#deductive logic

Deductive logic is a very careful method of drawing conclusions from premises. Many philosophers like to say deduction is "necessarily truth preserving." That is, given the premises, the conclusion must absolutely be true. If we take as our premise that Sherlock Holmes is a man, then we know with absolute certainty that either Sherlock Holmes is a man or he is a cucumber. This often strikes people as odd, but if we take Sherlock Holmes’ manhood as a given, then we can conclude that he is a man or anything else. The statement "Sherlock is a man or [insert something else]" will always be true given the premises because it will always be the case that Sherlock is a man.

It might be useful to think of deductive logic like math. A math problem takes an unsolved equation as its premise and arrives at a conclusion that must be true. If 2(x) = 4, then x must equal 2. Where math deals with numbers of things, deductive logic deals with the arrangement and existence of things. Deduction tells us that if the world is composed of x and y, and that x and y are connected by z, then a, b, and c, must also be true.

The necessity of deductive conclusions comes at a cost. Only so much is necessarily true given a set of premises, and often the possible conclusions are quite uninteresting. For instance, deduction’s reach is quite limited from the premises that Sherlock Holmes is a smart man, smokes a pipe, and wears a dearstalker hat.

With just a little work we could prove that "if Sherlock Holmes is a man, then he wears a dearstalker." This might at first look like an interesting conclusion, but it really says nothing. This is not the statement a detective might make: "if the person we are looking for, Sherlock Holmes, is a man, then he will surely be wearing a dearstalker hat." Rather, because the premises stipulate that Holmes is a man and wears a dearstalker, the if-then statement is just a strange way of putting what we already know.

Don’t assume from the examples that deduction is weak and always boring. The above examples use an extremely simplified version of deductive logic. Over the last 150 years, logicians have developed extremely complex versions of deductive logic that can deductively derive conclusions in all sorts of scenarios. For instance, there are versions of logic that can derive conclusions based on time. So-called tense logic can conclude from a premise like "it is currently the case that oranges are orange" that "at every point in the future, there was a point in the past when oranges were orange." Other versions of deductive logic include modal logic, which draws conclusions based on necessity and possibility, and doxastic logic, which attempts to deductively describe the relationships between our attitudes and beliefs. These more complex systems are capable of proving more interesting and surprising conclusions from a set of premises, even though the conclusions are still necessary.  

I’ve been belaboring the role of premises in deduction because understanding their role in the matter is essential for understanding another weakness of deduction. Instead of drawing conclusions about the world, deductive logic draws conclusions about the premises. The rules of deductive logic work whether or not the premises are true. We can stipulate any premises we want and still use deduction to arrive at a conclusion.

Why is this important? Let’s consider an example from Sherlock Holmes canon. In "The Adventure of Sherlock Holmes," Sherlock "deduces" that a man is a doctor from, among other things, the mark of a stethoscope upon his hat. We can reverse-engineer the deductive inference Sherlock probably used:

Premise 1: That man has a mark on his hat indicative of a stethoscope

Premise 2: Only doctors have marks on their hats indicative of a stethoscope

Conclusion: That man is a doctor

The proper use of deduction is almost inconsequential to the impressiveness of Sherlock's conclusion. Here is an equally valid argument I could sarcastically draw while looking at the same man with the same mark.

Premise 1: That man has a mark on his hat indicative of an overbearing landlord

Premise 2: Only people who are 29 years old and have a grandfather named Robert have marks on their hat indicative of an overbearing landlord

Conclusion: That man is 29 years old and has a grandfather named Robert.

My inference is as deductive as Sherlock’s—in both cases the conclusion follows from the two premises. So why is Sherlock right about the man being a doctor but I’m not right about the man having a grandfather named Robert? The answer, my dear Watson, is not deduction at all.