Bond villain Auric Goldfinger |

*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.