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.