Open Sentence, Statement

Date: 09/18/2001 at 21:24:40
From: Kristin
Subject: Math vocabulary - open sentence and statement

A statement is a mathematical sentence that is either true or false 
but not both. An open sentence is a mathematical sentence that has a 
variable.

My math teacher told me that y+x = x+y is a statement because it is 
always true, but I thought it was an open sentence because it has a 
variable. Then she siad x+0 = x was an open sentence, but that is 
always true, so wouldn't it be a statement?

Date: 09/18/2001 at 23:24:56
From: Doctor Peterson
Subject: Re: Math vocabulary - open sentence and statement

Hi, Kristin.

I think your definitions really come down to the question whether you 
can tell by looking at it whether the equation is true or false - not 
just whether there is a variable involved. It is a statement whether 
it is always true or always false, and "open" if its truth depends on 
the choice of a variable.

So y+x = x+y, since it is always true (we really call this an 
"identity"), would fall in the category of "statement" as you have 
defined it. So would x+0 = x; I can see no way to define the terms 
that would distinguish these equations. The other possibility would be 
that both equations are BOTH statements and open sentences, if "open" 
does mean merely that there is a variable in it, as you suggested.

Having said that, I did a web search to check for a clear definition 
of the terms. I found this:

   Open Sentences - Glencoe/McGraw-Hill Parent and Student Study 
   Guide, Algebra 1
   http://www.glencoe.com/sec/math/algebra/algebra1/algebra1_01/pdf/0105.pdf   

   Mathematical statements with one or more variables are called open
   sentences. An open sentence is neither true nor false until the
   variable has been replaced by a value. Finding a replacement for
   the variable that results in a true sentence is called solving the
   open sentence.

This is not exactly definitive, since although it defines an open 
sentence as one with a variable, it assumes that it can't be always 
true. So where does an identity fit?

Here's another with the same fault:

   Math Vocabulary - John S. Pitonyak
   http://wcvt.com/~tiggr/vocab.html   

   Open sentence
   A statement that contains at least one unknown. It becomes true
   or false when a quantity is substituted for the unknown. (e.g.,
   x + 5 = 9, y - 2 = 7).

Here's yet another, but this time the meaning of "open" is made more 
explicit:

   Mathematics Glossary N-Z 
   http://www.blc.edu/fac/rbuelow/MPS/glossaryn-z.htm   

   Open Sentence
   Sentences (equations) that have variables to be replaced. Open
   sentences cannot be labeled as true or false, their status is
   "open." 

This page doesn't mention that aspect:

   Algebra Glossary of Terms
   http://www.wtvl.net/honda/glossaryal.htm   

   open sentence: (10)  A sentence containing one or more variables.

But this one mentions only the "open" concept:

   Linear Equations - Vernon McBride
   http://www.usd.edu/~vmcbride/CollegeAlgebra/tsld004.htm   

   Whether an equation is true or false depends on the value
   substituted for the variable, and so is called an open sentence.
   The equation is neither true nor false until a value is chosen
   for the variable. 

On the whole, I think I'm right that the primary meaning of "open" is 
that no truth value can be assigned, so an identity is not properly an 
open sentence. But I wish educators were mathematically trained so 
they could give a decent definition of their terms, and be consistent 
in their usage! Without clear definitions, it just isn't mathematics.


After thinking about it, I realized that the term "open sentence" is 
in fact used by mathematicians, though in an area of math that I don't 
know well. It's not quite the same usage as at your level, but it 
probably is the source of the concept. This will go well beyond what 
you wanted to know, but it may be interesting.

In mathematical logic, an open sentence is defined as one that 
contains "free variables," variables that are not "bound" by being 
used in a phrase like "for all x" or "there exists an x such that..." 
(called a quantifier). You could think of them simply as variables 
that have not been given a specific meaning. If we say, "for all x and 
y, x+y = y+x", then x and y are bound variables and the sentence is 
not open. But if we just say "x+y = y+x" without saying anything about 
the variables, then they are presumably free and the sentence would be 
considered open. So although the primary meaning of "open" is "not 
having a definite truth value, because it depends on the unknown 
values of the variables," the fact that the equation happens to be 
an identity and is always true doesn't really change the fact that the 
variables are unspecified and could conceivably affect the validity of 
the sentence.

So if we take the question this way, it would seem that identities, 
when stated without explicitly saying that they are always true, might 
well be considered open sentences. I'm not sure we should go this far, 
though, because the definitions used at your level don't mention 
quantifiers at all. I still think the concept has been stretched a 
little too far by educators, and is not defined well enough when 
removed from its home in formal logic.

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