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.