Formal Logic, Part III

Today I'm going to depart from the textbook for a moment, and work out some consequences of what you've seen in the first two parts.  There's something very significant lurking here, and students of logic usually pass over it without forcing it out into the light.

In the last part we worked on the concept of logical equivalence, which is when two forms of an argument have exactly the same truth values in every case. This has an important consequence: because the two forms preserve each other's truth, you can substitute one for the other. Just as in mathematics, you can treat equalities as interchangable. If it is helpful in getting where you need to go in algebra, for example, you can divide both sides of an equation by two, or multiply them both by two. The truth of the equation is preserved:

ab=2
2(ab)=4
0.5(ab)=1

Likewise:

J ≡ M
M ≡ J

Once we get past the foundations of formal logic, and into advanced logic, this mathematical assumption becomes more and more important. Logical deductive systems have strict rules governing substitutions that are supposed to be truth-preserving. Various operators have different rules, so it is often important to be able to substitute one set of operators for another in order to reach the final result you are seeking.

Here is an example. In modal logic, the following two propositions are thought to be readily exchangable:

◊p ("Possibly p," or proposition p is possible)
~□~p ("Not-necessarily-not p," or, p is not necessarily forbidden -- and is, therefore, possible)

You can do the same thing with "necessarily P" and "not-possibly-not p."

Why does that matter? One reason is that there are rules for handling the box of necessity (□) that differ from the not-operator (~). You can only derive possibility from possibility (◊), but if you can switch to the not-necessarily-not and eliminate the first "not," you can then derive a necessary truth using the box forms.

Because all of these forms are thought to be proven to preserve truth, this means that you can use these advanced logical forms to move from a proposition known to be true to another very different proposition that you can treat as necessarily true also.

This is why, for more than a hundred years, this kind of logical philosophy has had pride of place in the Anglo-American world. It believes it is bringing something very much like mathematical precision to the wider world of human knowledge. If you also believe that, it is a very exciting field even today.

Nothing like this is true for Aristotle's philosophy. That is not to say that he didn't see a relationship between the mathematics of his day, and the logic of his day. He also saw a relationship between the logic of his day and the practical human problems of his day. However, he explicitly rejected the idea that you could create a deductive logic that applies directly to practical human problems. As he says in Nicomachean Ethics I.3:

Our discussion will be adequate if it has as much clearness as the subject-matter admits of, for precision is not to be sought for alike in all discussions, any more than in all the products of the crafts. Now fine and just actions, which political science investigates, admit of much variety and fluctuation of opinion, so that they may be thought to exist only by convention, and not by nature. And goods also give rise to a similar fluctuation because they bring harm to many people; for before now men have been undone by reason of their wealth, and others by reason of their courage. We must be content, then, in speaking of such subjects and with such premisses to indicate the truth roughly and in outline, and in speaking about things which are only for the most part true and with premisses of the same kind to reach conclusions that are no better. In the same spirit, therefore, should each type of statement be received; for it is the mark of an educated man to look for precision in each class of things just so far as the nature of the subject admits; it is evidently equally foolish to accept probable reasoning from a mathematician and to demand from a rhetorician scientific proofs.

Likewise in the Rhetoric I.1:

Persuasion is clearly a sort of demonstration, since we are most fully persuaded when we consider a thing to have been demonstrated. The orator's demonstration is an enthymeme, and this is, in general, the most effective of the modes of persuasion. The enthymeme is a sort of syllogism, and the consideration of syllogisms of all kinds, without distinction, is the business of dialectic, either of dialectic as a whole or of one of its branches. It follows plainly, therefore, that he who is best able to see how and from what elements a syllogism is produced will also be best skilled in the enthymeme, when he has further learnt what its subject-matter is and in what respects it differs from the syllogism of strict logic. The true and the approximately true are apprehended by the same faculty; it may also be noted that men have a sufficient natural instinct for what is true, and usually do arrive at the truth. Hence the man who makes a good guess at truth is likely to make a good guess at probabilities.

Aristotle preserves the idea that strict logic is closely related to practical decision-making, which is the proper subject matter of rhetoric and ethics and political science. But he explicitly rejects the idea that you can obtain a deduction, a demonstration, of the sort that contemporary analytic philosophy often seeks. His examples are on point: in general, courage is a praiseworthy quality, and most of the time it will lead you to greater success in life.

However, there are counterexamples. The poet Sydney Lanier, during his time as a Confederate officer aiding British blockade runners as a pilot, behaved courageously and honestly in refusing to disguise himself as British when overhauled by a Union naval vessel. As a result, he caught tuberculosis while interned as a prisoner of war and died before he was forty. His virtues caused him to produce some remarkable works of literature and music, but they also killed him.

Now a proper defense of contemporary logic might suggest that they have an out here. Truth-preserving forms can only preserve as much truth as was in the original proposition. Thus, if the original proposition you are starting from is -- as Aristotle says -- not necessarily true but only probably true, your conclusion can only be taken to be probably true as well.

But I think Aristotle's point is stronger than that. He's very much in favor of applying "a sort of syllogism" to the problems of everyday life, but he's also clear that there is a kind of double analogy at work. First of all, the earlier practical problem you are taking as an example is an analogy, and analogies are always a little imprecise. If we say "our current situation is like Washington at Valley Forge," we don't really mean that it's exactly like Washington's situation. There's an imprecision.

In addition, the kind of logic we can apply to these analogies isn't going to offer us truth-preservation in the same way as what Aristotle calls "strict logic." It's going to be a "sort of syllogism" we can actually bring to bear, and for a good reason: even if we should go as far as translating our problems into the mathematical language of formal logic, so we can apply a strict deduction according to rigorous forms, we will have introduced new ambiguities in the translation. That is, of course, just why Bertrand Russell and others preferred to symbolize propositions: they hoped to eliminate ambiguities of natural language. Finally, it simply cannot be done: strict logic doesn't admit of many of the elements we need to capture all the details of a real-world problem.

(A great example of this is the symbolized forms of St. Anslem's Ontological Proof for the Existence of God. It's a valid argument when symbolized... but it can't actually prove what Anslem was after, because it is necessary to formalize "best" or "better than" in a way that loses his sense of the term entirely.)

In any case, this is a major difference between the ancient and medieval understanding of logic, and the contemporary form. Whether you view the contemporary enchantment with mathematical logic a romance or a seduction depends on your view of the character of the logic itself. I believe Aristotle still has the best of this argument, even though he never got to see the development of algebra, or the subsequent similar refinements in formal logic.