Archive for September 30th, 2011

Basic logic — quantifiers

September 30, 2011

When I started writing about basic logic, I thought I was going to do the whole lot in one post. I’m quite taken aback by how long it has taken me just to deal with AND, OR, NOT and IMPLIES, because I thought that connectives were the easy part.

Anyway, I’ve finally got on to quantifiers, which are ubiquitous in advanced mathematics and which often cause difficulty to those beginning a university course. A linguist would say that there are many quantifiers, but in mathematics we normally make do with just two, namely “for all” and “there exists”, which are often written using the symbols \forall and \exists. (If it offends you that the A of “all” is reflected in a horizontal axis and the E of “exists” is reflected in a vertical axis, then help is at hand: they are both obtained by means of a half turn.)

Let me begin this discussion with a list of mathematical definitions that involve quantifiers. Some will be familiar to you, and others less so.