A peculiar question that has divided philosophers since the Middle
Ages (and possibly earlier) concerns the status of Conditional
Excluded Middle. CEM says that for any propositions A and C, either
C would be the case if A were the case, or ¬C would be the case if A
were the case. Some accept CEM as valid, others reject it. In the last
few decades, the CEM supporters have been gaining ground, especially in
the field of semantics. See, for example, Williams 2010, Moss 2013,
Mandelkern
2018, Shaffer and Beebe 2019, Cariani and
Goldstein 2020, Marty, Romoli, and Santorio
2020, and Ramotowska et al.
2025.
There's a lively literature on how graded beliefs (credences) relate
to ungraded beliefs. Some hold that ungraded beliefs are reducible to
credences, others that credences are reducible to ungraded beliefs; yet
others hold that both are real and irreducible to one another. I find
this debate confusing. By which of course I mean: confused. I'll try to
explain why. I'll also comment on one particular proposal, that
credences are beliefs about probability; an interestingly new version of
this has just been defended in Buchanan and Dogramaci
2026.
First, why I find the debate confusing. Imagine a parallel debate
about desire. Desires patently vary in strength: we desire some things
more than others. In decision theory, an agent's desires are represented
by a utility (or "desirability") function that assigns numbers to
propositions; the higher the number, the stronger the desire. Now
imagine someone asks how these graded desires relate to ungraded
desires: are graded desires reducible to ungraded ones, or the other way
round?