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:
- 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.
- 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:
- I(S) = T
- I(P%(S)) = T iff (I(C*(S)) = F and I(S) = T)
- 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).
Also, to clarify my Graham Priest reply to his very helpful criticism, musing, and question: "but isn't there still truth simplicater?"
- There are self-evidently many truth-predicates, whether this implies many truth-properties is a separate issue. Trivially, even the Predicate defined by the T-schema (the relevant one to the Liar debate apparently - at least insofar as it's the only conception that's been rigorously defended) can be broken into three distinct concepts (capture, release, biconditional).
- 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:
- Does there being multiple truth-predicates imply that there are multiple truth-properties? I think not. There's a deep question about how truth-predicates map to truth-properties (traditional worries about realism, epistemic access, representation, the limits of language, etc.). There's less of a question about the relationship between truth-predicates since their character is revealed through axioms - truth-predicates can typically be neatly mapped using familiar tools like lattices or the edifices of Kripke Frames/Possible World Semantics (how truth works at world, across worlds, via axioms and proof, KTB + S4, KT without S5, etc.).
- Are there wider questions about the nature of truth-values? What are they and how do they map to predicates? Presumably, they are 1-to-1 but the system I originally defended allows them to diverge into the scenario at the end of the point above. Defenders of the analyticity of the plain ole T-Schema tend to prefer this feature - that truth-values and predicates are 1-to-1. And, many seem to tacitly believe that such a characteristic of their conception implies that it is the correct conception. (That they don't diverge lends itself to truth monism and the traditional realist conception of truth: correspondance.)
- My previous articulation is more or less the same as the above but I’m drawing out some finer points a bit.
- So, the main thing perspective is that logic is a cognitive project subject to epistemic warrant and ensuring ongoing and helpfully iterative revision (Cantor broke math, Frege broke logic temporarily, Russell and Whitehead created non-logical monstrosities since they presuppose non-logical axioms and hence all their proving was non-logical that were nevertheless intellectually astounding, ZFC is questionably though fairly probably consistent). Logic proceeds through revision and change. Truth is part of logic and so is also revised over time. Is the concept correct, relatedly is the formulation even? (Will future eras look back at our crass formal tools and think us foolish.) So, the view I endorse is a placeholder (hopefully a best one).