The basic form of the slippery slope argument is that you concede that some policy x (lowercase) is prima facie acceptable, but that it sets a precedent for progressively stronger versions of that policy.  Ultimately, the strongest version of the policy, call it X (uppercase), at the extreme, will seem acceptable.  But X is clearly not.  The reasoning then goes that to stem the tide of precedent to X, we must not take that first step to accept x.  For a slippery slope argument to be acceptable, the slope must be genuinely slippery.  That is, there must be a relevant relation between x and X (namely, that x is not just a  preconditon for X, but that it must make X more acceptable), there must be no places where other considerations prevent the intermediary moves, and so on.  In cases where those desiderata fail, the slope isn’t slippery.  It’s more a bumpy staircase.

Some slippery slope arguments take the form of sorites (or vagueness) lines of reasoning.  And sorite reasoning is good only when there is a restricted range of considerations.  When there are other variables, vagueness arguments stink.

Here’s Ron Ross, over at the American Spectator, on President Obama’s recent proposal to raise the minimum wage.

When I taught economics, when possible I liked to use the “Socratic method,” which is essentially teaching by asking questions. The Socratic method helps the student deduce the answer by using what he already knows.

Most people, especially college freshmen and sophomores, feel that minimum wage laws are beneficial. When discussing the topic I would ask, “If a minimum wage of $8 is better than one of $5, why skimp? Why not make the minimum wage $10, or $20, or $30?” Passing minimum wage laws is relatively easy. If eliminating poverty is that easy, why not go all the way? Why be so miserly? It’s not your money you’re spending. Go big or go home!

He takes it that this is a full-on reductio of minimum wage proposals.  Ross’s argument is classic sorite version of slippery slope.  Here’s how I’d reformulate it:

P1. (Fact of case evaluation): $5 an hour isn’t enough.

P2. (Principle of tolerance): If a wage isn’t enough, then if we add 1 cent an hour to the wage, the new wage still isn’t enough.

Once we accept P2, the pile quickly accumulates.  Iterate modus ponens 500 times, 5,000 times on the inputs and products of P1 and P2, and you end up with Ross’s conclusion. (On the assumption that P1 and P2 are true, all those MP iterations will be sound.)  Go big or go home.

As I take it, Ross’s conclusion is that we should, to prevent the pile, reject P1.  But I think liberals, to prevent the pile, reject P2.  That’s what the concept of living wage is supposed to be — that there is an economically determinable line one passes where the one cent an hour makes a difference between having enough to pay all the bills (and perhaps save a small amount) and not.   And that’s why they want to raise the minimum wage.  Running a vagueness argument misses the point.  Not surprised that Ross ran it on his college undergrads.  They must not have taken a good logic class yet.