Weak interpretability
Encyclopedia
Weak interpretability is a special case of the concept of tolerance introduced by Giorgi Japaridze
in 1992.
Assume T and S are formal theories
. Slightly simplified, T is said to be weakly interpretable in S if, and only if, the language of T can be translate
d into the language
of S in such a way that the translation of every theorem
of T is consistent with S. Of course, there are some natural conditions on admissible translations here, such as the necessity for a translation to preserve the logic
al structure of formula
s.
This concept, together with interpretability
, was introduced by Alfred Tarski
in 1953.
See also Interpretability logic
.
Giorgi Japaridze
Giorgi Japaridze is a logician, at Villanova University in Villanova, Pennsylvania. In the past his contributions were primarily into the areas of provability logic and interpretability logic...
in 1992.
Assume T and S are formal theories
Formal theory
Formal theory can refer to:* Another name for a theory which is expressed in formal language.* An axiomatic system, something representable by symbols and its operators...
. Slightly simplified, T is said to be weakly interpretable in S if, and only if, the language of T can be translate
Translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. Whereas interpreting undoubtedly antedates writing, translation began only after the appearance of written literature; there exist partial translations of the Sumerian Epic of...
d into the language
Language
Language may refer either to the specifically human capacity for acquiring and using complex systems of communication, or to a specific instance of such a system of complex communication...
of S in such a way that the translation of every theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
of T is consistent with S. Of course, there are some natural conditions on admissible translations here, such as the necessity for a translation to preserve the logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
al structure of formula
Formula
In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....
s.
This concept, together with interpretability
Interpretability
In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other.-Informal definition:Assume T and S are formal theories...
, was introduced by Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
in 1953.
See also Interpretability logic
Interpretability logic
Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability and/or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance and arithmetic...
.