Watching the outrage over the DHS memos that purportedly target all Americans on the political right as potential enemies of the state, I have come to the realization that a great many political conspiracy theories are based on a trivial error in formal logic: namely, that the implication operator is not commutative.
The implication operator, A → B (A implies B) is true if A is false (B can be anything) or if both A and B are true. In other words, it is only false if A is true but B is false. However, A → B does not imply B → A; the former is true when A is false but B is true, but the latter isn’t.
Yet this is what is at the heart of many conspiracy theories. For instance, a DHS report might say, that those on the fringe of the political right are motivated by the Obama government’s more permissive stance on stem cell research. Some draw the conclusion that this report implies that all who are troubled by Obama’s stance on this issue must be right-wing extremists. I could write this symbolically as follows: we have
member(e, s) → prop(e, p)
where member(e, s) means that e is a member of set s, and prop(e, p) means that e has property p. This symbolic equation cannot be reversed: it does not follow that prop(e, p) → member(e, s).
A closely related mistake is the confusion of the universal and existential operators. The existential operator (usually denoted with an inverted E, but I don’t have an inverted E on my keyboard, so I’ll just use a regular E), E(s, p) says that the set s has at least one member to which property p applies. The universal operator (denoted with an inverted A; I’ll just use a plain A), A(s, p) says that all members of set s have property p. Clearly, the two do not mean the same. Yet all too often, people (on both sides of the political aisle, indeed a lot of the politically correct outrage happens because of this) make this error and assume that once it has been asserted that E(s, p), it is implied that A(s, p). (E.g., a logically flawless statement such as “some blacks are criminals” is assumed to imply the racist generalization that all blacks are criminals.)
One might wonder why formal logic is not taught to would be politicians. I fear that in actuality, the situation is far worse: that they do know formal logic, and use it to their best advantage assuming that you don’t.
