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Suppose you have a revealed preference analysis of preference. Note that I do NOT NOT NOT mean a revealed preference theory of evidence about preferences, but the idea that observed choice behaviour is what preference is. In that case, Completeness holds trivially, provided the choices in question are in fact made.
However, if you understand preferences as mental attitudes, then it is perfectly possible that someone does not have an attitude such that (1) they prefer A to B, (2) they prefer B to A, or (3) they are indifferent (in the technical sense) between A and B. For example, Duncan Luce did experiments that found that, under some conditions, people's choices in apparently repeatable choice-situations fluctuated probabilistically. IIRC, they preferred A or B to a random choice between the two, indicating that this was not indifference. Now, it's possible that they were interpreting those choice-situations as non-repeatable, but it could also be that their preferences with respect to A and B don't form a strict ordering.
There's no basis in decision theory or mathematics for that claim, AFAIK.
There is a cool literature on imprecise probabilities you might like to look at:
https://plato.stanford.edu/entries/imprecise-probabilities/
I haven't read any applications of this approach to Pascal's Wager, but since IP is arguably a more realistic model of human psychology than maximising expected utility (which assumes unique additive probabilities) someone should definitely do that.
Me too! I don't want to dox myself, but I think that Bayesian decision theory is similar to things like General Equilibrium Theory, neoclassical capital, and other concepts in economics, in that they can be useful tools to make decisions given idealised assumptions, but they shouldn't be taken too literally. Like any scientific model, their value comes not from their approximation of truth, but because of empirical and formal properties they possess, e.g. track-records and approximations of relevant features in the world (empirical) and tractability/computability properties (formal).
For more about the topics raised in the last paragraph in my comment above, the Stanford Encyclopedia page is pretty good - written by a very good young philosopher, Seamus Bradley, who has done other work on the topic worth reading. Much of the great work in this literature, e.g. by John Maynard Keynes, Teddy Seidenfeld, Peter Walley, Clark Glymour, Henry E. Kyburg, and Isaac Levi, is extremely technical, even for decision theorists. The SEP page covers most of their ideas at a more accessible level, sometimes in its appendix. The best young guy in the field is probably Richard Pettigrew, who has done some magnificent work that has still not been incorporated into the broader Bayesian consciousness, e.g. https://philarchive.org/rec/PETWIC-2 (see this video for a relatively easy introduction to that paper - https://youtube.com/watch?v=1W_wgQpZF2A ).
I find this topic very interesting, because (like you, I think?) I see Pascal's Wager (or something like it) as the best current case for religious belief. I actually like Arthur Balfour's variation of this type of reasoning, which avoids some of the features of Pascal's arguments that are awkward, e.g. regarding infinite expected payoffs:
https://archive.org/details/adefencephiloso01balfgoog/page/n12/mode/2up?view=theater
Here's John Passmore's summary of Balfour's position, from One Hundred Years of Philosophy (1968):
Basically, the idea would be that, at least assuming a common human nature, it is prudentially rational to believe in God, because one is permitted to do so in the absence of a refutation; there is no refutation of God's existence; and one can expect better consequences from such belief.
Whether that reasoning is sound is one of the most important questions of philosophy, in my view, and it's brought me very deep into epistemology/decision theory. Balfour's starting place is Hume, but Subjective Bayesianism (either with precise or imprecise probabilities) seems very apt for such reasoning. Indeed, on a Subjective Bayesian view, I don't think there is any reason to think that theism is less rational than belief in even our most supported scientific theories.
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