Husserl and Hilbert on Completeness and Husserl\u27s Term Rewrite-based Theory of Multiplicity (Invited Talk)

Hilbert and Husserl presented axiomatic arithmetic theories in different ways and proposed two different notions of \u27completeness\u27 for arithmetic, at the turning of the 20th Century (1900-1901). The former led to the completion axiom, the latter completion of rewriting. We look into the latter in comparison with the former. The key notion to understand the latter is the notion of definite multiplicity or manifold (Mannigfaltigkeit). We show that his notion of multiplicity is understood by means of term rewrite theory in a very coherent manner, and that his notion of \u27definite\u27 multiplicity is understood as the relational web (or tissue) structure, the core part of which is a \u27convergent\u27 term rewrite proof structure. We examine how Husserl introduced his term rewrite theory in 1901 in the context of a controversy with Hilbert on the notion of completeness, and in the context of solving the justification problem of the use of imaginaries in mathematics, which was an important issue in the foundations of mathematics in the period

Remarks on logic for process descriptions in ontological reasoning: A Drug Interaction Ontology case study

We present some ideas on logical process descriptions, using relations from the DIO (Drug Interaction Ontology) as examples and explaining how these relations can be naturally decomposed in terms of more basic structured logical process descriptions using terms from linear logic. In our view, the process descriptions are able to clarify the usual relational descriptions of DIO. In particular, we discuss the use of logical process descriptions in proving linear logical theorems. Among the types of reasoning supported by DIO one can distinguish both (1) basic reasoning about general structures in reality and (2) the domain-specific reasoning of experts. We here propose a clarification of this important distinction between (realist) reasoning on the basis of an ontology and rule-based inferences on the basis of an expert’s view

Semantics for "Enough-Certainty" and Fitting\u27s Embedding of Classical Logic in S4

In this work we look at how Fitting\u27s embedding of first-order classical logic into first-order S4 can help in reasoning when we are interested in satisfaction "in most cases", when first-order properties are allowed to fail in cases that are considered insignificant. We extend classical semantics by combining a Kripke-style model construction of "significant" events as possible worlds with the forcing-Fitting-style semantics construction by embedding classical logic into S4. We provide various examples. Our main running example is an application to symbolic security protocol verification with complexity-theoretic guarantees. In particular, we show how Fitting\u27s embedding emerges entirely naturally when verifying trace properties in computer security

Inductive-data-type Systems

In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed lambda-calculus enriched by pattern-matching definitions following a certain format, called the "General Schema", which generalizes the usual recursor definitions for natural numbers and similar "basic inductive types". This combined language was shown to be strongly normalizing. The purpose of this paper is to reformulate and extend the General Schema in order to make it easily extensible, to capture a more general class of inductive types, called "strictly positive", and to ease the strong normalization proof of the resulting system. This result provides a computation model for the combination of an algebraic specification language based on abstract data types and of a strongly typed functional language with strictly positive inductive types.Comment: Theoretical Computer Science (2002

Phase semantic cut-elimination and normalization proofs of first- and higher-order linear logic

AbstractWe give a natural extension of Girard's phase semantic completeness proof of the (first order) linear logic Girard (Theoret. Comput. Sci., 1987) to a phase semantic cut-elimination proof. Then we extend this idea to a phase semantic cut-elimination proof for higher order linear logic. We also extend the phase semantics for provability to a phase semantics-like framework for proofs, by modifying the phase space of monoid domain to that of proof-structures (untyped proofs) domain, in a natural way. The resulting phase semantic-like framework for proofs provides various versions of proof-normalization theorem