̶
ISU
NIU
CS MS TBD

# Fun Math Stuff and the Philosopher's Stone #2

A sketch in continuity with a previous post about alternative formulations to the standard methods used to tackle Incompleteness, the Liar Paradox, and the wider families of Semantic Paradox.

Previously, I discussed two, fairly original, formal approaches to address worries surrounding Second Incompleteness:

1. Using something like a non-diagonal predicate (which is mostly the same as restricting impredicative or self-referential expressions for certain classes of sentences - on such an approach the blocking mechanism is located at the level of a specific predicate rather than a language-wide ban on a certain kind of syntax): fixed-points are blocked in the definition of the predicate.
2. Considering some basic relationships between two techniques: (i.) axiomatizations and (ii.) defining a predicate or operator.

A third approach is to define a predicate P% such that when it’s evaluated by an interpretation assignment I it generates a truth value distinct from its assigned value. Given:

1. I(S) = T
2. I(P%(S)) = T iff (I(C*(S)) = F and I(S) = T)
3. I(P%(S)) = F iff (I(C*(S)) = F and I(S) = F)

Such an approach "adjusts" the truth predicate so that it matches 1-to-1 with the underlying atomic truth assignment. (To be clear, this is a metalogical semantic rule that's used instead of the "vanilla/default" model-theoretic semantics - it adds the above conditions when evaluating the left-hand side). It would only be used in the context of a restrictionist formulization (e.g. - C* would be the truth-grounded predicate).

2. So, being a pluralist of some variety is irresistible. So, the question of which truth theory is relevant is delimited partly by that (e.g.- it’s not necessarily a question of which one it’s a question of which) - the solution I offered is amenable to the latter question since KF allows for all four perspectives: `t`, `f`, `n`, `tf`.