## An institutional view on categorical logic and the Curry-Howard-Tait-isomorphism

Citations: | 1 - 1 self |

### BibTeX

@MISC{Mossakowski_aninstitutional,

author = {Till Mossakowski and Florian Rabe and Valeria De Paiva and Lutz Schröder and Joseph Goguen},

title = {An institutional view on categorical logic and the Curry-Howard-Tait-isomorphism },

year = {}

}

### OpenURL

### Abstract

We introduce a generic notion of propositional categorical logic and provide a construction of an institution with proofs out of such a logic, following the Curry-Howard-Tait paradigm. We then prove logic-independent soundness and completeness theorems. The framework is instantiated with a number of examples: classical, intuitionistic, linear and modal propositional logics. Finally, we speculate how this framework may be extended beyond the propositional case.

### Citations

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Citation Context ...puter science as a response to the population explosion among logicssin use, with the ambition of doing as much as possible at a level of abstraction independent of commitment to any particular logic =-=[16, 18]-=-. Its key idea is to focus on abstractly axiomatizing the satisfaction relation between sentences and models. A surprisingly large amount of meta-logical reasoning can be carried out in this abstract ... |

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Citation Context ...this context, a number of deep correspondences between categories and logical theories have been established (identifying ‘types’ with ‘objects in a category’): conjunctive logic cartesian categories =-=[19]-=- positive logic cartesian closed categories [19] intuitionistic propositional logic bicartesian closed categories [19] classical propositional logic bicartesian closed categories with [19] ¬¬-eliminat... |

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Citation Context ...s, we conclude with some speculations on how to extend this program to other logics. 2 Institutions and Logics We assume that the reader is familiar with basic notions from category theory; e.g., see =-=[1, 20]-=- for introductions to this subject. By way of notation, |C| denotes the class of objects of a category C, and composition is denoted by “◦”. Let CAT be the the quasi-category of all categories (quasi-... |

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Citation Context ...ies and programs [29, 10], and substantial portions of classical model theory can be lifted to the level of institutions [34, 7–9]. The notion of institution has been extended to cover also deduction =-=[23]-=- and proof theory [25]. Below, we give a general notion of propositional categorical logic, a construction of an institution with proofs out of such a logic, and logic-independent soundness and comple... |

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Citation Context ...ss [13], behavioural [2], coalgebraic [6], and object-oriented [17] logics have been shown to be institutions. 3 Partial Conditional Rewriting Logic Conditional rewriting logic has been introduced in =-=[24, 22]-=- as a model of concurrency that distinguishes equations from rewriting rules and hence naturally leads to notions of rewriting up to associativity, commutativity etc. Here, we will use conditional rew... |

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Citation Context ...es [19] intuitionistic propositional logic bicartesian closed categories [19] classical propositional logic bicartesian closed categories with [19] ¬¬-elimination linear logic ∗-autonomous categories =-=[33]-=- first-order logic hyperdoctrines [31] Martin-Löf type theory locally cartesian closed categories [32] Here, we present work aimed at casting these correspondences in a common framework based on the t... |

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Citation Context ...isingly large amount of meta-logical reasoning can be carried out in this abstract framework; e.g., institutions have been used to give general foundations for modularization of theories and programs =-=[29, 10]-=-, and substantial portions of classical model theory can be lifted to the level of institutions [34, 7–9]. The notion of institution has been extended to cover also deduction [23] and proof theory [25... |

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Citation Context ...isingly large amount of meta-logical reasoning can be carried out in this abstract framework; e.g., institutions have been used to give general foundations for modularization of theories and programs =-=[29, 10]-=-, and substantial portions of classical model theory can be lifted to the level of institutions [34, 7–9]. The notion of institution has been extended to cover also deduction [23] and proof theory [25... |

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Citation Context ...near logic, by taking judgements of the form ϕ1 . . . ϕn ⊢ ψ as sentences. Also, higher-order [4], polymorphic [30], temporal [13], process [13], behavioural [2], coalgebraic [6], and object-oriented =-=[17]-=- logics have been shown to be institutions. 3 Partial Conditional Rewriting Logic Conditional rewriting logic has been introduced in [24, 22] as a model of concurrency that distinguishes equations fro... |

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Citation Context ...puter science as a response to the population explosion among logicssin use, with the ambition of doing as much as possible at a level of abstraction independent of commitment to any particular logic =-=[16, 18]-=-. Its key idea is to focus on abstractly axiomatizing the satisfaction relation between sentences and models. A surprisingly large amount of meta-logical reasoning can be carried out in this abstract ... |

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Citation Context ...esian closed category) is strongly complete for Prop. For any dense-in-itself metric space X, the Heyting algebra O(X) of open sets in X (qua bicartesian closed category) is weakly complete for IProp =-=[27]-=-. We do not, at present, know whether O(X) is also strongly complete for IProp. Proposition 31. Let L be a categorical logic extending the theory of cartesian closed categories (i.e. minimal intuition... |

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Citation Context ...logic bicartesian closed categories with [19] ¬¬-elimination linear logic ∗-autonomous categories [33] first-order logic hyperdoctrines [31] Martin-Löf type theory locally cartesian closed categories =-=[32]-=- Here, we present work aimed at casting these correspondences in a common framework based on the theory of institutions. The notion of institution arose within computer science as a response to the po... |

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Citation Context ...he resulting tuple is in the predicate. This is extended to conditional formulas as usual. (A restricted version of this institution has been introduced under the names essentially algebraic theories =-=[14]-=- and HEP-varieties [28], and has been used in the meta-theory of categories.) Again, there are several proof systems for this institution; e.g. the system in [5] can easily be formalized in such a way... |

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Citation Context ...n the predicate. This is extended to conditional formulas as usual. (A restricted version of this institution has been introduced under the names essentially algebraic theories [14] and HEP-varieties =-=[28]-=-, and has been used in the meta-theory of categories.) Again, there are several proof systems for this institution; e.g. the system in [5] can easily be formalized in such a way that PHorn = becomes a... |

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Citation Context ...f excluded middle. Example 11 (Modal logic). Extend IP rop from Example 9 by adding operations ✷, ✸: Ob → Ob modeling the necessity and possibility operators from constructive modal S4, as defined by =-=[3]-=-. The necessity modality corresponds to a monoidal comonad, while the possibility modality corresponds to a monad, which is strong relative to the necessity comonad. Example 12 (Linear logic). For a d... |

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Citation Context ...tions. So can substructural logics like linear logic, by taking judgements of the form ϕ1 . . . ϕn ⊢ ψ as sentences. Also, higher-order [4], polymorphic [30], temporal [13], process [13], behavioural =-=[2]-=-, coalgebraic [6], and object-oriented [17] logics have been shown to be institutions. 3 Partial Conditional Rewriting Logic Conditional rewriting logic has been introduced in [24, 22] as a model of c... |

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Citation Context ...tional structure sheds light on the usual collapsing problem (classical bicartesian closed categories are boolean algebras); the collapsing can be avoided by using preorder-enriched categories, as in =-=[15]-=-. We have proved generic deduction, soundness, and completeness theorems. Moreover, while trying to recover the Curry-Howard-Tait isomorphism as an explicit isomorphism between institution, only a lim... |

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Citation Context ... names essentially algebraic theories [14] and HEP-varieties [28], and has been used in the meta-theory of categories.) Again, there are several proof systems for this institution; e.g. the system in =-=[5]-=- can easily be formalized in such a way that PHorn = becomes a preordcat/cat institution. Moreover, intuitionistic and modal logics can be formalized as institutions. So can substructural logics like ... |

19 |
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Citation Context ... and Boolvalued models. We have provided some general result about the relations between such models. An interesting question is whether the institutions I(L) have elementary diagrams in the sense of =-=[8]-=-. Our current notion of model is obviously too weak to ensure this; for ensuring elementary diagrams one would need “intensional models” over signatures containing proof variables, to be valuated with... |

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Citation Context ...perations ⊗, ⊸, ! modeling, respectively, multiplicative conjunction, linear implication and the exponential operator “of course!”. This corresponds to an autonomous category with a ’linear’ comonad, =-=[21]-=-. 6s5 The Institutional Curry-Howard-Tait Construction We now recast the Curry-Howard-Tait interpretation of categorical logic in an institutional setting. Given a categorical logic L, let C(L) be the... |

16 |
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Citation Context ...ogic bicartesian closed categories [19] classical propositional logic bicartesian closed categories with [19] ¬¬-elimination linear logic ∗-autonomous categories [33] first-order logic hyperdoctrines =-=[31]-=- Martin-Löf type theory locally cartesian closed categories [32] Here, we present work aimed at casting these correspondences in a common framework based on the theory of institutions. The notion of i... |

14 | Type class polymorphism in an institutional framework
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Citation Context ...c and modal logics can be formalized as institutions. So can substructural logics like linear logic, by taking judgements of the form ϕ1 . . . ϕn ⊢ ψ as sentences. Also, higher-order [4], polymorphic =-=[30]-=-, temporal [13], process [13], behavioural [2], coalgebraic [6], and object-oriented [17] logics have been shown to be institutions. 3 Partial Conditional Rewriting Logic Conditional rewriting logic h... |

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Citation Context ...ics can be formalized as institutions. So can substructural logics like linear logic, by taking judgements of the form ϕ1 . . . ϕn ⊢ ψ as sentences. Also, higher-order [4], polymorphic [30], temporal =-=[13]-=-, process [13], behavioural [2], coalgebraic [6], and object-oriented [17] logics have been shown to be institutions. 3 Partial Conditional Rewriting Logic Conditional rewriting logic has been introdu... |

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Citation Context ...er, intuitionistic and modal logics can be formalized as institutions. So can substructural logics like linear logic, by taking judgements of the form ϕ1 . . . ϕn ⊢ ψ as sentences. Also, higher-order =-=[4]-=-, polymorphic [30], temporal [13], process [13], behavioural [2], coalgebraic [6], and object-oriented [17] logics have been shown to be institutions. 3 Partial Conditional Rewriting Logic Conditional... |

11 | What is a logic
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Citation Context ...10], and substantial portions of classical model theory can be lifted to the level of institutions [34, 7–9]. The notion of institution has been extended to cover also deduction [23] and proof theory =-=[25]-=-. Below, we give a general notion of propositional categorical logic, a construction of an institution with proofs out of such a logic, and logic-independent soundness and completeness theorems. We th... |

8 |
Institutionalising many-sorted coalgebraic modal logic
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Citation Context ...structural logics like linear logic, by taking judgements of the form ϕ1 . . . ϕn ⊢ ψ as sentences. Also, higher-order [4], polymorphic [30], temporal [13], process [13], behavioural [2], coalgebraic =-=[6]-=-, and object-oriented [17] logics have been shown to be institutions. 3 Partial Conditional Rewriting Logic Conditional rewriting logic has been introduced in [24, 22] as a model of concurrency that d... |

8 |
Cut Elimination in Categories
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Citation Context ...l rewrites correspond to the various cases and subcases of the induction step of a constructive cut elimination proof. For the simple case of a cartesian logic a cut elimination proof can be found in =-=[12]-=-. An example for a rewrite that needs to be added to yield cut elimination is 〈f, g〉 ◦ h ❀ 〈f ◦ h, g ◦ h〉; this rewrite replaces one top-level cut with two lower-level cuts. Example 19. Consider the t... |

3 |
Conditional rewriting as a unified model of concurrency. Theoretical computer science 96(1
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Citation Context ...ss [13], behavioural [2], coalgebraic [6], and object-oriented [17] logics have been shown to be institutions. 3 Partial Conditional Rewriting Logic Conditional rewriting logic has been introduced in =-=[24, 22]-=- as a model of concurrency that distinguishes equations from rewriting rules and hence naturally leads to notions of rewriting up to associativity, commutativity etc. Here, we will use conditional rew... |