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Propositional logic
Published in Richard E. Neapolitan, Xia Jiang, Artificial Intelligence, 2018
Richard E. Neapolitan, Xia Jiang
Notice in the previous example that A ⇒ B and ¬A ∨ B have the same truth value in every possible world. This is a general result concerning logical equivalence, which we state in the following theorem.
Fuzzy intensional semantics
Published in Journal of Applied Non-Classical Logics, 2018
Intensional semantics was developed to provide a solution to the problem the history of which goes back (at least) to Frege: namely, to formalise the distinction between the meaning and denotation of linguistic expressions. It allows to represent an essential linguistic phenomenon that in different situations (or states, contexts, etc.), the same expression can have different truth values. The most widespread frameworks formalise this phenomenon using the notion of possible world. A (logical) expression has, in general, different truth values in different possible worlds; the truth value at a particular possible world is called an extension (at that possible world), while the set of worlds in which it is true is called an intension of that expression.
Indefinite abductive explanations
Published in Journal of Applied Non-Classical Logics, 2019
Luciano Caroprese, Ester Zumpano
Since a possible world W consists of definite atoms only, it can be regarded as a knowledge base and we can talk about truth values with respect to W. Due to the absence of indefinite facts in W, each atom is either true (if it belongs to ) or false (otherwise). Extending the notation we introduced earlier, for a possible world W and we write if and , otherwise.
A note on the complexity of S4.2
Published in Journal of Applied Non-Classical Logics, 2021
Aggeliki Chalki, Costas D. Koutras, Yorgos Zikos
Normal modal logics are interpreted over relational possible-worlds models, also called Kripke models. A Kripke model consists of a non-empty set of worlds (or states), a binary accessibility relation and a valuation which maps each propositional variable into the set of possible worlds which validate it. Inside each possible world, truth assignment to formulae formed with propositional connectives is recursively defined in the obvious way. For modal formulae: is true at a world s of a model (notation: ) iff ϕ is true at every world ‘seen’ by s via the accessibility relation : Validity in a model and validity in a class of models, are defined in the obvious way. The pair of a model is called a (relational, Kripke) frame. A logic is sound wrt a class of frames iff any model built on a frame from validates all theorems of . A logic is complete wrt a class of frames iff every formula valid through is a theorem of or equivalently, iff every non-theorem of can be refuted in a model built on a frame from . A normal modal logic is determined or characterised by a class of frames iff it is sound and complete wrt this class.