## A paraconsistent higher order logic (2004)

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Venue: | INTERNATIONAL WORKSHOP ON PARACONSISTENT COMPUTATIONAL LOGIC, VOLUME 95 OF ROSKILDE UNIVERSITY, COMPUTER SCIENCE, TECHNICAL REPORTS |

Citations: | 5 - 5 self |

### BibTeX

@INPROCEEDINGS{Villadsen04aparaconsistent,

author = {Jørgen Villadsen},

title = {A paraconsistent higher order logic},

booktitle = {INTERNATIONAL WORKSHOP ON PARACONSISTENT COMPUTATIONAL LOGIC, VOLUME 95 OF ROSKILDE UNIVERSITY, COMPUTER SCIENCE, TECHNICAL REPORTS},

year = {2004},

pages = {33--49},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledge-based systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order logic with countable infinite indeterminacy, where each basic formula can get its own indeterminate truth value (or as we prefer: truth code). The meaning of the logical operators is new and rather different from traditional many-valued logics as well as from logics based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine. Thus we try to build a bridge between the HOL and MVL communities. A sequent calculus is proposed based on recent work by Muskens.

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Citation Context ...ce there are many ways to change the meaning of these operators there are many different paraconsistent logics. We present a paraconsistent higher order logic ∇ based on the (simply) typed λ-calculus =-=[18,4]-=-. Although it is a generalization of ̷Lukasiewicz’s three-valued logic the meaning of the logical operators is new, but with relations to logics based on bilattices [13,22,7,9,10,25]. One advantage of... |

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Citation Context ...toms are not — for the sake of simplicity — allowed to be inconsistent. We also use the operator � in the “exclusion rule” of expert I; we discuss some variants later. The operator � is a S5 modality =-=[29]-=-. A formalization is as follows with Di for disease-i, Si for symptom-i, J for John and M for Mary: S1x ∧ S2x → D1x S1x ∧ S3x → D2x �(D1x ⊳⊲ D2x) S1x ∧ S4x → D1x ¬S1x ∧ S3x → D2x �S1J �¬S2J �S3J �S4J ... |

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Citation Context ...tures, functions and relations are available (for instance arithmetic). Another advantage is that there are several automated theorem provers for classical higher order logic, e.g. HOL [24], Isabelle =-=[33]-=-, and it should be possible to modify these to our paraconsistent logic. We are inspired by the notion of indeterminacy as discussed by Evans [19]. Even though the higher order logic ∇ is paraconsiste... |

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Citation Context ...ce there are many ways to change the meaning of these operators there are many different paraconsistent logics. We present a paraconsistent higher order logic ∇ based on the (simply) typed λ-calculus =-=[18,4]-=-. Although it is a generalization of ̷Lukasiewicz’s three-valued logic the meaning of the logical operators is new, but with relations to logics based on bilattices [13,22,7,9,10,25]. One advantage of... |

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Citation Context ...y generation. 8 Logical Semantics of Natural Language A paramount application of higher order logic is natural language semantics, in particular in the Montague grammar tradition of logical semantics =-=[35,17]-=-, where the grammar and meaning of natural language sentences are defined and the logical consequences of the sentences must in the end be tested against our intuition. A set of sentences provides a m... |

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Citation Context ...hematical structures, functions and relations are available (for instance arithmetic). Another advantage is that there are several automated theorem provers for classical higher order logic, e.g. HOL =-=[24]-=-, Isabelle [33], and it should be possible to modify these to our paraconsistent logic. We are inspired by the notion of indeterminacy as discussed by Evans [19]. Even though the higher order logic ∇ ... |

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Citation Context ...s, like ∇∆, are classical. We reuse the symbols ∇ and ∆ later for related purposes. We also propose a sequent calculus for the paraconsistent higher order logic ∇ based on the seminal work by Muskens =-=[32]-=-. In the sequent Θ ⊢ Γ we understand Θ as a conjunction of a set of formulas and Γ as a disjunction of a set of formulas. We use Θ � Γ as a shorthand for Θ, ω ⊢ Γ, where ω is an axiom which provides c... |

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Citation Context ...he (simply) typed λ-calculus [18,4]. Although it is a generalization of ̷Lukasiewicz’s three-valued logic the meaning of the logical operators is new, but with relations to logics based on bilattices =-=[13,22,7,9,10,25]-=-. One advantage of a higher order logic is that the logic is very expressive in the sense that most mathematical structures, functions and relations are available (for instance arithmetic). Another ad... |

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Citation Context ...y generation. 8 Logical Semantics of Natural Language A paramount application of higher order logic is natural language semantics, in particular in the Montague grammar tradition of logical semantics =-=[35,17]-=-, where the grammar and meaning of natural language sentences are defined and the logical consequences of the sentences must in the end be tested against our intuition. A set of sentences provides a m... |

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Citation Context ...e the axiom: ∆x The ∆ axiom is equivalent to ∇-non-existence, namely ∄x. ∇x, and with the axiom ∆ we extend ∇ to the classical propositional higher order logic ∇∆ which was thoroughly investigated in =-=[27,2]-=-. Finally we can combine the extensions ∇∆ and ∇ ι into the classical higher order logic ∇ ι ∆ , also known as Q0 based on the typed λ-calculus, and often seen as a restriction of the transfinite type... |

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Citation Context ...he (simply) typed λ-calculus [18,4]. Although it is a generalization of ̷Lukasiewicz’s three-valued logic the meaning of the logical operators is new, but with relations to logics based on bilattices =-=[13,22,7,9,10,25]-=-. One advantage of a higher order logic is that the logic is very expressive in the sense that most mathematical structures, functions and relations are available (for instance arithmetic). Another ad... |

22 | The Logical Foundations of Mathematics - Hatcher - 1984 |

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Citation Context ...= � and D2M = �� give �� (and also D1M = �� and D2M = � give �). 7 A Sequent Calculus 7.1 Preliminaries We base the paraconsistent higher order logic ∇ on the (simply) typed λ-calculus [18] (see also =-=[11]-=-, especially for the untyped λ-calculus and for the notion of combinators which we use later). Classical higher-order logic is often built from a very few primitives, say equality = and the selection ... |

18 |
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Citation Context ...classical higher order logic, e.g. HOL [24], Isabelle [33], and it should be possible to modify these to our paraconsistent logic. We are inspired by the notion of indeterminacy as discussed by Evans =-=[19]-=-. Even though the higher order logic ∇ is paraconsistent some of its extensions, like ∇∆, are classical. We reuse the symbols ∇ and ∆ later for related purposes. We also propose a sequent calculus for... |

15 |
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Citation Context ...we can combine the extensions ∇∆ and ∇ ι into the classical higher order logic ∇ ι ∆ , also known as Q0 based on the typed λ-calculus, and often seen as a restriction of the transfinite type theory Q =-=[3]-=- by removing the transfinite types. Q0 is implemented in several automated theorem provers with many active users [24,33]. Classical second order logic, first order logic, elementary logic (first orde... |

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Citation Context ...he (simply) typed λ-calculus [18,4]. Although it is a generalization of ̷Lukasiewicz’s three-valued logic the meaning of the logical operators is new, but with relations to logics based on bilattices =-=[13,22,7,9,10,25]-=-. One advantage of a higher order logic is that the logic is very expressive in the sense that most mathematical structures, functions and relations are available (for instance arithmetic). Another ad... |

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Citation Context ...e the axiom: ∆x The ∆ axiom is equivalent to ∇-non-existence, namely ∄x. ∇x, and with the axiom ∆ we extend ∇ to the classical propositional higher order logic ∇∆ which was thoroughly investigated in =-=[27,2]-=-. Finally we can combine the extensions ∇∆ and ∇ ι into the classical higher order logic ∇ ι ∆ , also known as Q0 based on the typed λ-calculus, and often seen as a restriction of the transfinite type... |

7 | Combinators for paraconsistent attitudes
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Citation Context ...positional attitudes of agents.sA Paraconsistent Higher Order Logic 15 We think that a robust treatment of propositional attitudes in natural language is critical for many AI applications. We show in =-=[39]-=- how to obtain a paraconsistent logic for the propositional attitudes of agents while retaining classical logic for the observer. The semantics of the natural language sentences can be tested when use... |

6 | Paraconsistent query answering systems - Villadsen - 2002 |

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Citation Context |

4 | Nabla: A Linguistic System based on Multi-dimensional Type Theory - Villadsen - 1995 |

3 | Meaning and partiality revised - Villadsen - 2001 |

2 | Logic based on semiotics - Villadsen |

2 | Paraconsistent knowledge bases and many-valued logic
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Citation Context ...ously used in a very different logic programming setting [8]. Originally the knowledge base was investigated by N. C. A. da Costa and V. S. Subrahmanian. We extend the analysis developed by Villadsen =-=[40]-=-. Three experts in medicine provided information related to the diagnosis of two diseases: disease-1 and disease-2. The information concerning John and Mary can be paraphrased as follows: — Expert I (... |

1 | Typed λ-calculus and automated mathematics - Andrews - 1987 |

1 |
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Citation Context ...nd ∇‡.s2 A Case Study A Paraconsistent Higher Order Logic 3 As a case study we consider a small knowledge base in the domain of medicine, previously used in a very different logic programming setting =-=[8]-=-. Originally the knowledge base was investigated by N. C. A. da Costa and V. S. Subrahmanian. We extend the analysis developed by Villadsen [40]. Three experts in medicine provided information related... |

1 |
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Citation Context ...we can combine the extensions ∇∆ and ∇ ι into the classical higher order logic ∇ ι ∆ , also known as Q0 based on the typed λ-calculus, and often seen as a restriction of the transfinite type theory Q =-=[3]-=- by removing the transfinite types. Q0 is implemented in several automated theorem provers with many active users [24,33]. Classical second order logic, first order logic, elementary logic (first orde... |

1 | and Ortwin Scheja. Higher-order multi-valued resolution - Kohlhase - 1999 |

1 | Supra-logic: Using transfinite type theory with type variables for paraconsistency - Villadsen - 2003 |