A.H. Lachlan’s research while affiliated with Simon Fraser University and other places

... One of the most important facts about the pure compositional truth theory CT − , shown by Lachlan in [12], is that the underlying arithmetical part of any model of CT − is recursively saturated. This theorem imposes a very important limitation on any possible conservativeness proof for pure CT − : in general, we cannot hope to simply patch together some locally defined fragments of a compositional truth predicate in a coherent manner. ...

Reference:

Saturation properties for compositional truth with propositional correctness

... There is also a notable methodological asymmetry between inductive satisfaction classes and full satisfaction classes: given an arbitrary model M of PA it is routine to construct an elementary extension of M that carries an inductive satisfaction class with the help of the standard tools of the trade (the ingredients are definable partial satisfaction classes, and compactness). However, an air of mystery has surrounded full satisfaction classes ever since the original tour de force Kotlarski-Krajewski-Lachlan construction [23] that was based on the ad hoc technology of M-logic (an infinitary logical system based on a nonstandard model M). The Kotlarski-Krajewski-Lachlan construction implies that the axiomatic theory of truth commonly known as CT -[PA] = CT -+ PA is conservative over PA, i.e., if an arithmetical sentence ϕ is provable in CT - [PA], then ϕ is already provable in PA. ...

Reference:

SATISFACTION CLASSES WITH APPROXIMATE DISJUNCTIVE CORRECTNESS