It seems odd that something which is meant to mirror the rules of good reasoning should radically change and evolve over time: to most, it seems natural to think there is a fairly intuitive way of telling valid from invalid, sound from unsound, and logic, once it can express this, is a satisfactory science without any inner tensions.
In the history of philosophy, however, this seems to have not been the case: even during the age of Antiquity, Aristotle’s systematised logic and the Stoic theories could seem incompatible. Although Aristotle seemed to dominate the philosophical field for years, most philosophers digress from his logic in some way or another; yet, it is only during the late 20th and 21st centuries that the whole category of non-classical logics – intuitionistic, modal, many-valued, relevance logic – attracted full attention and gained academic recognition.
It is within this atmosphere that ‘HYPE’, a logical system with new logical operators, whose name comes from ‘Hyperintensionality’, began to develop. Hyperintensional Logic is the concept in which there are situations where, for example, substituting a true part for an equivalent one in a true sentence makes it false (‘Oedipus wants to marry the queen’ versus ‘Oedipus wants to marry his mother’!), and ‘HYPE’ was recently created by Hannes Leitgeb at the Munich Centre of Mathematical Philosophy. What is significantly useful about this system is that it can be used as a general framework in which different logical systems can be studied, compared, and combined. This may help logicians immensely when trying to capture the most fine-grained and subtle aspects of our language and reasoning in a systematic manner.
Philosophy applicants could reflect on these new developments, on the one hand contemplating the philosophical meaning and purpose of formalised logic, and on the other, understanding the formal problems internal to old and new logical systems. Those applying for Mathematics could also find interest in this topic, considering the implications of these logical systems as applied to mathematical problems, as well as giving thought to the contributions mathematical methods can make in the advancement of logic.