Judea Pearl pioneered erasing this view by inventing Bayesian Networks. More here: https://ftp.cs.ucla.edu/pub/stat_ser/r476.pdf
That said, I think "quantitative measures of likelihood that we can't easily reduce to probabilities" deserve study.
I mean, it seems like, in rough, "fuzzy" sort of way, neural network activations are such measures of likelihood and certain they are used.
I enjoyed reading Probabilistic Logic Networks (https://www.amazon.com/Probabilistic-Logic-Networks-Comprehe...) as it introduces a possible framework to sanely deal with different kinds of truth values about different things which aren't necessarily a single number.
https://en.wikipedia.org/wiki/Fuzzy_logic discusses axiomitizations.
It took quite a while but Kolmogorov and company actually formalized what is really meant by a "random sequence". But now you have a strong description of the real world behaviors are expected asymptotically if a probability is assigned to a behavior. As far as I can tell, all that's being axiomatized here is how the arbitrary quantities people make up for the fuzzy logic of things get manipulated. IE, there's not description of the relation of fuzzy logic and "reality" (because if there were, it would map to either regular logic or probability).
AFAIK, fuzzy logic, and application in fuzzy control systems among other areas, is still a very active thing.
I thoroughly enjoyed his "The Book of Why", a lay introduction to this subject.
It is worth saying that there are still situations where any use of quantitative probabilities becomes something of an abuse. The most extreme example is Pascal's Wager; if you can assign a "small but meaingful" probability to any X you happen to mention in the discussion, you can assign a probability to the existence of the Great Old Ones or the Flying Spaghetti Monster or whatever implausible entity is going to create a hypothetical action of negative utility sufficient to counter it's unlikeliness.
And, of course, acknowledging some stuff outside of the domain of probability means you need a fuzzy border between two realms, which also can't be determined by probability.
Still doesn't change the problem of worshiping the wrong god. Is it better to worship no god than worship the wrong god?
While I have no issue at all with assigning a probability to the inconsistency mathematics, the value I'd assign varies with the branch and mathematics. For Zermelo–Fraenkel set theory, for example, I'd assign a probability very very very close to 0, but not equal to zero (because to be equal to zero, I'd need a proof).
That doesn't follow.
2. A v B
3. ~A & A v B
We start with A.
The union (logical-or) of true statement (here A) and any other statement (say B) is a true statement, thus A v B.
Then we introduce another true statement, ~A, via logical-and to get ~A & (A v B), which simplifies (disjunctive syllogism) to just B.
So we have proved B, but B was arbitrary.
It could be anything, including the statement that x = y for x and y two ostensible non-equal numbers.
Yes, but Pascal used a big prior of his cultural upbringing and 2.5+ millennia of Jewish+Christian religion to consider just one God as plausible, not every random possible entity.