Semantics

Consider this first-order formula:

$$\forall x \exists y Gyx$$

Is this formula -- let's call it `$\phi$' -- true? There is no way to know. Suppose I tell you that G means `greater than;' can you tell now whether or not $\phi$ is true? You might be tempted to make a guess, but you still can't tell. After all, you don't know what class of objects is being talked about here. Does the sentence refer to people (e.g., ``All people are such that there is at least on person greater than them"), or to numbers, or perhaps something else? -- unicorns, perhaps. Suppose we inform you that the objects $\phi$ is ``talking" about are numbers; now will you be able to ascertain whether or not the sentence is true? At this stage you might boldly guess -- but the fact of the matter is that you can't know for sure. By `numbers' do we mean N? R? Z⁺? {1, 2, 3}? For the first of these sets $\phi$ is true; for the last it is false. Now let's make some of these ideas more precise.