Logical Musings #4

One of the things things that logical arithmetic is good for is in proving logical formulas. An arbitrary expression in Boolean alebra, is usually conditional. It depends on the truth values of the variables it uses. However, for a certain class of formulas, the overall expression can be evaluated as true in every case. These are called truth-functionally true, or tautologies, and are the theorems, laws, or rules of boolean algebra. For instance, (A AND B) is equivalent to (B AND A). This is always true, regardless of whether A and B are true or false.

There are other ways of proving logical formulas. One of them is the axiomatic approach, in which one starts with a few axioms and a few rules of inference, and derives all the others. But wait, you may ask. Isn’t this a bit circular? Well, yes, it is, when applied to logic itself, and I much prefer the truth table approach when it can be used. For classical two-valued propositional logic, these give the same laws.

There are variants of standard logic which are examined for experimental purposes, in which some of the standard rules do not apply, and which do not or cannot use truth tables. These commonly use an axiomatic approach. It isn’t to my personal taste, but that’s just me.

Another topic that needs examination is the concept of a valid argument. Traditionally, rhetorical logic from the Greeks on was concerned with arguments that were logically valid. The goal is to start from true assumptions and reach true conclusions without introducing error in the reasoning itself. There are some forms of argument, commonly known as fallacies, which are considered invalid because they may introduce an error. By the time you boil things down to propositional logic, there is only one criterion that matters: Does the material conditional hold? If it does, your argument or proof is valid. If it doesn’t, then it isn’t. It’s that simple: Easy peasy. If you want to talk about classes of things, or talk about subjects where statements can be vague or uncertain, or fiddle with the laws of logic, it gets increasingly more complicated.

The other thing that needs to be considered is that from the time of Aristotle onward, logic was concerned with statements that were either true or false. No others were to be considered. This is easiest to do in mathematics, where we can control the definitions and decree certain things by convention. It also means that logic works less well in the real world. Take, for example “It is raining”. Aside from being dependent on time and place, there are certain states of the weather that make it hard to give that a definite true or false answer. If you are careful with definitions, you can control or limit the uncertainty…but then you have the problem of getting other people to agree with your definitions. Again, this is easiest to do in mathematics, unless perhaps you have engineers getting into two-fisted fights over Bessel functions or Laplace transforms.

As it turns out, uncertainty is a very slippery concept to grasp if all you have is words, and it’s easy to get mired in contradiction and confusion.

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