## Logic of Predicates With Explicit Substitutions (1996)

Venue: | Mathematical Foundations of Computer Science 1996, 21st Symposium |

Citations: | 3 - 3 self |

### BibTeX

@INPROCEEDINGS{Bednarczyk96logicof,

author = {Marek A. Bednarczyk},

title = {Logic of Predicates With Explicit Substitutions},

booktitle = {Mathematical Foundations of Computer Science 1996, 21st Symposium},

year = {1996},

pages = {192--206}

}

### OpenURL

### Abstract

This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one offered by Girard. The latter, cf. [9], translates every sequent of the usual propositional logic (classical, or intuitionistic) into a sequent of commutative linear logic. Then one shows that a sequent can be proved classically, resp., intuitionistically, iff its translation can be proved linearly. By contrast, our embedding only works on the level of predicate logic. We show that every theory of classical logic of predicates with equality lives as a theory within a non-commutative intuitionistic substructural logic: the logic of predicates with equality and explicit substitution. Also, our explanation does not require to call upon so called exponentials --- the modalities introduced by Girard just to facilitate his embedding. Our construction is also different from other proposals to move substitutions from the level of metatheory to the theory of logic, cf. [16]. They add substitutions as modal constructions. Here, substitutions are considered new atomic formulae.

### Citations

397 | Explicit substitutions
- Abadi, Cardelli, et al.
- 1991
(Show Context)
Citation Context ...process of performing a substitution. This calls for frameworks with substitution as a primitive operation. Indeed, a variety of -calculae with explicit substitutions have already been considered, cf =-=[1, 14]-=-. All of them are 2-sorted --- the old syntactic class of - terms is retained while a new class of substitutions is added. The logic of predicates has already two sorts: the sort of terms and, built o... |

332 |
Introduction to Mathematical Logic
- Mendelson
- 1987
(Show Context)
Citation Context ...e in mathematics and in computer science. From the perspective of the intended embedding it presence is, simply, indispensable. Our axiomatization is equivalent to the well-established tradition, cf. =-=[15, 8]-=-. (=) ` e = e (=s) e = e 0 ; '[e=x] ` '[e 0 =x] The first axiom schema asserts transitivity of equality. The second axiom schema relates substitution to equality. It captures the idea that equals may ... |

162 |
Logic for Computer Science. Foundation of Automatic Theorem Proving
- Gallier
- 1986
(Show Context)
Citation Context ...e consequence relation is at the same time a disjunction with respect to the dual consequence relation. Presentation of logics in terms of invertible rules offers many technical advantages, see e.g., =-=[8, 9]-=-. Avron, cf. [2], has made of it a dogma saying that the (proof-theoretic) meaning of a logical connective should always be given in terms of an invertible rule. The dogma limits the number of potenti... |

161 | logic: its syntax and semantics
- Linear
- 1995
(Show Context)
Citation Context ...y and explicit substitutions. Thus, the position of linear logic with respect to the usual logic is given a new explanation. 1 The world according to Girard A recent introduction to linear logic, cf. =-=[13]-=-, starts with the following explanation of the position of usual logic with respect to the linear. Linear logic is not an alternative logic ; it should rather be seen as an extension of usual logic. T... |

147 |
Proofs and Types. Volume 7 of Cambridge Tracts
- Girard
- 1989
(Show Context)
Citation Context ...e logic is given in Table 2 in the appendix. With one exception, the rules in Table 2 are the natural generalisations of the rules given by Girard for the commutative intuitionistic linear logic, cf. =-=[10, 11, 12]-=-, to the non-commutative case, cf [7]. The exceptional axiom is (?). Its expected generalisation is \Gamma; ?; \Delta ` A, as in [7]. However, the stronger axiom is not valid in our intended interpret... |

101 | Simple consequence relations
- Avron
- 1991
(Show Context)
Citation Context ...considered new atomic formulae. 1.1 Linear logic In classical or intuitionistic logics there are several equivalent ways of saying what conjunction or disjunction is. It has been argued by Avron, cf. =-=[2]-=-, that logical connectives do not exist outside the context provided by the underlying logic, i.e., the underlying consequence relation. In other words, only after defining a consequence relation it i... |

39 |
Linear logic and lazy computation
- Girard, Lafont
- 1986
(Show Context)
Citation Context ...e logic is given in Table 2 in the appendix. With one exception, the rules in Table 2 are the natural generalisations of the rules given by Girard for the commutative intuitionistic linear logic, cf. =-=[10, 11, 12]-=-, to the non-commutative case, cf [7]. The exceptional axiom is (?). Its expected generalisation is \Gamma; ?; \Delta ` A, as in [7]. However, the stronger axiom is not valid in our intended interpret... |

17 |
Basic category theory
- PoignÃ©
- 1992
(Show Context)
Citation Context ...odalities introduced by Girard just to facilitate his embedding. Our construction is also different from other proposals to move substitutions from the level of metatheory to the theory of logic, cf. =-=[16]-=-. They add substitutions as modal constructions. Here, substitutions are considered new atomic formulae. 1.1 Linear logic In classical or intuitionistic logics there are several equivalent ways of say... |

15 |
Relations and Non-commutative Linear Logic
- Brown, Gurr
- 1991
(Show Context)
Citation Context ...ith one exception, the rules in Table 2 are the natural generalisations of the rules given by Girard for the commutative intuitionistic linear logic, cf. [10, 11, 12], to the non-commutative case, cf =-=[7]-=-. The exceptional axiom is (?). Its expected generalisation is \Gamma; ?; \Delta ` A, as in [7]. However, the stronger axiom is not valid in our intended interpretation in quasi quantales as described... |

14 | Constructive logics . Part II : Linear logic and proof nets
- Gallier
- 1992
(Show Context)
Citation Context ...rather be seen as an extension of usual logic. This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one offered by Girard. The latter, cf. =-=[9]-=-, translates every sequent of the usual propositional logic (classical, or intuitionistic) into a sequent of commutative linear logic. Then one shows that a sequent can be proved classically, resp., i... |

10 | A Modular Presentation of Modal Logics in a Logical Framework
- Basin, Matthews, et al.
(Show Context)
Citation Context ...rately, see (2). Equivalent, Gentzen-style formulation of conditions (1) and (2) can be found in [2]. Thus, (1) is equivalent to the assumption that the consequence relation is closed under the rules =-=(3)-=-. Similarly, (2) is equivalent to the assumption that the consequence relation is closed under the rules (4). \Gamma; A; B ` \Delta \Gamma; A\Omega B ` \Delta \Gamma ` A; \Delta \Gamma 0 ` B; \Delta 0... |

5 |
Towards program development with Isabelle
- Bednarczyk, Borzyszkowski
- 1995
(Show Context)
Citation Context ... sort. All attempts known to the author use the idea that substitutions behave as modal operators, see e.g., [16]. We have good reasons to consider substitutions as a new kind of atomic formulae, cf. =-=[5]-=-. Predicates are eternal. They represent facts the truth of which does not depend on the context, or state, in which their truth is evaluated. Therefore, we call them Platonic here. Substitutions prov... |

3 |
Quasi quantales
- Bednarczyk
- 1994
(Show Context)
Citation Context ...m is (?). Its expected generalisation is \Gamma; ?; \Delta ` A, as in [7]. However, the stronger axiom is not valid in our intended interpretation in quasi quantales as described in section 4 and in. =-=[4]-=-. Embedding the usual logic into LP = oe gives a good reason for not assuming that having falsehood as one of the assumptions always logically implies anything. 3.2 Platonic Formulae The idea underlyi... |

2 | A.,(1995a), Logic of predicates versus linear logic
- Bednarczyk
- 1995
(Show Context)
Citation Context ... condition that all the assumptions and conclusions have the form !C, and ?D, respectively. The following logical equivalences hold in linear logic. !A\Omega !B a` !(A & B) and ?A # ?B a` ?(A \Phi B) =-=(6)-=- 1.3 Representing classical logics into linear logic The addition of exponentials helps --- classical and intuitionistic logics can now be represented in linear logic. The exposition given below is ba... |

2 |
From oe to v a journey through calculi of explicit substitutions
- Lescane
- 1994
(Show Context)
Citation Context ...process of performing a substitution. This calls for frameworks with substitution as a primitive operation. Indeed, a variety of -calculae with explicit substitutions have already been considered, cf =-=[1, 14]-=-. All of them are 2-sorted --- the old syntactic class of - terms is retained while a new class of substitutions is added. The logic of predicates has already two sorts: the sort of terms and, built o... |