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.

scholarly journals Property Preserving Embedding of First-order Logic

10.29007/18t1 ◽  
2020 ◽  
Author(s):  
Julian Parsert ◽  
Stephanie Autherith ◽  
Cezary Kaliszyk
Keyword(s):  
Training Evaluation ◽  
Logical Reasoning ◽  
Order Logic ◽  
First Order Logic ◽  
Data Sets ◽  
Training Set ◽  
First Order ◽  
Vector Representations

Logical reasoning as performed by human mathematicians involves an intuitive under- standing of terms and formulas. This includes properties of formulas themselves as well as relations between multiple formulas. Although vital, this intuition is missing when supplying atomically encoded formulae to (neural) down-stream models.In this paper we construct continuous dense vector representations of first-order logic which preserve syntactic and semantic logical properties. The resulting neural formula embeddings encode six characteristics of logical expressions present in the training-set and further generalise to properties they have not explicitly been trained on. To facilitate training, evaluation, and comparing of embedding models we extracted and generated data sets based on TPTP’s first-order logic library. Furthermore we examine the expressiveness of our encodings by conducting toy-task as well as more practical deployment tests.

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)

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.

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