Learner-Adaptive Pedagogical Model in SIAL, an Open-Ended Intelligent Tutoring System for First Order Logic

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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GAMES AND CARDINALITIES IN INQUISITIVE FIRST-ORDER LOGIC

The Review of Symbolic Logic ◽

10.1017/s1755020321000198 ◽

2021 ◽

pp. 1-28

Author(s):

IVANO CIARDELLI ◽

GIANLUCA GRILLETTI

Keyword(s):

Order Logic ◽

First Order Logic ◽

First Order

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Extensions in graph normal form

Logic Journal of IGPL ◽

10.1093/jigpal/jzaa054 ◽

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.

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Relational Chase Procedures Interpreted as Resolution with Paramodulation1

Fundamenta Informaticae ◽

10.3233/fi-1991-15203 ◽

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.

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