The concept of truth values is sometimes expressed in terms of "truth as an object vs. truth as a property." My in-a-slogan understanding of this alternative is "sentences being predicated of truth vs. truth being predicated of sentences."
My question is fairly simple: can we intelligibly speak of justification values, also? That is, of sentences being predicated of justification vs. justification being predicated of sentences? Or sentences' reference being their justification, then.
Motivation: again, this idea came to me while I was (over the last two or so years) studying set theory. After reading up on proof-theoretic ordinal analysis, I hit upon the idea of assigning a numerical value to a sentence's justification. This allowed me to devise what seems to be a novel defense of the initial axiom of infinity, as well as of large cardinal axioms: let S stand for a sentence and 𝔧(S) for the "justification function" that takes S for input and outputs a number representing the degree of justification this sentence has. Much as we assign 0 to falsity and 1 to truth in that context, I had it that 𝔧(S) = 0 means that S is completely unjustified and 𝔧(S) = 1 means that S is completely justified. And of course, we can imagine values between 0 and 1 being outputs, here. Now, however, what if we suppose that 𝔧(S) might output numbers less than 0 or greater than 1? Then there would be antijustified and hyperjustified S. So: let the initial axiom of infinity be expressed (this is rather impredicative, but it will have to do) as ∃S(𝔧(S) = ω), i.e. "there exists a sentence such that its justification value is the shortest infinite ordinal," and the sentence in question is ∃ω itself.† What about larger cardinal axioms? Why, if we are infinitely hyperjustified in asserting the existence of the smallest infinity, imagine that ∃S(𝔧(S) = ω1), ∃S(𝔧(S) = ω2), etc. Indeed, if the template of our justification, here, is to say that higher and higher infinities make higher and higher infinite hyperjustification possible, it seems that the larger the large cardinal axiom at issue, the larger its potential hyperjustification, which idea stands on its head the whole question of whether axioms of infinity are justifiable as such. (Note that the finitist cannot say that the axioms of infinity are infinitely antijustified, as the negative infinite surreal numbers like -ω presuppose their positive ordinal counterparts.)
What I'm not sure about: whether this talk of justification numbers is the same as the idea of justification values. Is the association of 0 and 1 with falsity and truth a matter of the theory of truth values? This seems like it might be the case: the SEP article on truth-values mentions that these are relevant to the notion of vagueness, and fuzzy logic (where there are real-valued truth values) handles vagueness fairly well, after all.
†There is no infelicity in this talk of, "There exists a sentence..." Gödel in a famous passage (forgive me, I have no idea about the exact reference right now) says something like, "There might exist axioms..." So my turn-of-phrase here is apropos.
