# Thread: Law of noncontradiction

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## Law of noncontradiction

http://en.wikipedia.org/wiki/Law_of_noncontradiction

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This article uses forms of logical notation. For a concise description of the symbols used in this notation, see List of logic symbols.

In classical logic, the law of non-contradiction (LNC) (or the law of contradiction (PM) or the principle of non-contradiction (PNC), or the principle of contradiction) is the second of the three classic laws of thought. It states that contradictory statements cannot both be true in the same sense at the same time, e.g. the two propositions "A is B" and "A is not B" are mutually exclusive.

The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

\mathbf{*3\cdot24}. \ \ \vdash. \thicksim(p.\thicksim p)[1]

The law of non-contradiction, along with its complement, the law of excluded middle (the third of the three classic laws of thought), are correlates of the law of identity (the first of the three laws). Because the law of identity partitions its logical Universe into exactly two parts, it creates a dichotomy wherein the two parts are "mutually exclusive" and "jointly exhaustive". The law of non-contradiction is merely an expression of the mutually exclusive aspect of that dichotomy, and the law of excluded middle, an expression of its jointly exhaustive aspect.
If you are going to use this as a basis of logical discussion, then make sure there is no ambiguity in your propositions.

http://en.wikipedia.org/wiki/Law_of_noncontradiction

Interpretations

One difficulty in applying the law of non-contradiction is ambiguity in the propositions. For instance, if time is not explicitly specified as part of the propositions A and B, then A may be B at one time, and not at another. A and B may in some cases be made to sound mutually exclusive linguistically even though A may be partly B and partly not B at the same time. However, it is impossible to predicate of the same thing, at the same time, and in the same sense, the absence and the presence of the same fixed quality.
David

2. Originally Posted by David M
If you are going to use this as a basis of logical discussion, then make sure there is no ambiguity in your propositions.
David
You mean like:

There would be a contradiction if P were true and yet P were not true?

3. ## Real world vs abstract example...

It helps me to understand better by comparing a real world example to an abstraction of the concept. Please give a real world example of an ambiguity in a proposition (i.e. one wherein P is true and yet P is not true?)

Thank you.

4. Originally Posted by Guido Fawkes
It helps me to understand better by comparing a real world example to an abstraction of the concept. Please give a real world example of an ambiguity in a proposition (i.e. one wherein P is true and yet P is not true?)

Thank you.
There is none, of course. I posted it because David has spent over two years insisting that that formulation is "ambiguous." It all started with this innocent little question I asked him on September 27, 2012:

Originally Posted by Richard Amiel McGough
Good morning David,

I don't understand your last post. Could you please tell me which of these two propositions you agree with?

P: There is a paradox if we say God's will is done in heaven and yet angels could sin.
Not P: There is NOT a paradox if we say God's will is done in heaven and yet angels could sin.

By the Law of the Excluded Middle (also known as the Law of Non-Contradiction), you must agree with one of those propositions, P or Not P.

The Law simply states that for any proposition P, either that proposition is true, or its negation Not P is true.

Thanks,

Richard
David answered, in part, by saying "The law that you now state (above) I do not disagree with. I am rejecting both of your statements, because the paradox that is at the center of this present discussion does not exist. I have explained to you that your premise is wrong. You are saying that angels can sin and I am saying that God's Angels do not sin." (link)

He says he doesn't disagree with the law of non-contradiction, and then directly contradicts both it and himself by rejecting both both P and Not P! Oy vey. The conversation went downhill from there, and remains unresolved to this day after many hundreds of posts on that one topic.

Richard

5. ## Aha!

Originally Posted by Richard Amiel McGough
There is none, of course. I posted it because David has spent over two years insisting that that formulation is "ambiguous." It all started with this innocent little question I asked him on September 27, 2012:

David answered, in part, by saying "The law that you now state (above) I do not disagree with. I am rejecting both of your statements, because the paradox that is at the center of this present discussion does not exist. I have explained to you that your premise is wrong. You are saying that angels can sin and I am saying that God's Angels do not sin." (link)

He says he doesn't disagree with the law of non-contradiction, and then directly contradicts both it and himself by rejecting both both P and Not P! Oy vey. The conversation went downhill from there, and remains unresolved to this day after many hundreds of posts on that one topic.

Richard
Richard,

Aha! I should have realized that there was some history there. Thanks for the explanation.

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David M

Law of non-contradiction

If you are going to use this as a basis of logical discussion, then make sure there is no ambiguity in your propositions.

Originally Posted by Richard Amiel McGough
You mean like:

There would be a contradiction if P were true and yet P were not true?
Originally Posted by Guido Fawkes
It helps me to understand better by comparing a real world example to an abstraction of the concept. Please give a real world example of an ambiguity in a proposition (i.e. one wherein P is true and yet P is not true?)

Thank you.
This is hilarious. Richard is missing the point. It appears the guru of logic (Richard) cannot apply the words "contradiction" and "ambiguity" correctly. No wonder that for two years I have been going round in circles with Richard. He has not taken notice of the warning I have quoted from Wikipedia.

Mark wants an example of a proposition which is true and not true. In that case, the example would not be ambiguous, it would be contradictory.

Note; Wikipedia says regarding the Law of non-contradiction; One difficulty in applying the law of non-contradiction is ambiguity in the propositions.

According to the dictionary, the word proposition as applied to logic, is defined as;
Logic. a statement in which something is affirmed or denied, so that it can therefore be significantly characterized as either true or false.

Also this;
proposition in Technology

logic
A statement in propositional logic which may be either true or false. Each proposition is typically represented by a letter in a formula such as "p => q", meaning proposition p implies proposition q.

P and Q only represent the propositions. The proposition has to be stated in words. The ambiguity resides in the words.

dictionary.com

ambiguity

noun, plural ambiguities.
1. doubtfulness or uncertainty of meaning or intention:
to speak with ambiguity; an ambiguity of manner.
2. an unclear, indefinite, or equivocal word, expression, meaning, etc.:
a contract free of ambiguities; the ambiguities of modern poetry.
http://muse.dillfrog.com/ambiguous_words.php

Ambiguous Words

Here's a bunch of words that, by themselves, have a handful of meanings. Because of this flexibility, they can be instrumental in titles for your songs/poems/stories/etc.

In other words, we cut you a break by breaking the breaks on "break" and a bunch of other ambiguous words, without taking a break!

break
cut
run
play
make
light
set
hold
clear

(DM. list is truncated here for brevity. One word not in the list on the website is "angel")

Word meaning database kindly provided by WordNet
Web site © 2009-2010 dillfrog.com except where noted.
David

7. Originally Posted by David M
This is hilarious. Richard is missing the point. It appears the guru of logic (Richard) cannot apply the words "contradiction" and "ambiguity" correctly. No wonder that for two years I have been going round in circles with Richard. He has not taken notice of the warning I have quoted from Wikipedia.

Mark wants an example of a proposition which is true and not true. In that case, the example would not be ambiguous, it would be contradictory.

Note; Wikipedia says regarding the Law of non-contradiction; One difficulty in applying the law of non-contradiction is ambiguity in the propositions.
Excellent. Please explain the "ambiguity" in this statement:

There would be a contradiction if God's will is done in heaven, and yet God's angels in heaven could sin.

Thanks!

Richard

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Originally Posted by Richard Amiel McGough

There would be a contradiction if God's will is done in heaven, and yet God's angels in heaven could sin.
Unless it is God's will that his subjects do sin.

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Originally Posted by sylvius
Unless it is God's will that his subjects do sin.
That is what you have to prove. Can you prove it from God's word?

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Originally Posted by Richard Amiel McGough
Excellent. Please explain the "ambiguity" in this statement:

There would be a contradiction if God's will is done in heaven, and yet God's angels in heaven could sin.

Thanks!

Richard
Please state your proposition.

I notice you have included the "and yet" again, which you pointed out to me you had dropped from your recent quotation of that sentence.

"and yet" is grammatically incorrect. As was found, the fact that the two words according to their placement can mean something different to what you intend. Different meanings lead to ambiguity. All you have to do, is re-word your statement (sentence) removing anything with more than one meaning, or first agreeing the meaning with the other party.

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