I always find it astonishing how bad we are, generally, at if-then statements. Even scientists make blunders with logic fundamentals—usually in life outside the lab, when we’re not paying close attention to those common pitfalls.

This idea came to mind recently when I heard about a podcast discussing the risks of alcohol and marijuana use in the context of the latter’s legalization. One of the most commonly cited reasons not legalize marijuana use is the idea of it being a gateway drug. There are plenty of articles that address the poor evidence of causation for this claim, but what I want to focus on today is something more basic: and that’s the pure logical problem with these arguments.

Essentially the argument is as follows:

(1)  Marijuana is a gateway drug:

A. If a person uses harder drugs, then they have used marijuana.[1]
B. If a person uses marijuana, then they will use harder drugs.

(2)  Therefore increasing marijuana usage will increase usage of harder drugs.

 So let’s take a step back and think about this logically, by working through some classic, old-school logic.[2] Consider the four scenarios diagrammed here:

And take the following two statements for each of the scenarios:

(1) If that person is female, then that person is wearing a yellow outfit
(2)  If a person is wearing a yellow outfit, then that person is female;

Now,—and this is the fun part!—let’s examine the veracity of the two statements in each of our four scenarios:

A cursory look at the True/False outcomes tells you that odd things happen when we switch the “if” and the “then” of our first statement to give our second statement (which is known as “the converse”). To be more precise, if Statement 1 is true (as it is in Scenarios A and B), then Statement 2 is true only in Scenario B—but in Scenario A, it’s false. And, similarly, if Statement 2 is true (Scenarios B and C), then Statement 1 is true only in Scenario B[3]—but in Scenario C, it’s false.

What’s the take-home? If you want to show that Statement 2 is true, you have to show that Statement 2 is true: Statement 1 won’t do anything for you.

So how does this apply to our “gateway drug” problem? Well, let’s go back to that original argument:

 (1)  Marijuana is a gateway drug:

A. If a person uses harder drugs, then they have used marijuana.
B. If a person uses marijuana, then they will use harder drugs.[4]

(2)  Therefore increasing marijuana usage will increase usage of harder drugs.

But, wait a sec, Part 1A and Part 1B are converses of each other, just like our yellow-outfit-gender example above! And so, just like our outfit example, from a pure logic viewpoint, the veracity of Part 1A has no bearing whatsoever on the veracity of Part 1B.

Unfortunately for the anti-legalization argument, because Part 2 relies entirely upon Part 1B, once we puncture that part of the chain, the rest of it just falls apart.

As I said, logic is tricky. Logical gaps, jumps and sleights-of-hand are easy to do, and easier to miss. But getting it right, even just the fundamentals, is important—in science and in life.

 

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[1] Often, there are some impressive statistics to support this first part. One anti-legalization website claimed that an astounding 99.9% of cocaine users used marijuana, cigarettes or alcohol first! (It’s a little unclear to me where this number came from.)

[2] If you want to dive into logic more thoroughly, a good place to start is, as always, Wikipedia.

[3] Scenario B thus represents a “special” type of scenario—which, in mathematics, comes with a whole host of nice characteristics.

[4] I recognize that the anti-legalization crowd would put some probability here— to make that statement something like “if ten people use marijuana, then one person will use harder drugs.” Of course, this weakening already demonstrates how little confidence they feel in the veracity of this if-then statement.