Subsection 5.1.1 Moving On From Representation to Proof
Recall that the point of our entire endeavor here is to get at truth.
What we’ve just spent a lot of effort on is simply representation. Our goal in doing that was to give ourselves a tool that lets us make claims that are unambiguous. We can’t ask whether a claim is true if we’re not sure what it means. No fighting about meaning allowed.
Approved Electronic Devices
Consider this sign. What does it mean?
Define:
A(x): True if x has been approved.
ED(x): True if x is an electronic device.
UIF(x): True if x may be used in flight.
Here’s one possible meaning for the sign:
If x is an electronic device that has been approved, it may be used in flight.
and
If it’s not true that x is an electronic device that has been approved, it may not be used in flight.
We can write that as this logical statement:
[1] \(\forall\) x (((ED(x) \(\wedge\) A(x)) \(\rightarrow\) UIF(x)) \(\wedge\) ((\(\neg\)(ED(x) \(\wedge\) A(x))) \(\rightarrow\) \(\neg\)UIF(x)))
But is that really what the sign means? It says that anything that isn’t an electronic device (whether approved or not) cannot be used in flight.
Surely the sign isn’t intended to ban straws and good old fashioned books. So here’s another possible meaning for the sign:
If x is an electronic device that has been approved, it may be used in flight.
and
If x is an electronic device that has not been approved, it may not be used in flight.
In other words, this is a sign only about electronic devices. It says nothing about books or straws (which, as it turns out, may be used). Nor does it say anything about guns or knives (which, as it turns out, may not be used).
We can write this second interpretation as this logical statement:
[2] \(\forall\)x (((ED(x) \(wedge\) A(x)) \(rightarrow\) UIF(x)) ((ED(x) A(x)) UIF(x)))
The point of our notation is that, while English signs are often ambiguous, logical statements are not.
Now it’s time to move on and see how to reason with the logical sentences that we write. In other words, we need to learn how to write proofs.
The job of a proof is to:
Assure us that some claim is true, and
(Ideally) give us some insight into why it is true.
We should be on firm ground here – this is exactly what we demanded of our proofs in Boolean logic.