Goldfinger Takes Fort Knox! (Propositional Fallacies, Part 2)

Bond villain Auric Goldfinger

In propositional calculus, we can describe certain arguments in mathematical terms. Some arguments are true if the component statements are true. The statement “It is raining here now, and it is raining where you are now as well” can be written as P⋀Q. It is true if both its component statements are true. On the other hand, “It is raining here now OR it is raining where you are now” (written as P⋁Q) is true as long as at least one of the statements is true.

Propositional fallacies involve fallacies of mathematical reasoning. They are fallacious regardless of the truth value of the component statements. Last time, we discussed affirming a disjunct, the fallacy of turning an inclusive OR into an exclusive one. The two remaining propositional fallacies are known as affirming the consequent and denying the antecedent.

Affirming the Consequent

If Auric Goldfinger owned Fort Knox, then he would be rich. Auric Goldfinger is rich. Therefore, Auric Goldfinger owns Fort Knox. Even if the first two statements are true, the conclusion is invalid because there are other ways to be rich besides owning Fort Knox.

Here's how to cast the argument in propositional calculus:

P→Q 

Q 

∴ P 

(If P, then Q. Q is true. Therefore, P.)

This is different from the argument "if and only if." If Auric Goldfinger is rich if and only if he owns Fort Knox, then the statement "Auric Goldfinger is rich" makes "Auric Goldfinger owns Fort Knox" necessarily true. But that's the case only if the first statement is true — which it isn't. In propositional calculus, we'd write that:

P⟷Q

Q

∴ P

Affirming the consequent is sometimes called converse error.

Denying the Antecedent

The opposite fallacy, denying the antecedent, is also known as inverse error.

If Auric Goldfinger owned Fort Knox, then he would be rich. Auric Goldfinger does not own Fort Knox. Therefore, Auric Goldfinger is not rich. This is wrong for the same reason as the previous argument was wrong: there are other ways to be rich.

In propositional calculus, this takes the form:

P→Q 

 ¬P

∴ ¬Q

If P, then Q. P is false (not-P). Therefore, Q is false (not-Q). As in the previous case, the rules for if and only if are different from if alone.

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.