In
proof theory
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Jon Barwise, Barwise (1978) consists of four correspo ...
, ludics is an analysis of the principles governing
inference rule
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of in ...
s of
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
. Key features of ludics include notion of compound connectives, using a technique known as ''focusing'' or ''focalisation'' (invented by the computer scientist
Jean-Marc Andreoli), and its use of ''locations'' or ''loci'' over a base instead of
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
s.
More precisely, ludics tries to retrieve known
logical connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary co ...
s and proof behaviours by following the paradigm of interactive computation, similarly to what is done in
game semantics
Game semantics (german: dialogische Logik, translated as ''dialogical logic'') is an approach to Formal semantics (logic), formal semantics that grounds the concepts of truth or Validity (logic), validity on game theory, game-theoretic concepts, su ...
to which it is closely related. By abstracting the notion of
formulae
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
and focusing on their concrete uses—that is distinct occurrences—it provides an
abstract syntax
In computer science, the abstract syntax of data is its structure described as a data type (possibly, but not necessarily, an abstract data type), independent of any particular representation or encoding. This is particularly used in the representa ...
for
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, as loci can be seen as
pointers on memory.
The primary achievement of ludics is the discovery of a relationship between two natural, but distinct notions of
type, or proposition.
The first view, which might be termed the proof-theoretic or
Gentzen
Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died o ...
-style interpretation of propositions, says that the meaning of a proposition arises from its introduction and elimination rules. Focalization refines this viewpoint by distinguishing between positive propositions, whose meaning arises from their introduction rules, and negative propositions, whose meaning arises from their elimination rules. In focused calculi, it is possible to define positive connectives by giving only their introduction rules, with the shape of the elimination rules being forced by this choice. (Symmetrically, negative connectives can be defined in focused calculi by giving only the elimination rules, with the introduction rules forced by this choice.)
The second view, which might be termed the computational or
Brouwer–Heyting–Kolmogorov interpretation In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kolmogorov. It is also sometimes called the real ...
of propositions, takes the view that we fix a computational system up front, and then give a
realizability In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas from a formal theory are "realized" by objects, known as "realizers", in a way t ...
interpretation of propositions to give them constructive content. For example, a realizer for the proposition "A implies B" is a computable function that takes a realizer for A, and uses it to compute a realizer for B. Realizability models characterize realizers for propositions in terms of their visible behavior, and not in terms of their internal structure.
Girard shows that for
second-order
Second-order may refer to:
Mathematics
* Second order approximation, an approximation that includes quadratic terms
* Second-order arithmetic, an axiomatization allowing quantification of sets of numbers
* Second-order differential equation, a di ...
affine
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine comb ...
linear logic
Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also be ...
, given a computational system with nontermination and error stops as effects, realizability and focalization give the same meaning to types.
Ludics was proposed by the
logician
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
Jean-Yves Girard
Jean-Yves Girard (; born 1947) is a French logician working in proof theory. He is the research director ( emeritus) at the mathematical institute of the University of Aix-Marseille, at Luminy.
Biography
Jean-Yves Girard is an alumnus of the ...
. His paper introducing ludics, ''Locus solum: from the rules of logic to the logic of rules'', has some features that may be seen as eccentric for a publication in
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
(such as illustrations of skunks). It has to be noted that the intent of these features is to enforce the point of view of
Jean-Yves Girard
Jean-Yves Girard (; born 1947) is a French logician working in proof theory. He is the research director ( emeritus) at the mathematical institute of the University of Aix-Marseille, at Luminy.
Biography
Jean-Yves Girard is an alumnus of the ...
at the time of its writing. And, thus, it offers to readers the possibility to understand ludics independently of their backgrounds.{{dubious, date=May 2012
External links
# Girard, J.-Y.
''Locus solum'': from the rules of logic to the logic of rules(.pdf), ''Mathematical Structures in Computer Science'', 11, 301–506, 2001.
Girard reading groupat
Carnegie-Mellon University
Carnegie Mellon University (CMU) is a private research university in Pittsburgh, Pennsylvania. One of its predecessors was established in 1900 by Andrew Carnegie as the Carnegie Technical Schools; it became the Carnegie Institute of Technology ...
(a wiki about Locus Solum)
Mathematical logic