# Is 1+101=110 a proposition?

There is a question in a book is: if 1+101=110 is a proposition or not.

The author said it's not a proposition, since it's true if the numbers are binaries, is false if the numbers are decimals. So we can't figure out if it's true or false, that it's not a proposition.

But in the book of "Discrete Mathematics and Its Applications", it says 1+1=2 is proposition:

If 1+101=110 is not a proposition, that 1+101=102 should not be a proposition, that 1+1=2 should not be a proposition.

Update

I tried to translate the original text to English, sorry for my poor English.

I will give you some examples to explain the concept of proposition.

1. Chinese people are great
2. Snow is black
3. 1+101=110
4. There are living things on the other stars
5. All of you, stand at attention！
6. Shall we have a meeting tomorrow?
7. What a nice day!
8. I'm lying.
9. I'll learn English, or Japanese.
10. If it's sunny, I will go for a walk.

For the examples above, (1)(2)(4)(9)(10) are propositions. For (4), although we don't know if it's true or false for now, but we will know it someday, so we can say it's a proposition. (3) is true if they are binaries, false if they are decimals, so it's truth value is depended on the context.

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Not sure but 1 + 1 = 2 in decimal and 1 + 1 = 10 in binary in both cases the number is still 2 regardless of it's representation. Whereas, 101 + 1 = 102 or 110 which is not a different representations of the same number. –  Tony Mar 7 '13 at 16:22
It's a little extreme to say $1+101=110$ is not a proposition. The general assumption is there is a clear line of communication going on -- that you know what language someone is speaking. If someone writes down a list of numerals you would expect some comment on how to interpret it. If that comment is not there, generally people make assumptions and you would consider it a proposition. If you're totally unassuming about everything the idea of a proposition is impossible, since we don't know exactly what people intend when they write things down, as we can't know another mind. –  Ryan Budney Mar 7 '13 at 16:25
If I do not speak Chinese, "wo shi zhongguoren" is not a proposition. If I do speak Chinese, it is. –  Neal Mar 7 '13 at 16:27
I think anyone who says Example 3 is not a proposition because of ambiguity should also say Example 1 is not a proposition because of ambiguity. Are Chinese people only those who live in China or also those who were born in China and live elsewhere? What about people with just one Chinese parent? One Chinese grandparent? Does the statement say that the Chinese people are collectively great, or that each Chinese person is individually great? Exercise: Attack Statement 4 similarly. –  Andreas Blass Mar 7 '13 at 16:55
Even aside from the ambiguity issues that Andreas brings up, I daresay that Example 1 is a matter of opinion, so will vary in truth-value from person to person. –  Cameron Buie Mar 7 '13 at 17:09

$1+101=102$ and $1+1=2$ are true in every single base in which $2$ is available to be used, and aren't even sensible in binary (so not false there). They are therefore context-independent (aside from the context they carry with them) in their truth-value.

By contrast, $1+101=110$ is sensible in all bases, but only true in binary, and false in all the others. The context is essential here, and not specified by the statement.

I suspect that's what is meant by "proposition" vs. "not a proposition."

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I would guess that as Cameron said, the book means that a proposition is a declarative sentence whose truth or falsity does not depend on the context in which it is evaluated. This is a pretty fuzzy definition of "proposition" and one could argue that the book is applying it incorrectly. For example:

• The truth of the sentence "Washington, D.C. is the capital of the United States of America" depends on the time at which it is evaluated. The US has had several other capitals in its history.

• The truth of the sentence "snow is black" depends on whether it is night-time and whether there are smoke-belching factories nearby.

• I don't think it is justified to say that we will someday know whether there are living things on other stars; also, even if we find something, it might be on the boundary of what we consider "life."

• The truth of the sentence "I'll learn English or Japanese" depends on who the speaker is, and also on information not available at the time it is spoken.

In any case, this definition of "proposition" is not sufficiently precise to be considered mathematical. I am not aware of any mathematically precise definition of "proposition" that applies to sentences in natural language.

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A proposition is an idea. Language is required to communicate ideas. If a language has not been agreed upon by two parties, they cannot communicate. Therefore, propositions require a language that has been agreed upon to be communicated.

In the case of "1 + 101 = 110", until a language is defined to communicate the idea it is impossible for this to be a proposition. As soon as a language is defined, it will be a proposition.

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