## Combining Conjunction with Disjunction (2005)

Venue: | Proceedings of the 2nd Indian International Conference on Artificial Intelligence (IICAI 2005 |

Citations: | 3 - 3 self |

### BibTeX

@INPROCEEDINGS{Béziau05combiningconjunction,

author = {Jean-yves Béziau and Marcelo E. Coniglio},

title = {Combining Conjunction with Disjunction},

booktitle = {Proceedings of the 2nd Indian International Conference on Artificial Intelligence (IICAI 2005},

year = {2005},

pages = {1648--1658},

publisher = {IICAI}

}

### OpenURL

### Abstract

Abstract. In this paper we address some central problems of combination of logics through the study of a very simple but highly informative case, the combination of the logics of disjunction and conjunction. At first it seems that it would be very easy to combine such logics, but the following problem arises: if we combine these logics in a straightforward way, distributivity holds. On the other hand, distributivity does not arise if we use the usual notion of extension between consequence relations. A detailed discussion about this phenomenon, as well as some possible solutions for it, are given. 1

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Citation Context ..., it seems reasonable that a matrix semantics based on a non-distributive lattice could falsified the distributivity law. So the idea is to consider one of the two simplest non-distributive lattices (=-=[8]-=- pp. 75), knowing in particular that a lattice which is not distributive contains one of them as a sublattice. Building a matrix semantics with these non-distributive lattices means having to choose t... |

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Citation Context ...ing (1, ∧) (5) a ∧ (b ∨ c)), b ⊢ (a ∧ b) ∨ c from (4) using (1, ∨) (6) c ⊢ c (identity) (7) c ⊢ (a ∧ b) ∨ c from (6) using (2, ∨) (8) a ∧ (b ∨ c)), (b ∨ c) ⊢ (a ∧ b) ∨ c from (5) and (7) using (3, ∨) =-=(9)-=- (a ∧ (b ∨ c)) ⊢ (a ∧ b) ∨ c from (8) using (1, ∧) ⊓⊔ 7 The combination of the logic of conjunction and disjunction is not necessarily distributive In the preceding section, we have considered the log... |

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Citation Context ...formal framework for combining consequence relations by means of meta-translations. The importance of meta-properties (such as those considered above) in the analysis of logics was already studied in =-=[4]-=-. There is still another solution in the case of the logic of disjunction, which is to consider logics as multiple-conclusion consequence relations. In this case we can state the law (3, ∨) without go... |

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Citation Context ...ven by a structural Tarskian consequence relation over the set For which extend both LC and LD by means of meta-translations) is exactly the logic LCD obtained in Theorem 2, which is distributive. In =-=[10]-=- was defined a formal framework for combining consequence relations by means of meta-translations. The importance of meta-properties (such as those considered above) in the analysis of logics was alre... |

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Citation Context ...he logic of disjunction, taken also as a consequence relation, is necessary distributive. Theorem 2. The logic of conjunction and disjunction, taken as a consequence relation, is distributive.sProof. =-=(1)-=- a ⊢ a (identity) (2) b ⊢ b (identity) (3) a, b ⊢ (a ∧ b) from (1) and (2) using (3, ∧) (4) a ∧ (b ∨ c)), b ⊢ (a ∧ b) from (3) using (1, ∧) (5) a ∧ (b ∨ c)), b ⊢ (a ∧ b) ∨ c from (4) using (1, ∨) (6) ... |

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Citation Context ...kian notion of semantical consequence. This general definition encompasses any matrix semantics finite or notsand also any truth-functional semantics. Moreover, we have shown in a previous paper (see =-=[2]-=-) that, if we are working within the framework of Tarskian semantical consequence, any semantics can be reduced to such a concept of semantics. This means that the following result really shows that t... |

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Citation Context ...nt (Consequence operators or consequence relations), but the following problem arises: if we combine these logics in a straightforward way, distributivity holds. We have introduced in a previous work =-=[5]-=- the suggestive terminology “copulation paradox” to describe this phenomenon, because conjunction and disjunction are interacting between each other generating distributivity. This is a paradox and a ... |