Distributing intersection and union types with splits and duality (functional pearl)

Subtyping with intersection and union types is nowadays common in many programming languages. From the perspective of logic, the subtyping problem is essentially the problem of determining logical entailment : does a logical statement follow from another one? Unfortunately, algorithms for deciding subtyping and logical entailment with intersections, unions and various distributivity laws can be highly non-trivial. This functional pearl presents a novel algorithmic formulation for subtyping (and logical entailment) in the presence of various distributivity rules between intersections, unions and implications (i.e. function types). Unlike many existing algorithms which first normalize types and then apply a subtyping algorithm on the normalized types, our new subtyping algorithm works directly on source types. Our algorithm is based on two recent ideas: a generalization of subtyping based on the duality of language constructs called duotyping ; and splittable types , which characterize types that decompose into two simpler types. We show that our algorithm is sound, complete and decidable with respect to a declarative formulation of subtyping based on the minimal relevant logic B + . Moreover, it leads to a simple and compact implementation in under 50 lines of functional code.

Essence, Propria and Essentialist Explanation

In this paper I propose a notion of propria inspired by Aristotle, on which propria are non-essential, necessary properties explained by the essence of a thing. My proposal differs from the characterization of propria by Kit Fine and Kathrin Koslicki: unlike Fine, the relation of explanation on my account can’t be assimilated to a notion of logical entailment. In disagreement with Koslicki, I suggest that the explanatory relation at issue needs not be necessary. My account of essence is conceptually parsimonious: it illuminates the contribution of essence to explanation without relying on obscure notions such as Aristotelian form or identity.

Explanations for Query Answers under Existential Rules

Ontology-mediated query answering is an extensively studied paradigm, which aims at improving query answers with the use of a logical theory. As a form of logical entailment, ontology-mediated query answering is fully interpretable, which makes it possible to derive explanations for query answers. Surprisingly, however, explaining answers for ontology-mediated queries has received little attention for ontology languages based on existential rules. In this paper, we close this gap, and study the problem of explaining query answers in terms of minimal subsets of database facts. We provide a thorough complexity analysis for several decision problems associated with minimal explanations under existential rules.

Informalizing Formal Logic

This paper presents a way in which formal logic can be understood and reformulated in terms of argumentation that can help us unify formal and informal reasoning. Classical deductive reasoning will be expressed entirely in terms of notions and concepts from argumentation so that formal logical entailment is equivalently captured via the arguments that win between those supporting concluding formulae and arguments supporting contradictory formulae. This allows us to go beyond Classical Logic and smoothly connect it with human reasoning, thus providing a uniform argumentation-based view of both informal and formal logic.

Halevi, Judah (before 1075–1141)

Physician, philosopher and perhaps the greatest Hebrew poet since the Psalms, Judah Halevi studied the Neoplatonic Aristotelianism widespread in Islamic Spain, but his loyalty to Judaic traditions, love of Israel and poetic empathy for the sufferings and aspirations of his people made him a powerful critic of that philosophical tradition. His philosophical masterpiece, the Kuzari, is a fictional dialogue set at the court of the king of the Khazars, a people of the Volga basin whose leaders had converted to Judaism in the early ninth century. Reports of the Khazar realm sparked Halevi’s imagination and gave him the backdrop for this effort to celebrate and shape his ancestral faith. Beyond heartening his fellow Jews in times of upheaval, Halevi confronted philosophical questions that conventional thinkers often begged or ignored. He found the erudite Neoplatonism of his day too confining to God, too speculative and a priori. Tellingly, he condemns Neoplatonism for cultural vacuity, moral sterility and spiritual escapism: while Christians and Muslims, with the highest spiritual intentions, earnestly set about one another’s murder, the philosophers fail to differentiate one faith from another. What is needed, Halevi reasons, is not a still more spiritual intellectualism but a historically and geographically rooted tradition concretely directed by God’s love. Halevi did not, as romantics often suppose, simply turn his back on reason, or on philosophy generically. Rather, he used his own philosophical gifts and poetic tact to retune philosophy to the ground notes of Jewish experience. He retained but structurally adapted the Neoplatonic linkage of God to the world via emanation, replacing the elaborate hierarchy of star souls with the simple manifestation of God’s word, the ’Amr. Like Philo’s Logos, Halevi’s ’Amr was at once an attribute of God, his wisdom and a manifestation of God immanent in nature. Since the ’Amr is an imperative, it connotes power, volition and command, not just logical entailment or necessitation. Since it is immanent, it allows fuller appropriation than was possible for many philosophers and many of the pietists and mystics in their wake, of the material side of nature, including human nature: language, material culture – including agriculture and other economic activities – law and politics belong to realm of God’s expression. Particularity is not isolated from God. Poetry and works of imagination can be expressions of the divine, not just stepchildren mediating the ever more abstract and abtruse flights of the intellect. Zion could be acknowledged as the land where the divine afflatus was most clearly articulated as a way of life. Longing for Zion need no longer be sublimated in prayer; rather, Israel’s songs of longing for the robust life of the land of God’s grace would voice a spiritual imperative that demanded practical expression and historical realization.

A Reasoning System for a First-Order Logic of Limited Belief

Logics of limited belief aim at enabling computationally feasible reasoning in highly expressive representation languages. These languages are often dialects of first-order logic with a weaker form of logical entailment that keeps reasoning decidable or even tractable. While a number of such logics have been proposed in the past, they tend to remain for theoretical analysis only and their practical relevance is very limited. In this paper, we aim to go beyond the theory. Building on earlier work by Liu, Lakemeyer, and Levesque, we develop a logic of limited belief that is highly expressive but remains decidable in the first-order and tractable in the propositional case and exhibits some characteristics that make it attractive for an implementation. We introduce a reasoning system that employs this logic as representation language and present experimental results that showcase the benefit of limited belief.

Confirmation Theory

We first discuss several qualitative properties of confirmation as analyzed in a probabilistic framework. Some of these properties are classical, while others are relatively novel; some are shared by absolute and incremental confirmation, others are distinctive for each kind. We then proceed to address axiomatic characterizations of major classes of probabilistic measures of incremental confirmation. This treatment includes an original result displaying how conditions which single out the traditional probability difference measure up to ordinal equivalence. Finally, we argue that the longstanding project of a compelling confirmation-theoretic generalization of logical entailment (and refutation) can be achieved, provided that the right explicatum is adopted (to wit, a relative distance measure). This conclusion, we submit, dispels concerns that have been aired in the literature up to recent times.

The liberty of the moderns: Market freedom and democracy within the EU

Taking its cue from Benjamin Constant’s famous comparison of the liberty of the ancients with that of the moderns, this article examines the compatibility of democracy with free markets within the EU. Constant argued that commerce had replaced the political liberty of the ancients with the civil liberties of the moderns. Nevertheless, he contended a degree of political liberty remained necessary to guarantee these civil liberties. The difficulty was whether the political system could operate in the interest of all if modern citizens had ceased to identify with the public interest in the manner of the ancients and preferred to pursue their private interests. Constant believed representative democracy offered a form of political liberty that was compatible with modern liberty. It involved a less demanding view of civic virtue to ancient liberty and a different conception of the public interest as promoting rather than in conflict with private interests. However, for it to operate as Constant expected required certain social and cultural conditions that emerged in European nation states but are not themselves the products of commerce and may even be undermined by it: namely, a national identity; a social contract; and political parties. The EU involves a further deepening of modern commercial liberty beyond the nation state. This article explores three main issues raised by this development. First, have any of the three elements that facilitated the operation of representative democracy within the member states evolved at the EU level? Second, if not, is it possible to create an effective form of representative democracy on a post-national basis as the logical entailment of the liberties of the moderns? Third, if neither of these is possible, can we simply detach modern liberty from political liberty and see social rights as attributes of free movement, and efficient and equitable economic regulations as the products of technocratic governance? All three questions are answered in the negative.