## On Extensions of Intermediate Logics by Strong Negation (1996)

Venue: | Journal of Philosophical Logic |

Citations: | 13 - 0 self |

### BibTeX

@ARTICLE{Kracht96onextensions,

author = {Marcus Kracht},

title = {On Extensions of Intermediate Logics by Strong Negation},

journal = {Journal of Philosophical Logic},

year = {1996},

volume = {27},

pages = {49--73}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we will study the properties of the least extension n() of a given intermediate logic by a strong negation. It is shown that the mapping from to n() is a homomorphism of complete lattices, preserving and reflecting finite model property, frame--completeness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n(). This summarizes results that can be found already in [13, 14] and [4]. Furthermore, we determine the structure of the lattice of extensions of n(LC). Introduction Constructive logic is an extension of intuitionistic logic by another connective, the strong negation. 1 Basically, this additional connective is motivated by the fact that we can not only verify a simple proposition such as This door is locked. by direct inspection, but also falsify it. An intuitionist is forced to say that the falsity of this sentence is seen only indirectly, namely by seeing that it is impossible for this s...

### Citations

1499 | The stable model semantics for logic programming
- Gelfond, Lifschitz
- 1988
(Show Context)
Citation Context ...ich is so characteristic in logic programming. The present essay has been sparked off by a question raised by David Pearce concerning the so--called answer set semantics of Gelfond and Lifschitz, see =-=[2, 3]-=-. In [8], Pearce shows that the constructive logic of the two--element chain is a deductive base for the nonmonotonic logic derived from stable models `a la Gelfond & Lifschitz. While the knowledge ab... |

336 |
Logic programs with classical negation
- Gelfond, Lifschitz
(Show Context)
Citation Context ...ich is so characteristic in logic programming. The present essay has been sparked off by a question raised by David Pearce concerning the so--called answer set semantics of Gelfond and Lifschitz, see =-=[2, 3]-=-. In [8], Pearce shows that the constructive logic of the two--element chain is a deductive base for the nonmonotonic logic derived from stable models `a la Gelfond & Lifschitz. While the knowledge ab... |

274 | A course in universal algebra
- Burris, Sankappanavar
- 1981
(Show Context)
Citation Context ... congruence distributive if the lattice of congruences of any algebra from that variety is distributive. Any variety containing at least the two lattice operationssandsis congruence distributive (see =-=[1]-=-). Hence the variety of Nelson algebras is congruence distributive. J' onsson has proved that in such varieties the subdirectly irreducible members of HSP(K) are already Marcus Kracht, On Extensions o... |

75 | Constructible falsity - Nelson - 1949 |

26 |
Reasoning with Negative Information I: Strong Negation
- Pearce, Wagner
- 1990
(Show Context)
Citation Context ...low ourselves some relaxed use of terminology. 1 Marcus Kracht, On Extensions of Intermediate Logics by Strong Negation 2 Constructive logic has attracted attention in logic programming recently, see =-=[9, 10]-=- and [5]. Its main advantage is in admitting the possibility of a direct statement of the falsity of a proposition while retaining the `negation as failure', which is so characteristic in logic progra... |

21 |
Logic programming with strong negation
- Pearce, Wagner
- 1989
(Show Context)
Citation Context ...low ourselves some relaxed use of terminology. 1 Marcus Kracht, On Extensions of Intermediate Logics by Strong Negation 2 Constructive logic has attracted attention in logic programming recently, see =-=[9, 10]-=- and [5]. Its main advantage is in admitting the possibility of a direct statement of the falsity of a proposition while retaining the `negation as failure', which is so characteristic in logic progra... |

3 |
The craig interpolation theorem for propositional logics with strong negation
- Goranko
- 1985
(Show Context)
Citation Context ...leteness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n(). This summarizes results that can be found already in [13, 14] and =-=[4]-=-. Furthermore, we determine the structure of the lattice of extensions of n(LC). Introduction Constructive logic is an extension of intuitionistic logic by another connective, the strong negation. 1 B... |

3 |
Disjunctive logic programming, constructivity and strong negation
- Herre, Pearce
- 1992
(Show Context)
Citation Context ...s some relaxed use of terminology. 1 Marcus Kracht, On Extensions of Intermediate Logics by Strong Negation 2 Constructive logic has attracted attention in logic programming recently, see [9, 10] and =-=[5]-=-. Its main advantage is in admitting the possibility of a direct statement of the falsity of a proposition while retaining the `negation as failure', which is so characteristic in logic programming. T... |

1 |
Craig's Theorem in superintionistic logics and amalgamable varieties of Pseudo--boolean algebras
- Maksimova
- 1977
(Show Context)
Citation Context ...lation ( i!n q i ! :p i ): ! :OE !s[~p; ~q] 2s: A fortiori we have ( i!n q i $ :p i ): ! :OE !s[~p; ~q] 2s: From this we conclude that OE !s[~p; :~p] = OE !s2 . Likewise it is shown thats! / 2 . a In =-=[6]-=- it was shown that there are exactly seven intermediate logics with interpolation. (By definition, intermediate logics are consistent. The inconsistent logic has interpolation as well. Also, for the t... |

1 |
A New Characterization of Stable Models
- Pearce
- 1995
(Show Context)
Citation Context ...haracteristic in logic programming. The present essay has been sparked off by a question raised by David Pearce concerning the so--called answer set semantics of Gelfond and Lifschitz, see [2, 3]. In =-=[8]-=-, Pearce shows that the constructive logic of the two--element chain is a deductive base for the nonmonotonic logic derived from stable models `a la Gelfond & Lifschitz. While the knowledge about inte... |