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Many of the criticisms mentioned in the above comments have in fact been addressed by Bartley in his conception of pan-critical rationalism. See his book "The Retreat to Commitment".

Bayesian methods can be considered useful within such an epistemological system, however one cannot justify that one fact is more true than another merely based on Bayesian probabilities.

Both justificationist and falsificationist outlooks are stated with respect to something else. That is why philosophers played all those language games. They soon realised that you couldn't reduce everything down to language, be it natural or symbolic without losing something. Axiomatic systems don't make any sense on their own. It is sad that many commit to a justificationist position without realising that they are doing so.