Type does not play a special role in Integral Relational Logic other than to liberate the art and science of reason from the theory of types, which Bertrand Russell introduced in an attempt to eliminate the fundamental law of the Universe from human inquiries.
He did so after paradoxes were found in set theory, in the foundations of mathematics, spending many years, some with A. N. Whitehead, writing the indigestible Principia Mathematica. Yes, if the axioms of mathematics are contradictory, you can prove anything from them.
But rather than create a view of the world, as we would like it to be, we need to develop a radically new approach to mathematics to see the world, as it is. For, we live in an inherently paradoxical Universe. So, if we ignore self-contradictions, we are inevitably led into delusion.
We can give mathematics, and hence the entire body of human knowledge, a sound foundation, by turning the mechanistic horizontal dimension of time into the creative vertical dimension, arising directly from the Divine Origin of the Universe.
By thus invoking Self-reflective Intelligence to understand what causes us to behave as we do, self-referencing statements, like ‘This sentence is false’, are no longer a problem. They are simply expressions of the Cosmic Equation, as the primal axiom of human learning.
Type can then be viewed as a synonym of class, a primal concept in the system of thought we all use every day to form concepts and organize our ideas. By thus taking the abstractions of mathematics to the utmost level of generality in a meta-algebra, we can solve the ultimate problem of human learning, as Panosophy or the Unified Relationships Theory.
About 1470, ‘symbol, emblem’, from Latin typus ‘figure (on a wall), image; pattern, model, symbol; form, kind’, from Greek tupos ‘stroke, blow; impression, stamp, mark; figure, image, statue; sketch, model, type; system, character’, from root of typtein ‘to strike, beat; poke, stab’, from PIE base *tup-, variant of root *(s)teu- ‘to push, stick, knock, beat’.
“The meaning of the general form or character of some kind, class, or group, is first recorded in English in 1843, in John Stuart Mill’s System of Logic, and that of kind, class, or group having common characteristics, in 1854, both senses evolving from the original sense in English and also from the same senses in Latin and Greek.” (Barnhart)
The verb type was first used in 1596 to mean ‘to foreshadow as a type, prefigure’ in theology. The meaning ‘to symbolize, typify’ is from 1836 and ‘to write with a typewriter’ is from 1888.