Anti-Inductive Logic
As part of my continuing interest in logic, here’s a rather curious result.
Some of you are probably familiar with so-called “inductive logic”, which is usually framed more or less as something like:
If something is true for the first n instances, then it is true for the n+1 th instance.
Now, such a rule cannot be part of a formal deductive system of logic, because there are clearly counterexamples; just because something is true for the first few instances, doesn’t mean that it must be true for all of them.
Nevertheless, inductive logic does serve as a useful “logical heuristic” of sorts; for example,
For the past 4 billion years in the existence of earth, the sun has risen and set every day without fail. Therefore, it will rise and set tomorrow.
While not exactly sound deductive reasoning, it works relatively well.
However, since we’re abandoning formality, we might as well try a new, different inductive rule, which I refer to as “Anti-Inductive Logic”:
If something is true for the first n instances, then it is false for the n+1 th instance.
While this may seem odd (and blatantly incorrect), it’s not entirely unworkable; in one sense, it’s loosely related to the Gambler’s Fallacy:
The result of 1000 flips of an unbiased coin is 1000 heads. Therefore, the next flip will result in a tails.
A rather curious fact about Anti-Inductive Logic is that, to a certain extent, it is self consistent. (Again, technically, if we were working in a formal system of logic, it is not consistent, but we’re not considering formal systems right now)
Consider walking up to a practitioner of Anti-Inductive Logic, who claims the following:
For the past 4 billion years in the existence of earth, the sun has risen and set every day without fail. Therefore, it will not rise and set tomorrow.
In an effort to convince the poor soul that his Anti-Inductive Logic has led him astray, we might argue:
For the past 4 billion years, since the sun first rose and set on earth, Anti-Inductive Logic has predicted that the sun will not rise and set the next day. Therefore, for every day in the past 4 billion years, Anti-Inductive Logic has been incorrect.
Unfortunately, without regular inductive logic, the poor soul is forced to conclude:
For the past 4 billion years, Anti-Inductive Logic has not worked once in predicting whether the sun will rise or set the following day. Therefore, Anti-Inductive Logic will correctly predict whether the sun will rise and set tomorrow.
And we are forced into an endless loop, until someone actually manages to introduce formal deductive logic into the discussion. Or perhaps encourage the poor soul to run for political office.
Note: I’m not entirely sure if this was a completely original idea, or whether I read something similar somewhere – if you know this result from elsewhere, please comment below! Thanks.
