Truth Values

TRUTH VALUES

Mathematicians might seem like they’re trying to make things more complicated all the time, but really they’re trying to make things simpler so that we can try to understand them better. There’s an important difference between simple and simplistic though, and I think it’s to do with illumination. If you make things simplistic you are probably ignoring crucial details that actually are illuminating. Whereas the key to a good simplification is to keep some pertinent illuminating details and forget everything else – at least for now. Another key is to be aware of what you’re forgetting all the time, rather like deliberately leaving your umbrella at home if the weather forecast is fine, rather than accidentally forgetting it and also forgetting to look at the forecast. If you’re aware of what you’re forgetting then you can also be aware of the limitations of what you’re doing, and the situations you shouldn’t get yourself into.

In a way the law of the excluded middle is a simplification. We will later see that this means there are certain situations we really can’t deal with without modifying our logic. Another way of viewing this law is that we are viewing truth as binary, just “yes” or “no”. Mathematicians push this further and give truth a value: 0 if something is false, and 1 if it is true. You might think mathematicians just love turning things into numbers, but remember: maths isn’t just about numbers, it’s about many other things too. However, numbers are so familiar and easy to reason with that if we can turn a situation into some numbers it can be very helpful.

In this classical two-valued system, there is no truth value in between 0 and 1. You might cry foul at this point and say doesn’t this mean we have thrown away the middle? There are, after all, a lot of decimal numbers between 0 and 1. These might try to encapsulate partial truth. Perhaps if something has truth value 0.5 that means it is half true? There is a form of logic that takes this approach, called “fuzzy logic”, and we’ll come back to it in Chapter 12 when we discuss more ways to deal with grey areas.

Using 0 and 1 as the only possible truth values is like allowing only the answers “Yes” and “No” in court, a favourite tool of lawyers when they’re pressurising someone to say something that casts them in a bad light (at least, this is what fictional lawyers do). In this classical model, statements are either true or not true, 1 or 0. This can sound draconian, which is why it’s important to remember that “not true” does include all possible shades of grey.

If a statement is true then its negation must be false. Also, if something is false then its negation must be true. We can sum this up in this little truth table. Here A is any statement, and “not A” is its negation:

• A = 0 → not A = 1
• A = 1 → not A = 0

Truth tables, like Venn diagrams, are another useful way to encapsulate logic. They get a little bit further away from intuition, but sometimes when you move away from intuition two things happen. One is that you are better able to engage your logical brain. The other is that you actually develop new intuition, intuition about logic rather than anything else. If logical intuition sounds like a contradiction then I apologise. (And that’s a promise rather than a logical implication.)

Finally I should note that there is one other possibility for truth values: it is possible that the truth value for something cannot be determined. We’ll come back to this in Chapter 9 on paradoxes. Some paradoxes are caused by a statement that is so self-contradictory that it can’t be given any truth value – true or false – without causing a contradiction. Such a paradox cannot be assigned either classical truth value consistently. That differs from an ordinary statement whose truth value we do not yet know.

The passage offers these examples to distinguish questions about the world from internal logical contradictions. Their status depends on the available evidence:

1. The universe is finite.
2. One day we will be able to cure all cancer.
3. A meteor caused the extinction of the dinosaurs.