Subsection 1.4.6 Remember the Critical Role of the Premises
The tools that we are about to describe, powerful as they are, cannot pull truth out of thin air. What they are designed to do is to preserve truth. If we begin with true statements and then reason logically, we will conclude with statements that must also be true.
But if we start with junk, we can easily create more junk.
Activity 1.4.16.
Suppose that we start with the following premises:
If there’s a ladder that reaches from A to B, then it is possible to go from A to B by climbing the ladder one rung at a time.
The Fountain of Youth is on the moon.
There is a ladder from Austin to the moon.
There are people in Austin.
Then there is a perfectly logical argument that it is possible for someone to reach the Fountain of Youth.
Unfortunately, that doesn’t appear to accord with the facts. The problem is that premises [2] and [3] haven’t been chosen well if we want to describe the world in which we live.
Logic can’t insulate us from bad premise choices and the junk that can result.
Sometimes though it’s not a question of junk. It’s simply the case that different premises make sense in different situations. Because this can happen, it’s important that we clearly articulate all of our possibly controversial premises.
Activity 1.4.17.
Suppose that I want to reason about going to the beach. I’ll take [1] and [1] as premises:
People like going to the beach when it’s warm, not when it’s cold.
The beaches will be crowded at times that people like to go.
Then I can argue: The beaches will be crowded in August because people like to go then.
My argument makes sense to me. I live in North America. It’s summer (and thus warm) in August. But our friends Down Under may laugh at me. Their beaches are crowded in December. The issue is that I have one additional premise (as yet unstated), while they have another:
[mine] It’s warm in June, July and August.
[theirs] It’s warm in December, January and February.
So I can prove that the beaches will be crowded in August. They can prove that the beaches will be crowded in January.
When we want to reason about the real world, we typically choose premises that correspond to our observations and that lead us to conclusions that also accord with the facts. So there’s some notion of reasonable premises and unreasonable ones.
In mathematics, however, there is no single “reasonable” set of premises. Each set of premises leads to a theory: a collection of provable claims. Some theories may be more “interesting” than others. But it can easily happen that two theories that start with competing premises can both be useful, although in different contexts.
Activity 1.4.18.
Probably the most well-known example of this is in geometry.
In plane geometry, we generally accept all five of Euclid’s postulates (premises). The fifth of these is often called the parallel postulate:
Given a line \(L_1\) and a point P not on \(L_1 \text{,}\) there is exactly one line \(L_2\) that is parallel to \(L_1\) contains \(P \text{.}\)
Using all five of the postulates, we can prove claims such as, “Any two lines, if they intersect at all, must intersect at exactly one point.” Thus we get claims that correspond to what we observe on a plane.
But now consider geometry on a sphere (interesting because, for example, we can approximately model the Earth as a sphere). Now our observations are different. We want a theory that predicts them. So spherical geometry starts with a different notion of a line. Rather than straight lines, we define a line as the shortest distance (on the surface of the sphere) between two points. And we begin with different premises. So, for example, the sum of the interior angles of a triangle is not \(180^{\circ} \text{.}\)
Premise switching also plays a key role in many kinds of fiction. For example, in alternative history, it is assumed that one thing (or maybe a small number of things) happened differently. From then on, everything that happens follows the usual rules of causality and event sequencing. But outcomes can be completely different. Fantasy worlds and games do the same thing. Logic stays the same. Just a few premises change.
Problems 1.4.19.
(a)
Consider the following argument, which will begin with four premises, [1] – [4]. It will conclude [5]:
Morgan is in Pflugerville without a car.
Morgan has a class at UT in an hour.
Morgan can walk 20 miles in an hour.
The distance between UT and Pflugerville is 15 miles.
Morgan can get to class on time.
The logic of this argument is good but the conclusion is crazy. There’s one premise that is in blatant contradiction to the facts of the world in which we live. We should scrap it if we want to make it impossible to conclude [5] while still letting us reason sensibly about the world. Which premise should we scrap
(b)
Consider the following “proof” that 2 = 0. We’ll take [1] – [5] as premises.
For any numbers \(a \text{,}\)\(b \) ,and \(c \text{,}\) if \(a = b \text{,}\) and \(b = c \) then \(a = c.\)
For any numbers \(a \) and \(b\text{,}\) if \(a \lt b \) is false, then \(a = b. \)
For any numbers \(a \) and \(b \text{,}\) if \(a > b \) , then \(a \lt b\) is false.
For any number \(a, a + 1 > a. \)
For any number \(a, a = a.\)
Since \(2 = 1 + 1, 2 > 1 \text{.}\) So it’s false that \(2 \lt 1\text{.}\) Then \(2 = 1 \text{.}\) By a similar argument, since \(1 = 0 + 1, 1 > 0\text{.}\) So it’s false that \(1 \lt 0\text{.}\) So \(1 = 0\text{.}\) By premise [1] we then have that \(2 = 0\)
.The logic in this proof is sound. But one of our premises has made it possible to conclude nonsense. Which one: