Concept
Interpretation (model theory)
Overview
In model theory, interpretation of a structure M in another structure N (typically of a different signature) is a technical notion that approximates the idea of representing M inside N. For example, every reduct or definitional expansion of a structure N has an interpretation in N.
Many model-theoretic properties are preserved under interpretability. For example, if the theory of N is stable and M is interpretable in N, then the theory of M is also stable.
Note that in other areas of mathematical logic, the term "interpretation" may refer to a structure, rather than being used in the sense defined here. These two notions of "interpretation" are related but nevertheless distinct. Similarly, "interpretability" may refer to a related but distinct notion about representation and provability of sentences between theories.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
This latitude in interpretation is a characteristic of mathematical physics. What we know is certain very abstract logical relations, which we express in mathematical formulæ; we know also that, at certain points, we arrive at results which are capable of being tested experimentally.
Bertrand Russell, The A B C of Relativity
States what Interpretation is; names Interpretation several times.
As may be seen, it leads to a revival of the dream interpretation by means of symbolism, which the ancients used, except that the source from which the interpretation is to be taken is limited to the human body. The lack of a technique of interpretation which is scientifically comprehensible must seriously limit the applicability of Scherner’s theory. Arbitrariness in dream interpretation seems in no wise excluded…
Sigmund Freud, The Interpretation of Dreams
Names Interpretation several times; states what Interpretation is.
They ought to be told that things of his character are parabolical, and that we should pause and consider the saying of God: “Yet none knoweth the interpretation thereof except God.” Although learned men agree that these are to be interpreted, they differ in the interpretation according to their knowledge of principles of philosophy. There is a third part of the Law which occupies an intermediate position, on…
Averroës, The Philosophy and Theology of Averroes
Names Interpretation several times; states what Interpretation is.
Related ideas
- First-order logic
- Integer
- Mathematical logic
- Rational number
- Model theory
- Topological space
- Structure (mathematical logic)
- Map (mathematics)
- Signature (logic)
Place in the tradition
Part of Philosophy of logic.
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