1,002 research outputs found
Predicativity, the Russell-Myhill Paradox, and Church's Intensional Logic
This paper sets out a predicative response to the Russell-Myhill paradox of
propositions within the framework of Church's intensional logic. A predicative
response places restrictions on the full comprehension schema, which asserts
that every formula determines a higher-order entity. In addition to motivating
the restriction on the comprehension schema from intuitions about the stability
of reference, this paper contains a consistency proof for the predicative
response to the Russell-Myhill paradox. The models used to establish this
consistency also model other axioms of Church's intensional logic that have
been criticized by Parsons and Klement: this, it turns out, is due to resources
which also permit an interpretation of a fragment of Gallin's intensional
logic. Finally, the relation between the predicative response to the
Russell-Myhill paradox of propositions and the Russell paradox of sets is
discussed, and it is shown that the predicative conception of set induced by
this predicative intensional logic allows one to respond to the Wehmeier
problem of many non-extensions.Comment: Forthcoming in The Journal of Philosophical Logi
Computer Science and Metaphysics: A Cross-Fertilization
Computational philosophy is the use of mechanized computational techniques to
unearth philosophical insights that are either difficult or impossible to find
using traditional philosophical methods. Computational metaphysics is
computational philosophy with a focus on metaphysics. In this paper, we (a)
develop results in modal metaphysics whose discovery was computer assisted, and
(b) conclude that these results work not only to the obvious benefit of
philosophy but also, less obviously, to the benefit of computer science, since
the new computational techniques that led to these results may be more broadly
applicable within computer science. The paper includes a description of our
background methodology and how it evolved, and a discussion of our new results.Comment: 39 pages, 3 figure
First-Order Stable Model Semantics with Intensional Functions
In classical logic, nonBoolean fluents, such as the location of an object,
can be naturally described by functions. However, this is not the case in
answer set programs, where the values of functions are pre-defined, and
nonmonotonicity of the semantics is related to minimizing the extents of
predicates but has nothing to do with functions. We extend the first-order
stable model semantics by Ferraris, Lee, and Lifschitz to allow intensional
functions -- functions that are specified by a logic program just like
predicates are specified. We show that many known properties of the stable
model semantics are naturally extended to this formalism and compare it with
other related approaches to incorporating intensional functions. Furthermore,
we use this extension as a basis for defining Answer Set Programming Modulo
Theories (ASPMT), analogous to the way that Satisfiability Modulo Theories
(SMT) is defined, allowing for SMT-like effective first-order reasoning in the
context of ASP. Using SMT solving techniques involving functions, ASPMT can be
applied to domains containing real numbers and alleviates the grounding
problem. We show that other approaches to integrating ASP and CSP/SMT can be
related to special cases of ASPMT in which functions are limited to
non-intensional ones.Comment: 69 page
Constrained Query Answering
Traditional answering methods evaluate queries only against positive
and definite knowledge expressed by means of facts and deduction rules. They do
not make use of negative, disjunctive or existential information. Negative or indefinite
knowledge is however often available in knowledge base systems, either as
design requirements, or as observed properties. Such knowledge can serve to rule out
unproductive subexpressions during query answering. In this article, we propose an
approach for constraining any conventional query answering procedure with general,
possibly negative or indefinite formulas, so as to discard impossible cases and to
avoid redundant evaluations. This approach does not impose additional conditions
on the positive and definite knowledge, nor does it assume any particular semantics
for negation. It adopts that of the conventional query answering procedure it
constrains. This is achieved by relying on meta-interpretation for specifying the
constraining process. The soundness, completeness, and termination of the underlying
query answering procedure are not compromised. Constrained query answering
can be applied for answering queries more efficiently as well as for generating more
informative, intensional answers
Models of Type Theory Based on Moore Paths
This paper introduces a new family of models of intensional Martin-L\"of type
theory. We use constructive ordered algebra in toposes. Identity types in the
models are given by a notion of Moore path. By considering a particular gros
topos, we show that there is such a model that is non-truncated, i.e. contains
non-trivial structure at all dimensions. In other words, in this model a type
in a nested sequence of identity types can contain more than one element, no
matter how great the degree of nesting. Although inspired by existing
non-truncated models of type theory based on simplicial and cubical sets, the
notion of model presented here is notable for avoiding any form of Kan filling
condition in the semantics of types.Comment: This is a revised and expanded version of a paper with the same name
that appeared in the proceedings of the 2nd International Conference on
Formal Structures for Computation and Deduction (FSCD 2017
- …