## Categorical Proof Theory of Classical Propositional Calculus (2005)

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Citations: | 9 - 1 self |

### BibTeX

@MISC{Bellin05categoricalproof,

author = {Gianluigi Bellin and Martin Hyland and Edmund Robinson and Christian Urban},

title = {Categorical Proof Theory of Classical Propositional Calculus},

year = {2005}

}

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### Abstract

We investigate semantics for classical proof based on the sequent calculus. We show that the propositional connectives are not quite well-behaved from a traditional categorical perspective, and give a more refined, but necessarily complex, analysis of how connectives may be characterised abstractly. Finally we explain the consequences of insisting on more familiar categorical behaviour.

### Citations

337 |
An algorithmic interpretation of classical natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ...or classical proof. There are a number of immediate problems. The established term languages for classical proofs are either incompatible with the symmetries apparent in the sequent calculus (Parigot =-=[16]-=-) or in reconciling themselves to that symmetry at least make evaluation deterministic (cf Danos et al [5,21]). Either way the ideas, which derive from analyses of continuations in programming (Griffi... |

250 | A formulae-as-types notion of control
- Griffin
- 1990
(Show Context)
Citation Context ...or in reconciling themselves to that symmetry at least make evaluation deterministic (cf Danos et al [5,21]). Either way the ideas, which derive from analyses of continuations in programming (Griffin =-=[9]-=-, Preprint submitted to Elsevier Science 13 April 2005sMurthy [15]) can be thought of as reducing classical proof to constructive proof via a double negation translation. (A categorical semantics is d... |

175 | The duality of computation
- Curien, Herbelin
- 2000
(Show Context)
Citation Context ...ples of models sensitive to the issues on which we focus here. The connection with established work on polarised logic, modelling both call-by-name and call-by-value reduction strategies ([23], [26], =-=[10]-=-), is also problematic. Even if one considers a system (as in [5]) that mixes the two and considers all the normal forms reachable from representations in it of a proof, one still does not exhaust all... |

129 |
Ideas and results in proof theory
- Prawitz
- 1971
(Show Context)
Citation Context ...een deductions in minimal logic and terms of a typed lambda calculus. Deductions in minimal logic (as in most constructive systems) reduce to a unique normal form, and around 1970 Per Martin-Löf (see =-=[18]-=-) suggested using equality of normal forms as the identity criterion for proof objects in his constructive Type Theories: normal forms serve as the semantics of proof. But βη-normal forms for typed la... |

116 | Weakly distributive categories
- Cockett, Seely
- 1997
(Show Context)
Citation Context ...Szabo [22]). Rather than being definitive, in the way that the notion of an ordinary category is definitive, there are any number of variants adapted to particular contexts (recent treatments include =-=[4]-=- and [2]). Definition 2.1. A symmetric polycategory (henceforth just polycategory) P consists of • A collection obP of objects of P; and for each pair of finite sequences Γ and ∆ of objects, a collect... |

96 | Control categories and duality: on the categorical semantics of the lambda-mu calculus
- Selinger
(Show Context)
Citation Context ...to Elsevier Science 13 April 2005sMurthy [15]) can be thought of as reducing classical proof to constructive proof via a double negation translation. (A categorical semantics is described in Selinger =-=[23]-=-.) There are term calculi associated directly with the sequent calculus (Urban [25]) but it is not clear how to formulate mathematically appealing criteria for identity of such terms. What we do here ... |

66 | Natural deduction and coherence for weakly distributive categories
- Blute, Cockett, et al.
- 1996
(Show Context)
Citation Context ...2]). Rather than being definitive, in the way that the notion of an ordinary category is definitive, there are any number of variants adapted to particular contexts (recent treatments include [4] and =-=[2]-=-). Definition 2.1. A symmetric polycategory (henceforth just polycategory) P consists of • A collection obP of objects of P; and for each pair of finite sequences Γ and ∆ of objects, a collection P(Γ;... |

63 |
Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads
- Kelly, Power
- 1993
(Show Context)
Citation Context ...paper. One simple thought is as follows. Suppose that C is a model for classical logic in the general sense, freely generated by some category of objects and 23smaps. (This makes sense by Kelly-Power =-=[13]-=-.) We can inductively define idempotents eA on objects A of C: we set eA = idA for atomic objects (which includes the duals A∗ ) and then set eA∧B = eA ∧eB and eA∨B = eA ∨eB. (Implicitly we have taken... |

63 | Classical Logic and Computation
- Urban
- 2000
(Show Context)
Citation Context ...cal proof to constructive proof via a double negation translation. (A categorical semantics is described in Selinger [23].) There are term calculi associated directly with the sequent calculus (Urban =-=[25]-=-) but it is not clear how to formulate mathematically appealing criteria for identity of such terms. What we do here suggests many commutative conversions for Urban’s terms, but the matter is not stra... |

55 |
Call-by-value is dual to call-by-name, in
- Wadler
- 2003
(Show Context)
Citation Context ...g examples of models sensitive to the issues on which we focus here. The connection with established work on polarised logic, modelling both call-by-name and call-by-value reduction strategies ([23], =-=[26]-=-, [10]), is also problematic. Even if one considers a system (as in [5]) that mixes the two and considers all the normal forms reachable from representations in it of a proof, one still does not exhau... |

29 |
Proof Nets for Classical Logic
- Robinson
(Show Context)
Citation Context ...ly, the axioms of [11] entail full naturality of logical operations contrary to the clear intentions of the paper. Here we make that good and analyse the issue. Since then, another of us suggested in =-=[19]-=- basing analysis of classical proof on a simple (box-free) notion of proof net. Such systems have implicit naturalities built in so this is in contrast with [11]. In [6] Führmann and Pym analyse Robin... |

27 |
The Noble Art of Linear Decorating
- Schellinx
- 1994
(Show Context)
Citation Context ...proofs are either incompatible with the symmetries apparent in the sequent calculus (Parigot [16]) or in reconciling themselves to that symmetry at least make evaluation deterministic (cf Danos et al =-=[5,21]-=-). Either way the ideas, which derive from analyses of continuations in programming (Griffin [9], Preprint submitted to Elsevier Science 13 April 2005sMurthy [15]) can be thought of as reducing classi... |

21 | Order-enriched Categorical Models of the Classical Sequent Calculus
- Führmann, Pym
(Show Context)
Citation Context ...n, another of us suggested in [19] basing analysis of classical proof on a simple (box-free) notion of proof net. Such systems have implicit naturalities built in so this is in contrast with [11]. In =-=[6]-=- Führmann and Pym analyse Robinson’s proposal further. They give categorical combinators, add η-equalities to the implicit naturalities and succeed in axiomatizing reduction. The interaction between t... |

17 |
Premonoidal categories as categories with algebraic structure
- Power
- 2000
(Show Context)
Citation Context ...re of functoriality of the logical operations. We have not troubled with natural refinements (linearity in the domain or codomain). In a properly algebraic formulation we would expect to follow Power =-=[17]-=- and take this explicitly as part of the structure. Before doing that we should probably decide just how much use to make of it. In [11] where already an explicit notion of linearity is proposed, the ... |

11 | Duplication of directed graphs and exponential blow up of proofs
- Carbone
- 1999
(Show Context)
Citation Context ...mselves: so we end up with commutative comonoid structure for ⊤, ∧ and commutative monoid structure for ⊥, ∨. However not a great deal rides on this choice. We note that the optical graphs of Carbone =-=[3]-=- provide free models for a notion of abstract interpretation in which this choice is not made. Computation of values. We take something from ideas of non-determinism: a classical proof has a non-deter... |

9 |
Abstract Interpretation of Proofs: Classical Propositional Calculus
- Hyland
(Show Context)
Citation Context ... so ignoring the difference between ∧ and ∨). We think of these as abstract interpretations, allowing one in particular to associate a variety of invariants to proofs. Preliminary observations are in =-=[12]-=-, [7]. • Categorical models: that is models satisfying the Führmann-Pym equality axioms [6]. We know some examples of these, and have a little theory, but there is more to do. • General models: that i... |

7 |
LKT and LKQ: sequent calculi for second order logic based upon dual linear decomposition of classical implication
- Danos, Joinet, et al.
- 1995
(Show Context)
Citation Context ...proofs are either incompatible with the symmetries apparent in the sequent calculus (Parigot [16]) or in reconciling themselves to that symmetry at least make evaluation deterministic (cf Danos et al =-=[5,21]-=-). Either way the ideas, which derive from analyses of continuations in programming (Griffin [9], Preprint submitted to Elsevier Science 13 April 2005sMurthy [15]) can be thought of as reducing classi... |

7 |
Classical Proofs as Programs
- Murthy
- 1992
(Show Context)
Citation Context ...ation deterministic (cf Danos et al [5,21]). Either way the ideas, which derive from analyses of continuations in programming (Griffin [9], Preprint submitted to Elsevier Science 13 April 2005sMurthy =-=[15]-=-) can be thought of as reducing classical proof to constructive proof via a double negation translation. (A categorical semantics is described in Selinger [23].) There are term calculi associated dire... |

7 |
Non-commutative logic I: the multiplicative fragment, Ann
- Abrusci, Ruet
- 2003
(Show Context)
Citation Context ... setting ⊤ ∗ = ⊥ ⊥ ∗ = ⊤, (A ∧ B) ∗ = B ∗ ∨ A ∗ (A ∨ B) ∗ = B ∗ ∧ A ∗ , that is, more or less, by de Morgan duality. The cyclic choice of order may be familiar from non-commutative linear logic (Ruet =-=[20]-=-). It is not strictly necessary here, but serves as there to preserve a strict duality at the level of proofs. Exact duality permits a purely one-sided sequent calculus as in Girard [8], but we prefer... |

6 |
On the Geometry of Interaction for Classical Logic (Extended Abstract
- Fuhrman, Pym
- 2004
(Show Context)
Citation Context ...istic, normal forms do not readily provide a criterion for identity of such proofs. There are problems at the level of semantics. There are more or less degenerate models giving invariants of proofs (=-=[7]-=- and [12]) and we know how to construct some more general models. But all that is parasitic on experience with Linear Logic. We lack convincing examples of models sensitive to the issues on which we f... |

5 |
Naming Proofs in Classical Logic
- Lamarche, Strassburger
(Show Context)
Citation Context ...ot expect that in the classical case. At the very least different systems of proof can be expected to lead to different semantics. A compelling example is the recent work of Lamarche and Strassburger =-=[14]-=-. 2s2 Modelling classical proofs 2.1 Sequent Calculus and Polycategories It is a familiar idea that what the sequent calculus provides is not a collection of ideal proofs-in-themselves, but something ... |

5 | On the Geometry of Interaction for Classical Logic (Extended Abstract
- Führmann, Pym
(Show Context)
Citation Context ...istic, normal forms do not readily provide a criterion for identity of such proofs. There are problems at the level of semantics. There are more or less degenerate models giving invariants of proofs (=-=[7]-=- and [12]) and we know how to construct some more general models. But all that is parasitic on experience with Linear Logic. We lack convincing examples of models sensitive to the issues on which we f... |

3 | Two paradigms of logical computation in Affine Logic
- Bellin
(Show Context)
Citation Context ... some kind of non-determinism: the computation or reduction process is in principle non-deterministic. But we do not for example have primitives for non-deterministic choice. In particular in view of =-=[1]-=- we should investigate an approach to the idea of non-deterministic choice in proofs using the MIX rule. Idempotents. While it is not clear whether our formulation of semantics for classical proof is ... |

3 |
Strong Normalisation of Cut-Elimination
- Urban, Bierman
- 2000
(Show Context)
Citation Context ...re is no easy way to extract models for our system from categorical models in the style of Selinger. The project on which we report here was motivated by Urban’s strong normalisation result ([25] and =-=[24]-=-) for a formulation of classical proof. In [11], one of us then outlined a proposal for a semantics. Unfortunately, the axioms of [11] entail full naturality of logical operations contrary to the clea... |