Nominal terms extend first-order terms with binding. They lack some properties of first- and higher-order terms: Terms must be reasoned about in a context of ‘freshness assumptions’; it is not always possible to ‘choose a fresh variable symbol’ for a nominal term; it is not always possible to ‘α-convert a bound variable symbol’ or to ‘quotient by α-equivalence’; the notion of unifier is not based just on substitution.Permissive nominal terms closely resemble nominal terms but they recover these properties, and in (...) particular the ‘always fresh’ and ‘always rename’ properties. In the permissive world, freshness contexts are elided, equality is fixed, and the notion of unifier is based on substitution alone rather than on nominal terms’ notion of unification based on substitution plus extra freshness conditions.We prove that expressivity is not lost moving to the permissive case and provide an injection of nominal terms unification problems and their solutions into permissive nominal terms problems and their solutions.We investigate the relation between permissive nominal unification and higher-order pattern unification. We show how to translate permissive nominal unification problems and solutions in a sound, complete, and optimal manner, in suitable senses which we make formal. (shrink)

Nominal terms extend first-order terms with binding. They lack some properties of first- and higher-order terms: Terms must be reasoned about in a context of ‘freshness assumptions’; it is not always possible to ‘choose a fresh variable symbol’ for a nominal term; it is not always possible to ‘α-convert a bound variable symbol’ or to ‘quotient by α-equivalence’; the notion of unifier is not based just on substitution. Permissive nominal terms closely resemble nominal terms but they recover these properties, and (...) in particular the ‘always fresh’ and ‘always rename’ properties. In the permissive world, freshness contexts are elided, equality is fixed, and the notion of unifier is based on substitution alone rather than on nominal terms’ notion of unification based on substitution plus extra freshness conditions. We prove that expressivity is not lost moving to the permissive case and provide an injection of nominal terms unification problems and their solutions into permissive nominal terms problems and their solutions. We investigate the relation between permissive nominal unification and higher-order pattern unification. We show how to translate permissive nominal unification problems and solutions in a sound, complete, and optimal manner, in suitable senses which we make formal. (shrink)

Why is there something rather than nothing? This form of the metaphysical question adopted by Heidegger has become quite popular. And that is very understandable; for in a world where man has it within his power to destroy the whole human race, the contingency of creatures is brought home to us with brutal clarity. Strictly speaking, however, the question is metaphysical nonsense: I may ask why I exist, why others exist, but not why being, reality, exists—it has no meaning. An (...) absolute must exist. The real question is: what kind is this absolute? Can we get any real knowledge about it, and if so, how? (shrink)

Evil, like the poor, we have always with us—because it is perennial and universal, it presents a problem to be solved, a mystery to be probed. Like the great mystery of being—of which it is but an aspect—the mystery of evil has engaged the minds of thinking men down through the ages. Charles Journet is one of the most recent to grapple with it. His book is a very complete study—philosophical and theological—of all the various aspects of this question: the (...) nature of evil, the different kinds of evil, God and evil, punishment for evil, evil in history. He bases his work on the teaching of two great Doctors, St Augustine and St Thomas. Of the moderns, his preference goes, as he expressly declares to Jacques Maritain whose works contain ‘the most penetrating teaching on evil written in our own times from the Christian viewpoint’. The result is a very clear and penetrating exposition of the most classical and traditional teaching. (shrink)