Propositions

Assertions that are either true or false are called statements, or propositions. Most of the propositions in this book are mathematical ones, but others may arise in daily life. "All individuals who breathe are alive" is an example of a true proposition, whereas the assertion "all individuals who breathe are healthy" is an example of a false proposition. It should be noted that if the words used to express such assertions lack a precise meaning, it will often be difficult to distinguish between a true and a false proposition.

Suppose an assertion such as "x2 —1=0" includes one or more variables. By substituting various real numbers for the variable x, we can generate many different propositions, some true and some false. For this reason we say that the assertion is an open proposition. In fact, the proposition x1 — 1 =0 happens to be true if x = 1 or —1, but not otherwise. Thus, an open proposition is not simply true or false. It is neither true nor false until we choose a particular value for the variable. In practice we are somewhat careless about this distinction between propositions and open propositions; instead, we simply call both types propositions.

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