my point is that in order to break the law of identity, one would have to dissolve the entire concept of identity altogether in order to get to anything remotely resembling a counter example. This isn't really the game logic is designed to play.

One thought I had regarding contradiction would be reality as a whole necessarily operating independently of our human concepts. For example, there is the law of impermanence in which change is unchanging. At that point how do we distinguish between reality being constant change and constant persistence? If there is no real difference between change and not change without human concepts creating ground rules and limitations, then doesn't existence itself have this paradoxical nature which can't be cleanly pinned down without human concepts ultimately collapsing under their self imposed limitations, thus leading to contradictions?

What do you think about "everything except change is changing?"

[Ulax]

Epistemic regress argument makes me say no.

I love the epistemic regress argument so much. There should be a statue dedicated to it in every town and city worldwide.

[trenton]

2 hours ago, Ulax said:

Epistemic regress argument makes me say no.

I love the epistemic regress argument so much. There should be a statue dedicated to it in every town and city worldwide.

That is where I explored when I considered this as well.

If logic can't tell if the premises are true, then logically that applies to the laws of logic as well. From that standpoint it can't be known if the laws of logic are true or false.

I don't think the epistemic regress problem demonstrates that the laws are false, but rather that they are unknowable in their truth value. Logic does concede that it can't tell you if it's premises are true. It therefore limits itself to the specific domain of defined propositions which in turn are structured such that they either are true or false, thus following the laws of logic.

This looks like a circular argument if logic is thus used to justify logic. A proposition which enters that system will not break the system due to the rules that system set up for what counts as a proposition. Logic is therefore a system that succeeds on its terms, but not necessarily on other terms like fuzzy logic among other logic types. Succeeding on its terms does not mean it is true according to logic.

[Joseph Maynor]

The laws of logic are a collective creation, or an artform.  This artform has the relevance it has but no more too.  There is a rhetoric and power aspect to using laws of anything to consider.  And it depends on context too.  Sometimes logic is very clear -- someone got busted with their hand in the cookie jar for example.  Or there is x amount of money in someone's bank account right now, or not.  That is an exclusive or that comes from logic.  

[something_else]

I think Gödel's incompleteness theorem is useful here.

Quote

Secondly, that law of non-contradiction seems to hold in most cases. The closest thing to a test would be "this statement is false." Some argue that it is genuinely both true and false at the same time. Dialetheism is something I would like to check for those who claim contradiction is not an error. The counterargument is that "this statement is false" doesn't give any truth value at all and is completely vacuous. But then would that make this neither true nor false? That is still a third category of truth.

This was Gödel's path to his theorem. He essentially took the statement "this statement cannot be proven" and showed that:

• if the statement is true, then there must exist logical statements which cannot be proven (logic is incomplete)
• if the statement is false, then it means the statement can be proven but that would logic has just proved a false theorem (logic is inconsistent)

The way he did this is really cool, basically getting maths/logic to eat its own tail by creating an encoding system, representing a formula in that system that when decoded says "The formula with code number X has no proof.". But then if you decode what X is, it is the encoding of the exact same formula.

Edited yesterday at 10:14 AM by something_else

[Leo Gura]

Law of excluded middle is a stupid one. The other two are good.

[trenton]

1 hour ago, Leo Gura said:

Law of excluded middle is a stupid one. The other two are good.

Wait.

Are you being clever with this statement? This might be a genuine counter example to the laws of logic.

If the laws of logic claim that all three are true, A, B, and C, but C is false, then by the law of the excluded middle, it must be false to claim that A, B, and C are true.

If you don't intend to break logic with that proposition, then I would like to see the precise reasoning for why the law of the excluded middle breaks. I am skeptical of it because it looks like it holds given a very narrow framework for what counts as a proposition. Otherwise, in formal logic a very specific set of criteria are demanded in order to define what counts as a genuine counter example.

The argument for the law of the excluded middle is that if a claim is not true, then it is false. There isn't really a middle ground between true or false. There are logical systems that use other truth values though, meaning it isn't really universal or inherent to logic itself. It only works within a very narrow domain of structured propositions. Beyond that it would collapse.

I think the problem is that the law of the excluded middle assumes a mind independent reality. If reality is mind dependent, then there could easily be more truth values.

I think 'indeterminate' would be a clean counter example to the law of the excluded middle. For example in sorites paradox, there is no objective heap without an arbitrary line. In that example, it is not the case that a thing either is or is not a heap. Instead a heap is indeterminate. It is therefore neither true nor false that a thing is a heap.

The criteria required to defeat the law of the excluded middle is to create a proposition that is neither true nor false. I believe the heap paradox is a clean counter example.

"A heap has exactly 5000 grains." This is genuinely neither true nor false. It is an arbitrary line, or a self constructed truth with no mind independent reality as to what a heap is.

Does this succeed in refuting the law of the excluded middle? Otherwise, the laws of logic are narrowly defined such that they only operate within cleanly defined terms which themselves can be neither true nor false by their constructed nature.

[Xonas Pitfall]

They are all "stable/true" because they all stem from the Law of Identity. The other two are simply derived from it.

1. The Law of Identity

This one is the fundamental law of everything, to be honest. You can derive what God is from just this identity. In fact, God is this law. This identity is truth itself.

Things are what they are. If you are defined as A, then you are A, and as long as you are A, you can never not be A. You acquire everything that makes you A and everything that distinguishes you from B, C, D, E, F, G, and everything else.

2. The Law of Non-Contradiction

This is simply derived from the Law of Identity. You cannot claim that A is not A, or that A is B, etc. The only way these things may appear to happen is through linguistic errors, or because the context and definitions have not been properly defined. Reality, language, and the paradigms we work with can sometimes be tricky and contain implicit meanings that may or may not be properly defined.

For example, sometimes A can be B if B is simply defined as A with more steps. For example:

• All unmarried men are A.
• A bachelor is B.
• Therefore, all unmarried men are bachelors. A = B.

This would appear contradictory, but the reason A = B is that B is secretly just A expressed through a different definition or way of describing it. To be a bachelor means to be an unmarried man. So A is already contained within the definition of B. B is not something fundamentally different from A. B is still, in essence, A. A = A; nothing is broken.

The Liar's Paradox:

"This sentence is false."

It seems contradictory because if the sentence is true, then it is false, and if it is false, then it is true. But the problem is that nothing definite has actually been defined for us to evaluate.

What exactly does "this sentence" refer to? If it refers to the words "this sentence," those words are not themselves a truth claim. If it refers to the entire statement "this sentence is false," then we are simply defining the statement in terms of itself and never arrive at a definite proposition to evaluate.

The paradox does not show that something can be both true and false. The statement fails to establish a definite truth claim in the first place. The apparent contradiction comes from an improperly defined proposition.

3. The Law of the Excluded Middle

The Law of the Excluded Middle is also completely fine and can be derived from the Law of Identity. If A is A, then A is A, and A is not everything else but A. Therefore, it cannot be something else at the same time.

The ways people try to challenge this law usually come from imprecise contextual, linguistic, or conceptual definitions. For example, someone might ask:

"How can something be bittersweet?"
"How can I be both sad and happy?"

But there is no fundamental contradiction here. "Bittersweet" simply means that something has both a bitter and a sweet aspect. Likewise, you can be happy about one aspect of a situation while being sad or unhappy about another aspect of the same situation.

The apparent contradiction only arises when we fail, deliberately or accidentally, to define precisely what aspect, context, or sense we are talking about. Once everything is properly defined, the contradiction disappears.

Edited 17 hours ago by Xonas Pitfall

[Leo Gura]

Nonduality means your excluded middle will collapse.

Excluded middle assumes that category A is inherently different from B. But all categories are duality.

Edited 16 hours ago by Leo Gura

[Xonas Pitfall]

7 minutes ago, Leo Gura said:

Nonduality means your excluded middle will collapse.

Excluded middle assumes that category A is inherently different from B. But all categories are duality.

Yes. That's why I said it must be defined properly.

Ultimately, the only definition you have is God = God, and everything else is a "cut" or definition derived from it.

That's why it is extremely important to define the observed object, definition, or sentence, as well as the context within which it is being defined.

But when you enter a nondual context, you must define it as a nondual context, and then all "dualities" are understood as one.

The law does not necessarily fail, because you would never have defined A as separate from B in the first place.

Whereas, within a dual framework, you would define A as separate from B.

[Leo Gura]

As a historical example: in order to make sense of quantum mechanics, some scientists proposed doing away with the law of excluded middle.

It works for common phenomena, but fails for uncommon phenomena where Unity plays a role.

[Xonas Pitfall]

54 minutes ago, Leo Gura said:

As a historical example: in order to make sense of quantum mechanics, some scientists proposed doing away with the law of excluded middle.

It works for common phenomena, but fails for uncommon phenomena where Unity plays a role.

I'd love to hear more about this! 

[Joseph Maynor]

Naturally the dominant thinkers are going to want logic to be true on some level.  

[Leo Gura]

1 hour ago, Xonas Pitfall said:

I'd love to hear more about this! 

Quine talked about it in his paper.

[Xonas Pitfall]

44 minutes ago, Leo Gura said:

Quine talked about it in his paper.

Appreciated!

[trenton]

Is it possible to actually have a stable definition?

If I wanted to, couldn't I ask for a definition for every word with which you defined another word? This would create an infinite regress or a circular definition.

Once again, it looks like the laws of logic should survive still within the domain of propositions, so long as it is considered to be on a deeper layer than language.

Also I would like to check out Quine.

[Jirh]

True and false are logical entities. The question is self-referential.