Propositional Fallacies, Part 1

There’s a branch of math known as propositional calculus that treats arguments like mathematical propositions. Using propositional calculus, you can demonstrate the truth or falsity of certain arguments.

Take the statement “It is raining here now.” Depending on when you make the statement, it can be either true or false. In propositional calculus, you’d represent the statement as “P,” and the opposite, “It is not raining here now” as “¬P.” If P is true, then ¬P has to be false; if ¬P is true, then P has to be false.

You can link together statements with connectors. Common connectors are AND, OR NOT, ONLY IF, and IF AND ONLY IF. If we say “It is raining here now, and it is raining where you are now as well,” we can label the second statement as Q. Represent AND with the symbol ⋀, and we can write “It is raining here now, and it is raining where you are now as well” as P⋀Q.

Of course, maybe it is raining here or it isn’t; maybe it’s raining at your house and maybe it isn’t. Because the individual statements can be true or false, we can prepare a truth table.

P                    Q                    P⋀Q
True         True            True
True         False              False
False             True            False
False             False           False

With and as a connector, the proposition P⋀Q is only true if both statements are true.

The connector OR (represented as “⋁”), on the other hand, makes the proposition true as long as at least one of the statements are true. “It is raining here now OR it is raining where you are now” results in the following truth table.

P                    Q                    P⋁Q
True         True           True
True         False          True
False          True              True
False           False             False

Notice that OR is used here inclusively rather than exclusively. That is, P doesn’t exclude Q from being true. If it’s raining at my house, that doesn’t mean it’s not raining at yours.

Given the idea of propositional logic, it's easy to conclude that there are fallacies to go with it. The first of these is known as affirming a disjunct.

Affirming a Disjunct

Also known as the fallacy of the alternative disjunct, or the false exclusionary disjunct, this particular fallacy occurs when you change an inclusive OR into an exclusive one. “It is raining here now or it is raining where you are now” gets interpreted as “If it is raining here now, then it isn’t raining where you are now.”

In our symbolic structure, that gets represented as the following argument (with “therefore” represented by ∴).

P⋁Q
P
∴¬Q

That’s a fallacy because it could be raining both places. One doesn’t preclude the other.

While OR in logic always means an inclusive “or,” that doesn’t mean you don’t sometimes want to be more concrete. The logical operator XOR is an exclusive or. When you use it, you’re saying “one or the other, but not both.” The symbol for that is ⊻.

More next week.

Who Was That Masked Man? (Formal Fallacies Part 3)

Formal fallacies are arguments that are always wrong, regardless whether the argument's premises (statements claimed as fact) are true or false. For example, in the appeal to probability, someone makes a claim that because something could happen, therefore it will happen. That’s false even if it's true that the something in question could indeed happen.

Masked Man Fallacy

I know who Bruce Wayne is.

I do not know who Batman is.

Therefore, Bruce Wayne is not Batman.

In the masked man fallacy, a substitution of identical designators in a true statement can lead to a false one. The statement "I do not know who Batman is" gets treated as if it excludes Bruce Wayne simply because I do know who he is. Of course, as long as I don’t know that Bruce is actually Batman, both statements can be absolutely true, and yet the conclusion does not follow logically.

The general form of the argument is:

X is known.
Y is unknown.
Therefore, X is not Y.

A similar argument, however, is valid.

Clark Kent is Superman (X is Z).

Batman is not Superman (Y is not Z).

Therefore, Clark Kent is not Batman (therefore, X is not Y).

That’s because being something is different from knowing something. Lack of proof of one proposition doesn’t serve as proof of the counter proposition.

Fallacy Fallacy (Formal Fallacies Part 2)

Formal fallacies are arguments that are always wrong, regardless whether the argument's premises (statements claimed as fact) are true or false. In the previous installment, the appeal to probability, a claim that because something could happen, therefore it will happen is false even if it's true that the something in question could indeed happen.

Argument from Fallacy

If an argument contains a fallacy, what does that say about the conclusion? Actually, it doesn’t say very much. Excessively pointing to fallacies can itself trigger a fallacy of its own: the argument from fallacy, or the fallacy fallacy.

The argument from fallacy is the error of concluding that if an argument can be shown to be fallacious, that means its conclusion necessarily must be false. The form of the argument is:

If P, then Q
P is a fallacious argument.
Therefore, Q is false.

Take, for example, the following claim: “I speak English, therefore I am an American citizen.” That’s a fallacious argument, because many people who speak English are not American citizens. To conclude, however, that because the argument is fallacious, you must not be an American citizen, is taking the claim a step too far. A conclusion can be right even if the argument supporting it happens to be wrong.

If you can show that a particular argument is fallacious, the only thing that means is that the particular argument can’t be used to prove the proposition. The opposite argument, that the fallacious argument itself disproves the proposition, is also a fallacy.

The argument from fallacy is also known as the argument to logic (argumentum ad logicam) and the fallacist’s fallacy. It’s part of a group of fallacies known as fallacies of relevance.

Base Rate Fallacy
Conjunction Fallacy

The base rate fallacy and the conjunction fallacy also fall into the category of cognitive bias, and were both treated earlier in this blog and in my compilation of cognitive biases, published separately.

The Drake Equation (Formal Fallacies, Part 1)

Frank Drake

In February, I completed a 25-part series on red herrings, a category of argumentative fallacies that are intended to distract from the argument, rather than address it directly. That's only one category of argumentative fallacy. In this series, we'll look at formal fallacies. Formal fallacies are errors in basic logic. You don't even need to understand the argument to know that it is fallacious. Let's start with the appeal to probability.

Appeal to Probability

If I play the lottery long enough, I'm bound to win, and I can live on the prize comfortably for the rest of my life! Yes, it's possible that if you play the lottery, you'll win. Somebody has to. The logical fallacy here is to confuse the possibility of winning with the inevitability of winning. Of course, that doesn't follow.

In our study of cognitive bias (also available in compiled form here), we learned that numerous biases result from the misapplication or misunderstanding of probability in a given situation. Examples include the base rate effect, the gambler's fallacy, the hindsight bias, the ludic fallacy, and overall neglect of probability. Use the tag cloud to the right to learn more about each. We are, as a species, generally bad at estimating probability, especially when it affects us personally.

Various arguments about the Drake equation can fall into this trap. The Drake equation, developed by astrophysicist Frank Drake in 1961, provides a set of guidelines for estimating the number of potential alien civilizations that might exist in the Milky Way galaxy. Here's the formula:

in which:

N = the number of civilizations in our galaxy with which communication might be possible;
R* = the average rate of star formation per year in our galaxy
fp = the fraction of those stars that have planets
ne = the average number of planets that can potentially support life per star that has planets
fℓ = the fraction of the above that actually go on to develop life at some point
fi = the fraction of the above that actually go on to develop intelligent life
fc = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space
L = the length of time for which such civilizations release detectable signals into space

There are various arguments about the Drake equation. Some argue for additional terms in the equation, others point out that the value of many of the equation's terms are fundamentally unknown. There's a reasonable argument to be made that "N" has to be a fairly low number, on the simple grounds that we have not yet detected any extraterrestrial civilizations. Depending on the assumed values of the terms in the equation, you can derive conclusions that range from the idea that we're alone in the galaxy (see the Fermi Paradox) to an estimate that there may be as many as 182 million alien civilizations awaiting our discovery (or their discovery of us).

From a fallacies basis, however, the problem comes when people argue that the vast number of stars makes it certain that alien civilizations exist. As much as I'd personally prefer to believe this, the logic here is fallacious. Probable — even highly probable — doesn't translate to certainty.

That's not an argument against the Drake equation per se, but merely a problem with an extreme conclusion drawn from it. The Drake equation was never intended to be science, but rather a way to stimulate dialogue on the question of alien civilizations.