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Subsection 4.4.1 Introduction

Recall that a statement is a wff that has a truth value. All of the following are statements:

  1. Bear(Smokey)

  2. ∀x (Bear(x) → Animal(x))

  3. ∀x (Animal(x) → Bear(x))

  4. ∀x (Animal(x) → ∃y (MotherOf(y, x)))

  5. ∀x ((Animal(x) ∧ ¬Dead(x)) → Alive(x))

The following wffs are not statements because they contain unbound variables:

[6] Bear(x)

[7] Animal(x) ∨ Vegetable(x)

We don’t have any way to assign truth values to these wffs without changing them in some way. (We could substitute a particular value, such as Smokey, for x. Or we could enclose the statement inside a quantifier.) But then we’d be assigning meaning to a different wff.

So we’ll focus here just on statements. Returning to [1] – [5] above, we observe that [1], [2], [4], and [5] are true in our everyday world (assuming the obvious referent for the constant Smokey and the obvious meanings of the predicates Bear, Animal, and MotherOf). On the other hand, [3] is not true.