1,441 research outputs found
Predicativity, the Russell-Myhill Paradox, and Church's Intensional Logic
This paper sets out a predicative response to the Russell-Myhill paradox of
propositions within the framework of Church's intensional logic. A predicative
response places restrictions on the full comprehension schema, which asserts
that every formula determines a higher-order entity. In addition to motivating
the restriction on the comprehension schema from intuitions about the stability
of reference, this paper contains a consistency proof for the predicative
response to the Russell-Myhill paradox. The models used to establish this
consistency also model other axioms of Church's intensional logic that have
been criticized by Parsons and Klement: this, it turns out, is due to resources
which also permit an interpretation of a fragment of Gallin's intensional
logic. Finally, the relation between the predicative response to the
Russell-Myhill paradox of propositions and the Russell paradox of sets is
discussed, and it is shown that the predicative conception of set induced by
this predicative intensional logic allows one to respond to the Wehmeier
problem of many non-extensions.Comment: Forthcoming in The Journal of Philosophical Logi
Towards a unified theory of intensional logic programming
AbstractIntensional Logic Programming is a new form of logic programming based on intensional logic and possible worlds semantics. Intensional logic allows us to use logic programming to specify nonterminating computations and to capture the dynamic aspects of certain problems in a natural and problem-oriented style. The meanings of formulas of an intensional first-order language are given according to intensional interpretations and to elements of a set of possible worlds. Neighborhood semantics is employed as an abstract formulation of the denotations of intensional operators. Then we investigate general properties of intensional operators such as universality, monotonicity, finitariness and conjunctivity. These properties are used as constraints on intensional logic programming systems. The model-theoretic and fixpoint semantics of intensional logic programs are developed in terms of least (minimum) intensional Herbrand models. We show in particular that our results apply to a number of intensional logic programming languages such as Chronolog proposed by Wadge and Templog by Abadi and Manna. We consider some elementary extensions to the theory and show that intensional logic program clauses can be used to define new intensional operators. Intensional logic programs with intensional operator definitions are regarded as metatheories
Intensional Logic and Topology
This thesis is concerned with mathematical logic, in particular it is an investigation of a branch of mathematical logic called modal logic. This branch of mathematical logic extends the propositional calculus by adding two unary operators â–¡ and 0 to the standard set of logical operators. This extension of classical logic has many interpretations; traditionally it is said to be the logic of necessity, denoted by the box operator, and possibility, denoted by the diamond operator. The notion of necessity within modal logic is ubiquitous and lends itself to a vast sea of metaphysics. For example, if X is necessarily true, denoted O X , then it is said to be true in all possible worlds. This way of understanding modalities gave imputes for a semantics that provided fodder for the first completeness proofs in modal logic.
Modalities in logic have its roots in philosophy and dates back as far as Aristotle’s M etaphysics, but was brought into the limelight with the work of the philosopher mathematician Saul Kripke who in 1959, as a high school student, published the first completeness proof for a class of modal logics [Kripke]. His method used the so-called semantic-tableaux which was introduced by B eth’s The foundations of mathematics to obtain quick completeness proof for the propositional and predicate calculus. In this thesis, we are also interested in completeness for modal logics, but will use a more modern method known in the.literature as canonical model constructions . Moreover, we wish to provide a semantics for modal logics that is not the traditional possible world semantics. Our models will be topological in nature. Our goal is to provide a completeness proof for a particular modal logic called S4 which interprets the modal operators as the interior and closure operators on topological spaces. We will also prove that the logic S4 is complete with respect to the class of transitive and reflexive trees. This gives us two new completeness proof for the modal logic S4
Reasoning About a Simulated Printer Case Investigation with Forensic Lucid
In this work we model the ACME (a fictitious company name) "printer case
incident" and make its specification in Forensic Lucid, a Lucid- and
intensional-logic-based programming language for cyberforensic analysis and
event reconstruction specification. The printer case involves a dispute between
two parties that was previously solved using the finite-state automata (FSA)
approach, and is now re-done in a more usable way in Forensic Lucid. Our
simulation is based on the said case modeling by encoding concepts like
evidence and the related witness accounts as an evidential statement context in
a Forensic Lucid program, which is an input to the transition function that
models the possible deductions in the case. We then invoke the transition
function (actually its reverse) with the evidential statement context to see if
the evidence we encoded agrees with one's claims and then attempt to
reconstruct the sequence of events that may explain the claim or disprove it.Comment: 18 pages, 3 figures, 7 listings, TOC, index; this article closely
relates to arXiv:0906.0049 and arXiv:0904.3789 but to remain stand-alone
repeats some of the background and introductory content; abstract presented
at HSC'09 and the full updated paper at ICDF2C'11. This is an updated/edited
version after ICDF2C proceedings with more references and correction
Formally Specifying and Proving Operational Aspects of Forensic Lucid in Isabelle
A Forensic Lucid intensional programming language has been proposed for
intensional cyberforensic analysis. In large part, the language is based on
various predecessor and codecessor Lucid dialects bound by the higher-order
intensional logic (HOIL) that is behind them. This work formally specifies the
operational aspects of the Forensic Lucid language and compiles a theory of its
constructs using Isabelle, a proof assistant system.Comment: 23 pages, 3 listings, 3 figures, 1 table, 1 Appendix with theorems,
pp. 76--98. TPHOLs 2008 Emerging Trends Proceedings, August 18-21, Montreal,
Canada. Editors: Otmane Ait Mohamed and Cesar Munoz and Sofiene Tahar. The
individual paper's PDF is at
http://users.encs.concordia.ca/~tphols08/TPHOLs2008/ET/76-98.pd
- …