115,248 research outputs found
A logical basis for constructive systems
The work is devoted to Computability Logic (CoL) -- the
philosophical/mathematical platform and long-term project for redeveloping
classical logic after replacing truth} by computability in its underlying
semantics (see http://www.cis.upenn.edu/~giorgi/cl.html). This article
elaborates some basic complexity theory for the CoL framework. Then it proves
soundness and completeness for the deductive system CL12 with respect to the
semantics of CoL, including the version of the latter based on polynomial time
computability instead of computability-in-principle. CL12 is a sequent calculus
system, where the meaning of a sequent intuitively can be characterized as "the
succedent is algorithmically reducible to the antecedent", and where formulas
are built from predicate letters, function letters, variables, constants,
identity, negation, parallel and choice connectives, and blind and choice
quantifiers. A case is made that CL12 is an adequate logical basis for
constructive applied theories, including complexity-oriented ones
Definite Formulae, Negation-as-Failure, and the Base-extension Semantics of Intuitionistic Propositional Logic
Proof-theoretic semantics (P-tS) is the paradigm of semantics in which
meaning in logic is based on proof (as opposed to truth). A particular instance
of P-tS for intuitionistic propositional logic (IPL) is its base-extension
semantics (B-eS). This semantics is given by a relation called support,
explaining the meaning of the logical constants, which is parameterized by
systems of rules called bases that provide the semantics of atomic
propositions. In this paper, we interpret bases as collections of definite
formulae and use the operational view of the latter as provided by uniform
proof-search -- the proof-theoretic foundation of logic programming (LP) -- to
establish the completeness of IPL for the B-eS. This perspective allows
negation, a subtle issue in P-tS, to be understood in terms of the
negation-as-failure protocol in LP. Specifically, while the denial of a
proposition is traditionally understood as the assertion of its negation, in
B-eS we may understand the denial of a proposition as the failure to find a
proof of it. In this way, assertion and denial are both prime concepts in P-tS.Comment: submitte
Negation-as-Failure in the Base-extension Semantics for Intuitionistic Propositional Logic
Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and use the operational view of them as provided by uniform proof-searchâthe proof-theoretic foundation of logic programming (LP)âto establish the completeness of IPL for the B-eS. This perspective allows negation, a subtle issue in P-tS, to be understood in terms of the negation-as-failure protocol in LP. Specifically, while the denial of a proposition is traditionally understood as the assertion of its negation, in B-eS we may understand the denial of a proposition as the failure to find a proof of it. In this way, assertion and denial are both prime concepts in P-tS
Some Concerns Regarding Ternary-relation Semantics and Truth-theoretic Semantics in General
This paper deals with a collection of concerns that, over a period of time, led the author away from the RoutleyâMeyer semantics, and towards proof- theoretic approaches to relevant logics, and indeed to the weak relevant logic MC of meaning containment
A map of dependencies among three-valued logics
International audienceThree-valued logics arise in several fields of computer science, both inspired by concrete problems (such as in the management of the null value in databases) and theoretical considerations. Several three-valued logics have been defined. They differ by their choice of basic connectives, hence also from a syntactic and proof-theoretic point of view. Different interpretations of the third truth value have also been suggested. They often carry an epistemic flavor. In this work, relationships between logical connectives on three-valued functions are explored. Existing theorems of functional completeness have laid bare some of these links, based on specific connectives. However we try to draw a map of such relationships between conjunctions, negations and implications that extend Boolean ones. It turns out that all reasonable connectives can be defined from a few of them and so all known three-valued logics appear as a fragment of only one logic. These results can be instrumental when choosing, for each application context, the appropriate fragment where the basic connectives make full sense, based on the appropriate meaning of the third truth-value
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
Theoretical Setting of Inner Reversible Quantum Measurements
We show that any unitary transformation performed on the quantum state of a
closed quantum system, describes an inner, reversible, generalized quantum
measurement. We also show that under some specific conditions it is possible to
perform a unitary transformation on the state of the closed quantum system by
means of a collection of generalized measurement operators. In particular,
given a complete set of orthogonal projectors, it is possible to implement a
reversible quantum measurement that preserves the probabilities. In this
context, we introduce the concept of "Truth-Observable", which is the physical
counterpart of an inner logical truth.Comment: 11 pages. More concise, shortened version for submission to journal.
References adde
(Co-)Inductive semantics for Constraint Handling Rules
In this paper, we address the problem of defining a fixpoint semantics for
Constraint Handling Rules (CHR) that captures the behavior of both
simplification and propagation rules in a sound and complete way with respect
to their declarative semantics. Firstly, we show that the logical reading of
states with respect to a set of simplification rules can be characterized by a
least fixpoint over the transition system generated by the abstract operational
semantics of CHR. Similarly, we demonstrate that the logical reading of states
with respect to a set of propagation rules can be characterized by a greatest
fixpoint. Then, in order to take advantage of both types of rules without
losing fixpoint characterization, we present an operational semantics with
persistent. We finally establish that this semantics can be characterized by
two nested fixpoints, and we show the resulting language is an elegant
framework to program using coinductive reasoning.Comment: 17 page
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