## Finitary Partial Inductive Definitions as a General Logic (1994)

Citations: | 11 - 1 self |

### BibTeX

@TECHREPORT{Eriksson94finitarypartial,

author = {Lars-henrik Eriksson},

title = {Finitary Partial Inductive Definitions as a General Logic},

institution = {},

year = {1994}

}

### OpenURL

### Abstract

. We describe how the calculus of partial inductive definitions is used to represent logics. This calculus includes the powerful principle of definitional reflection. We describe two conceptually different approaches to representing a logic, both making essential use of definitional reflection. In the deductive approach, the logic is defined by its inference rules. Only the succedent rules (in a sequent calculus setting -- introduction rules in a natural deduction setting) need be given. The other rules are obtained implicitly using definitional reflection. In the semantic approach, the logic is defined using its valuation function. The latter approach often provides a more straightforward representation of logics with simple semantics but complicated proof systems. 1 Introduction: Finitary Partial Inductive Definitions We will describe how to use the calculus of partial inductive definitions as a general logic. That is, as a framework for representing various logics. Following common...

### Citations

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- Kleene
- 1952
(Show Context)
Citation Context ...of the next example is to show how the semantic approach can be extended in a natural way to logics with more than two truth values. We will represent the threevalued logics of Kleene and Lukasiewicz =-=[15]-=-. The example was inspired by the representation of three-valued logic in LF described in [1]. Kleene's logic has three truth values: t, f, and u. The first two are the usual truth and falsity values,... |

709 | A framework for defining logics
- Harper, Honsell, et al.
- 1993
(Show Context)
Citation Context ...r using the calculus of finitary partial inductive definitions as a general logic. Expressions of the object logic will be represented by expressions of typed lambda calculus in the standard way (see =-=[14]-=- for a detailed presentation in the context of the Edinburgh Logical Framework). Free and bound variables of the metalogic will typically be used to represent free and bound variables of the object lo... |

430 | The foundation of a generic theorem prover
- Paulson
- 1989
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Citation Context ...a logic. By introducing additional judgments, they can often be worked around. Also, they are not unique to our framework. E.g. the Edinburgh Logical Framework (LF) [14] and the metalogic of Isabelle =-=[18]-=- share -- apart from the lack of connection between antecedent and succedent rules -- the same basic constraints. Still, many nontrivial logics can be represented in these frameworks -- possibly using... |

290 | A logic programming language with lambda-abstraction, function variables, and simple unification
- Miller
- 1991
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Citation Context ...ers (mgus) exist whenever two expressions are unifiable. That is not always the case with higher-order expressions. In this paper, all higher-order expressions will be so-called higher-order patterns =-=[16]-=- where mgus always exist. In the more general case, there must be one premise for each clause and each unifier in some complete set of unifiers for a and the head of that clause -- see [6] for details... |

177 |
An overview of >'Prolog
- Nadathur, Miller
- 1988
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Citation Context ...es higher-order resolution. A third interesting approach is that of Felty [7, 8]. Felty's work intends to specify theorem provers using a higher-order logic programming language (specifically .Prolog =-=[17]-=-). In that respect, her approach is quite similar to that of Isabelle in that languages of the .Prolog family implement higher-order resolution. There are, however, important differences in how the re... |

83 | Using Typed Lambda Calculus to Implement Formal Systems on a Machine
- Avron, Mason, et al.
- 1992
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Citation Context ...ecedent and succedent rules -- the same basic constraints. Still, many nontrivial logics can be represented in these frameworks -- possibly using extra machinery to overcome the constraints (see e.g. =-=[1] for the case of LF). The following partial inductive-=- definition (which we will call FOL) defines first-order (intuitionistic) logic. A��B �� A, B A��B �� A A��B �� B A��B �� A��B A �� A��^ "x A(x) ��... |

71 |
Specifying theorem provers in a higher-order logic programming language
- Felty, Miller
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Citation Context ...Logical Framework (LF) [14] which represents logics using a dependent type system, and the metalogic of Isabelle [18] which uses higher-order resolution. A third interesting approach is that of Felty =-=[7, 8]-=-. Felty's work intends to specify theorem provers using a higher-order logic programming language (specifically lProlog [17]). In that respect, her approach is quite similar to that of Isabelle in tha... |

71 |
A proof-theoretic approach to logic programming. II. Programs as definitions
- Hallnäs, Schroeder-Heister
- 1991
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Citation Context ...elated Work The first presentation of a finitary deductive system based on the same ideas as those underlying the theory of partial inductive definitions was given by Halln��s and SchroederHeister=-= in [11]-=-. That system did not include parameters or universal quantification (but did include logical variables). In [4], the present author used an informal finitary presentation of partial inductive definit... |

59 | Rules of definitional reflection
- Schroeder-Heister
- 1993
(Show Context)
Citation Context ...is the corresponding most general unifier, e.g. s i * =mgu(a,H i ). The clauses must not have any parameters in common with G, a or C. The D -- rule expresses the principle of definitional reflection =-=[20]-=-. An important special case of this rule is when no clause heads unify with a. In that case n=0 so the inference step has no premises. That is, the conclusion sequent is immediately proved. We say tha... |

50 | Specifying and Implementing Theorem Provers in a Higher-Order Logic Programming Language
- Felty
- 1989
(Show Context)
Citation Context ...Logical Framework (LF) [14] which represents logics using a dependent type system, and the metalogic of Isabelle [18] which uses higher-order resolution. A third interesting approach is that of Felty =-=[7, 8]-=-. Felty's work intends to specify theorem provers using a higher-order logic programming language (specifically lProlog [17]). In that respect, her approach is quite similar to that of Isabelle in tha... |

28 | A finitary version of the calculus of partial inductive definitions
- Eriksson
- 1991
(Show Context)
Citation Context ...this can be seen as a precursor of the w-rule of [20]). Formalising the idea of case analysis and combining it with the D -- rule resulted in the present finitary system, which was first presented in =-=[5]-=-, and further developed in [6]. Another finitary presentation is that of Hanschke [13], using Skolemisation rather than parameters to represent universally quantified variables. Hanschke's approach is... |

28 |
Partial inductive definitions
- Hallnäs
- 1991
(Show Context)
Citation Context ...a finitary version of it since the proper calculus is infinitary, and thus unsuitable for use as a metalogic. The finitary version is described in detail in [6], and the original infinitary theory in =-=[12]-=-. The presentation given here is slightly different from that of [6]. Formulae of the metalogic (called conditions) will be the following: t where t is an expression of the simply typed lambda calculu... |

11 |
A constructive presentation for the modal connective of necessity
- Benevides, Maibaum
- 1992
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Citation Context ...are notorious for having complicated proof theories -- in particular natural deduction systems -- which makes them less well suited to a representation based on inference rules. Benevides and Maibaum =-=[2]-=- have proposed a elegant and general natural deduction treatment of modal logic. It is interesting to see that although they do not use explicit accessibility relations, their treatment does make use ... |

9 |
Structural frameworks, substructural logics, and the role of elimination inferences
- Schroeder-Heister
- 1991
(Show Context)
Citation Context ... are proper inference rules, the last one is implicit in viewing the antecedent of a sequent as a multiset. A sequent calculus with these features is called a structural framework by SchroederHeister =-=[19]. Thi-=-s puts some constraints on what logics we can represent. Any logics that are represented directly must admit general contraction, weakening and permutation. Logics that do not -- so called "subst... |

8 |
1993], What is the status of pattern matching in type theory
- Coquand, Smith
(Show Context)
Citation Context ...itly using the principle of definitional reflection (D -- rule). This use of definitional reflection is unique among general logics, although it has recently been applied to MartinL ��f's type the=-=ory [3]-=-. The other approach will be called the semantic approach. Here, the emphasis is on representing the truth of a judgment in an interpretation according to some semantics. The semantics of the logic is... |

6 |
Partial Inductive Definitions
- Halln��s
- 1991
(Show Context)
Citation Context ...a finitary version of it since the proper calculus is infinitary, and thus unsuitable for use as a metalogic. The finitary version is described in detail in [6], and the original infinitary theory in =-=[12]-=-. The presentation given here is slightly different from that of [6]. Formulae of the metalogic (called conditions) will be the following: t where t is an expression of the simply typed lambda calculu... |

6 |
An overview of lProlog
- Nadathur, Miller
- 1988
(Show Context)
Citation Context ...es higher-order resolution. A third interesting approach is that of Felty [7, 8]. Felty's work intends to specify theorem provers using a higher-order logic programming language (specifically lProlog =-=[17]-=-). In that respect, her approach is quite similar to that of Isabelle in that languages of the lProlog family implement higher-order resolution. There are, however, important differences in how the re... |

4 |
On normalization of proofs in set theory
- Hallnäs
- 1983
(Show Context)
Citation Context ... are made “undefined”.s4 Deductive Approach Example: Naive Set Theory As another example, we modify the definition FOL for reasoning in naive set theory. Our formalisation is based on that of Hallnäs =-=[10]-=-, and the example in [12]. To represent formulae of set theory, we extend first-order logic with a logical constant for set membership. The characteristic of naive set theory as opposed to axiomatic s... |

3 |
A Programming Calculus Based on
- Eriksson, Halln��s
- 1988
(Show Context)
Citation Context ... theory of partial inductive definitions was given by Halln��s and SchroederHeister in [11]. That system did not include parameters or universal quantification (but did include logical variables).=-= In [4]-=-, the present author used an informal finitary presentation of partial inductive definitions including parameters, but without unification of parameters in the D -- rule. Instead, parameters were inst... |

2 |
Terminological Reasoning and Partial Inductive Definitions
- Hanschke
- 1992
(Show Context)
Citation Context ...analysis and combining it with the D -- rule resulted in the present finitary system, which was first presented in [5], and further developed in [6]. Another finitary presentation is that of Hanschke =-=[13]-=-, using Skolemisation rather than parameters to represent universally quantified variables. Hanschke's approach is unsound in general (two different Skolem constants cannot be unified as two parameter... |

1 |
Investigations into Logical Deduction (translation of Untersuchungen ��ber das logische Schlie��en
- Gentzen
- 1969
(Show Context)
Citation Context ...th B as succedent and A in the antecedent, i.e. the rule G,A - B G - A��B -�� which is the rule for implication in the antecedent in Gentzen's system LJ for intuitionistic first-order predicat=-=e logic [9]. Instantiating -=-the --D rule with the clause and using the --�� rule to derive the premise of the --D rule, we get the derivation schema: G -A��B ________ -D G -A��B ________ -�� G, A -B which admits ... |

1 |
On Normalisation of Proofs in Set Theory
- Halln��s
- 1983
(Show Context)
Citation Context ...are made "undefined". 4 Deductive Approach Example: Naive Set Theory As another example, we modify the definition FOL for reasoning in naive set theory. Our formalisation is based on that of=-= Halln��s [10]-=-, and the example in [12]. To represent formulae of set theory, we extend first-order logic with a logical constant for set membership. The characteristic of naive set theory as opposed to axiomatic s... |