Degenerate Relations

Both math and logic have the notion of “relation”, an open sentence that takes two variables. For example, “x is greater than y“.  The set of ordered pairs <x, y> that satisfy a relation is said to be its extension.  Functions are relations, but clearly not all relations are functions.

Relations can be generalized to have an arbitrary number of variables.  In this case, the extension would be tuples.

Generalizing in the other direction, at some strain to English we could say that the extension of a unary “relation” (called a property) is a set of scalars.  For example, “z is a positive, even integer” has the extension {2, 4, 6, 8, …}, while the extension of “y is Red” is the set of red things.

Is a property the most degenerate form of relation, or can we take this further?  What about an “open sentence” with no variables? I.e., a statement?  Something like “All integers are real numbers”?  What would its extension be?

Beats me, but the empty set is all that comes close to making sense — there are no scalars or tuples that can be plugged in.  So there’s nothing in the extension.

But this presents a pretty big problem:  the extension of a false statement is clearly the empty set — no scalar or tuple of any value could make a false statement true.

Some decades ago, I was sympathetic to the idea that a relation (or property) could be considered identical to its extension. The above line of reasoning makes me think that maybe that’s not so…

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