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# 1.2 Propositions

1.11   Definition (Proposition.) A proposition is a statement that is either true or false.

1.12   Examples. Both

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 unlucky number).

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.

1.13   Definition (and, or, not.) If and are propositions, then

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.

1.14   Example.

are all true propositions.

1.15   Notation ( .) We abbreviate

and we abbreviate

1.16   Notation ( ) If and are propositions, we write
 (1.17)

to denote the proposition  implies ".

1.18   Example. If are integers then all of the following are true:
 (1.19) (1.20) (1.21)

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

 (1.22)

We denote property (1.22) by saying that is transitive.

1.23   Example. The meaning of a statement like
 (1.24)

or
 (1.25)

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 .

1.26   Example. For every , the proposition

is true. Hence all three of the statements below are true:
 (1.27) (1.28) (1.29)

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
Thus a true statement does not imply a false one. All other sorts of implications are valid.

1.30   Notation ( .) Let , , , be propositions. Then
 (1.31)

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.

1.32   Definition (Equivalence of propositions, .) Let be propositions. We say that and are equivalent and write

(read this  is equivalent to " or  if and only if ") to mean
 (1.33)

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.

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