2,069 research outputs found
Transparent quantification into hyperpropositional contexts de re
This paper is the twin of (Duží and Jespersen, in submission), which provides a logical rule for transparent quantification into hyperprop- ositional contexts de dicto, as in: Mary believes that the Evening Star is a planet; therefore, there is a concept c such that Mary be- lieves that what c conceptualizes is a planet. Here we provide two logical rules for transparent quantification into hyperpropositional contexts de re. (As a by-product, we also offer rules for possible- world propositional contexts.) One rule validates this inference: Mary believes of the Evening Star that it is a planet; therefore, there is an x such that Mary believes of x that it is a planet. The other rule validates this inference: the Evening Star is such that it is believed by Mary to be a planet; therefore, there is an x such that x is believed by Mary to be a planet. Issues unique to the de re variant include partiality and existential presupposition, sub- stitutivity of co-referential (as opposed to co-denoting or synony- mous) terms, anaphora, and active vs. passive voice. The validity of quantifying-in presupposes an extensional logic of hyperinten- sions preserving transparency and compositionality in hyperinten- sional contexts. This requires raising the bar for what qualifies as co-denotation or equivalence in extensional contexts. Our logic is Tichý’s Transparent Intensional Logic. The syntax of TIL is the typed lambda calculus; its highly expressive semantics is based on a procedural redefinition of, inter alia, functional abstraction and application. The two non-standard features we need are a hyper- intension (called Trivialization) that presents other hyperintensions and a four-place substitution function (called Sub) defined over hy- perintensions
Quotient completion for the foundation of constructive mathematics
We apply some tools developed in categorical logic to give an abstract
description of constructions used to formalize constructive mathematics in
foundations based on intensional type theory. The key concept we employ is that
of a Lawvere hyperdoctrine for which we describe a notion of quotient
completion. That notion includes the exact completion on a category with weak
finite limits as an instance as well as examples from type theory that fall
apart from this.Comment: 32 page
W-types in setoids
W-types and their categorical analogue, initial algebras for polynomial
endofunctors, are an important tool in predicative systems to replace
transfinite recursion on well-orderings. Current arguments to obtain W-types in
quotient completions rely on assumptions, like Uniqueness of Identity Proofs,
or on constructions that involve recursion into a universe, that limit their
applicability to a specific setting. We present an argument, verified in Coq,
that instead uses dependent W-types in the underlying type theory to construct
W-types in the setoid model. The immediate advantage is to have a proof more
type-theoretic in flavour, which directly uses recursion on the underlying
W-type to prove initiality. Furthermore, taking place in intensional type
theory and not requiring any recursion into a universe, it may be generalised
to various categorical quotient completions, with the aim of finding a uniform
construction of extensional W-types.Comment: 17 pages, formalised in Coq; v2: added reference to formalisatio
Towards an extensional calculus of hyperintensions
In this paper I describe an extensional logic of hyperintensions, viz. Tichý's Transparent Intensional Logic (TIL). TIL preserves transparency and compositionality in all kinds of context, and validates quantifying into all contexts, including intensional and hyperintensional ones. The received view is that an intensional (let alone hyperintensional) context is one that fails to validate transparency, compositionality, and quantifying-in; and vice versa, if a context fails to validate these extensional principles, then the context is 'opaque', that is non-extensional. We steer clear of this circle by defining extensionality for hyperintensions presenting functions, functions (including possible-world intensions), and functional values. The main features of our logic are that the senses of expressions remain invariant across contexts and that our ramified type theory enables quantification over any logical objects of any order into any context. The syntax of TIL is the typed lambda calculus; its semantics is based on a procedural redefinition of, inter alia, functional abstraction and application. The only two non-standard features of our logic are a hyperintension called Trivialization and a fourplace substitution function (called Sub) defined over hyperintensions. Using this logical machinery I propose rules of existential generalization and substitution of identicals into the three kinds of context.Web of Science191452
Guarded Cubical Type Theory: Path Equality for Guarded Recursion
This paper improves the treatment of equality in guarded dependent type
theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an
extensional type theory with guarded recursive types, which are useful for
building models of program logics, and for programming and reasoning with
coinductive types. We wish to implement GDTT with decidable type-checking,
while still supporting non-trivial equality proofs that reason about the
extensions of guarded recursive constructions. CTT is a variation of
Martin-L\"of type theory in which the identity type is replaced by abstract
paths between terms. CTT provides a computational interpretation of functional
extensionality, is conjectured to have decidable type checking, and has an
implemented type-checker. Our new type theory, called guarded cubical type
theory, provides a computational interpretation of extensionality for guarded
recursive types. This further expands the foundations of CTT as a basis for
formalisation in mathematics and computer science. We present examples to
demonstrate the expressivity of our type theory, all of which have been checked
using a prototype type-checker implementation, and present semantics in a
presheaf category.Comment: 17 pages, to be published in proceedings of CSL 201
Deduction in TIL: from simple to ramified hierarchy of types
Tichý’s Transparent Intensional Logic (TIL) is an overarching logical
framework apt for the analysis of all sorts of discourse, whether colloquial, scientific,
mathematical or logical. The theory is a procedural (as opposed to denotational) one, according
to which the meaning of an expression is an abstract, extra-linguistic procedure
detailing what operations to apply to what procedural constituents to arrive at the product
(if any) of the procedure that is the object denoted by the expression. Such procedures
are rigorously defined as TIL constructions. Though TIL analytical potential is
very large, deduction in TIL has been rather neglected. Tichý defined a sequent calculus
for pre-1988 TIL, that is TIL based on the simple theory of types. Since then no
other attempt to define a proof calculus for TIL has been presented. The goal of this
paper is to propose a generalization and adjustment of Tichý’s calculus to TIL 2010.
First I briefly recapitulate the rules of simple-typed calculus as presented by Tichý.
Then I propose the adjustments of the calculus so that it be applicable to hyperintensions
within the ramified hierarchy of types. TIL operates with a single procedural semantics
for all kinds of logical-semantic context, be it extensional, intensional or hyperintensional.
I show that operating in a hyperintensional context is far from being technically
trivial. Yet it is feasible. To this end we introduce a substitution method that
operates on hyperintensions. It makes use of a four-place substitution function (called
Sub) defined over hyperintensions.Web of Science20suppl 236
Extensional and Intensional Strategies
This paper is a contribution to the theoretical foundations of strategies. We
first present a general definition of abstract strategies which is extensional
in the sense that a strategy is defined explicitly as a set of derivations of
an abstract reduction system. We then move to a more intensional definition
supporting the abstract view but more operational in the sense that it
describes a means for determining such a set. We characterize the class of
extensional strategies that can be defined intensionally. We also give some
hints towards a logical characterization of intensional strategies and propose
a few challenging perspectives
Metamodel for Tracing Concerns across the Life Cycle
Several aspect-oriented approaches have been proposed to specify aspects at different phases in the software life cycle. Aspects can appear within a phase, be refined or mapped to other aspects in later phases, or even disappear.\ud
Tracing aspects is necessary to support understandability and maintainability of software systems. Although several approaches have been introduced to address traceability of aspects, two important limitations can be observed. First, tracing is not yet tackled for the entire life cycle. Second, the traceability model that is applied usually refers to elements of specific aspect languages, thereby limiting the reusability of the adopted traceability model.We propose the concern traceability metamodel (CTM) that enables traceability of concerns throughout the life cycle, and which is independent from the aspect languages that are used. CTM can be enhanced to provide additional properties for tracing, and be instantiated to define\ud
customized traceability models with respect to the required aspect languages. We have implemented CTM in the tool M-Trace, that uses XML-based representations of the models and XQuery queries to represent tracing information. CTM and M-Trace are illustrated for a Concurrent Versioning System to trace aspects from the requirements level to architecture design level and the implementation
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