and

are propositions. The first is true and the second is false. I will consider

to be a proposition, because I expect that you know what a prime number is. However, I will not consider

to be a proposition (unless I provide you with a definition for

The proposition

is true, and the proposition

is false, but

is not a proposition but rather a meaningless statement (cf (1.8)). Observe that ``'' makes sense whenever and are sets, and `` '' makes sense when is a set, and is any object. Similarly

is meaningless rather than false, since division by zero is not defined., i.e. I do not consider to be a name for any object.

are propositions, and ( or ) is true if and only if at least one of is true; ( and ) is true if and only if both of are true; (not ) is true if and only if is false.

are all true propositions.

and we abbreviate

The three main properties of implication that we will use are:

We denote property (1.22) by saying that is

or

may not be obvious. I claim that both (1.24) and (1.25) should be true.

``Proof'' of (1.24):

and

so by transitivity of ,

``Proof'' of (1.25):

so

so

The previous example is supposed to motivate the following assumption:

A false proposition implies everything,

i.e.
If is false, then
is true for all propositions .

is true. Hence all three of the statements below are true:

Proposition (1.28) is an example of a false statement implying a true one, and
proposition (1.29) is an example of a false statement implying a false one.
Equations (1.27) and (1.28) together provide motivation for the
assumption.

Every statement implies a true statement;

i.e.

If is true then
is true for all propositions .

The following table shows the conditions under which is true.

true | true | true |

true | false | false |

false | true | true |

false | false | true |

is an abbreviation for

It follows from transitivity of that if (1.31) is true, then is true.

Note that (1.31) is *not* an abbreviation for

i.e., when I write (1.31), I do not assume that is true.

(read this `` is equivalent to " or `` if and only if ") to mean

If either ( are both true) or ( are both false), then
is
true. If one of is false, and the other is true,
then one of
,
has the form true
false, and hence in this case
is false. Thus

is true if and only if are both true or both false.