scholarly journals 4. Reasoning with Partial Knowledge

2002 ◽  
Vol 32 (1) ◽  
pp. 133-181 ◽  
Author(s):  
László Pólos ◽  
Michael T. Hannan
Keyword(s):  
Logical Consequence ◽  
Order Logic ◽  
First Order Logic ◽  
Partial Knowledge ◽  
Age Dependence ◽  
Consequence Relation ◽  
First Order ◽  
Correct Execution ◽  
Organizational Mortality ◽  
Monotonicity Principle

We investigate how sociological argumentation differs from classical first-order logic. We focus on theories about age dependence of organizational mortality. The overall pattern of argument does not comply with the classical monotonicity principle: Adding premises overturns conclusions in an argument. The cause of nonmonotonicity is the need to derive conclusions from partial knowledge. We identify metaprinciples that appear to guide the observed sociological argumentation patterns, and we formalize a semantics to represent them. This semantics yields a new kind of logical consequence relation. We demonstrate that this new logic can reproduce the results of informal sociological theorizing and lead to new insights. It allows us to unify existing theory fragments, and it paves the way toward a complete classical theory. Observed inferential patterns which seem “wrong” according to one notion of inference might just as well signal that the speaker is engaged in correct execution of another style of reasoning. —Johan van Benthem (1996)

Frege’s Begriffsschrift Theory of Identity Vindicated

2019 ◽  
pp. 122-147 ◽  
Author(s):  
Ulrich Pardey ◽  
Kai F. Wehmeier
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Consequence Relation ◽  
First Order ◽  
Michael Dummett

In the Begriffsschrift, Frege held that identity is a relation between names, to wit, the relation of co-reference. The verdict of Frege scholarship on this conception of identity has not been favorable, to say the least; indeed, commentators including Alonzo Church, Michael Dummett, and Richard Heck have claimed that it is incompatible with ordinary first-order quantification. We show that these commentators are mistaken. The Begriffsschrift conception of identity is perfectly consistent with ordinary quantification over objects, and moreover generates the same consequence relation between sentences as does standard first-order logic with identity.

scholarly journals A Knowledge Level Account of Forgetting

2017 ◽  
Vol 60 ◽  
pp. 1165-1213 ◽  
Author(s):  
James P. Delgrande
Keyword(s):  
Belief Change ◽  
Description Logics ◽  
Knowledge Bases ◽  
Order Logic ◽  
First Order Logic ◽  
Consequence Relation ◽  
First Order ◽  
Formal Systems ◽  
High Level ◽  
Main Definition

Forgetting is an operation on knowledge bases that has been addressed in different areas of Knowledge Representation and with respect to different formalisms, including classical propositional and first-order logic, modal logics, logic programming, and description logics. Definitions of forgetting have been expressed in terms of manipulation of formulas, sets of postulates, isomorphisms between models, bisimulations, second-order quantification, elementary equivalence, and others. In this paper, forgetting is regarded as an abstract belief change operator, independent of the underlying logic. The central thesis is that forgetting amounts to a reduction in the language, specifically the signature, of a logic. The main definition is simple: the result of forgetting a portion of a signature in a theory is given by the set of logical consequences of this theory over the reduced language. This definition offers several advantages. Foremost, it provides a uniform approach to forgetting, with a definition that is applicable to any logic with a well-defined consequence relation. Hence it generalises a disparate set of logic-specific definitions with a general, high-level definition. Results obtained in this approach are thus applicable to all subsumed formal systems, and many results are obtained much more straightforwardly. This view also leads to insights with respect to specific logics: for example, forgetting in first-order logic is somewhat different from the accepted approach. Moreover, the approach clarifies the relation between forgetting and related operations, including belief contraction.

scholarly journals A Tractable, Expressive, and Eventually Complete First-Order Logic of Limited Belief

2019 ◽  
Author(s):  
Gerhard Lakemeyer ◽  
Hector J. Levesque
Keyword(s):  
Knowledge Representation ◽  
Expressive Language ◽  
Logical Consequence ◽  
Logical Reasoning ◽  
Research Topic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order

In knowledge representation, obtaining a notion of belief which is tractable, expressive, and eventually complete has been a somewhat elusive goal. Expressivity here means that an agent should be able to hold arbitrary beliefs in a very expressive language like that of first-order logic, but without being required to perform full logical reasoning on those beliefs. Eventual completeness means that any logical consequence of what is believed will eventually come to be believed, given enough reasoning effort. Tractability in a first-order setting has been a research topic for many years, but in most cases limitations were needed on the form of what was believed, and eventual completeness was so far restricted to the propositional case. In this paper, we propose a novel logic of limited belief, which has all three desired properties.

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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