It is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered:

1. There exist sentences that are neither paradoxical nor grounded.

2. There are fixed points.

3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n ∣ A(n) is true in the minimal fixed point}, where A(x) is a formula of AR + T) are precisely the sets.

4. In the minimal fixed point the totally defined sets (sets weakly defined by formulae all of whose instances are true or false) are precisely the sets.

5. The closure ordinal for Kripke's construction of the minimal fixed point is .

In contrast, we show that under the weak Kleene scheme, depending on the way the Gödel numbering is chosen:

1. There may or may not exist nonparadoxical, ungrounded sentences.

2. The number of fixed points may be any positive finite number, ℵ0, or .

3. In the minimal fixed point, the sets that are weakly definable may range from a subclass of the sets 1-1 reducible to the truth set of AR to the sets, including intermediate cases.

4. Similarly, the totally definable sets in the minimal fixed point range from precisely the arithmetical sets up to precisely the sets.

5. The closure ordinal for the construction of the minimal fixed point may be ω, , or any successor limit ordinal in between.

In addition we suggest how one may supplement AR + T with a function symbol interpreted by a certain primitive recursive function so that, irrespective of the choice of the Gödel numbering, the resulting language based on the weak Kleene scheme has the five features noted above for the strong Kleene language.

In monomodal logic there are a fair number of high-powered results on completeness covering large classes of modal systems; witness for example Fine [74], [85] and Sahlqvist [75]. Monomodal logic is therefore a well-understood subject in contrast to polymodal logic, where even the most elementary questions concerning completeness, decidability, etc. have been left unanswered. Given that in many applications of modal logic one modality is not sufficient, the lack of general results is acutely felt by the “users” of modal logics, contrary to logicians who might entertain the view that a deep understanding of one modality alone provides enough insight to be able to generalize the results to logics with several modalities. Although this view has its justification, the main results we are going to prove are certainly not of this type, for they require a fundamentally new technique. The results obtained are called transfer theorems in Fine and Schurz [91] and are of the following type. Let L ∌ ⊥ be an independently axiomatizable bimodal logic and L⎕ and L∎ its monomodal fragments. Then L has a property P iff L⎕ and L∎ have P. Properties which will be discussed are completeness, the finite model property, compactness, persistence, interpolation and Halldén-completeness. In our discussion we will prove transfer theorems for the simplest case when there are just two modal operators, but it will be clear that the proof works in the general case as well.