In classic use the term "hypothesis" marked propositions from which deductions were to be made.
In particular, the set \ \Omega\), on which the preference ordering \ \preceq\) is defined, has to be an atomless Boolean algebra of propositions, from which the impossible propositions, denoted \ \bot\), have been removed.
The first edition of Book 3 begins with a list of nine hypotheses that were clearly intended to be propositions from which the deductions of the Book were to be made, and not theoretical conjectures.
An axiom is simply an indemonstrable proposition from which other propositions may be deduced.
What's required before an a priori entitlement can be recognized cannot be that one has to entertain some prior knowledge of a proposition from which the entitlement can be deduced.
This is the second task of a logic of sense: "to combine the sterility of sense in relation to the proposition from which it was extracted with its power of genesis in relation to the dimensions of the proposition" (LS 32).
If some such view is correct, it might avoid certain (unpleasant) questions prompted by earlier Fregean views: how can a sentence indicate a proposition with a different structure; and if grammar is deeply misleading, why think that our intuitions concerning impeccability provide reliable evidence about which propositions follow from which?
By the supreme moral law Bolzano means "a practical truth [i.e., a true ought proposition] from which every other practical truth can be derived objectively, i.e., as the consequence from its 'ground'" (RW I, 228; cf. also RW I, 44, 244, 256, RW IV, 27, 217, 221, and WL II, 348, WL IV, 178).
Let us say that one has inferential or nonfoundational knowledge that p when one's knowledge that p depends on one's knowledge of some other proposition(s) from which one can legitimately infer p; and one has foundational or noninferential knowledge that p when one's knowledge that p does not depend on any other knowledge one has in this way.
Roughly, one has nonfoundational or inferential knowledge that p when one's knowledge that p depends on one's knowledge of some other proposition(s) from which one can legitimately infer p; and one has foundational or noninfrential knowledge that p when one's knowledge that p does not depend on any other knowledge one has.
Let us say that one has an inferentially justified belief that p when one's belief that p depends for its justification on justified belief in some other proposition(s) from which one can legitimately infer p; and one has a foundationally or noninferentially justified belief that p when one's belief that p is justified but does not depend on any other beliefs for its justification.
