ALL the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain.  
David Hume: An Enquiry Concerning Human Understanding, SECTION IV Part I 
HyperRationality
a standard of rationality for
 

What is HyperRationality?  

Hyperrationality adds to the analytic/synthetic distinction a modern appreciation of just how hard it is to use deductive reasoning in a coherent way.
Since Russell's discovery of the inconsistency of Frege's logical foundations there have been many other demonstrations of how easy it is for experts to put together formal logical systems which turn out to be inconsistent.
Despite the difficulties, it is reasonable to say that we do now know how to do it. Hyperrationalism is essentially the refusal to countenance deductive reasoning in any context which does not meet modern standards for ensuring logical consistency. Though the basic principles are now well understood, this cannot in practice be done without machine support much better than is currently available. This is one of the reasons why hyperrationality is put forward as a standard for the globally networked computers of the future, rather than for us people in the present. Once you start worrying seriously about consistency, the best framework for deductive reasoning is a foundation system. This is because a foundation system is designed to take all the risks about consistency off your hands. 
A foundation system incorporates sufficient ontological premises that the development of elaborate theories (such as are required in mathematics) can be undertaken using only extensions to the logical system which are known to be conservative, and hence not to compromise consistency (in Frege's day the talk was of definitions, but conservative extension is a more general approach).
Hyperrationality involves the view that no claim which has not been proven in the context of a logical foundation system (and which is therefore known with a high degree of confidence to be analytic and necessary) can be asserted without risk of incoherence, and therefore, hyperrational agents refuse to assert any other statements. They are nevertheless able to make affirmations which have the highest relevance to science, engineering and commerce. For example, a hyperrational agent will happily support the construction of a complete formal model of any coherent TOE together with a definition of suitable initial conditions for the universe and then to affirm demonstratively a claim about what the Universe will do if it starts in the initial state and conforms to the TOE. With suitable methods, formal analytic truths such as can be hyperrationally asserted, suffice for a large tranche of mathematics science and engineering. 