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Let's put it on the line -- ordinary logic is that common, garden-variety logic that was begun centuries ago by the well-known philosopher, Aristotle, of ancient Greece. You might say he's the father of logic. Does that make it ordinary, or what?

Well, maybe not quite ordinary. In fact, a better choice of words might be extraordinary.

Yes, that's better, because Aristotle was no ordinary mortal and he was trying to clean up the language of the streets. And from what I hear of street language, today, it probably needed a lot of heavy cleaning -- no offense, guys!

Even now, logic doesn't seem to enter conversation at all. So if ordinary logic has to do with the logic we use every day, we have to admit it's obscure. More delusion than logic. More confusion than clarity. Even so, Aristotle managed to find it -- or should I say, uncover it? Remove the debris?

But what does it really mean to 'clean up' a language? What is logic, anyway?

Have you ever stopped to ask yourself how or why language got started in the first place and what it's supposed to do? Not only that, but how did people get started arguing? And when did they start drawing conclusions from arguments?

It's the arguing part that's especially fascinating. It may be easy enough to appreciate the use, early on, of words (or grunts) to warn about dangers, i.e., to provide information or to draw attention to a source of food, or other practical things. (That alone had to be a powerful boost to empowerment!) But when did people start getting into discussions, and trying to persuade each other with words to do this or that, go here or there, do something this way or that way? That's when "logic" gets interesting.

You can certainly use emotions to persuade people to do what's correct, say by scaring the wits out of them, or trying to convince them they like something even when they don't. We see it all the time -- in politics, TV ads,. ... You name it! It's called salesmanship. You might even convince them through sound reasoning -- a tactic you don't often see on television or find in politics. It's a matter of directing the affairs of others, and you can use either emotional or rational arguments to do the directing. Rational argument produces good information. Too often, though, sheer sophistry seems to carry the day!

But how do you deal with the rightness, or the correctness of the way you apply the language parts?  What does it mean to be rational or reasonable -- to be correct in your arguing? What does it mean to render a correct chain of reasoning?

Finding answers to questions like these is what logic is all about. It tries to determine what is and what isn't correct inference, what is and what isn't valid. It is argument validity that Aristotle was trying to figure out -- how to get order out of messy or chaotic use of language.

Mind you, logic didn't just appear to Aristotle out of thin air. Ordinary people already had to have been arguing, and the arguments had to have reached a certain level of sophistication to become a subject of importance and something worth discussing. This means that the value or utility of argument had to have been recognized. And people had to start thinking about how language could or should be used in the discussions. They had to become aware of the usefulness of argument -- logical inference -- and the problems associated with it and then start to pay more attention to the principles. Only then could they begin to make sense of what was going on. And only then could logic really start to get formalized.

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What is logic, after all? Isn't it just the art of being more careful in how you converse, or the art of drawing conclusions from what you say -- i.e., implications? Doesn't that mean paying more attention to the parts of language that have to do with the reasoning process? It might even have something to do with truth. After all, isn't it desirable for the rational process to lead to true statements? Let's see what this might be.

First of all, what parts would you need in order to exhibit a bit of logic?

Well, of course, if you were using logic, you'd at least have to be saying something. It is sentences that provide or express information. ('Information' has been defined in terms of computer bits, but I believe that approach is more appropriate for the sentence variable interpretation of 'information,' rather than sentences) So the first thing you'd need would be sentences. Plain old simple sentences. Like

After all, it is sentences that carry meaning. The idea behind 'simplicity' is that the sentences are so simple they can't be split apart without losing their meaning -- simple sentences are the units of information, so to say.

That's simple enough, I should think, and about as simple as you can get, even though it's not very interesting logically. The real logic -- and the difficulties -- begin when you select sentences from all of the available sentence forms and string them together to generate arguments, on the basis of which you draw conclusions or persuade others. (Note I said sentence forms -- more about what they mean in just a moment.) One group of such forms comes straight from Aristotle. These are the traditional sentence forms.

In Aristotle's time, four specific forms of sentences made up what we know in logic today to be the categorical forms, and they turn on four common and often used expressions, namely all, no, some, and some not.

If I may be allowed to use the letters x and y as variables to represent all the possible items or events that might be the subjects of the statements, the forms can be presented as follows:

The forms are like blank income tax forms, but they're not nearly so complicated! You generate specific sentences when you "fill them in."

For example, if x stands for men, and y stands for mortal, then the four categorical forms become the categorical statements:

Substituting your own values for x and y, you can make up your own long list of examples.

These categorical sentence forms can be organized to generate the kind of arguments known as syllogisms. Here, though, we look at a more up-to-date version of ordinary logic and consider compound forms.

Since the days of ancient Greece, the field of logic has grown considerably, and many new argument varieties have evolved. One group of arguments is grounded on certain kinds of compound sentences. That is, we use connecting words like 'and' and 'or,' for instance, to join simple sentences together to form longer, compound sentences. Two examples of compound sentences are:

Not only that, but compound sentences are important enough to have special names. Those formed by using 'and' to join sentences are called conjunctions. And those formed by using 'or' are called disjunctions.

Another thing we do with sentences is to negate them. Here we use 'not,' for instance. So we might say:

This sentence is referred to as the negation of the original sentence. You can also negate the sentence by saying:

And finally there's the form of compound sentences known as the conditional. This is where it really gets interesting, because this is where the notion of a rule comes into focus in the picture I'm trying to paint. You'll recognize this form as the one involving the If condiition:

That is, if such and such is the case, then this or that is the result. For example:

Or:

So in general we have:

You may have a bit of trouble keeping in mind that this form is really a sentence, even though it may not seem to be one and even though we use it all the time! But it does carry its own information, which you can usually grasp.

Here we examine the arguments generated by this group of compound sentences.

You should keep in mind that nothing said about the sentence parts of logic gives you any kind of clue about the truth of the sentences. (Finding the truth is the job of science; to establish the truth you have to make observations, take measurements.) For instance, from the point of view of logic, the statement:

could be a complete fabrication as far as you're concerned -- as a logician, that is! And so could the conditional:

In logic you don't need to know scientific truths to draw conclusions. In fact, you don't even have to deal with explicit sentences. You can derive conclusions from statements that you merely assume to be true. You might simply use sentence representations, like P or Q. It's the validity that really counts -- i.e., the relationship between the sentences. The truth will take care of itself.

In normal conversation we pretty much generate the truth of a compound sentence based on the truth of its parts. But you see, in logic the actual truth-value of a sentence doesn't matter all that much, except for what you conclude from it, more or less as a function. Truth itself isn't high on the logician's agenda. We are more concerned with truth functions, which deal with combinations of truth-value. Even so, you might question whether the statement:

expresses any truth at all. I.e., is it meaningful?

To repeat, the literal truth or falsity of sentences is not part of logic. Rather, it falls in the domain of science, of empirical observation, or ordinary experience -- not logic.

In logic it isn't the bare-bones sentences themselves that you have to worry about so much as the way they are strung together as an argument. What you do is to assume the starting situation of the string to be true and decide whether the end result is true based on the assumption. Logic is concerned with what you conclude from your premises. It is concerned with inferences from the conditions being assumed.

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To see how a conditional can have a truth-value, let's take one apart.

For our example, let's use the statement:

The idea is that whenever something is identified as a man, that something has to be mortal. The reason is that mortality expresses an essential property of being a man. (That's 'man' in the generic sense, ladies, so it includes you, too.) So, if that something is not mortal, then it can't be a man. In this case we have what is called strict implication, principally because the conclusion is already in the condition. In effect, the conclusion comes out of the definition of man. Thus the expression has the equivalent form:

You might also say, equivalently, that:

Or:

Strictly speaking, these sentences are equivalent because they have the same truth-value function. They express the same information.

Now consider in more detail the nature of a truth-value function. What does it mean? Is a truth function the same as an ordinary numerical function, only different?

First, though, I should say that the example:

presents a different problem from that posed by the statement:

In the "food" statement there's nothing in the conditional that has anything to do with grass being green. I.e., the conditional doesn't strictly imply the consequent -- i.e., the conclusion that the grass is green. In fact, if you were to use expressions like that in ordinary conversations, people would begin to cast sidelong glances, wondering if you were in possession of your faculties. For that matter, you won't often spout truisms like All men are mortal, either, but even if you did, nobody would be likely to pelt you with stones. The latter expression is at least reasonable, whereas the former is just nonsense.

Even so, the two sentences have the same logical form and therefore the same truth values, or what is commonly called the truth-value function. In other words, it is the truth function that carries the importance of the statement form and it doesn't matter whether or not the compound sentence has any meaning in specific instances.

The truth-value function of the conditional, as expressed using letter symbols, is simply:

In this form, the letters A and B are sentence variables, because you can substitute any sentence for them. The individual sentences themselves can be true or false. Now the truth of the conditional depends on, or is a function of, the truth-values assigned to the variables.

Besides the letter symbol form of the conditional, we can also use a tabular form. Expressed as a truth table, the truth-value function of the conditional is as follows:

Note that the tabular form has its own structure. Only when you understand its nature can you read it. In case you haven't looked at tables in this light, let me say that in this representation, each row identifies a specific condition of the truth-value function:

In other words, the value of the conditional, If P, then Q, is a function of the values of the two variables, P and Q, each of which can have one or the other of the truth values T or F. This condition identifies a two-valued logic. Other logical systems, such as fuzzy logic, which we review here, have more than two truth-values.

In this tabular structure, the first row shows that both P and Q are true and that If P, then Q is also true. That's the sense of the tabular structure. So, in English, it tells us that, if both the antecedent, P, and the consequent, Q, are true -- i.e., if both the condition and the conclusion in the statement are true, regardless of their meaning -- then the statement itself is true. The function is true when both of its variables have the value T. In fact, the only time that the conditional is false is when P is true and Q is false (second row, gray).

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Logic has to do with proving that one or another statement is true, or not, based on something else assumed to be true. If such and so, then this or that. So it's reasonable to think that truth is what the subject is all about. Well, truth does have a certain degree of importance. However, while it has a part to play, it's not correct to say that truth has the central role. Rather, the key idea is argument validity.

The main job in logic is to determine whether or not one or another statement follows from, or can be legitimately deduced from, other statements. The legitimacy of a conclusion depends on the nature of the argument -- whether it is valid or not. For example, how would you tell if the following simple, often-illustrated argument is valid or not?

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