Concept
Interpretation (logic)
Overview
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.
The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of an object language. For example, an interpretation function could take the predicate symbol
T
{\displaystyle T}
and assign it the extension
{
(
a
)
}
{\displaystyle \{(\mathrm {a} )\}}
. All our interpretation does is assign the extension
{
(
a
)
}
{\displaystyle \{(\mathrm {a} )\}}
to the non-logical symbol
T
{\displaystyle T}
, and does not make a claim about whether
T
{\displaystyle T}
is to stand for tall and
a
{\displaystyle \mathrm {a} }
for Abraham Lincoln. On the other hand, an interpretation does not have anything to say about logical symbols, e.g. logical connectives "
a
n
d
{\displaystyle \mathrm {and} }
", "
o
r
{\displaystyle \mathrm {or} }
" and "
n
o
t
{\displaystyle \mathrm {not} }
". Though we may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function.
An interpretation often (but not always) provides a way to determine the truth values of sentences in a language. If a given interpretation assigns the value True to a sentence or theory, the interpretation is called a model of that sentence or theory.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
In the original texts
Passages from public-domain books where Interpretation (logic) is discussed.
If for the “arbitrary decision” of the interpreter you will substitute his skill, his experience and his understanding, then I am with you. This kind of personal factor is of course indispensable, especially when interpretation is difficult; it is just the same in other scientific work, however; it can’t be helped that one man will use any given technique less well, or apply it better, than another. The impression…
Sigmund Freud, Introductory lectures on psycho-analysis
names the idea; states what it is; opens on it
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.
Sigmund Freud, The Interpretation of Dreams
names the idea; states what it is; the book is about it; opens on it
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 the idea; states what it is; opens on it
For in the act of Judicature, the Judge doth no more but consider, whither the demand of the party, be consonant to naturall reason, and Equity; and the Sentence he giveth, is therefore the Interpretation of the Law of Nature; which Interpretation is Authentique; not because it is his private Sentence; but because he giveth it by Authority of the Soveraign, whereby it becomes the Soveraigns Sentence; which is Law…
Thomas Hobbes, Leviathan
names the idea; states what it is; opens on it
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. Take, for example, the eclipse observations by which Einstein’s theory as to the bending of light was established.
Bertrand Russell, The A B C of Relativity
states what it is; names the idea; opens on it
Related ideas
- Empiricism
- Formal language
- First-order logic
- Journal of Symbolic Logic
- Isomorphism
- Propositional variable
- Truth value
- Intuitionistic logic
- Formal grammar
- Logical connective
- Propositional logic
- Function (mathematics)
- Mathematics
- Model theory
- Natural number
- Domain of discourse
- Primitive notion
- Symbol (formal)
- Peano axioms
- Rule of inference
- Syntax
- Set theory
- Predicate (logic)
- Semantics
Place in the tradition
Part of First-order logic.