Modal Logic For Computability
Kleene's original realizability interpretation has received much attention among those who study connections between computability and logic. It was extended to full higher-order intuitionistic logic by Martin Hyland in 1982 who constructed the effective topos. In 2002, Steven Awodey, Lars Birkedal, and Dana Scott formulated a modal logic for computability which extended the usual realizability interpretation with two modal operators expressing the notion of being "computably true".
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“Though living is a dreadful thing
And a dreadful thing is it
Life the niggard will not thank,
She will not teach who will not sing,
And what serves, on the final bank,
Our logic and our wit?”
—Philip Larkin (1922–1986)