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The Categorical Converter: Why I Constructed It|
There are two major reasons why I constructed the Categorical Converter, one practical and one perverse.
The Practical Reason
In 1989 I was given the task of teaching first year logic students. In particular, I had to teach them the rules of contradiction, conversion, and the rest.
The rules as represented on the Square of Opposition were easy to teach; students to easily see which rule or sequence of rules would lead from premise to conclusion. However, once we got to obversion, conversion and transposition, it got a lot harder.
In particular, the following question perplexed both myself and my students: how do we know which rules to try first? It seemed like a matter of trial and error, and the logic texts were no help whatsoever.
I began by tracing the routes from SP to PS propositions. Once I realized that it could be done, constructing the remainder of the diagram was just work.
The rule for selecting which rule to start with which emerged was the following: Begin by using obversion and conversion until the order of the S and P in the premise is the same as the conclusion. Then apply the Square of Opposition.
Now why couldn't any of the dozens of logic texts I consulted say that? I ask you!
The Perverse Reason
Stripped of all words - including such suggestive words as All, No, Some and non-, the Categorical Converter is nothing more than an empty structure.
Or - as I picture it - it is a map without any place names.
When viewed that way, it is entirely arbitrary where we place our lines. For example, there is no line from SP under the A column to the PS under the A column. Why not?
The arrangement of the lines seems entirely arbitrary.
Even if we use All, No, Some and non-, the arrangement is arbitrary. There is no line between All S are P and All P are S. Viewed strictly as a structure or map, there is no reason why we shouldn't place a line there.
The reason why the lines are where they are, of course, is that inferences from one proposition to another should be truth preserving. But until we introduce truth into the system, the requirement of truth preservation cannot be used to dictate where we should place our lines.
In the Categorical Converter, however, truth is introduced by means of the T and F tokens. And indeed, it is not until we stipulate that T stands for Truth that we have introduced truth into the system.
But even that is not enough to tell us where to place the lines. Given only the structure and the truth tokens, there is still nothing which prevents me from drawing a line from All S are P to All P are S.
Such a system would be different from categorical reasoning as we know it, to be sure. But there is no a priori argument which may be given which states that it is wrong.
The only reason we use the structure we actually use is that the world resembles this structure. For example, in the world, even though it may be true that All dogs are mammals, there are counterexamples to the proposition that All mammals are dogs. But were the world constructed differently, there might not be any counterexamples. Our rules of inference would then be different.
So: my perverse reason is that this is one more way of showing that logic does not represent a priori necessary truth. Rather, logic follows from emprical observation. Logic works the way it does because the world works the way it does, and not the other way around.
10 August 1996