Wednesday, January 25, 2006

Facts and Forms, or, Elementary my dear Clouseau

Copyright © 2006 by Joel Marks

Inspector Jacques Clouseau: Facts, Hercule, facts! Nothing matters but the facts. Without them the science of criminal investigation is nothing more than a guessing game. So consider the facts in the case at hand. Fact: Maria Gambrelli was found standing over the dead man with a smoking gun in her hand. Fact: She had a big smile on her face. Fact: No other fingerprints were found on the gun. So what do we conclude, Hercule?
Hercule LaJoy: Why, that Maria Gambrelli committed the murder.
Inspector Jacques Clouseau: No, you fool! You are forgetting the most important fact: motive.
Hercule LaJoy: He beat her.
Inspector Jacques Clouseau: He was Spanish!
Hercule LaJoy: He tore her dress off.
Inspector Jacques Clouseau: Oh, don't be ridiculous. Would you kill someone who tore your dress off?

-- from the 1964 movie, A Shot in the Dark, starring Peter Sellers as Inspector Clouseau

Hilarity aside, the cited scene from the classic comedy serves as a perfect example of how logic works, or doesn’t.

Clouseau begins by touting facts as the key to solving a murder, or any crime. And indeed, what could be more noble and useful than a total allegiance to the facts? Not only the detective, but the scientist and any genuine seeker will want to become adept at discerning and uncovering facts. It seems to be practically a tautology. But, in fact, the facts fail him utterly when Clouseau draws his final conclusion, which is that the beautiful Maria Gambrelli is not the murderer. Alternatively we might say that the only fact that really moves the inspector is the beauty of the accused.

The logical conclusion to draw about Clouseau’s reasoning, or about any reasoning, is that it is not facts that decide an inference by themselves but also logic. But what is logic if not the compilation of facts? There is a very precise answer to that question, and its name is validity. Its meaning is that the facts adduced suffice to assure the truth of the conclusion. But how do they do that? Is it just a matter of strong belief? Clearly not, for Clouseau certainly has a conviction that Maria Gambrelli’s physical charms guarantee that she should not be convicted; but of course that is absurd. So the assurance provided by the facts, or as they are technically known, the premises of a logical argument must have a more secure source than the subjective psychology of the person who is presenting the argument.

There are many ways to describe the essential logical element of validity. One simple way is to point to various rules. Perhaps the most fundamental rule is this: If it is true that p and it is also true that q is true if p is true, then q is also true. That rule is traditionally called modus ponens. When it obtains, one can say that the conclusion of the argument follows from the premises; that is, they “follow,” not sequentially, but with logical necessity. The conclusion must be true, given the truth of the premises. The rule given has two premises: (1) p is true and (2) if p is true, then q is true. Note that both premises are presumed to be facts. For example, an argument might be that (1) (it is true that) today is Monday and (2) (it is true that) if today is Monday, then tomorrow is Tuesday; therefore tomorrow is (or will be) Tuesday. This argument is valid because, if both premises were true, the conclusion would also be true -- guaranteed.

Note also, however, that validity by itself does not guarantee the truth of a conclusion. Even though the above argument is surely valid, and hence is a logical argument, its conclusion would not be true if one of its premises happened to be false. Thus, if today is Tuesday, then even though Premise 2 remains true – since it is only a hypothetical assertion – the conclusion is false, because tomorrow would be Wednesday. I should also point out that the conclusion of a valid argument that contains a false premise could also still be true. For example: (1) The Sun revolves around the Earth and (2) If the Sun revolves around the Earth, then (provided the Earth’s rotation is not in unitary synchrony with the Sun’s revolution and there are clear skies, etc.) the Sun will appear to move in Earth’s sky; therefore (provided the Earth’s rotation is not in unitary synchrony with the Sun’s revolution and there are clear skies, etc.) the Sun appears to move in Earth’s sky. In this argument Premise 1 is false; however the conclusion is true anyway. Thus, both facts and logic are required for a sound argument. My point in this essay is only to emphasize that facts alone will not suffice to yield sound reasoning.

Inspector Clouseau, therefore, may have been right about all of his facts; but his inference was faulty because of its illogic. The mere fact of Maria Gambrelli’s beauty does not, by any rule of logic, guarantee that she is not a murderer. But suppose Clouseau insisted upon a second premise, namely: If someone is beautiful, then they couldn’t be a murderer. Fine; logic has been restored to the inference. But of course now the argument is unsound since at least that second premise is surely a false generalization. Thus, Clouseau would have won the logical battle, but lost the rational war. Here again we see facts and logic working in tandem; each by itself is insufficient to guarantee sound reasoning.

The reason I wish to stress the importance of logic over facts, however, is that it is the underappreciated sibling of this pair. People generally are comfortable talking about facts and thinking in terms of facts. But this gets us only so far, and produces a lot of mischief by people who present facts and then draw conclusions willy-nilly. For example: “The destruction of the World Trade Center was an atrocity; Saddam Hussein is an evil dictator; therefore we should invade Iraq.” Why does the crucial component of logic (or illogic!) escape notice so often? Perhaps because it is such an abstract notion; furthermore its nature is relational, a “ghostly” connection between one set of “solid” facts (the premises) and another (purported) fact (the conclusion). Yet this is also wherein resides its power. The way logicians capture these phenomena is with the use of symbols or forms; for example, as we have seen, by the use of lower-case letters like p and q, which can function as variables that stand for any statement whatsoever. The resultant “formal” logical rules have universal application, whether they be about Maria Gambrelli or the Sun or a pig in a poke.

Therefore Inspector Clouseau needs to add one more fact to his repertoire, to wit: In order to crack the case, the facts must fit the right forms.

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