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!
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