Formal Logic, Part I

I told Tom I would accompany his early posts on Aristotle's logic with a companion series on formal logic as it is practiced today. It will be very introductory, and will take as its textbook Richard C. Jeffrey's Formal Logic: Its Scope and Limits.

I'm going to be attempting no more than to introduce the basic concepts, with an eye toward showing how modern logic and Aristotle's logic differ. It's mostly for Tom, because he said he was interested, but perhaps some of you may find it interesting as well. Formal logic is not my chief area of interest in philosophy, of course, so if any of you are well-versed you will probably find points of disagreement which you are free and welcome to argue. For those of you who haven't studied it, I hope I can provide a solid enough introduction to make it worth your while.

The online text is the Fourth Edition, and it has lost what I think is a helpful explanation from earlier editions -- I suppose he thought on reflection that it was too basic. The earlier piece talks about truth, and it points out two things very helpful to keep in mind when doing formal logic:
It is the job of pure logic to point out that if it is true to say
Tom stole it,
then it is equally true to say
Tom stole it or Dick stole it.
He goes on to point out that in natural language we would find the second statement to be much LESS true, assuming we know for a fact that Tom stole whatever it was. Pure logic doesn't deal in greater or lesser degrees of truth. It is purely binary: a statement is true, or it is not true. Since it is true that "Tom stole it or Dick stole it," the statement evaluates as true.

He has a second example on the other side. There exist albino crows, which means that the following statement is false:
All crows are black.
But of course the next statement is true:
Nearly all crows are black.
That means we'd be inclined to say that "All crows are black" is NEARLY true. But there is no 'nearly true' in pure logic, any more than there is 'less true.' It is true, or it is false. So "All crows are black" is false. (Examples are from the 1967 edition, pages 3-4.)

The Fourth Edition begins with validity. That gives us a good early point of contrast with Aristotle, who had a notion of validity that I think is actually better than the one in use in modern logic. See section 3.2 of this article, and contrast with what Jeffrey says. What is the difference, exactly? Do you agree with me that Aristotle is on stronger ground, or do you disagree? What are the strengths of each approach?


E Hines said...

I have some questions, and maybe I'm just reading too much into this stuff. All engineers, after all, over analyze....

Nearly all crows are black.

How is this necessarily true? If all we know about albino crows is that they exist, on what basis do we conclude that their number is sufficiently small that "nearly all" is a legitimate qualifier? It seems to me that the only true statement in this context is "Some crows are black."

From the Stanford article: The plural "certain things having been supposed" was taken by some ancient commentators to rule out arguments with only one premise.

Why should I accept that ruling out just because ancient commentators claim it?

More importantly, though, on what basis do those commentators make the claim? Other than a First Principle, don't all premises have antecedents--that are subsumed into the stated one(s)? If so than even "only one premise" would seem to be the several prior premises, also, summarized by the one being explicitly articulated.

As to which is better, the only thing I see Aristotle doing is bypassing some elements of what might be foolishness--apparently trivial arguments and irrelevancies.

Eric Hines

Grim said...

As to which is better, the only thing I see Aristotle doing is bypassing some elements of what might be foolishness--apparently trivial arguments and irrelevancies.

You aren't familiar with how much work is currently being done on these trivialities and irrelevancies. :)

But there is a serious point, perhaps -- at least, I'm told that there is, and I'm willing to entertain the argument. The argument is that logic's structure is worth working out even where it seems trivial, because logical truths are supposed to be necessary truths. Thus we can sometimes gain surprising insights into what is necessarily true from the cracks and corners of the structure of logic.

How is this necessarily true? If all we know about albino crows is that they exist, on what basis do we conclude that their number is sufficiently small that "nearly all" is a legitimate qualifier?

That's a good point, and one that he goes on to address in another section on the basics that didn't survive until the 4th Edition. It's not one of the usual quantifiers, because it's susceptible to vagueness. (His actual example is "Socrates was bald when X." That statement is clearly true if Socrates had no hair at all; it is clearly false if he had lots of hair. But there's a huge vague middle in which we can't say if it is true or not, which logic really can't speak to because we can't get to a binary T/F).

There are ways of doing it, of course: you could stipulate that "nearly all" would be satisfied if 95% or more were. Perhaps you could count; or, more likely, perhaps you could deduce something from the structure of the crow's genetics that will let you know how probable albinism is, which would give you an expected percentage.

Still, the usual quantifiers in formal logic are "all" and "some" (or "at least one"). These are called the universal quantifier (the inverse "A" I used in the comment below, for 'all') and the existential quantifier (a reversed "E," for 'there exists'). "Nearly All" isn't normally used.

Why should I accept that ruling out just because ancient commentators claim it?

You don't. We're talking about the differences between Aristotle and modern logic. So it's OK if you settle on a difference you'd prefer, either with Aristotle himself or with one of his interpreters.

Eric Blair said...

Logical ORs will definitely ruin your day in programming if you haven't thought through what you are trying to code.

Grim said...

Yeah, that's right. There's a huge difference between inclusive and exclusive OR (aka OR and XOR).

Eric Blair said...

And don't get me started on nested IF ELSE statements.

Tom said...

Thanks for doing this, Grim. This should be interesting.

It seems that one difference in the ideas of validity is that Aristotle insists a valid argument must be productive.

I'm curious how the two play out in practice.

Grim said...

Well, it's not just that it be productive, because it turns out you can produce very effectively from some of the arguments he disallows. Here's a famous paradox of modern logic, the raven paradox. (It's close to, but distinct from, the crow thing we were just talking about.) We discover that formal logic produces evidence for the truth of the proposition "If x is a raven, then it is black," if we encounter an x that is a shoe.

This is because modern logic insists that an argument of the "if/then" form is TRUE if the antecedent (the 'if..') part is false. So, "If Elvis is currently president, then..." is TRUE for logical purposes no matter what follows "then...".

Even more powerful, as you will see, is that you can derive anything from a contradiction. The proof of this will take a little longer, but it follows this form:

1) Realize that you are in a position to derive a logical contradiction (both A and ~A).
2) Assume the contrary of the thing you want to prove (~P).
3) Working under that assumption, go ahead and derive the contradiction.
4) Since contradictions are impossible, the assumption ~P is impossible because you could derive the contradiction while assuming it.
5) Therefore, P, since its opposite has been proven to result in a logical contradiction. QED.

This is a very nasty habit of modern logic, which you simply can't do with Aristotle. But we will see that his system really is lacking in some desirable machinery that modern (and medieval) systems have.

Anonymous said...

This can be proven much more simply. A&-A => P is a tautology. P then follows from A&-A by modus ponens.
Aristotle's logic is only of antiquarian interest.

Grim said...

Even if that were true, it would still be of greater interest than anonymous commentary. However, it's not at all true; as the section 3.2 mentioned demonstrates, the debate over relevance logic is still quite active, and a discussion of the relative strengths of the form is significant. A review of the older models often produces new insights, and not just Aristotle's.

In any case, modus ponens requires you to show not just Av~A, but that there is a material implication to P. If that is only an assumption, then you have proven no more than that you can get to P if the assumption is true.

What you can do with the contradiction trick is to get to both P and ~P because you can get to A and ~A. You can pick which one you want to prove and prove it, because you've already proven something impossible.

That's a real problem. But it's not Aristotle's problem -- it's modern logic's problem.

Tom said...

"If x is a raven, then it is black," if we encounter an x that is a shoe.

Well, this seems only productive of silliness so far. Can this assumption tell us anything about the world?

Aristotle's logic is only of antiquarian interest.

Given that Aristotle's logic and its derivatives were the engine of Western scholarship well into the modern period, it's of great interest to anyone who wants to understand Western history.

Grim said...

Can this assumption tell us anything about the world?

Yes, actually: it tells us that our intuition (yours too -- your reaction is the usual one) is not in line with the rules of inductive logic.

Now our intuitions are presumably products of the world to some degree. Insofar as there is an evolutionary basis for them, they would be. So we have learned something about the world -- we have learned that it doesn't obey the rules of our inductive logic.

We haven't started to talk about induction in Aristotle or modern logic yet, but it has its own set of problems. It may be that we're wrong to believe we can rely on it, and Aristotle doesn't do so outside of the natural sciences (where he thinks there is no choice, since we only have empirical evidence and not access to universal truths). In modern logic, we use proof by induction for a vast array of arguments.

Maybe we'll try one of those proofs later, but there's a lot of groundwork to get through first.

Tom said...

So we have learned something about the world -- we have learned that it doesn't obey the rules of our inductive logic.

I've had to let this sit in my mind for a while.

I'm not sure this tells me something about the world. I would say it tells me something about this kind of inductive logic, not about the world, per se.

(Although, that may not make sense. I'll have to think about this some more as we go.)

Grim said...

Well, you've learned something about both of them. You've learned that they don't gear up just right, or at least apparently not -- when you become a little more grounded in modern logic, you can work through the various ways people have tried to solve the raven paradox.

This is an interesting alignment with Aristotle as well, although not with his logic (since in his purely logical works he makes little of induction). Aristotle, in De Anima, assumes that you are able to extract the form that is in the world by a kind of mental activity.

(Not to get into it too deeply, first you have a sense perception of a thing in the world, and then you use your imagination to try to decide what part of that thing is essential and which parts could be different without changing the thing. E.g., a table could be a different color and still be a table, but if you lost the ability to hold stuff you'd lose the table. Thus, the form of the table -- the thing that makes it a table -- has to do with the capacity to hold things up off the ground/floor.)

By the time of Aquinas, that aspect of Aristotle is very much called into question. Aquinas has a different model for what kinds of things you can know -- he no longer believes you can have the actual form of the object in your mind. It's related, but it's something else.

We have a very similar problem with inductive logic, which by the way underlies all empirical science (for Aristotle as for us). If it doesn't quite line up with the world we're trying to learn about, our scientific ideas about the world are less reliable than we often suppose.

Scientists are usually better on this point than logicians, as the former know that a scientific theory is only as good as its predictive power. A logician will often write as if the logically proven thing really has to be true, because logical truths are necessary truths.

Ymar Sakar said...

Some of it is useful in law, at least practiced by Obama.