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A Judgmental Reconstruction of Modal Logic
 Mathematical Structures in Computer Science
, 1999
"... this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deduction for ..."
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Cited by 160 (38 self)
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this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deduction for intuitionistic modal logic which does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic [FM97] and find that it is already contained in modal logic, using the decomposition of the lax modality fl A as
Coercive Subtyping in Type Theory
 Proc. of CSL'96, the 1996 Annual Conference of the European Association for Computer Science Logic, Utrecht. LNCS 1258
, 1996
"... We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as MartinLof's type theory [NPS90] and the type theory UTT [Luo94]. In this approach, subtyping with specified implicit coercions is treated as a feature at the level of the logical framework; ..."
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Cited by 26 (14 self)
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We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as MartinLof's type theory [NPS90] and the type theory UTT [Luo94]. In this approach, subtyping with specified implicit coercions is treated as a feature at the level of the logical framework; in particular, subsumption and coercion are combined in such a way that the meaning of an object being in a supertype is given by coercive definition rules for the definitional equality. It is shown that this provides a conceptually simple and uniform framework to understand subtyping and coercion relations in type theories with sophisticated type structures such as inductive types and universes. The use of coercive subtyping in formal development and in reasoning about subsets of objects is discussed in the context of computerassisted formal reasoning. 1 Introduction A type in type theory is often intuitively thought of as a set. For example, types in MartinLof's type theory [ML84, NPS90...
Intensionality, Extensionality, and Proof Irrelevance in Modal Type Theory
 Pages 221–230 of: Symposium on Logic in Computer Science
, 2001
"... We develop a uniform type theory that integrates intensionality, extensionality, and proof irrelevance as judgmental concepts. Any object may be treated intensionally (subject only to #conversion), extensionally (subject also to ##conversion), or as irrelevant (equal to any other object at the sam ..."
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Cited by 5 (3 self)
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We develop a uniform type theory that integrates intensionality, extensionality, and proof irrelevance as judgmental concepts. Any object may be treated intensionally (subject only to #conversion), extensionally (subject also to ##conversion), or as irrelevant (equal to any other object at the same type), depending on where it occurs. Modal restrictions developed in prior work for simple types are generalized and employed to guarantee consistency between these views of objects. Potential applications are in logical frameworks, functional programming, and the foundations of firstorder modal logics.
Chapter 7 Equality
"... Reasoning with equality in first order logic can be accomplished axiomatically. That is, we can simply add reflexivity, symmetry, transitivity, and congruence rules for each predicate and function symbol and use the standard theorem proving technology developed in the previous chapters. This approac ..."
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Reasoning with equality in first order logic can be accomplished axiomatically. That is, we can simply add reflexivity, symmetry, transitivity, and congruence rules for each predicate and function symbol and use the standard theorem proving technology developed in the previous chapters. This approach, however, does not take strong advantage of inherent properties of equality and leads to a very large and inefficent search space. While there has been a deep investigation of equality reasoning in classical logic, much less is known for intuitionistic logic. Some recent references are [Vor96, DV99]. In this chapter we develop some of the techniques of equational reasoning, starting again from first principles in the definition of logic. We therefore recapitulate some of the material in earlier chapters, now adding equality as a new primitive predicate symbol. 7.1 Natural Deduction We characterize equality by its introduction rule, which simply states that s. = s for any term s. =I ⊢ s. = s We have already seen this introduction rule in unification logic in Section 4.3. In the context of unification logic, however, we did not consider hypothetical judgments, so we did not need or specify elimination rules for equality. If we know s. = t we can replace any number of occurrences of s in a true proposition and obtain another true proposition. ⊢ s. = t ⊢ [s/x]A.=E1 ⊢ [t/x]A
Chapter 5 The Inverse Method
, 1999
"... After the definition of logic via natural deduction, we have developed a succession of techniques for theorem proving based on sequent calculi. We considered asequentΓ= ⇒ C as a goal, to be solved by backwardsdirected search which was modeled by the bottomup construction of a derivation. The criti ..."
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After the definition of logic via natural deduction, we have developed a succession of techniques for theorem proving based on sequent calculi. We considered asequentΓ= ⇒ C as a goal, to be solved by backwardsdirected search which was modeled by the bottomup construction of a derivation. The critical choices were disjunctive nondeterminism (resolved by guessing and backtracking) and existential nondeterminism (resolved by introducing existential variables and unification). The limiting factor in more refined theorem provers based on this method is generally the number of disjunctive choices which have to be made. It is complicated by the fact that existential variables are global in a partial derivation, which means that choices in one conjunctive branch have effects in other branches. This effects redundancy elimination, since subgoals are not independent of each other. The diametrically opposite approach would be to work forward from the initial sequents until the goal sequent is reached. If we guarantee a fair strategy in the selection of axioms and inference rules, every goal sequent can be derived this way. Without further improvements, this is clearly infeasible, since there are too many derivations for us to hope that we can find one for the goal sequent in this manner. The inverse method is based on the property that in a cutfree derivation of a goal sequent, we only need to consider subformulas of the goal and their substitution instances. For example, when we have derived both A and B in the forward direction, we only derive their conjunction A ∧ B if A ∧ B is a subformula of the goal sequent. The nature of forward search under these restrictions is quite different from the backward search. Since we always add new consequences to the sequents already derived, knowledge grows monotonicallyand no disjunctive nondeterminism arises. Similarly for existential nondeterminism, if we keep sequents in their maximally general form. On the other hand, there is a potentially very large amount of conjunctive nondeterminism, since we have to apply all applicable rules to all sequents in a fair manner in order to guarantee completeness. The critical factor in forward search is to limit conjunctive nondeterminism. We
unknown title
"... The inference rules so far only model intuitionistic logic, and some classically true propositions such as A ∨¬A (for an arbitrary A) are not derivable, as we will see in Section??. There are three commonly used ways one can construct a system of classical natural deduction by adding one additional ..."
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The inference rules so far only model intuitionistic logic, and some classically true propositions such as A ∨¬A (for an arbitrary A) are not derivable, as we will see in Section??. There are three commonly used ways one can construct a system of classical natural deduction by adding one additional rule of inference. ⊥C is called Proof by Contradiction or Rule of Indirect Proof, ¬¬C is the Double Negation Rule, and XM is referred to as Excluded Middle.
.2 Classical Logic
"... Since hypotheses and their restrictions are critical for linear logic, we give here a formulation of natural deduction for intuitionistic logic with localized hypotheses, but not parameters. For this we need a notation for hypotheses which we call a context. Contexts \Gamma ::= \Delta j \Gamma; u:A ..."
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Since hypotheses and their restrictions are critical for linear logic, we give here a formulation of natural deduction for intuitionistic logic with localized hypotheses, but not parameters. For this we need a notation for hypotheses which we call a context. Contexts \Gamma ::= \Delta j \Gamma; u:A Here, "\Delta" represents the empty context, and \Gamma; u:A adds hypothesis ` A labelled u to \Gamma. We assume that each label u occurs at most once in a context in order to avoid ambiguities. The main judgment can then be written as \Gamma ` A, where \Delta; u 1 :A 1 ; : : : ; un :An ` A stands for u 1 ` A 1 : : :<F43.12
.1 Natural Deduction
"... 108 Equality Symmetrically, we can also replaces of occurrences of t by s. ` s : = t ` [t=x]A : = E 2 ` [s=x]A It might seem that this second rule is redundant, and in some sense it is. In particular, it is a derivable rule of the calculus with only : = E 1 : ` s : = t : = I ` s : = s : ..."
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108 Equality Symmetrically, we can also replaces of occurrences of t by s. ` s : = t ` [t=x]A : = E 2 ` [s=x]A It might seem that this second rule is redundant, and in some sense it is. In particular, it is a derivable rule of the calculus with only : = E 1 : ` s : = t : = I ` s : = s : = E 1 ` t : = s ` [t=x]A : = E 1 ` [s=x]A However, this deduction is not normal (as defined below), and without the second elimination rule the normalization theorem would not hold and cut elimination in the sequent calculus would f