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...

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...

The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order predicate logic. In this paper we prove that the lattice of lambda theories is not modular and that the variety generated by the term algebra of a semi-sensible lambda theory is not congruence modular. A...

Analyzing coverage of a student’s utterance or essay (completeness) and diagnosing errors (correctness) can be treated as a diagnosis problem and solved using a well-known technique for model-based diagnosis: an assumption-based truth maintenance system (ATMS). The function-free first-order predicate logic (FOPL) representation of the essay is matched with nodes of the ATMS that are then analyz...

We observe that removing contraction from a standard sequent calculus for first-order predicate logic preserves completeness for the modal fragment.