Setting-up early computer programs: D. H. Lehmer's ENIAC computation

A complete reconstruction of Lehmer's ENIAC set-up for computing the exponents of p modulo two is given. This program served as an early test program for the ENIAC (1946). The reconstruction illustrates the difficulties of early programmers to find a way between a man operated and a machine operated computation. These difficulties concern both the content level (the algorithm) and the formal level (the logic of sequencing operations)

Goal Translation for a Hammer for Coq (Extended Abstract)

Hammers are tools that provide general purpose automation for formal proof assistants. Despite the gaining popularity of the more advanced versions of type theory, there are no hammers for such systems. We present an extension of the various hammer components to type theory: (i) a translation of a significant part of the Coq logic into the format of automated proof systems; (ii) a proof reconstruction mechanism based on a Ben-Yelles-type algorithm combined with limited rewriting, congruence closure and a first-order generalization of the left rules of Dyckhoff's system LJT.Comment: In Proceedings HaTT 2016, arXiv:1606.0542

Stable Normative Explanations: From Argumentation to Deontic Logic

This paper examines how a notion of stable explanation developed elsewhere in Defeasible Logic can be expressed in the context of formal argumentation. With this done, we discuss the deontic meaning of this reconstruction and show how to build from argumentation neighborhood structures for deontic logic where this notion of explanation can be characterised. Some direct complexity results are offered.Comment: 15 pages, extended version of the short paper accepted at JELIA 202

Consequences and Design in General and Transcendental Logic

In this article, I consider Kant’s dichotomy between general and transcendental logic in light of a retrospective reconstruction of two approaches originating in 14th century scholasticism that are used to demarcate formal and material consequences. The first approach (e. g., John Buridan, Albert of Saxony, Marsilius of Inghen) holds that a consequence is formal if it is valid — because of its form only — for any matter. Since the matter of a consequence is linked to categorematic terms, its formal validity is defined as being invariant under substitutions for such terms. According to the second approach (e. g., Richard Billingham, Robert Fland, Ralph Strode, Richard Lavenham), the validity of a formal consequence stems from the formal understanding of the consequent in the consequence’s antecedent. I put forward the hypothesis that in his logical taxo­nomy, Kant attempted to reconcile the substitutional interpretation of formal consequences and a formal analysis of the transcendental relations of objects of experience. However, if we interpret the limi­tations imposed by transcendental logic on the power of judgement in the spirit of the scholastic ontology of transcendental relations, it would contradict Kant’s critique of dogmatic ontology. Following in Luciano Floridi’s path, I thus propose to consider transcendental logic, not as a system of consequences equipped with ontologically grounded transcendental limitations, but rather as the logic of design. The logic of design has the benefit of enriching traditional logical tools with a series of notions borrowed primarily from computer programming. A conceptual system designer sets out feasibility requirements and defines a system’s functions that make it possible to achieve the desired outcome using available resources. Kant’s project forbids a dogmatic appeal to the transcendental relations and eternal truths of scholasticism. However, the constitutive nature of the rules of transcendental logic in regard to the power of judgement precludes the pluralism of conceptual systems that can be interpreted within possible experience. Thus, the optimisation problem of finding the best conceptual design from all feasible designs is beyond the competence of transcendental logic

Per Se Modality and Natural Implication – an Account of Connexive Logic in Robert Kilwardby

We present a formal reconstruction of the theories of the medieval logician Robert Kilwardby, focusing on his account of accidental and natural inferences and the underlying modal logic that gives rise to it. We show how Kilwardby’s use of an essentialist modality underpins his connexive account of implication

Topological Models of Columnar Vagueness

This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a “translation” of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfitt’s recent topological reconstruction of Sainsbury’s theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzien’s account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenfors’ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamson’s “logic of clarity” is explicated in terms of a generalized topology (“locology”) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamson’s logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a “slim boundary” are (stably) columnar. Thus, Williamson’s logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness

Proof Certificates for Equality Reasoning

International audienceThe kinds of inference rules and decision procedures that one writes for proofs involving equality and rewriting are rather different from proofs that one might write in first-order logic using, say, sequent calculus or natural deduction. For example, equational logic proofs are often chains of replacements or applications of oriented rewriting and normal forms. In contrast, proofs involving logical connectives are trees of introduction and elimination rules. We shall illustrate here how it is possible to check various equality-based proof systems with a programmable proof checker (the kernel checker) for first-order logic. Our proof checker's design is based on the implementation of focused proof search and on making calls to (user-supplied) clerks and experts predicates that are tied to the two phases found in focused proofs. It is the specification of these clerks and experts that provide a formal definition of the structure of proof evidence. As we shall show, such formal definitions work just as well in the equational setting as in the logic setting where this scheme for proof checking was originally developed. Additionally, executing such a formal definition on top of a kernel provides an actual proof checker that can also do a degree of proof reconstruction. We shall illustrate the flexibility of this approach by showing how to formally define (and check) rewriting proofs of a variety of designs

A one-valued logic for non-one-sidedness

Does it make sense to employ modern logical tools for ancient philosophy? This well-known debate2 has been re-launched by the indologist Piotr Balcerowicz, questioning those who want to look at the Eastern school of Jainism with Western glasses. While plainly acknowledging the legitimacy of Balcerowicz's mistrust, the present paper wants to propose a formal reconstruction of one of the well-known parts of the Jaina philosophy, namely: the saptabhangi, i.e. the theory of sevenfold predication. Before arguing for this formalist approach to philosophy, let us return to the reasons to be reluctant at it