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In orthodox first-order logic, variables and expressions are only allowed to take one value at a time; a variable {x}, for instance, is not allowed to equal {+3} and {-3} simultaneously. We will call such variables completely specified. If one really wants to deal with multiple values of objects simultaneously, one is encouraged to use the language of set theory and/or logical quantifiers to do so.

However, the ability to allow expressions to become only partially specified is undeniably convenient, and also rather intuitive. A classic example here is that of the quadratic formula:

\displaystyle  \hbox{If } x,a,b,c \in {\bf R} \hbox{ with } a \neq 0, \hbox{ then }

\displaystyle  ax^2+bx+c=0 \hbox{ if and only if } x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}. \ \ \ \ \ (1)

Strictly speaking, the expression {x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}} is not well-formed according to the grammar of first-order logic; one should instead use something like

\displaystyle x = \frac{-b - \sqrt{b^2-4ac}}{2a} \hbox{ or } x = \frac{-b + \sqrt{b^2-4ac}}{2a}


\displaystyle x \in \left\{ \frac{-b - \sqrt{b^2-4ac}}{2a}, \frac{-b + \sqrt{b^2-4ac}}{2a} \right\}


\displaystyle x = \frac{-b + \epsilon \sqrt{b^2-4ac}}{2a} \hbox{ for some } \epsilon \in \{-1,+1\}

in order to strictly adhere to this grammar. But none of these three reformulations are as compact or as conceptually clear as the original one. In a similar spirit, a mathematical English sentence such as

\displaystyle  \hbox{The sum of two odd numbers is an even number} \ \ \ \ \ (2)

is also not a first-order sentence; one would instead have to write something like

\displaystyle  \hbox{For all odd numbers } x, y, \hbox{ the number } x+y \hbox{ is even} \ \ \ \ \ (3)


\displaystyle  \hbox{For all odd numbers } x,y \hbox{ there exists an even number } z \ \ \ \ \ (4)

\displaystyle  \hbox{ such that } x+y=z

instead. These reformulations are not all that hard to decipher, but they do have the aesthetically displeasing effect of cluttering an argument with temporary variables such as {x,y,z} which are used once and then discarded.

Another example of partially specified notation is the innocuous {\ldots} notation. For instance, the assertion

\displaystyle \pi=3.14\ldots,

when written formally using first-order logic, would become something like

\displaystyle \pi = 3 + \frac{1}{10} + \frac{4}{10^2} + \sum_{n=3}^\infty \frac{a_n}{10^n} \hbox{ for some sequence } (a_n)_{n=3}^\infty

\displaystyle  \hbox{ with } a_n \in \{0,1,2,3,4,5,6,7,8,9\} \hbox{ for all } n,

which is not exactly an elegant reformulation. Similarly with statements such as

\displaystyle \tan x = x + \frac{x^3}{3} + \ldots \hbox{ for } |x| < \pi/2


\displaystyle \tan x = x + \frac{x^3}{3} + O(|x|^5) \hbox{ for } |x| < \pi/2.

Below the fold I’ll try to assign a formal meaning to partially specified expressions such as (1), for instance allowing one to condense (2), (3), (4) to just

\displaystyle  \hbox{odd} + \hbox{odd} = \hbox{even}.

When combined with another common (but often implicit) extension of first-order logic, namely the ability to reason using ambient parameters, we become able to formally introduce asymptotic notation such as the big-O notation {O()} or the little-o notation {o()}. We will explain how to do this at the end of this post.

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