The law of noncontradiction

Christoffer · #1 (**permalink**) 03-03-2009, 04:22 AM

I was browsing through Clarks "Thales to Dewey" yesterday, the chapter on Aristotle.

I understand that the LNC is a necessary assumption for all reasoning, but Clark also says that if you deny the LNC then you refute yourself the moment you utter a statement.

My reasoning so far: the LNC says that a statement is either true or false. When you utter a stamement, you assert that what you are saying is true and not false.

But even if I were to deny the LNC, it still seems I could assert that some statement is true. I would have to agree that my statement can be false even if it is true, but my statement could still have the value "true".

I don't see how a denial of the LNC leads to insignificant speech or how it is a necessary assumption for making a statement. Any logicians on the PB board?

Help would be appreciated!

ChristianTrader · #2 (**permalink**) 03-03-2009, 05:25 AM

If one denies the LNC, then one loses the ability to make distinctions. LNC is necessary to retain meaning when one says that something is true. You have to be able to something is X and other stuff is not X. Otherwise meaning breaks down.

In your example, without LNC "true" no longer has any meaning because you cannot differentiate it from anything else. "true" = "false", "true"="no" etc. True means everything and therefore it means nothing.

CT

Christoffer · #3 (**permalink**) 03-03-2009, 07:44 AM

Thanks, CT, for your response. I hope you don't mind a few questions, I'd like to grasp it

Quote:

In your example, without LNC "true" no longer has any meaning because you cannot differentiate it from anything else.

How does that follow?

Quote:

"true" = "false", "true"="no" etc. True means everything and therefore it means nothing.

I don't understand. Why would a denial of the LNC entail that the word "true" gets several different meanings?

Whitefield · #4 (**permalink**) 03-03-2009, 07:55 AM

Under the LNC, providing "x" is not equivocated, "x" cannot be both "x" and "~x" (not x). Remove the restriction of LNC and "x" can be both "x" and "~x". In other words the LNC says "a dog cannot be both a dog and not a dog"; remove LNC and "a dog can be both a dog and not a dog", or "a dog can be a dog and a cat". Substitute "true" for "x" and "false" for "~x", remove LNC and you get "true can be both true and false". Without the LNC there are no truth statements.

sotzo · #5 (**permalink**) 03-03-2009, 08:45 AM

Quote:

Originally Posted by Christoffer

Thanks, CT, for your response. I hope you don't mind a few questions, I'd like to grasp it

Quote:

In your example, without LNC "true" no longer has any meaning because you cannot differentiate it from anything else.

How does that follow?

Quote:

"true" = "false", "true"="no" etc. True means everything and therefore it means nothing.

I don't understand. Why would a denial of the LNC entail that the word "true" gets several different meanings?

The best way to see why LNC cannot be denied is to simply put a denial of the LNC into sentence form.

Denial of LNC:
"The PB is a web discussion board and is not a web discussion board."

Now, assuming we are not using "web discussion board" in 2 different ways in the above sentence, then what we are saying is that the PB belong to the class of things known as a web discussion board while at the same time being outside the class of things known as a discussion board. If such is the case, then there is no difference between the definitions of what makes something a web discussion board and what does not...that is the classes collapse into one making it impossible to distinguish what is a web discussion board and what is not. Apply this denial universally to all truths / falsehoods and you eventually wind up with truth = falsehoods (again, all classes collapse) which renders language and thought meaningless.

As Bahnsen said, ultimately the LNC reflects the character of God and His law against lying.

Sorry...just saw I posted the same sort of response as Lance!

Christoffer · #6 (**permalink**) 03-03-2009, 08:45 AM

Quote:

Originally Posted by Whitefield

Under the LNC, providing "x" is not equivocated, "x" cannot be both "x" and "~x" (not x). Remove the restriction of LNC and "x" can be both "x" and "~x". In other words the LNC says "a dog cannot be both a dog and not a dog"; remove LNC and "a dog can be both a dog and not a dog", or "a dog can be a dog and a cat". Substitute "true" for "x" and "false" for "~x", remove LNC and you get "true can be both true and false". Without the LNC there are no truth statements.

Thanks.

Ok, so if we remove the LNC x can be both x and ~x

Or, by extension, a statement can be both true and false. But I don't see how it follows that there are no truth statements.

Maybe we could illustrate by example. Assume that I deny the LNC then later go on to say that "Socrates is a man"

How would you respond?

Skyler · #7 (**permalink**) 03-03-2009, 08:49 AM

Quote:

Originally Posted by Christoffer

Quote:

Originally Posted by Whitefield

Under the LNC, providing "x" is not equivocated, "x" cannot be both "x" and "~x" (not x). Remove the restriction of LNC and "x" can be both "x" and "~x". In other words the LNC says "a dog cannot be both a dog and not a dog"; remove LNC and "a dog can be both a dog and not a dog", or "a dog can be a dog and a cat". Substitute "true" for "x" and "false" for "~x", remove LNC and you get "true can be both true and false". Without the LNC there are no truth statements.

Thanks.

Ok, so if we remove the LNC x can be both x and ~x

Or, by extension, a statement can be both true and false. But I don't see how it follows that there are no truth statements.

Maybe we could illustrate by example. Assume that I deny the LNC then later go on to say that "Socrates is a man"

How would you respond?

Socrates is not a man.

Whitefield · #8 (**permalink**) 03-03-2009, 08:56 AM

Quote:

Originally Posted by Christoffer

Quote:

Originally Posted by Whitefield

Under the LNC, providing "x" is not equivocated, "x" cannot be both "x" and "~x" (not x). Remove the restriction of LNC and "x" can be both "x" and "~x". In other words the LNC says "a dog cannot be both a dog and not a dog"; remove LNC and "a dog can be both a dog and not a dog", or "a dog can be a dog and a cat". Substitute "true" for "x" and "false" for "~x", remove LNC and you get "true can be both true and false". Without the LNC there are no truth statements.

Thanks.

Ok, so if we remove the LNC x can be both x and ~x

Or, by extension, a statement can be both true and false. But I don't see how it follows that there are no truth statements.

Maybe we could illustrate by example. Assume that I deny the LNC then later go on to say that "Socrates is a man"

How would you respond?

Suspending LNC then everything is true AND everything is false. So "Socrates is a man" is a true statement and as Skyler said: "Socrates is not a man" is equally true. Hence we have nonsense and not a truth statement. Blue is green, up is down, bad is good, left is right ... ad infinitum.

Christoffer · #9 (**permalink**) 03-03-2009, 08:58 AM

Quote:

The best way to see why LNC cannot be denied is to simply put a denial of the LNC into sentence form.

Denial of LNC:
"The PB is a web discussion board and is not a web discussion board."

Now, assuming we are not using "web discussion board" in 2 different ways in the above sentence, then what we are saying is that the PB belong to the class of things known as a web discussion board while at the same time being outside the class of things known as a discussion board. If such is the case, then there is no difference between the definitions of what makes something a web discussion board and what does not...that is the classes collapse into one making it impossible to distinguish what is a web discussion board and what is not. Apply this denial universally to all truths / falsehoods and you eventually wind up with truth = falsehoods (again, all classes collapse) which renders language and thought meaningless.

As Bahnsen said, ultimately the LNC reflects the character of God and His law against lying.

Some questions:

If I deny the LNC, can't I still see that the PB is a web discussion board?

IOW, doesn't a denial of the LNC only mean this:

"The PB can be a web discussion board and not a web discussion board at the same time".

As far as I can see, a denial of the LNC only means that contradictory state-of-affairs are possible. Not necessarily actual

Whitefield · #10 (**permalink**) 03-03-2009, 09:03 AM

Quote:

Originally Posted by Christoffer

Quote:

The best way to see why LNC cannot be denied is to simply put a denial of the LNC into sentence form.

Denial of LNC:
"The PB is a web discussion board and is not a web discussion board."

Now, assuming we are not using "web discussion board" in 2 different ways in the above sentence, then what we are saying is that the PB belong to the class of things known as a web discussion board while at the same time being outside the class of things known as a discussion board. If such is the case, then there is no difference between the definitions of what makes something a web discussion board and what does not...that is the classes collapse into one making it impossible to distinguish what is a web discussion board and what is not. Apply this denial universally to all truths / falsehoods and you eventually wind up with truth = falsehoods (again, all classes collapse) which renders language and thought meaningless.

As Bahnsen said, ultimately the LNC reflects the character of God and His law against lying.

Some questions:

If I deny the LNC, can't I still see that the PB is a web discussion board?

IOW, doesn't a denial of the LNC only mean this:

"The PB can be a web discussion board and not a web discussion board at the same time".

As far as I can see, a denial of the LNC only means that contradictory state-of-affairs are possible. Not necessarily actual

But you are using LNC to deny LNC .. because without LNC it is the case that the situation is both possible and not possible ... actual and not actual ... so in answer your statement, "a denial of the LNC only means that contradictory state-of-affairs are possible", without LNC, I would say you are right .. no, you are wrong .. no, you are right ... no, you are wrong ... etc.

sotzo · #11 (**permalink**) 03-04-2009, 06:12 PM

Quote:

Originally Posted by Christoffer

Quote:

The best way to see why LNC cannot be denied is to simply put a denial of the LNC into sentence form.

Denial of LNC:
"The PB is a web discussion board and is not a web discussion board."

Now, assuming we are not using "web discussion board" in 2 different ways in the above sentence, then what we are saying is that the PB belong to the class of things known as a web discussion board while at the same time being outside the class of things known as a discussion board. If such is the case, then there is no difference between the definitions of what makes something a web discussion board and what does not...that is the classes collapse into one making it impossible to distinguish what is a web discussion board and what is not. Apply this denial universally to all truths / falsehoods and you eventually wind up with truth = falsehoods (again, all classes collapse) which renders language and thought meaningless.

As Bahnsen said, ultimately the LNC reflects the character of God and His law against lying.

Some questions:

If I deny the LNC, can't I still see that the PB is a web discussion board?

IOW, doesn't a denial of the LNC only mean this:

"The PB can be a web discussion board and not a web discussion board at the same time".

As far as I can see, a denial of the LNC only means that contradictory state-of-affairs are possible. Not necessarily actual

To say "a denial of the LNC only means that contradictory state-of-affairs are possible" but not "actual" is not dealing with a LNC as a law. As a law of logic, LNC circumscribes, not only what is actual, but what is possible such that the possible=actual. If we were talking about valid versus true reasoning, then yes, possible does not necessarily mean actual. Reasoning can be valid, but not true. All dogs are purple, I have a dog, therefore, my dog is purple is valid but not true (ie, possible, but not actual). However, the LNC isn't in the category of reason, but law..without the latter, you don't have the former.

JohnGill · #12 (**permalink**) 03-04-2009, 07:26 PM

Quote:

Originally Posted by Christoffer

I was browsing through Clarks "Thales to Dewey" yesterday, the chapter on Aristotle.

I understand that the LNC is a necessary assumption for all reasoning, but Clark also says that if you deny the LNC then you refute yourself the moment you utter a statement.

My reasoning so far: the LNC says that a statement is either true or false. When you utter a stamement, you assert that what you are saying is true and not false.

But even if I were to deny the LNC, it still seems I could assert that some statement is true. I would have to agree that my statement can be false even if it is true, but my statement could still have the value "true".

I don't see how a denial of the LNC leads to insignificant speech or how it is a necessary assumption for making a statement. Any logicians on the PB board?

Help would be appreciated!

There is no law of non-contradiction. It is raining outside. It is not raining outside. (Same location, same time.)

Poimen · #13 (**permalink**) 03-04-2009, 07:28 PM

I talked about this in my latest blog post:

Precious Words

panta dokimazete · #14 (**permalink**) 03-04-2009, 07:32 PM

isn't antinomy a type of LNC denial?

**Prufrock** · #15 (**permalink**) 03-04-2009, 07:45 PM

Christoffer,

This is certainly a deep issue with which you're wrestling. If you'll permit my overly-simple answer, even if discussion can proceed at an ontological level regarding the LNC, at a practical, everyday level we need it to be true, lest both discourse and even thought become wholly devoid of meaning. When I say "God is x," I mean to say by this that "God is ~(~x)." If such statements aren't equivalent, then x has just lost all meaning.

JohnGill · #16 (**permalink**) 03-04-2009, 07:48 PM

Quote:

Originally Posted by Prufrock

Christoffer,

This is certainly a deep issue with which you're wrestling. If you'll permit my overly-simple answer, even if discussion can proceed at an ontological level regarding the LNC, at a practical, everyday level we need it to be true, lest both discourse and even thought become wholly devoid of meaning. When I say "God is x," I mean to say by this that "God is ~(~x)." If such statements aren't equivalent, then x has just lost all meaning.

Yeah just read his post and ignore mine. It's a better explanation.

To play off of it:

God is omniscient.
God is not omniscient.

Both are true without LNC.

**Prufrock** · #17 (**permalink**) 03-04-2009, 07:49 PM

To dumb this down to my level, I have a proposal which will serve as an illustration. I propose that you and I start a debate in this thread about anything. The only rule guiding the debate is that LNC is assumed to be false by either one or both of us. Care to try it and see what happens?

JohnGill · #18 (**permalink**) 03-04-2009, 08:05 PM

Quote:

Originally Posted by Prufrock

To dumb this down to my level, I have a proposal which will serve as an illustration. I propose that you and I start a debate in this thread about anything. The only rule guiding the debate is that LNC is assumed to be false by either one or both of us. Care to try it and see what happens?

You talking to me? Don't know if I can dumb down that far.

Assuming no LNC, if you say yes, then you mean no, and if you say no then you mean yes.

OWW! Headache! I hurt my spleen.

Confessor · #19 (**permalink**) 03-04-2009, 08:18 PM

Quote:

Originally Posted by Christoffer

Or, by extension, a statement can be both true and false. But I don't see how it follows that there are no truth statements.

The problem is not that there would be no truth statements. The problem is that, if the LNC were not in effect, any proposition -- notably including the proposition, "No truth statements exist" -- is utterly meaningless.

Brian Withnell · #20 (**permalink**) 03-05-2009, 01:13 AM

Quote:

Originally Posted by Christoffer

I was browsing through Clarks "Thales to Dewey" yesterday, the chapter on Aristotle.

I understand that the LNC is a necessary assumption for all reasoning, but Clark also says that if you deny the LNC then you refute yourself the moment you utter a statement.

My reasoning so far: the LNC says that a statement is either true or false. When you utter a stamement, you assert that what you are saying is true and not false.

But even if I were to deny the LNC, it still seems I could assert that some statement is true. I would have to agree that my statement can be false even if it is true, but my statement could still have the value "true".

I don't see how a denial of the LNC leads to insignificant speech or how it is a necessary assumption for making a statement. Any logicians on the PB board?

Help would be appreciated!

Post modernism in full array denies truth exists. Reject God and religion, reject philosophy, reject science, reject logic and reason and there isn't anything left.

"How does it make you feel?" (Which from me is sarcasm ... it is the only thing left.)

Confessor · #21 (**permalink**) 03-05-2009, 01:18 AM

Quote:

Originally Posted by Brian Withnell

Post modernism in full array denies truth exists. Reject God and religion, reject philosophy, reject science, reject logic and reason and there isn't anything left.

"How does it make you feel?"

Both angry and not angry, at the same time and in the same respect, of course.

Christoffer · #22 (**permalink**) 03-05-2009, 06:21 AM

Quote:

Originally Posted by Prufrock

To dumb this down to my level, I have a proposal which will serve as an illustration. I propose that you and I start a debate in this thread about anything. The only rule guiding the debate is that LNC is assumed to be false by either one or both of us. Care to try it and see what happens?

Ok

Here is my statement (assume I am an atheist)

Logic is just a formal language that we have made up. Without humans there would be no logic. The LNC doesn't necessarily tell us anything about reality.

Theogenes · #23 (**permalink**) 03-05-2009, 10:51 AM

The LNC is not just a construct of thought or language. It has an ontological basis in the Being of the Trinity. The Father is the Father and is not the Son nor the Spirit; The Son is the Son and not the Father nor the Spirit; The Spirit is the Spirit and not the Father nor the Son. So the LNC is a characteristic of THE FINAL reality, the Trinity.

Chew on that for a while...

Christoffer · #24 (**permalink**) 03-06-2009, 08:22 AM

ok, now I've thought about this.

If the LNC does not hold then cats can be dogs and cars. The term "cat" refers not only to cats but also to an infinite number of other things.

So is the problem then epistemological? If the LNC does not hold then it is impossible to know the meaning of a statement? So it seems to me. It is impossible to know what my statements mean and what other people mean by their statements, since the terms we use have an infinite amount of meanings.

Am I on the right track?

jwithnell · #25 (**permalink**) 03-06-2009, 08:40 AM

I'm surprised my mathematician husband didn't jump in with a basic premise in math: that A cannot be non-A. Same idea, different language. And a 1800s mathematician, George Boole, further refined Aristotelian logic by addressing whether or not you can presume that something exists in certain categories. (I'm going back 25 years in my memory on this one, so excuse the fuzziness.)

What is important about the law of contradictions is that it allows you to make logical assertions about other type of statements both in terms of universal statements (All-A) and particular statements (Some-A). I.e., if all A's are B's than you cannot say that some A's are not B's and so forth ....

sotzo · #26 (**permalink**) 03-06-2009, 08:54 AM

Quote:

Originally Posted by Christoffer

Quote:

Originally Posted by Prufrock

To dumb this down to my level, I have a proposal which will serve as an illustration. I propose that you and I start a debate in this thread about anything. The only rule guiding the debate is that LNC is assumed to be false by either one or both of us. Care to try it and see what happens?

Ok

Here is my statement (assume I am an atheist)

Logic is just a formal language that we have made up. Without humans there would be no logic. The LNC doesn't necessarily tell us anything about reality.

A couple options for a response:

"If it's just made up, then I'm going to go ahead and change the so-called made-up rules for this conversation: the LNC is false. Now since it is false, there are no longer any definitions for the words in this conversation....even the words in this sentence are unable to be distinguished from "dogs piece once for apple treat small can". But such a rule change reduces its application to absurdity...there is no getting around this and, hence, it is not made up but part of the composition of this world...an ontological reality. If you can find a way to change this "made-up" rule and show how application of its converse makes sense, I'm all ears."

"If it is possible for a universe to exist where absent humans there would be no logic, then please give an example of such a possible universe. You can't simply conjecture this because it assumes what is yet to be proven...that logic is an ontological reality versus a mere convention of man. Conventions can be changed, so please show how changing the LNC is possible. Also, this anthropomorphic approach is subject to the same criticism as "if a tree falls in the forest and nobody hears it, does it make a sound?". In such scenarios, we would necessarily be required to deny that the physical laws of nature cease where man is not present to detect them at work. Similarly, by saying the LNC's existence depends on man's use of it, results in the belief that laws of logic cease to operate in cases where man is absent."

In both instances, the interlocutor is beyond faith...he is in territory that renders all of the world absurd....the very words he used to ask the question are uninterpretable.

I remember being a new believer and hearing the Bahnsen / Stein debate where the former did a masterful job of showing how logic is inherent in reality, not a posteriori to it. Once the truth of the LNC sinks in and you take it through a few rounds of examples of trying to deny it, you end up truly getting a glimpse of what insanity is....it's like watching Fear and Loathing in Las Vegas or Requiem for a Dream. (If you haven't seen those movies I don't recommend it...just trying to drive home the point of the true feeling of insanity that comes over you when you try to deny LNC.)

***greenbaggins*** · #27 (**permalink**) 03-06-2009, 10:12 AM

Quote:

Originally Posted by Christoffer

ok, now I've thought about this.

If the LNC does not hold then cats can be dogs and cars. The term "cat" refers not only to cats but also to an infinite number of other things.

So is the problem then epistemological? If the LNC does not hold then it is impossible to know the meaning of a statement? So it seems to me. It is impossible to know what my statements mean and what other people mean by their statements, since the terms we use have an infinite amount of meanings.

Am I on the right track?

Right. Incidentally, this is the "logical" conclusion to Medieval nominalism, which said that distinctions don't exist in reality, they only exist in the mind. This is the problem of whether terms refer to things outside the term. Is a word (and consequently the whole language) a symbol that refers to something, or can I willy-nilly apply the term to anything I want?

Confessor · #28 (**permalink**) 03-06-2009, 01:32 PM

Quote:

Originally Posted by Christoffer

So is the problem then epistemological?

It definitely is, but I think it might also apply to all branches of philosophy. It would apply to metaphysics in that, if the law were suspended, dogs would actually be not-dogs, and if applies to ethics, murder would actually be both wrong and permissible.

Christoffer · #29 (**permalink**) 03-12-2009, 06:38 AM

Now I understand why language needs the LNC.

If x can be not-x, then cars can be houses and an infinite amount of other things. Thus calling something a car says nothing definite about it.

But what when you are standing right in front of, say, a house. You think to yourself "this is a house". Isn't that a true thought then even though the LNC does not hold?

Or what if I point to a car and say "that is a car". If the LNC wouldn't hold, wouldn't the statement still be true?

Or would the above statements/beliefs be trivially true? Ie. it would be equally true to call houses cars and cars dogs?

hmmm

sotzo · #30 (**permalink**) 03-12-2009, 08:58 AM

Quote:

Originally Posted by Christoffer

Now I understand why language needs the LNC.

If x can be not-x, then cars can be houses and an infinite amount of other things. Thus calling something a car says nothing definite about it.

But what when you are standing right in front of, say, a house. You think to yourself "this is a house". Isn't that a true thought then even though the LNC does not hold?

Or what if I point to a car and say "that is a car". If the LNC wouldn't hold, wouldn't the statement still be true?

Or would the above statements/beliefs be trivially true? Ie. it would be equally true to call houses cars and cars dogs?

hmmm

Per above thread, if LNC doesn't hold "that is a car" or "that is a house" would not be true...neither would it be false...it wouldn't have any content whatsoever.

I still think where you may be needing more clarity is in differentiating the conventional nature of language with the non-conventional nature of LNC. Yes, we could agree to say "that is a stourd" rather than "that is a car"...but so long as by "stourd" we both understood the referent was a car, the LNC would absolutely hold and if it didn't there would be no content whatsoever in the thought. So, we CAN decide to call houses cars and cars dogs....however, notice what you are doing...you are simply taking a convention (language), changing it, then saying "see the LNC doesn't hold because we can call a car a dog"....but all that does is exchange words to refer to the same object. Ironically, one would need to invoke LNC to even do that because now what one would say is "that is no longer called car (A) , rather it is called dog (not-A)".

LNC is unescapable.

louis_jp · #31 (**permalink**) 03-12-2009, 09:07 AM

How could one say that the LNC is not true? To make that statement is to say that it could be true. So you've basically refuted yourself with your own proposition.