The Difference between Open Sentences and Statements

Date: 06/28/2008 at 13:45:25
From: Eric
Subject: What if an open sentence is true/false for all x?

If an open sentence is true or false for all x, is it still considered 
an open sentence or is it considered a statement?  For example:

  x is greater than 4 or x is less than 7.

Because the example has a variable in it, it may be considered an open 
sentence because sometimes whether an open sentence is true or false 
depends on the x.  However, a statement is either true or false.  My 
example is true for all x.

Before I start my logic unit with my tutor, she asked me to find out 
the answer to the question above.  I have tried different websites, 
but none of them helped.  She gave me your website and I decided to 
try it.  I know the different types of sentences, which of them are 
logic sentences and which of them are math sentences.  This, however, 
is something that I not only need to find out, but it is something 
that I am interested in.

Date: 06/28/2008 at 22:58:59
From: Doctor Peterson
Subject: Re: What if an open sentence is true/false for all x?

Hi, Eric.

Did you try searching our site for the phrase "open sentence" and find
this page?

  Open Sentence, Statement
    http://mathforum.org/library/drmath/view/53280.html 

That discusses your exact question.  But even though I wrote it, I'm
not entirely satisfied with it, and probably you aren't either.  The
problem is that I do not teach at your level, and this terminology is
not used by mathematicians except in a narrow field, so I'm not really
sure how YOU are supposed to be using it.  Moreover, like all 
definitions, these terms vary in use from author to author, and my
answer might not fit your particular context--languages just work like 
that!

Really, the best answer is to point you back to whatever definitions
you have been given; mathematicians always define any terms that might
be used in different ways before they use them, to make sure that 
their readers know what they mean, and we don't try to use our own
definitions when we read what someone else is writing.  So I'd like to
see what definitions you were given for "open sentence" and
"statement", and where they came from.

Looking through what I wrote there, and specifically the sample
definitions I found, I've more or less changed my mind.  I think, 
first, that an open sentence IS a statement, and, second, that any
statement with a variable is an open sentence, regardless of whether
it is always true or always false.

A statement is, primarily, anything you say that is unambiguous, so
that it can't be considered both true and false under ANY
circumstances.   I don't think the main idea is whether it is ALWAYS
true or ALWAYS false, just that it is never BOTH at the same time, or
NEITHER.  So, for example, "I'm tall" is not properly considered a
statement because it depends on whom you compare me to.  In reality,
I'm neither tall nor short, though in some places I'd be called tall.
Do you see why that's not a statement?  It's not the fact that
different people might be either tall or short, but that one person
might not be clearly either, or could be both depending on how you
look at it.  So in this sense, a statement that involves a variable
can be called a statement even though it is not always true or always
false.

On the other hand, the main idea of "open sentence" is that it has no
value until you put in a value for the variable.  That is, ANYTHING
with a variable is "open", even if it turns out that it is always
true; in order to determine the latter, you have to (at least
hypothetically) give the variable a value and see that it is true.  So
by my definitions, something like x + y = y + x is both a statement
and an open sentence--because it is unambiguous but involves a 
variable.  The fact that it is always true, FOR all values you take
for x and y, is irrelevant.  Until you assign values, it remains "open".

Now, I repeat, your definitions may be different, because you may be
working in a different context than I am.  In particular, it sounds
like texts often treat "statement" and "open sentence" as mutually
exclusive (they can't both apply to the same thing).  The only way to
know for sure is to look at the exact wording YOU are given for the
definitions (and perhaps examples), and deduce the answer from that.
If the definitions aren't clear enough to determine the answer, then
it is the definitions' fault!

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/29/2008 at 08:07:43
From: Eric
Subject: Thank you (What if an open sentence is true/false for all x?)

Thanks for replying to my question!  You really helped me!