This week Friday falls on Wednesday.

I've already taught one cycle of this week's material to my Monday evening class, and now I'm trying to make sure my Monday-Wednesday-Friday version of the same material is consistent with it, and preparing other material. So the look backwards function of a blog can be exercised from this vantage point as well as any other.

I learned (or perhaps re-learned) just how thoroughly counter-intuitive the notion of vacuous truth can be this week. The name sounds as though the variety of truth is silly, trivial, or empty, but it's an important technique in logic. If we have an implication that claims that if P is true, then Q follows, then then implication as a whole is true whenever P is false. The notion rests on the fact that an implication claims that Q follows from P, and the only time that "following from" rule is broken is when P is true and Q breaches faith by being false.

In particular, whenever P is false, no breach of faith is possible.

I've often fired up vast and powerful pedagogical machinery to convey this point, being sure to make it several different ways since it registers with different people in differently. There are always some people who resist the first onslaught, so I back the machinery up and take another run at it. Usually nobody is willing to admit that vacuous truth still seems weird after two or three runs at it.

Monday night, though, I presented a particularly twisted exemplar of vacuous truth to an audience that wasn't shy about admitting that it seemed weird. Some idea of the dynamics are seen in the annotated slide where I present the claim that for every real number x, if x^2-2x+2 = 0, then x > x+5. The unnerving thing is that the entire if-then claim is true, since you'll never find a real number that satisfies the first part, but falsifies the second (there are no real roots of that quadratic equation). There's no magic: the false conclusion doesn't become true, but the claim that it follows from the antecedent is true.

In response to my question about whether this implication is true for all real numbers (it is), the Monday night crowd was split. Once we thrashed out the point that any real number substituted for x gives us a false antecedent (hence a true implication), I asked whether the claim true if it asked where there existed some real number where the antecedent implied the consequent. The majority disagreed, and asked me to come up with examples. I started spinning off various real numbers: 17.98, 13.532, ... (any real number would work), and we eventually agreed that the "there exists" form of the statement is also true. Then somebody asked what happens if we change the set x is a member of from the real numbers (where there is no solution to the quadratic equation), to the complex numbers (where all solutions exist). That made the question more interesting, and I found myself regretting that I hadn't thought of this twist first.

I had the feeling that several people were coming to grips with vacuous truth in real time during the lecture. Impressive.