scholarly journals Connecting a Logical Framework to a First-Order Logic Prover

2005 ◽  
pp. 285-301 ◽  
Author(s):  
Andreas Abel ◽  
Thierry Coquand ◽  
Ulf Norell
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Logical Framework ◽  
First Order

scholarly journals A universal inductive inference machine

1991 ◽  
Vol 56 (2) ◽  
pp. 661-672 ◽  
Author(s):  
Daniel N. Osherson ◽  
Michael Stob ◽  
Scott Weinstein
Keyword(s):  
Inductive Inference ◽  
Scientific Discovery ◽  
Order Logic ◽  
First Order Logic ◽  
Logical Framework ◽  
First Order

AbstractA paradigm of scientific discovery is defined within a first-order logical framework. It is shown that within this paradigm there exists a formal scientist that is Turing computable and universal in the sense that it solves every problem that any scientist can solve. It is also shown that universal scientists exist for no regular logics that extend first-order logic and satisfy the Löwenheim-Skolem condition.

scholarly journals Formalization Of Theories of Design Using: First Order Logic Augmented With A Causal Calculus; And Entropy of a Design Defined In Terms Of The Function, Behavior and Structure Ontology.

2021 ◽  
Author(s):  
KARTHIK GURUMURTHI
Keyword(s):  
High Probability ◽  
Formal Proof ◽  
Order Logic ◽  
First Order Logic ◽  
Logical Framework ◽  
Seamless Integration ◽  
Good Design ◽  
First Order ◽  
Individual Variables ◽  
And Behavior

A symbolic logical framework (L) consisting of first order logic augmented with a causal calculus has been provided to formalize, axiomatize and integrate theories of design. L is used to represent designs in the Function-Behavior-Structure (FBS) ontology in a single, widely applicable language that enables the following: seamless integration of representations of function, behavior and structure; and generality in the formalization of theories of design. FRs, constraints, structure and behavior are represented as sentences in L. FRs are represented (as abstractions of behavior) in the form of existentially quantified sentences, the instantiation of whose individual variables yields the representation of behavior. This enables the logical implication of FRs by behavior, without recourse to apriori criteria for satisfaction of FRs by behavior. Functional decomposition is represented to enable lower level FRs to logically imply the satisfaction of higher level FRs. The theory of whether and how structure and behavior satisfy FRs and constraints is represented as a formal proof in L. Important general attributes of designs such as solution-neutrality of FRs, probability of satisfaction of requirements and constraints (calculated in a Bayesian framework using Monte Carlo simulation), extent and nature of coupling, etc. have been defined in terms of the representation of a design in L. The entropy of a design is defined in terms of the above attributes of a design, based on which a general theory of what constitutes a good design has been formalized to include the desirability of solution-neutrality of (especially higher level) FRs, high probability of satisfaction of requirements and constraints, wide specifications, low variability and bias, use of fewer attributes to specify the design, less coupling (especially circular coupling at higher levels of FRs), parametrization, standardization, etc..

SKEPTICAL ABDUCTION

1993 ◽  
Vol 02 (04) ◽  
pp. 511-540 ◽  
Author(s):  
P. MARQUIS
Keyword(s):  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Real Problem ◽  
Logical Framework ◽  
First Order ◽  
Selective Generation ◽  
The Given ◽  
Reasonable Prospect

Abduction is the process of generating the best explanation as to why a fact is observed given what is already known. A real problem in this area is the selective generation of hypotheses that have some reasonable prospect of being valid. In this paper, we propose the notion of skeptical abduction as a model to face this problem. Intuitively, the hypotheses pointed out by skeptical abduction are all the explanations that are consistent with the given knowledge and that are minimal, i.e. not unnecessarily general. Our contribution is twofold. First, we present a formal characterization of skeptical abduction in a logical framework. On this ground, we address the problem of mechanizing skeptical abduction. A new method to compute minimal and consistent hypotheses in propositional logic is put forward. The extent to which skeptical abduction can be mechanized in first—order logic is also investigated. In particular, two classes of first-order formulas in which skeptical abduction is effective are provided. As an illustration, we finally sketch how our notion of skeptical abduction applies as a theoretical tool to some artificial intelligence problems (e.g. diagnosis, machine learning).

Equivalences between logics and their representing type theories

1995 ◽  
Vol 5 (3) ◽  
pp. 323-349 ◽  
Author(s):  
Philippa Gardner
Keyword(s):  
Natural Deduction ◽  
Order Logic ◽  
First Order Logic ◽  
Logical Framework ◽  
Simple Formulation ◽  
First Order ◽  
Relevant Logics ◽  
First Time ◽  
New Framework

We propose a new framework for representing logics, called LF+, which is based on the Edinburgh Logical Framework. The new framework allows us to give, apparently for the first time, general definitions that capture how well a logic has been represented. These definitions are possible because we are able to distinguish in a generic way that part of the LF+ entailment corresponding to the underlying logic. This distinction does not seem to be possible with other frameworks. Using our definitions, we show that, for example, natural deduction first-order logic can be well-represented in LF+, whereas linear and relevant logics cannot. We also show that our syntactic definitions of representation have a simple formulation as indexed isomorphisms, which both confirms that our approach is a natural one and provides a link between type-theoretic and categorical approaches to frameworks.

First-Order Logic Reasoning Support for the Semantic Web

2009 ◽  
Vol 19 (12) ◽  
pp. 3091-3099 ◽  
Author(s):  
Gui-Hong XU ◽  
Jian ZHANG
Keyword(s):  
Semantic Web ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Logic Reasoning

Boolean-valued structures

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
Mathematical Concepts ◽  
Semantic Framework ◽  
First Order ◽  
Standard Models ◽  
Boolean Valued Model ◽  
Semantic Values

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.

scholarly journals Extensions in graph normal form

2020 ◽  
Author(s):  
Michał Walicki
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Special Cases

Abstract Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.

Relational Chase Procedures Interpreted as Resolution with Paramodulation1

1991 ◽  
Vol 15 (2) ◽  
pp. 123-138
Author(s):  
Joachim Biskup ◽  
Bernhard Convent
Keyword(s):  
Alternative Interpretation ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Problem ◽  
First Order ◽  
Inference Problems ◽  
The One ◽  
The Relationship ◽  
Theoretical Foundations ◽  
Insight Into

In this paper the relationship between dependency theory and first-order logic is explored in order to show how relational chase procedures (i.e., algorithms to decide inference problems for dependencies) can be interpreted as clever implementations of well known refutation procedures of first-order logic with resolution and paramodulation. On the one hand this alternative interpretation provides a deeper insight into the theoretical foundations of chase procedures, whereas on the other hand it makes available an already well established theory with a great amount of known results and techniques to be used for further investigations of the inference problem for dependencies. Our presentation is a detailed and careful elaboration of an idea formerly outlined by Grant and Jacobs which up to now seems to be disregarded by the database community although it definitely deserves more attention.

scholarly journals Formula size games for modal logic and μ-calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1311-1344 ◽  
Author(s):  
Lauri T Hella ◽  
Miikka S Vilander
Keyword(s):  
Modal Logic ◽  
Optimal Strategy ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
First Order ◽  
Modal Operators ◽  
Crucial Difference ◽  
Definition Of ◽  
Person Game

Abstract We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\textrm{FO}$ and (basic) modal logic $\textrm{ML}$. We also present a version of the game for the modal $\mu $-calculus $\textrm{L}_\mu $ and show that $\textrm{FO}$ is also non-elementarily more succinct than $\textrm{L}_\mu $.

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