Expanding Belnap: dualities for a new class of default bilattices

Bilattices provide an algebraic tool with which to model simultaneously knowledge and truth. They were introduced by Belnap in 1977 in a paper entitled How a computer should think. Belnap argued that instead of using a logic with two values, for ‘true’ (t) and ‘false’ (f), a computer should use a logic with two further values, for ‘contradiction’ (⊤) and ‘no information’ (⊥). The resulting structure is equipped with two lattice orders, a knowledge order and a truth order, and hence is called a bilattice. Prioritised default bilattices include not only values for ‘true’ (t), ‘false’ (f), ‘contradiction’ and ‘no information’, but also indexed families of default values, t₁, ⋯ , t_n and f₁, ⋯ , f_n, for simultaneous modelling of degrees of knowledge and truth. We focus on a new family of prioritised default bilattices: J_n, for n∈ ω. The bilattice J is precisely Belnap’s seminal example. We obtain a multi-sorted duality for the variety [InlineEquation not available: see fulltext.] generated by J_n, and separately a single-sorted duality for the quasivariety [InlineEquation not available: see fulltext.] generated by J_n. The main tool for both dualities is a unified approach that enables us to identify the meet-irreducible elements of the appropriate subuniverse lattices. Our results provide an interesting example where the multi-sorted duality for the variety has a simpler structure than the single-sorted duality for the quasivariety.