G. Sambin, Pretopologies and completeness proofs. Journal of Symbolic Logic, 60 (1995), pp. 861-878.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....logic. It encodes linear proofs as well as affine linear and relevant proofs (without distribution) This report focuses on a bimodal version of intuitionistic linear logic. We prove a soundness and completeness theorem for the pretopology semantics for Linear Logic developed by G. Sambin in [17, 18, 19]. Our main contribution is in modeling Bimodal Intuitionistic Linear Logic, commutative or not, and proving it sound and complete in an extension of Sambin s pretopologies. The construction of a canonical model and the completeness theorem we propose will work just as well in the limit case where ....

....and intuitive semantic framework (perhaps more so than the phase space semantics) So we prefer to present our models as pretopologies of a certain kind, namely modal pretopologies as described in Definition 3.3. For intuitions behind the pretopology semantics we refer the reader to G. Sambin [17, 18, 19], which is also the source of the following definition. Definition 3.1 A pretopology is a quadruple X = X; e; Delta) such that 1. X; e) is a monoid, called the formal base of the pretopology 2. Delta is a relation (the precover relation of the pretopology X ) between elements and ....

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G. Sambin, "Pretopologies and Completeness Proofs", Journal of Symbolic Logic 60, No 3, 1995.

....monoidal (i.e. associative with a unit) it comes up that the set of possibility distributions endowed with such an operation forms an algebraic structure known as a quantal. It is then natural to define a local possibilistic logic, which turns out to belong to the class of substructural logics [Sambin 1995] and in particular a specialization of linear logic [Girard 1987] We provide a sequent calculus LPL for this logic 1 , which is parameterized by the t norm . This calculus is parametrically proved sound and complete with respect to the corresponding algebraic structure. We represent the ....

G. Sambin. Pretopologies and completeness proofs. J. Symbolic Logic, 60:861--878, 1995.

.... disjunction we require x A # B if and only if x # # ( A] # [ B] Sambin and others have used the notion of a pretopology (in our language, a set with a closure operator) not only as a model of substructural logics but also as a constructive generalisation of a topological space [131, 244, 245, 246]. Do sen [68, 69] Ono and Komori [205] and Ono [206] have also used given semantics involving a closure operation This is not the only way to avoid distribution. In a model without a notion of inclusion, we can get by with a negation to define a closure operator: EXAMPLE 59 (GOLDBLATT FRAMES) ....

GIOVANNI SAMBIN. "Pretopologies and Completeness Proofs". Journal of Symbolic Logic, 60(3):861--878, 1995.

....points. This was not possible, so to dene abstract formal topologies we instead used contexts. But note that the context TOP is one arbitrary formal topology and not a template for formal topologies. This formalisation of pointfree topology was used by Persson [Per96] Following a proof by Sambin [Sam95], Persson developed a machine assisted, constructive completeness proof for intuitionistic predicate logic, using models based on formal topology. In the second paper the continuum is dened as a formal topology from a base of rational intervals, using only nitary inductive denitions. Then a ....

G. Sambin. Pretopologies and completeness proofs, The Journal of Symbolic Logic 60, pp. 861878, 1995.

....exploring completeness of various logical systems by means of canonical embedding to locales. It should be interesting to develop richer tools for this purposes, in order to handle additional logical operators. The well known Henkin construction for instance, has been investigated in this setting [14] for linear logic. 15 Acknowledgment. We would like to thank the anonymous referees for insightful comments which lead to the improved presentation. ....

G. Sambin. Pretopologies and completeness proofs. J. Symbolic Logic 60 (1995), no. 3, 861-878.

....exploring completeness of various logical systems by means of canonical embedding to locales. It should be interesting to develop richer tools for this purposes, in order to handle additional logical operators. The well known Henkin construction for instance, has been investigated in this setting [12]. ....

G. Sambin. Pretopologies and completeness proofs. J. Symbolic Logic 60 (1995), no. 3, 861--878.

.... , #1 is a commutative monoid. Moreover, the entailment (#) satisfies the adjoint condition: ## ## # ## if and only if ## # ## # ##. Thus we have a commutative monoid, S, with as product, # as its adjoint operator, and # as order relation, hence using Dedekind MacNeille theorem [ Sambin, 1994 ] the above structure can be embedded into a Quantale Q. To construct Q we need the following closure operation: if X # S, then the closure of X is defined as X = ## : # # #) X # ## # ## # ##) Using the above closure operator, we have that Q = X # S : X = X . It can be endowed ....

G. Sambin. Pretopologies and completeness proof. Journal of Symbolic Logics, 1994.

.... which has a zero function z(x) together with the transfer principle that if z(x) holds in the lattice, then x 2 radM (I) holds constructively in the standard sense (in Set) Two possible ways to construct such a distributive lattice is to use the notion of phase space [Gir87] or formal topology [Sam86,Sam95]. 5 3.1 Phase Semantics Definition 8. Let R be an Abelian monoid. A phase space over R is a subset Z R, such that 1. if x 2 Z and y 2 R, then xy 2 Z, and 2. if x 2 2 Z, then x 2 Z. Any phase space gives rise to a distributive lattice 1 . Define a negation, U , as the set f r 2 R j rU ....

G. Sambin. Pretopologies and completeness proofs. Journal of Symbolic Logic, 60:861--878, 1995. 19

....for mathematical notions is too restricted. We can look for entirely new ways to do mathematics that are especially suited for formalisation. An example of this is the use of formal topological models that has recently received interest in the type theory community. Following the ideas of Sambin (1995), Persson (1996) describes the formalisation of a completeness theorem for intuitionistic first order logic using formal topological models. Coquand (1995) proposes a program of prooftheoretic analysis of non effective arguments using formal topological models. Such problems seemed impossible to ....

Sambin, G. (1995). Pretopologies and completeness proofs, Journal of Symbolic Logic 60: 861--878.

....logic. It encodes linear proofs as well as affine linear and relevant proofs (without distribution) This report focuses on a bimodal version of intuitionistic linear logic. We prove a soundness and completeness theorem for the pretopology semantics for Linear Logic developed by G. Sambin in [17, 18, 19]. Our main contribution is in modeling Bimodal Intuitionistic Linear Logic, commutative or not, and proving it sound and complete in an extension of Sambin s pretopologies. The construction of a canonical model and the completeness theorem we propose will work just as well in the limit case where ....

....and intuitive semantic framework (perhaps more so than the phase space semantics) So we prefer to present our models as pretopologies of a certain kind, namely modal pretopologies as described in Definition 3.3. For intuitions behind the pretopology semantics we refer the reader to G. Sambin [17, 18, 19], which is also the source of the following definition. Definition 3.1 A pretopology is a quadruple X = X; e; Delta) such that 1. X; e) is a monoid, called the formal base of the pretopology 2. Delta is a relation (the precover relation of the pretopology X ) between elements and ....

[Article contains additional citation context not shown here]

G. Sambin, "Pretopologies and Completeness Proofs", Journal of Symbolic Logic 60, No 3, 1995.

....systems. An example of this is the use of formal topological models that has recently received interest in the Type Theory community. Persson [Per96] describes the formalization of a completeness theorem for intuitionistic first order logic, using formal topological models as suggested by Sambin [Sam95]. Coquand [Coq] proposes a program of proof theoretic analysis of non effective arguments using formal topological models. Such problems seemed infeasible for Constructive Type Theory until this approach was developed. 4.3 How to believe a mechanical proof checker Every computer program is a ....

Giovanni Sambin. Pretopologies and completeness proofs. Journal of Symbolic Logic, 60:861--878, 1995.

.... to the case w = in g i (y) and u i = g i (y: 1 Gamma x) v i = y:g i (x) We introduce next a covering relation on the set of clopen of the space X : x Delta U = 8z) 8y 2 U)Z I (z:y) Z I (z:x) This defines a formal space M I , following Sambin s definition of a formal topology [16]. Proposition 2: The relation x Delta U satisfies ffl if x 2 U; then x Delta U; ffl if x Delta U; and u Delta V for all u 2 U; then x Delta V; ffl if x Delta U and x Delta V; then x Delta U:V; ffl if x Delta U then x:y Delta U; ffl x Delta g(x) ffl 1 Delta 1 Gamma x; W n in ....

G. Sambin. Pretopologies and completeness proofs. J. Symbolic Logic 60, (1995), p. 863-878

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G. Sambin, Pretopologies and completeness proofs. Journal of Symbolic Logic, 60 (1995), pp. 861-878.

....to a maximally consistent set, satisfying the following existence property: if it contains (9x)OE it also contains some substitution OE(y=x) of a variable y for x. In Feferman s review [5] of [13] an improvement, due to Tarski, is given by which the proof gets a simple algebraic form. Sambin [15] used the same method in the setting of formal topology [16] thereby obtaining a constructive completeness proof. This proof is elementary and can be seen as a constructive and predicative version of the one in Feferman s review. It is a typical, and simple, example where the use of formal ....

....definition, while the definition of Delta DM is elementary. Given that Sambin s completeness proof can be seen as a constructive version of the Henkin Rasiowa Sikorski proof, it was natural to conjecture that the points of this topology correspond to Henkin sets; this conjecture appears in [15]. For the inductive topology, it is easy to see that the points correspond to Henkin sets. Hence, the natural question: do these two topologies coincide We show in this paper that the question has a simple negative answer. This raised further natural questions on what can be said about the points ....

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G. Sambin. Pretopologies and completeness proofs. Journal of Symbolic Logic, 60:861--878, 1995.

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