A Philosopher's Dozen
Selections from the writings of Stephen Yablo
The Tale of A-B, in which A is (gasp) Stripped of the Implication that B
—excerpt from "Extrapolation and its Limits"
IT WOULD BE NICE IF I could explain the topic with examples, but we're going to have to make do again with anecdotes. The first comes from a conversation Einstein is supposed to have had with a puzzled citizen, or maybe it's just a Jewish folk tale:
"How does the telegraph system work?
I don't understand how they can make a message go down a wire."
"Simple, imagine a giant dog, with his head in Moscow and his tail in Leningrad.
Pull the tail in Leningrad and the head barks in Moscow."
"OK, but then, what about the wireless telegraph? How does that work?"
"The same way, but without the dog."
Another comes from the 1980 presidential debates between Ronald Reagan and Walter Mondale. Reagan had been showing colossal ignorance about world affairs; someone asked him about Valéry Giscard D'Estaing, the president of France, and he said, "I don't believe I have heard that name." The moderator wanted Mondale's thoughts on this. Did it bother him that there were so many gaps in Reagan's knowledge? "No," Mondale said, "it's not what he doesn't know that bothers me; it's what he knows for sure that just isn't true." (Borrowed from Will Rogers, apparently.)
Scaling a hypothesis back (so it no longer implies what it used to)
In both these stories you've got a hypothesis, A, that implies another one, B: pulling the dog's tail to get its head to bark implies there's a dog there; and knowing that trees cause more pollution than cars (to mention one of the untruths that Reagan "knew for sure") implies that they do cause more pollution than cars.
In both cases, A and B are supposed to determine a weaker hypothesis that is, as we might put it, A stripped of its implication that B, or the remainder when B is subtracted from A, or simply A-B. Mondale, for instance, alleges that Reagan only quasi-knows various things, where quasi-knowing that p is knowing that p, stripped of its implication that p.
This kind of implication-stripping, or scaling a hypothesis back so that it no longer implies what it used to, presents something of a challenge to analytic philosophy’s traditional self-image. Frege, Russell, and Moore sought where possible to analyze contents “from below” by showing how they could be built up out of weaker contents. They never, as far as I know, tried the opposite approach, approaching a content “from above” by first overshooting the target and then stripping away unwanted extras.
The mystery of logical subtraction
One could see this as just an oversight on their part, but the fact is that logical addition is pretty well understood—it’s just conjunction—while logical subtraction is a bit of a mystery. I suspect it’s no accident that Wittgenstein, as he began tearing himself free of the analytic paradigm, also began to wonder about logical subtraction, asking what is left if we subtract a man’s arm going up from his raising that arm, and what is left if we subtract from the fact that 'It hurts!' the fact that the sufferer is me.
Logical subtraction is a mystery, but that’s not to say philosophers don’t sometimes give it a try. Nelson Goodman considered a statement lawlike if it was a law, except it might not be true. Parfit, in his work on personal identity, explains quasi-memory in something like the way we explained quasi-knowledge on behalf of Will Rogers. A theory is empirically adequate if it is true, ignoring what it says about unobservables. The scare quotes sense of a term (as in, "careful, that joke is 'politically incorrect'") is the regular sense, minus any evaluative implication. But although philosophers are sometimes caught in the act of subtraction, they’re also nervous about it, perhaps without always knowing why.
Some instances of the genre are fine, like this one taken from math: triangles are similar if they're congruent, except they might not be the same size. With others, though, there is at best the illusion of understanding. Consider Wittgenstein's "explanation" of what is involved in its being 5 o'clock on the sun; it's just like 5 o'clock here, except it's on the sun! Is subtracting truth from knowledge like subtracting "same size" from "congruent"? Or is it like subtracting "in Greenwich" from "It's 5 o'clock in Greenwich"? No wonder we're nervous; it's just very hard to tell. •
Photograph: Jon Sachs