Sunday, December 23, 2007

Argument Analysis: From Casual Logic to Formal Analysis

An article from Jan Korger and Florian Leber's Spacezone published on the Website Linux Today compares Ubuntu Linux with Microsoft (c) Windows in the following argument:
Ubuntu is not Linux. Linux is not Windows. Then, Ubuntu is ...

Penguin Pete told us Ubuntu is not Linux in an infamous post he later deleted. An unrelated article tells you Linux is not Windows. Anyone bad enough at math will conclude a relationship between Ubuntu and Windows and secretly that’s the real subject of both articles...

You can interpret this as Ubuntu is (like / similar to) Windows or Ubuntu is not Windows depending on your point of view if you're into such questions. However, this doesn't tell us whether Ubuntu is actually any good! (
What I find interesting about this article is that the logic used, although not mistaken in an informal sense, illustrates several common confusions of many persons beginning their study of formal logic:
(1) expression in mathematics and logic are interchangeable,
(2) the "is" of identity and the "is" of predication are the same kind of thing, and
(3) reasoning using the principles of logic are a matter of opinion.
In observing these student misconceptions, I am reminded here of Ernst Haeckel's generalization of the similarity of embryonic development of human beings to the evolutionary development of species: ontogeny recapitulates phylogeny. Ofttimes, in mastering a discipline such as logic, there seems to be a similarity of the stages of learning the discipline to the historical development of the subject.

For example with regard to (1) the relation of mathematics to logic, historically Pythagoras, in his geometrical demonstrations, used logical reasoning, but it was Aristotle who later formalized and distinguished logical reasoning in the Organon from geometrical arguments. (Of course, there is the similarity that syllogistic logic and geometrical reasoning are both deductive forms, but that is insufficient to imply the logistic thesis.)

With respect to item (2), in the Sophist, Plato addressed Parminides' confusion of the "is" of predication:
Well, when we speak of a man we give him many additional names--we attribute to him colors and shapes and sizes and defects and good qualities, and in all these and countless other statements we say he is not merely a 'man' but also 'good' and any number of other things. And so with everything else. We take any given thing as one and yet speak of it as many and by many names.
(Plato, Sophist, trans. F.M. Cornford in Plato: Collected Dialogues, ed.,Edith Hamilton and Huntington Cairns (1969 rpt. Princeton: Princeton University Press, 1961), 251 a-b.

And with Aristotle, there was little confusion left:
...(1) an attribute is predicated of some subject, so that the subject to which 'being' is attributed will not be, as it is something different from 'being.' Something, therefore, which is not will be. Hence 'substance' will not be a predicate of anything else. For the subject cannot be a being, unless 'being' means several things, in such a way that each is something. But ex hypothesi 'being' means only one thing,
(Aristotle, Physica in The Basic Works of Aristotle, ed. Richard McKeon (1970 rpt. New York: Random House, 1941), I,3 (186a35-186b1).

The third misunderstanding, that logic is "the last refuge of a scoundrel" is reflected in the conclusion (of the quoted passage above):
You can interpret this as Ubuntu is (like / similar to) Windows or Ubuntu is not Window depending on your point of view if you're into such questions.
The implication from these authors is that you can draw whatever conclusion you wish from the premises provided in accordance with your own opinions. In truth, however, no conclusion validity follows from a syllogistic argument with two negative premises.

Translating Korger and Leber's argument into standard form yields ...
No [M Linux OSs] are [P OSs similar to Windows].
No [S versions of Ubuntu] are [M Linux OSs].
All [S versions of Ubnutu] are [P OSs similar to Windows].
No [S versions of Ubuntu] are [P OSs similar to Windows].
The reason why no conclusion validly follows from two negative premises is negative statements exclude partly or wholly the subject class from the predicate class. So what is being asserted in in the premises relating S to P through M is being asserted regardless of the kind of statements being used.

By referring to the mnemonic of the mechanism of the syllogism sketched here, we can surmise that the basis of the syllogism is captured by noting that two things related to the same thing should be somehow related to each other, if at least one of them is completely related in some manner.

However, when both premises are negative, our mnemonic shows the classes are not related in some way to each other, and this information is of no use to see how the terms in the conclusion are related. From the observation that two things are not related to a third thing, it's impossible to tell whether or not they would be related to each other. This state of affairs can be illustrated in the diagram to the right.

So it's simply not true that we may conclude whatever we wish; in fact, whenever both premises of a syllogism show an exclusive relationship between the subject and predicate classes, no conclusion can be correctly drawn.

Saturday, September 29, 2007

Once Again, the Hemline Theory

The occasion of longer hemlines in fashion houses together with the current mortgage crisis is described in the "Dashboard" section of Time as follows:
Longer hemlines are showing up in fashion designs for spring 2008, indicating that U.S. economic woes could worsen. ... [T]he hemline theory has proved correct at times. Hemlines were short in the Roaring Twenties but fell before the 1929 stock-market crash. In the '60s miniskirts were en vogue, and stocks rose. In the summer of 2006, designers showed short hems for spring, and in May the Standard & Poor's 500 Index hit a seven-year high.
("Hemline Theory," Time 170 No. 13 (24 September 2007), 19.)

Unless some kind of causal mechanism is hypothesized and tested, several observations of similar conjunctions of events are insufficient evidence for prediction of future connections.

Normally, one would simply label a prediction based on the theory that shorter hemlines are a causal predictor of a better economy and longer hemlines are causally related to a poorer economy (as reflected in the performance of the U.S. stock market), as a case of the fallacy of false cause or non causa pro causa. In particular, as stated in the Time excerpt above, the fallacy would seem to be the more specific post hoc ergo propter hoc--the fallacy of arguing that one kind of event was caused by another kind of event merely because the second kind occurred after the first kind of event.

Yet, what is the precise distinction between an "accidental" constant conjunction of events and and an "empirically necessary" or causal connection between events? Supposing that there were no disconfirming instances of this observed conjunction, how many confirming instances of a change in hemlines and the relevant kinds of consequent stock market movement are sufficient to conclude that there is a causal connection between these kind of events?

On the one hand, first and foremost, the problem of event-description is paramount. Just as it is arguable that the various instances of changes in hemlines are not similar enough to describe these various instances of these events as the same kind of events, so likewise it is arguable that the various instances of changes in the stock market are not similar enough to describe those kinds of events as the same kind of events. But notice how, with a bit of effort, we could tailor our definitions to cover just those cases of hemline change and just those relevant stock market movements in such a way that there could be no logically-possible counterexamples. (For example, this is how Alexander Fleming in his famous paper described the discovery that penicillin is effective against penicillin-susceptible bacteria. If penicillin weren't so effective, then the bacteria would not have been penicillin-susceptible.)

These kinds of uses of the heuristic of affirming consequential results in a general statement are a common practice for hypothesis-generation by research scientists. In the case of hemline theories, if we were able to define hemline lengths and stock market movement in such a manner that these events were empirically or logically necessarily connected, then we would presumably not have a viable testable theory. If a future event conforming to our definitions were somehow to unexpectedly disconfirm our theory because of an error in formulation, then we could always revise the theory to exclude that event. Whenever the revision becomes patently viciously circular, the so-called causal theory would, in all good conscience, have to be abandoned.

On the other hand, presuming we could construct reasonably neutral definitions of our key terms of hemline lengths and stock market movements, then, of course, our first task would be to try to falsify the theory by using something like Mill's Methods.

But, again, suppose we were to find there are no disconfirming instances? Does the fact that hemline lengths and stock market movements belong to different categories of events, as stated in ordinary language, permit us to conclude causal irrelevancy? Although there surely are specific reasons based on the entrenchment of past language use for assuming there is no causal or necessary connection, that, in itself, does not allow us to conclude that a fallacy has occurred solely on that account. After all, proposed new theories, by their very nature, are not grounded in the ordinary implications of the use of the words themselves.

Assuming that we could select data to show there are no disconfirming instances, how then can we justify calling the hemline theory a case of post hoc ergo propter hoc? The answer has to do with the burden of proof. If we claim an argument is fallacious, we have not thereby proved the conclusion false--we have only claimed the conclusion does not follow from the proposed premises. Therefore, the fallacy of false cause, in this case, is accurately noted, because no cogent or testable causal mechanism is proposed to account for the conjunction of events.

This too-short analysis, of course, does not disprove the hemline theory from a statistical point of view. One can always theorize about the probabilities of coincidence. But that is a whole different subject.

Monday, September 10, 2007

Validity and Soundness in a Modus Tollens: A Vital Difference

Mother Teresa: Come Be My Light, a book in which Teresa's letters and correspondence are presented by Brian Kolodiejchuk, has generated much interest in the spiritual life of one of the most remarkable women of the past century. From that book, David Van Biema of Time quotes from a letter Teresa wrote in 1959:
What do I labour for? If there be not God--there can be no soul--if there is no Soul then Jesus--You also are not true.
(David Van Biema, "Her Agony," Time 170 No. 10 (3 September 2007): 40)

Of course, this passage as a question and a series of hypothetical or conditional statements makes no assertion of deistic, agnostic, or atheistic belief. A conclusion could be entailed if the additional statement "Jesus is not true" (via modus tollens) or "There be not God" (via modus ponens) were present. As neither of these latter statements are asserted, we cannot validly infer an expression of any definite belief.

Nevertheless, a reader comments on the Time article by supporting the view that a purportedly atheistic Teresa would be a far more courageous individual than a purportedly doubting Teresa:
. . . people point to Teresa's lack of feeling the presence of Jesus as proof of God's existence. These people note her courage in persevering despite severe doubts. If she had been even more courageous, she would have admitted she was an atheist. Helping the poor without a belief in a heavenly reward is one of the greatest aspects of secular humanism.

(August Berkshire, "Inbox," Time 170 No. 12 (17 September 2007): 8).

Note that the structure of this argument (with an overly generous invocation of the principle of charity, neglect of the subjunctive tense, and the introduction of an inferred subconclusion), is modus tollens:

If p is true, then q is true.
q is not true.
Therefore, p is not true.

(Where p represents something like "Teresa is more courageous than someone who has doubts," and q represents "Teresa states she is an atheist.")

Since helping the poor without a belief in heavenly reward contextually implies such a person would be an atheist, and Teresa helped the poor with (presumably) having doubts about heaven--we have the vague appearance of a (valid) modus tollens.

Well, of course, all modus tollens arguments are valid. But since there is no evidence adduced for the statement that she admits to being an agnostic, much less an atheist, by any interpretation or translation of p and q, we have at least one false premise.

Consequently, even if we were to agree somehow to the questionable and murky translations, the argument in the Time letter is unsound. The best that Mr. Berkshire can hope for in this excerpt printed in Time is the assertion of a contrary-to-fact conditional statement.