## General logics (1989)

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Venue: | In Logic Colloquium 87 |

Citations: | 11 - 5 self |

### BibTeX

@INPROCEEDINGS{Mossakowski89generallogics,

author = {Till Mossakowski and Joseph Goguen and Răzvan Diaconescu},

title = {General logics},

booktitle = {In Logic Colloquium 87},

year = {1989},

pages = {113--133},

publisher = {North}

}

### OpenURL

### Abstract

theory, categorical logic. model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representing proofs as usual. A natural equivalence relation on institutions is defined such that its equivalence classes are logics. Several invariants are defined for this equivalence, including a Lindenbaum

### Citations

515 |
Institutions: Abstract model theory for specification and programming
- Goguen, Burstall
- 1992
(Show Context)
Citation Context ...ience in response to the population explosion among the logics in use there, with the ambition of doing as much as possible at a level of abstraction independent of commitment to any particular logic =-=[17, 31, 19]-=-. The soundness aspect of sound reasoning is addressed by axiomatizing the notion of satisfaction, and the reasoning aspect is addressed by calling on categorical logic, which applies category theory ... |

449 |
Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...t is a Logic? 13 Proof theoretic institutions include both proofs and sentences. Categorical logic usually works with categories of sentences, where morphisms are (equivalence classes of) proof terms =-=[22]-=-. But this only captures provability between single sentences, whereas logic traditionally studies provability from a set of sentences. The following overcomes this limitation by considering categorie... |

266 |
Abstract and Concrete Categories
- Adámek, Herrlich, et al.
(Show Context)
Citation Context ... use this feature to resolve a problem about cardinality raised in [3]; see Example 2.2. 2. Institutions and Logics We assume the reader is familiar with basic notions from category theory; e.g., see =-=[1, 23]-=- for introductions to this subject. By way of notation, |C| denotes the class of objects of a category C, and composition is denoted by “◦”. Categories are assumed by convention to be locally small (i... |

179 |
General Logics
- Meseguer
- 1989
(Show Context)
Citation Context ...itutions are sound, which in particular implies the following: Proposition 5.2. Any cat/cat equivalence is a set/cat equivalence. � Proposition 5.3. ⊢ satisfies the properties of an entailment system =-=[25]-=-, i.e. it is reflexive, transitive, monotonic and stable under translation along signature morphisms. In fact, entailment systems are in bijective correspondence with proof theoretic institutions havi... |

143 |
Categories for the working mathematician”, second edition
- Lane
- 1997
(Show Context)
Citation Context ... use this feature to resolve a problem about cardinality raised in [3]; see Example 2.2. 2. Institutions and Logics We assume the reader is familiar with basic notions from category theory; e.g., see =-=[1, 23]-=- for introductions to this subject. By way of notation, |C| denotes the class of objects of a category C, and composition is denoted by “◦”. Categories are assumed by convention to be locally small (i... |

94 | Specifications in an arbitrary institution
- Sannella, Tarlecki
- 1988
(Show Context)
Citation Context ...ience in response to the population explosion among the logics in use there, with the ambition of doing as much as possible at a level of abstraction independent of commitment to any particular logic =-=[17, 31, 19]-=-. The soundness aspect of sound reasoning is addressed by axiomatizing the notion of satisfaction, and the reasoning aspect is addressed by calling on categorical logic, which applies category theory ... |

90 | Towards an algebraic semantics for the object paradigm
- Goguen, Diaconescu
- 1994
(Show Context)
Citation Context ...s, like linear logic, where judgements of the form ϕ1 . . . ϕn ⊢ ψ are sentences. Higher-order [7], polymorphic [32], temporal [16], process [16], behavioural [4], coalgebraic [9] and object-oriented =-=[18]-=- logics also form institutions. Many familiar basic concepts can be defined over any institution: Definition 2.6. Given a set of Σ-sentences Γ and a Σ-sentence ϕ, then ϕ is a semantic consequence of Γ... |

65 | Institution morphisms
- Goguen, Rosu
(Show Context)
Citation Context ...ience in response to the population explosion among the logics in use there, with the ambition of doing as much as possible at a level of abstraction independent of commitment to any particular logic =-=[17, 31, 19]-=-. The soundness aspect of sound reasoning is addressed by axiomatizing the notion of satisfaction, and the reasoning aspect is addressed by calling on categorical logic, which applies category theory ... |

57 | Many-valued logics
- Urquhart
- 1986
(Show Context)
Citation Context ...ariables: 6 By determining Ξ I in a purely model-theoretic way, we avoid the need to deal with different signatures of Lindenbaum algebras of equivalent logics, as it is necessary in the framework of =-=[28]-=-. 7 A cryptomorphism is a homomorphism between algebras of different signatures linked by a signature morphism; the homomorphism goes from the source algebra into the reduct of the target algebra. 8 L... |

50 |
Bits and pieces of the theory of institutions
- Tarlecki
- 1986
(Show Context)
Citation Context ...rve as models in institutions lacking (non-trivial) models.s10 Till Mossakowski, Joseph Goguen, Răzvan Diaconescu and Andrzej Tarlecki Definition 4.2. An institution has external semantic conjunction =-=[34]-=- if for any pair of sentences ϕ1, ϕ2 over the same signature, there is a sentence ψ such that ψ holds in a model iff both ϕ1 and ϕ2 hold in it. ψ will also be denoted ϕ1 ∧○ ϕ2, a meta-notation which m... |

48 |
Axioms for abstract model theory
- Barwise
- 1974
(Show Context)
Citation Context ...n institution independent way, such that the expected theorems hold under reasonable assumptions. All this is very much in the spirit of “abstract model theory,” in the sense advocated by Jon Barwise =-=[2]-=-, but it goes much further, including even some new results for known logics, such as many sorted first order logic [14, 21]. The faithful functors to Set make it possible to consider cardinalities fo... |

34 | An implementation-oriented semantics for module composition
- Goguen, Tracz
- 2000
(Show Context)
Citation Context ...Sen I (Σ). We denote the reduct functor Mod I (σ) by ↾σ and the sentence translation Sen I (σ) by σ( ). When M = M ′ ↾σ we say that M ′ is a σ-expansion of M. 1 A more concrete definition is given in =-=[20]-=-, which avoids category theory by spelling out the conditions for functoriality, and assuming a set theoretic construction for signatures. Though less general, this definition is sufficient for everyt... |

31 |
On the Existence of Free Models in Abstract Algebraic Institutions
- Tarlecki
- 1985
(Show Context)
Citation Context ... the first term refers to sentences, and the second to models); the table in Thm. 5.20 summarizes many of their properties. See [19] for a general treatment of the variant notions of institution, and =-=[33, 35, 11, 12, 14]-=- for some non-trivial results in abstract model theory done institutionally. This paper adds to the literature on institutions a notion of equivalence, such that a logic is an equivalence class of ins... |

30 | Behavioural satisfaction and equivalence in concrete model categories
- Bidoit, Tarlecki
- 1996
(Show Context)
Citation Context ...d Andrzej Tarlecki Set to categories of many sorted sets Set S , where the sets S range over sort sets of an institution’s signatures. The following enriches institutions with carrier sets for models =-=[5]-=-: Definition 2.7. A concrete institution is an institution I together with a functor sorts I : Sign I → Set and a natural transformation | | I : Mod I → Set (sortsI ) op ( ) between functors from Sign... |

23 |
Quasi-Varieties in Abstract Algebraic Institutions
- Tarlecki
- 1986
(Show Context)
Citation Context ... the first term refers to sentences, and the second to models); the table in Thm. 5.20 summarizes many of their properties. See [19] for a general treatment of the variant notions of institution, and =-=[33, 35, 11, 12, 14]-=- for some non-trivial results in abstract model theory done institutionally. This paper adds to the literature on institutions a notion of equivalence, such that a logic is an equivalence class of ins... |

22 | On the integration of observability and reachability concepts
- Bidoit, Hennicker
- 2002
(Show Context)
Citation Context ... S4 or S5, as well as substructural logics, like linear logic, where judgements of the form ϕ1 . . . ϕn ⊢ ψ are sentences. Higher-order [7], polymorphic [32], temporal [16], process [16], behavioural =-=[4]-=-, coalgebraic [9] and object-oriented [18] logics also form institutions. Many familiar basic concepts can be defined over any institution: Definition 2.6. Given a set of Σ-sentences Γ and a Σ-sentenc... |

21 |
Grothendieck institutions. Applied categorical structures
- Diaconescu
- 2002
(Show Context)
Citation Context ...atural in Σ. I is equivalent to J if there is an institution equivalence from I to J . � This definition is very natural; it is 2-categorical equivalence in the appropriate 2-category of institutions =-=[10]-=-. The requirement for a set/set institution comorphism to be a set/set equivalence is weaker: each βΣ need only have an inverse up to elementary equivalence β ′ Σ . Definition 3.6. A concrete institut... |

21 |
An institution-independent proof of Craig Interpolation Property
- Diaconescu
(Show Context)
Citation Context ... the first term refers to sentences, and the second to models); the table in Thm. 5.20 summarizes many of their properties. See [19] for a general treatment of the variant notions of institution, and =-=[33, 35, 11, 12, 14]-=- for some non-trivial results in abstract model theory done institutionally. This paper adds to the literature on institutions a notion of equivalence, such that a logic is an equivalence class of ins... |

19 |
Elementary diagrams in institutions
- Diaconescu
(Show Context)
Citation Context |

18 |
Institution-independent ultraproducts
- Diaconescu
(Show Context)
Citation Context |

15 |
A simple algebraic proof of the equational interpolation theorem
- Rodenburg
- 1991
(Show Context)
Citation Context ...ted first order logic has it only for those where one component is injective on sorts [8, 6, 21], and EQ and Horn clause logic only have it for pushout squares where R consists of injective morphisms =-=[30, 14]-=-. Using sets of sentences rather than single sentences accommodates interpolation results for equational logic [30] as well as for other institutions having Birkhoff-style axiomatizability properties ... |

14 |
Specification in an arbitrary institution with symbols
- Mossakowski
- 2000
(Show Context)
Citation Context ..., a model M with the carrier |M| being a family of finite sets). A concrete institution admits free models if all carrier functors for model categories have left adjoints. � The following notion from =-=[26]-=- also provides signatures with underlying sets of symbols, by extending sorts I ; essentially all institutions that arise in practice have this structure: Definition 2.8. A concrete institution with s... |

14 | Type class polymorphism in an institutional framework
- Schröder, Mossakowski, et al.
- 2004
(Show Context)
Citation Context ...logics restricting K by further axioms, such as S4 or S5, as well as substructural logics, like linear logic, where judgements of the form ϕ1 . . . ϕn ⊢ ψ are sentences. Higher-order [7], polymorphic =-=[32]-=-, temporal [16], process [16], behavioural [4], coalgebraic [9] and object-oriented [18] logics also form institutions. Many familiar basic concepts can be defined over any institution: Definition 2.6... |

12 | Mirror, mirror in my hand: A duality between specifications and models of process behaviour
- Fiadeiro, Costa
- 1996
(Show Context)
Citation Context ...ing K by further axioms, such as S4 or S5, as well as substructural logics, like linear logic, where judgements of the form ϕ1 . . . ϕn ⊢ ψ are sentences. Higher-order [7], polymorphic [32], temporal =-=[16]-=-, process [16], behavioural [4], coalgebraic [9] and object-oriented [18] logics also form institutions. Many familiar basic concepts can be defined over any institution: Definition 2.6. Given a set o... |

11 | Moving specification structures between logical systems
- Borzyszkowski
- 1999
(Show Context)
Citation Context ...s do other modal logics restricting K by further axioms, such as S4 or S5, as well as substructural logics, like linear logic, where judgements of the form ϕ1 . . . ϕn ⊢ ψ are sentences. Higher-order =-=[7]-=-, polymorphic [32], temporal [16], process [16], behavioural [4], coalgebraic [9] and object-oriented [18] logics also form institutions. Many familiar basic concepts can be defined over any instituti... |

11 |
Herbrand theorems in arbitrary institutions
- Diaconescu
(Show Context)
Citation Context ...cludes second order quantification by taking D to be all extensions of signatures by operation and relation symbols. First order quantification is modeled with D the representable signature morphisms =-=[11, 13]-=- defined below, building on the observation that an assignment for a set of (first order) variables corresponds to a model morphism from the free (term) model over that set of variables: 6 By determin... |

11 |
An Institution-Independant Proof of Robinson Consistency Theorem’, Studia Logica
- Găină, Popescu
- 2006
(Show Context)
Citation Context ...atures to arbitrary classes of pushout squares. While FOL has interpolation for all pushout squares [15], many sorted first order logic has it only for those where one component is injective on sorts =-=[8, 6, 21]-=-, and EQ and Horn clause logic only have it for pushout squares where R consists of injective morphisms [30, 14]. Using sets of sentences rather than single sentences accommodates interpolation result... |

10 |
On a Generalized Modularization Theorem
- Dimitrakos, Maibaum
(Show Context)
Citation Context ...is generalizes the conventional formulation of interpolation from intersection/union squares of signatures to arbitrary classes of pushout squares. While FOL has interpolation for all pushout squares =-=[15]-=-, many sorted first order logic has it only for those where one component is injective on sorts [8, 6, 21], and EQ and Horn clause logic only have it for pushout squares where R consists of injective ... |

8 |
Institutionalising many-sorted coalgebraic modal logic
- Cirstea
(Show Context)
Citation Context ...l as substructural logics, like linear logic, where judgements of the form ϕ1 . . . ϕn ⊢ ψ are sentences. Higher-order [7], polymorphic [32], temporal [16], process [16], behavioural [4], coalgebraic =-=[9]-=- and object-oriented [18] logics also form institutions. Many familiar basic concepts can be defined over any institution: Definition 2.6. Given a set of Σ-sentences Γ and a Σ-sentence ϕ, then ϕ is a ... |

6 |
Functorial semantics of elementary theories
- Lawvere
- 1966
(Show Context)
Citation Context ... institutions I and J , J is a cat/cat skeleton of I if it is like a set/cat skeleton, but such that Sen J (Σ) = Sen I (Σ)/ ∼ =, and such that Pr J (Σ) = LC ⊢⊣ (Σ), the Lindenbaum category. � Lawvere =-=[24]-=- defined quantification as adjoint to substitution. Here we define quantification as adjoint to sentence translation along a class D of signature morphisms, which typically introduce new constants to ... |

5 |
Generalized interpolation in first-order logic
- Borzyszkowski
- 2005
(Show Context)
Citation Context ...atures to arbitrary classes of pushout squares. While FOL has interpolation for all pushout squares [15], many sorted first order logic has it only for those where one component is injective on sorts =-=[8, 6, 21]-=-, and EQ and Horn clause logic only have it for pushout squares where R consists of injective morphisms [30, 14]. Using sets of sentences rather than single sentences accommodates interpolation result... |

5 | Generalized interpolation in CASL
- Borzyszkowski
(Show Context)
Citation Context ...atures to arbitrary classes of pushout squares. While FOL has interpolation for all pushout squares [15], many sorted first order logic has it only for those where one component is injective on sorts =-=[8, 6, 21]-=-, and EQ and Horn clause logic only have it for pushout squares where R consists of injective morphisms [30, 14]. Using sets of sentences rather than single sentences accommodates interpolation result... |

4 |
Homeomorphism and the equivalence of logical systems
- Pollard
- 1998
(Show Context)
Citation Context ... is not. This means it makes too fine-grained distinctions; for example, in FOL, (true • ) • is infinite, while in a skeleton of FOL, ([true] • ) • is just the singleton {[true]}. As already noted in =-=[29]-=-, the closure operator at the same time is too coarse for determining the identity of a logic: while e.g. proof theoretic falsum in a sound and complete logic (see Section 5) is preserved by homeomorp... |

3 |
What is classical propositional logic? (a study in universal logic
- Béziau, Freitas, et al.
(Show Context)
Citation Context ...nterpretants,” which allow for context dependency of denotation in his semiotics, as opposed to Tarski’s dyadic satisfaction. We also use this feature to resolve a problem about cardinality raised in =-=[3]-=-; see Example 2.2. 2. Institutions and Logics We assume the reader is familiar with basic notions from category theory; e.g., see [1, 23] for introductions to this subject. By way of notation, |C| den... |