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Posted 15 January 2009

INTENT OF ASSERTIONS

An assertion in mathematical writing can be a claim, a definition or a constraint.  It may be difficult to determine the intent of the author.  That is discussed briefly here.

English sentences can state facts, ask question, give commands, and other things.  The intent  of an English sentence is often obvious, but sometimes it can be unexpectedly different from what is apparent in the sentence.  For example, the statement “Could you turn the TV down?” is apparently a question expecting a yes or no answer, but in fact it may be a request.  (See Wikipedia about this phenomenon.)  Such things are normally understood by people who know each other, but people for whom English is a foreign language or who have a different culture have difficulties with them.

There are some problems of this sort in math English and the symbolic language, too.  An assertion can have the intent of being a claim, a definition, or a constraint.  Sentences in math, as in ordinary English, can also be questions or commands.   

Most of the time the intent of an assertion in math  is obvious.  But there are conventions and special formats that newcomers to abstract math may not recognize, so they misunderstand the point of the assertion.   This section takes a brief look at some of the problems.

Terminology

The way I am using the words “assertion”, “claim”, “constraint” is not standard usage in math, logic or linguistics. 

Claims

In most circumstances, you would expect that if a lecturer or author makes a math assertion they are claiming that it is a true statement, and you would be right.  

Examples

a)    “The 240th digit of  after the decimal point is 4.”

b)    “An integer divisible by 4 must be even.”

c)    “If a function is differentiable, it is continuous.”

d)     “  ”

Remarks

¨  Note that you don’t have to know whether these statements are true or not to recognize them as claims.  An incorrect claim is still a claim. 

¨  The assertion in (a) is a statement, in this case a false one.  If it claimed the googolth digit was 4 you would never be able to tell whether it is true or not, but it still would be an assertion intended as a claim.

¨  The assertions in (b) and (c) both use the standard math convention that an indefinite noun phrase (such as “a widget”) in the subject of a sentence is universally quantified (see also here).  In other words, “An integer divisible by 4 must be even” claims that any integer divisible by 4 must be even. This statement is a claim, and it is true.  (Similar remarks for (c).) 

¨  (d) is a (true) claim in the symbolic language.  Note that “3 + 4” is not an assertion at all, much less a claim. 

Definitions

Definitions are discussed primarily in the chapter on definitions.  A definition is not the same thing as a claim. 

Example

The definition

“An integer is even if it is divisible by 2”

implies that the claim

“An integer is even if and only if it is divisible by 2”

is true. The definition makes the claim true. 

Newcomers may not understand that the intent of an assertion they are looking at is that it is a definition.  This can be the author’s fault; some texts are very sloppy about this.  You can also write down a statement that a lecturer made and fail to write down that they said it was a definition, causing you no end of confusion later when you review your notes.

Examples

¨  Suppose that the concept of “even integer” was new to you and the book said, “A number is even if it is divisible by 4.”  Later the book refers to 6 as even and you pull your hair out wondering why.   The statement is a correct claim but an incorrect definition.  A GOOD writer would write something like “Recall that a number is even if it is divisible by 2, so that in particular it is even if it is divisible by 4.”

¨  Suppose you had heard the word “even” but took overly sketchy notes.  For example, perhaps the lecturer said, “By definition, an integer is even if it is divisible by 2.” and you wrote in your notes “An integer is even if it is divisible by 2.”  Later you get all panicky wondering How Did She Know That?? 

¨  The confusion in the previous example can also occur if a book says “An integer is even if it is divisible by 2” and you don’t know about the convention that when an author puts a word or phrase in boldface or italics it may mean they are defining it.

Constraints

Here are two assertions that contain variables. 

¨   “n is even”. 

¨     “  ”.

Such an assertion is a constraint or a condition if the intent is that the assertion will hold in that part of the text (the scope of the constraint).  The part of the text in which it holds is usually the immediate vicinity unless the authors explicitly says it will hold in a larger part of the text such as “this chapter” or “in the rest of the book”.

Examples

¨   Sometimes the wording makes it clear that the phrase is a constraint.  Thus a statement such as  “Suppose  ” is a constraint on the possible values of x.  Note

¨  The statement “Suppose n is even”  is an explicit requirement that n be even and an implicit requirement that n be an integer.

¨  A condition for which you are told to find the solution(s) is a constraint.  For example:  “Solve the equation .”  This equation is a constraint on the variable x.  “Solving” the equation means saying explicitly which two number make the equation true.   You may have been given that inequality with instructions to “solve” or “find the truth set”. 

Postconditions

The constraint may appear in parentheses as a postcondition on an assertion, for example

“ .                                                          (  )”

which means that if the constraint “x > 1” holds, then “  ” holds.  In other words, for all , the statement  is true. In this statement “  ” is not a constraint, but a claim.   More about this notation here.

Usage

Mathematicians usually use the word “condition”.  “Constraint” is used mostly in a few specific fields, notably differential equations.  I am using “constraint” here because “condition” sounds too much like conditional. 

Note

You may have been given the inequality  with instructions to “solve” or “find the truth set”.  This usually means express it as linear inequalities, in this case “  ”.  Still, it is correct to say that the truth set is the set of x for which .  Return