Logic through pictures - introduction
This page is about fundamental building blocks of classical logic – true, false, and functions that have these values as inputs and outputs. We will focus on the functions and, or, not, and implies.
This page uses pictures of mathematical worlds or universes that have things in them. The things can have properties. A property is something like being red, being new, or being a member of some group. Numbers can have properties such as being even, odd, positive, or prime. In classical logic, a thing either has a particular property or it does not, and there is no middle ground.
The logic concepts described here through pictures are enough to make precise virtually all of the reasoning needed by students of high school algebra and trigonometry. (Calculus needs a bit more.) Adding suitable rules of inference and axioms for logic and numbers gives a precise system that can be used to solve a great many math problems at this level.
This is also the kind of logic used in the Prooftoys automated proof assistant.
Summary for the impatient
The basic logical concepts of and, or, not, and implies can all be expressed as functions from boolean values to boolean values, and truth tables capture their essence as functions.
Implication in particular is more familiar in statements that also include predicates, such as “all multiples of 10 are multiples of 5”, or “if it rains the street will be wet”.
A mathematical statement is considered true if it
is true in all possible cases. Imagine assigning all possible
combinations of values to the variables in the statement and checking
the truth of the statement in each of these cases. We say the statement is
true if it is true in every case. For example if we take
x to be an integer, the statement
x is even or x is odd is true, because it is true for every possible value
x can have.
If you are satisfied with this summary and don’t like looking at pictures about abstract mathematics you might prefer to skip reading the rest of the page.
About the pictures
In the pictures a mathematical world is shown as a circle. The world has various things in it, each of those things having a particular spot in the picture.
Mathematical writing, instead of saying a point of a picture is blue,
or has the property of being blue, often says it is a member of the
set of points that are blue. In mathematics, properties are also
often referred to as predicates. These are all just different
ways of describing the same situation. (A predicate can also
express a relationship between two or more things, as in
x < y. A predicate like
< is also commonly called a relation, because it expresses a relationship between
Collections of things with some common property are shown as a circle, part of a circle, or other region of this world, shaded with a particular color or pattern. Each set is labeled with a name. Each contains the things inside its shaded circle or part circle, but nothing ouside of that. In the diagrams below, areas where some statement is true are generally colored. Areas where a statement is not true are usually white or gray.
In our version of mathematical notation we are going to write
(green x) to mean that x is green,
(blue x) to mean that x is blue, and so on.
Some of the pictures are near tables that define key functions. Moving the mouse over one of these pictures or touching it on a touch screen will cause the relevant table entries to highlight themselves.