In this paper we present a powerful uniform first-order framework for representing and reasoning with complex forms of default knowledge. This is achieved by extending first-order predicate logic with a new generalized quantifier, anchored in the quasi-probabilistic ranking measure paradigm [Weydert 94], which subsumes and refines the original, propositional notion of a default conditional [Del...

There is a range of modelling notations, both textual and diagrammatic, whose semantics are based on first-order predicate logic. This paper presents a visual framework for organizing models of systems which allows a mixture of notations, diagrammatic or textbased, to be used. The framework is based on the use of templates for connective operations which can be nested and sometimes flattened. I...

Regular path expressions represent the navigation core of the XPath query language for semi-structured data (XML), and it has been characterized as the First Order Logic with Two Variables (FO). Data tests refers to (dis)equality comparisons on data tree models, which are unranked trees with two kinds of labels, propositions from a finite alphabet, and data values from a possibly infinite alpha...

W-JS is a first-order predicate calculus on the modal theory of knowledge. It is based on natural deduction rules and accompanied by possible-world-accessibility semantics. As an example, the famous "Mr. S and Mr. P" puzzle is solved in W-JS.

We introduce two logics for partiality, a call-by-value logic and a call-by-name logic. Both logics are variants of the logic of partial terms, an extension of the first-order predicate calculus by a definedness predicate. In our logics, however, quantifiers may only be instantiated to value terms and not to arbitrary, defined terms as in the logic of partial terms. We show that the call-by-val...

A comparison of Landin’s form of lambda calculus with Church’s shows that, independently of the lambda calculus, there exists a mechanism for converting functions with arguments indexed by variables to the usual kind of function where the arguments are indexed numerically. We call this the “lambda mechanism” and show how it can be used in other calculi. In first-order predicate logic it can be ...

Permissive-Nominal Logic (PNL) extends first-order predicate logic with term-formers that can bind names in their arguments. It takes a semantics in (permissive-)nominal sets. In PNL, the ∀-quantifier or λ-binder are just term-formers satisfying axioms, and their denotation is functions on nominal atoms-abstraction. Then we have higher-order logic (HOL) and its models in ordinary (i.e. ZermeloF...

We present an extension of first-order predicate logic with a novel predicate ‘at t’ meaning intuitively “this term is a variable symbol”. We give simple sequent proof-rules for it, we demonstrate cut-elimination for the resulting logic, and we give a semantics for which the logic is sound and complete. Because we can now make assertions about what would normally be considered an intensional pr...