Abstract
Reasoning in the presence of inconsistencies and in the absence of complete knowledge has long been a major challenge in artificial intelligence. In this paper, we revisit the classical semantics of propositional logic by generalising the notion of world (valuation) so that it allows for propositions to be both true and false, and also for their truth values not to be defined. We do so by adopting neither a many-valued stance nor the philosophical view that there are ‘real’ contradictions. Moreover, we show that satisfaction of complex sentences can still be defined in a compositional way. Armed with our semantic framework, we define some basic notions of semantic entailment generalising the classical one and analyse their logical properties. We believe our definitions can serve as a springboard to investigate more refined forms of non-classical entailment that can meet a variety of applications in knowledge representation and reasoning.
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Notes
- 1.
We could, in principle, also have used
in the definition of satisfaction, but here we shall adopt the (possibly superfluous) stance that truth of a fact within the ‘actual’ world and truth of a sentence at given worlds are notions sitting at different levels, or, at the very least, are notions of subtly different kinds. - 2.
We do not rule out the remaining combinations; space and time constraints prevent us from assessing them here.
in the definition of satisfaction, but here we shall adopt the (possibly superfluous) stance that truth of a fact within the ‘actual’ world and truth of a sentence at given worlds are notions sitting at different levels, or, at the very least, are notions of subtly different kinds.References
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Acknowledgements
I would like to thank the anonymous referees for their comments and helpful suggestions. This work was partially supported by the National Research Foundation (NRF) of South Africa.
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Varzinczak, I. (2020). An Exercise in a Non-classical Semantics for Reasoning with Incompleteness and Inconsistencies. In: Gerber, A. (eds) Artificial Intelligence Research. SACAIR 2021. Communications in Computer and Information Science, vol 1342. Springer, Cham. https://doi.org/10.1007/978-3-030-66151-9_16
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