For weakest implicit belief, I think I would have nominated "That I have the slightest idea how to properly calculate the probability of the mugger following through on his/her threat".

Also, Torture vs. Specks seems like another instance where many of us are willing to sacrifice apparent consistency. Most coherent formulations of utilitarianism must choose torture, yet many utilitarians are hesitant to do so.

In both cases, it seems like what we're doing isn't abandoning consistency, but admitting to the possibility that our consistent formula (e.g. naive utilitarianism) isn't necessarily the optimal / subjectively best / most reflectively equilibrial one. We therefore may choose to abandon it in favor of the intuitive answer (don't pay the mugger, choose specks, etc), not because we choose to be inconsistent, but because we predict the existence of a Better But Still Consistent Formula not yet known to us.

Of course, as Yvain notes, we can take pretty much any set of arbitrary preferences and create a "consistent" formula by adding enough terms to the equation. The difference is that the Better But Unknown formula above is both consistent and something we'd be in reflective equilibrium about.