Wednesday, December 31, 2014

Vacuously true statements

You can think of a vacuously true statement as a sort of mathematical joke. For example, if Santa Claus does not have any pigs, it is mathematically true that “All of Santa’s pigs can fly”! Vacuously true statements can also play a role in mathematical fallacies, which are incorrect arguments that may look reasonable until you take a closer look at them.

You might want to review the previous posting on the meaning of "implies". The key to understanding vacuous statements is that if the “antecedent” (left side) of an implication is false, the whole statement is true no matter whether the right side (“consequent”) is reasonable or not. In our little example, it is true that all of Santa’s pigs can fly, even if no pig in the entire world can fly. And I guess all purple cows are math whizzes, too — unless of course there really is a purple cow somewhere.

In the truth table for the implication operator (P ⇒ Q), if you look at the rows where P is false, you will see that the whole implication is true regardless of whether Q is true or not. So any implication with "false" on the left side is true; and we say "false implies anything".

The definition of "⇒" gives just the results we need to do mathematics — just watch out for statements that are vacuously true!

The meaning of “implies”

Nothing is more essential to mathematical thinking than deduction, the art of working out the consequences — the implications — of precise assumptions.

This posting takes a closer look at the mathematical meaning of “implies” through an example in basic algebra. It assumes that you know just a little bit about propositional logic and its “implication” operator, as explained for example in Logic through Pictures.

Let’s look at the statement that

if x is equal to 1 then x squared is also equal to 1.

In mathematical notation we write this as

x = 1 ⇒ x2 = 1

and this is a true mathematical statement just as you would expect. We typically read the arrow here in English as “implies”, reading the whole statement as “x equals 1 implies x squared equals 1”. We are going to look more closely at the precise mathematical meaning of this implication arrow.

The mathematical meaning of a true statement with variables is that any possible value can be used as the value of each variable, giving a true statement in every case. So in this example the statement should be true for x = 0, x = 1, x = 2, and so on. And if we are talking about the real numbers, it should also be true for ½, the square root of 2, and so on.

For each possible value of x we consider the value of x = 1 and the value of x2 = 1. Each of these will be either true or false. The implication arrow is a function that takes two “truth values” (either T or F) as inputs, and gives a truth value as its result.

Since x = 1 ⇒ x2 = 1 is a true statement, its value must be T (true) for all possible values of x. Let’s take a look at the truth table for the ⇒ operator.

P Q P ⇒ Q

Its value is T for all inputs except P = T and Q = F. In our true statement we will see that all three of the cases that give a value of T occur for certain values of x.

Suppose the value of x is 1. Then the expression x = 1 is true and the expression x2 = 1 is also true. The value of T ⇒ T is T, and 1 = 1 ⇒ 12 = 1 is true as expected.

Next suppose x is -1. The expression x = 1 is false, but the expression x2 = 1 is true. The value of F ⇒ T is T, so -1 = 1 ⇒ (-1)2 = 1 is also true.

Now suppose x is 0. Then the expression x = 1 is false and the expression x2 = 1 is also false. The truth value of F ⇒ F is T, and 0 = 1 ⇒ 02 = 1 is true, also as expected. The inputs to “⇒” are false for x = 2, x = 3, x = ½, and all other values of x as well.

As long as there is no value of x that causes x = 1 to be true and x2 = 1 to be false, the implication is true. For this statement there are no such cases, and we see in detail how our original statement is true, looking at all possible cases and using the “truth table” for the “⇒“ operator.

What is high school algebra for?

Basic algebra is a core part of virtually all high school level educational programs. There are areas of controversy over teaching of math, but the value of studying basic algebra itself is not one of them. The skills and understanding of high school algebra provide a base for much practical and important mathematics, and advanced areas of math build on basic algebra, including probability, statistics, calculus, real and complex analysis to mention only a few.

Still, algebra is only one face of mathematics. Ever since the ancient Greeks, math has had other important faces as well. Euclidean geometry is recognized to this day as a shining example of systematic mathematical reasoning and taught in standard school programs. With the arrival of the computer and the Internet, new kinds of mathematics have come to have great practical importance, including discrete mathematics and mathematics of computation. And mathematics itself has exploded in many directions, especially over the last century or so.

With the computer have also come questions about the math students learn in school. For example now that we have calculators, how important is it to learn to calculate by hand? Powerful computer algebra software now can solve many kinds of math problems, often beyond the abilities of all but the most capable humans. Textbook problems in basic algebra are among the simplest problems of these kinds.

In spite of all this there is still a consensus in favor of basic algebra in schools. It does not seem to be just conservatism or inertia. Where does this continued belief in high school algebra come from? High school algebra can contribute in a number of areas, including:

  • Problem-solving skills. Math has always been a tool for solving problems, and algebra provides excellent arenas in which to state problems, consider the means available, and systematically solve those problems.
  • Fundamental concepts of mathematics. Among these are variables, constants, terms, functions, domain, range, relations and equations.
  • Interpreting mathematical expressions and statements. This means understanding for example the structure of a term, how it transforms its inputs to its outputs, the graphs of equations, and the conditions when mathematical statements are true or false.
  • Understanding and using properties of numbers and operations. These include ordering of numbers, operations such as commutativity, associativity, and distributivity; symmetry, transitivity, and so on.
  • Describing the world mathematically. Typically this means transforming questions about the world into mathematical statements that can allow the questions to be solved or analyzed.
  • Reasoning mathematically. Proper mathematical reasoning is deductive, and the fundamental rules of deductive reasoning apply to all branches of mathematics, both simple and advanced.

Some of the posts on this blog discuss how basic algebra can illustrate principles of mathematical reasoning often overlooked in textbooks and courses in the subject. These principles can also be readily implemented in interactive computer software such as Mathtoys that can support its users in correct reasoning.