## A Complete Axiomatization of Higher-Order Intuitionistic Logic (2001)

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Venue: | CLE e-Prints |

Citations: | 2 - 2 self |

### BibTeX

@TECHREPORT{Coniglio01acomplete,

author = {Marcelo E. Coniglio and Cristina Sernadas},

title = {A Complete Axiomatization of Higher-Order Intuitionistic Logic},

institution = {CLE e-Prints},

year = {2001}

}

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### Abstract

Two Hilbert calculi for higher-order logic (or theory of types) are introduced.

### Citations

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Citation Context ... by Church in [4]. By Godel's second theorem, it is immediate to show that there is no (reasonable) proof system complete for the standard (set) semantics of hol. On the other hand, Henkin proves in [=-=-=-8] that it is possible to give an axiomatization of hol sound and complete w.r.t. a wider class of models, called general models, in which types of the form ( ! 0 ) are interpreted as subsets of the ... |

140 |
P.J.: Introduction to Higher-Order Categorical Logic
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Citation Context ...) 0 . From the works of Lawvere (see for example [14, 15]) it was proved that the usual axiomatizations of hol are sound and complete w.r.t. an extremely elegant topos semantics (see, for instance, [=-=12, 3, 16, 9, 17]-=-). In this research report we introduce two very simple Hilbert-style axiomatizations of hol, which are sound and complete w.r.t. topos semantics. Thesrst one is obtained by adapting the sequent calcu... |

117 |
Mathematical logic as based on the theory of types
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Citation Context ...ory of types) is dened in a very rich language which permits to express most of mathematics reasoning. Theory of types was introduced by Russell in 1908 as a solution to paradoxes in set theory (see [=-=19-=-]) and was reformulated by Church in [4]. By Godel's second theorem, it is immediate to show that there is no (reasonable) proof system complete for the standard (set) semantics of hol. On the other h... |

55 |
Equality in hyperdoctrines and the comprehension schema as an adjoint functor
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Citation Context ...called general models, in which types of the form ( ! 0 ) are interpreted as subsets of the set of maps from (the carrier of) to (the carrier of) 0 . From the works of Lawvere (see for example [14,=-= 15]-=-) it was proved that the usual axiomatizations of hol are sound and complete w.r.t. an extremely elegant topos semantics (see, for instance, [12, 3, 16, 9, 17]). In this research report we introduce t... |

51 | Fibring of logics as a categorial construction
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Citation Context ... syntactical inference, one for each concept of semantic entailment. In several contexts it is useful to maintain the distinction between the two notions of (semantic and syntactical) inferences (cf. =-=[21, 22, 5-=-]). 3 Hilbert Calculi In this section we recall (a simplied version of) the notion of Hilbert calculus introduced in [5]. In order to represent arbitrary terms in rules of Hilbert calculi we will use ... |

49 | Sheaves and logic - Fourman, Scott - 1979 |

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Toposes and Local Set Theories
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Citation Context ...f types) are introduced. Thesrst is dened in a language that uses just exponential types of power type, and it is obtained by adapting the sequent calculus for local set theory introduced by Bell in [=-=3-=-]. The second one, originally introduced in [5], is dened in a language with arbitrary functional types. Using usual topos semantics we show that both systems are sound and complete. Introduction High... |

45 | Fibring: Completeness preservation
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Citation Context ... syntactical inference, one for each concept of semantic entailment. In several contexts it is useful to maintain the distinction between the two notions of (semantic and syntactical) inferences (cf. =-=[21, 22, 5-=-]). 3 Hilbert Calculi In this section we recall (a simplied version of) the notion of Hilbert calculus introduced in [5]. In order to represent arbitrary terms in rules of Hilbert calculi we will use ... |

39 |
Categorical Logic and Type Theory, volume 141
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Citation Context ...) 0 . From the works of Lawvere (see for example [14, 15]) it was proved that the usual axiomatizations of hol are sound and complete w.r.t. an extremely elegant topos semantics (see, for instance, [=-=12, 3, 16, 9, 17]-=-). In this research report we introduce two very simple Hilbert-style axiomatizations of hol, which are sound and complete w.r.t. topos semantics. Thesrst one is obtained by adapting the sequent calcu... |

22 | 1969] Adjointness in Foundations
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Citation Context ...called general models, in which types of the form (θ → θ ′ ) are interpreted as subsets of the set of maps from (the carrier of) θ to (the carrier of) θ ′ . From the works of Lawvere (see for example =-=[14, 15]-=-) it was proved that the usual axiomatizations of hol are sound and complete w.r.t. an extremely elegant topos semantics (see, for instance, [12, 3, 16, 9, 17]). In this research report we introduce t... |

8 | Topological completeness for higher-order logic - Awodey, Butz - 2000 |

7 |
Adjointness in foundations. Dialectica
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Citation Context ...called general models, in which types of the form ( ! 0 ) are interpreted as subsets of the set of maps from (the carrier of) to (the carrier of) 0 . From the works of Lawvere (see for example [14,=-= 15]-=-) it was proved that the usual axiomatizations of hol are sound and complete w.r.t. an extremely elegant topos semantics (see, for instance, [12, 3, 16, 9, 17]). In this research report we introduce t... |

4 | Notes on Categorical Logic, Tutorials of Wollic '97 - Johnstone - 1997 |

2 | First-Order Categorical Logic, volume 611 - Makkai, Reyes - 1977 |

2 |
Topos-theoretic semantics of fibring. Research report
- Coniglio, Sernadas, et al.
- 2002
(Show Context)
Citation Context ...a language that uses just exponential types of power type, and it is obtained by adapting the sequent calculus for local set theory introduced by Bell in [3]. The second one, originally introduced in =-=[5]-=-, is defined in a language with arbitrary functional types. Using usual topos semantics we show that both systems are sound and complete. Introduction Higher-order logic (a.k.a. hol, or theory of type... |

1 |
Topos-theoretic semantics of Research report
- Coniglio, Sernadas, et al.
- 2002
(Show Context)
Citation Context ...a language that uses just exponential types of power type, and it is obtained by adapting the sequent calculus for local set theory introduced by Bell in [3]. The second one, originally introduced in =-=[5-=-], is dened in a language with arbitrary functional types. Using usual topos semantics we show that both systems are sound and complete. Introduction Higher-order logic (a.k.a. hol, or theory of types... |