**Abstract:**

In this talk, we draw together two strands of our recent research. The

first, joint work with David Miller (Warwick), concerns the history of

dual-intuitionistic logic. The second aim is to develop a

non-monotonic version of the constructive logic that is known as Nelson

logic.

Somewhat surprisingly, the history of dual intuitionistic logic is, in a

sense, as old as the history of intuitionistic logic itself. It goes

back to the famous prize question Mannoury asked and which Heyting

answered by axiomatizing intuitionistic logic. While this axiomatization

was clearly the main result the prize question asked for, it also

explicitly addressed the possibility of constructing a dual system, a

sub-task Heyting was unable to accomplish (we will disclose who should

be the winner of this part of the competition in the talk).

In an attempt to speculate on the question why Mannoury was interested

in such a dualization in the first place, we turn to the main

interpretation that has been offered for dual intuitionistic logic as a

logic of falsification. We quickly review some arguments against dual

intuitionistic logic as a satisfactory logic of falsification from

Kapsner's book "Logics and Falsifications" (Springer, 2014) and the case

made therein for versions of Nelson logic as an alternative for this

task.

From there, we present the philosophical motivation to move even further

away from dual intuitionistic logic, towards a non-monotonic version of

the Nelson systems. To this end, we present an adaptive logic that

combines the explosive (N3) and the paraconsistent (N4) version of

Nelson logic. For each specific set of premises, the adaptive logic

looks for an optimum between N3 and N4. This is realized by allowing the

law of non-contradiction only where it does not lead to triviality in

view of the premises. As a result, the adaptive logic combines N3’s

inferential strength with N4’s ability to cope with inconsistencies. In

the talk we will give a short general introduction to adaptive logics

and show how these ideas can be applied to Nelson logic. We wrap up the

talk by arguing why this adaptive logic approach is useful for the

project of defining a logic of falsification.