45,897 research outputs found
Cut-Simulation and Impredicativity
We investigate cut-elimination and cut-simulation in impredicative
(higher-order) logics. We illustrate that adding simple axioms such as Leibniz
equations to a calculus for an impredicative logic -- in our case a sequent
calculus for classical type theory -- is like adding cut. The phenomenon
equally applies to prominent axioms like Boolean- and functional
extensionality, induction, choice, and description. This calls for the
development of calculi where these principles are built-in instead of being
treated axiomatically.Comment: 21 page
Hilbert-Post completeness for the state and the exception effects
In this paper, we present a novel framework for studying the syntactic
completeness of computational effects and we apply it to the exception effect.
When applied to the states effect, our framework can be seen as a
generalization of Pretnar's work on this subject. We first introduce a relative
notion of Hilbert-Post completeness, well-suited to the composition of effects.
Then we prove that the exception effect is relatively Hilbert-Post complete, as
well as the "core" language which may be used for implementing it; these proofs
have been formalized and checked with the proof assistant Coq.Comment: Siegfried Rump (Hamburg University of Technology), Chee Yap (Courant
Institute, NYU). Sixth International Conference on Mathematical Aspects of
Computer and Information Sciences , Nov 2015, Berlin, Germany. 2015, LNC
Logical consequence in modal logic II: Some semantic systems for S4
ABSTRACT: This 1974 paper builds on our 1969 paper (Corcoran-Weaver [2]). Here we present three (modal, sentential) logics which may be thought of as partial systematizations of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of these three logics coincide with one another and with those of standard formalizations of Lewis's S5. These logics, when regarded as logistic systems (cf. Corcoran [1], p. 154), are seen to be equivalent; but, when regarded as consequence systems (ibid., p. 157), one diverges from the others in a fashion which suggests that two standard measures of semantic complexity may not be as closely linked as previously thought.
This 1974 paper uses the linear notation for natural deduction presented in [2]: each two-dimensional deduction is represented by a unique one-dimensional string of characters. Thus obviating need for two-dimensional trees, tableaux, lists, and the likeâthereby facilitating electronic communication of natural deductions.
The 1969 paper presents a (modal, sentential) logic which may be thought of as a partial systematization of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of this logic coincides those of standard formalizations of Lewisâs S4. Among the paper's innovations is its treatment of modal logic in the setting of natural deduction systems--as opposed to axiomatic systems.
The authorâs apologize for the now obsolete terminology. For example, these papers speak of âa proof of a sentence from a set of premisesâ where today âa deduction of a sentence from a set of premisesâ would be preferable.
1. Corcoran, John. 1969. Three Logical Theories, Philosophy of Science 36, 153â77. J P R
2. Corcoran, John and George Weaver. 1969. Logical Consequence in Modal Logic: Natural Deduction in S5 Notre Dame Journal of Formal Logic 10, 370â84. MR0249278 (40 #2524).
3. Weaver, George and John Corcoran. 1974. Logical Consequence in Modal Logic: Some Semantic Systems for S4, Notre Dame Journal of Formal Logic 15, 370â78. MR0351765 (50 #4253)
From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions
Permissive-Nominal Logic (PNL) extends first-order predicate logic with
term-formers that can bind names in their arguments. It takes a semantics in
(permissive-)nominal sets. In PNL, the forall-quantifier or lambda-binder are
just term-formers satisfying axioms, and their denotation is functions on
nominal atoms-abstraction.
Then we have higher-order logic (HOL) and its models in ordinary (i.e.
Zermelo-Fraenkel) sets; the denotation of forall or lambda is functions on full
or partial function spaces.
This raises the following question: how are these two models of binding
connected? What translation is possible between PNL and HOL, and between
nominal sets and functions?
We exhibit a translation of PNL into HOL, and from models of PNL to certain
models of HOL. It is natural, but also partial: we translate a restricted
subsystem of full PNL to HOL. The extra part which does not translate is the
symmetry properties of nominal sets with respect to permutations. To use a
little nominal jargon: we can translate names and binding, but not their
nominal equivariance properties. This seems reasonable since HOL---and ordinary
sets---are not equivariant.
Thus viewed through this translation, PNL and HOL and their models do
different things, but they enjoy non-trivial and rich subsystems which are
isomorphic
Hybrid type theory: a quartet in four movements
This paper sings a song -a song created by bringing together the work of four great names in the history of logic: Hans Reichenbach, Arthur Prior, Richard Montague, and Leon Henkin. Although the work of the first three of these authors have previously been combined, adding the ideas of Leon Henkin is the addition required to make the combination work at the logical level. But the present paper does not focus on the underlying technicalities (these can be found in Areces, Blackburn, Huertas, and Manzano [to appear]) rather it focusses on the underlying instruments, and the way they work together. We hope the reader will be tempted to sing along
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