|Fixed point: a point that satisfies the equation f (x) = x is a fixed point i.e. x is mapped to itself. S.A. Kripke based his alternative theory of truth from 1975 on fixed points in order to resolve the problem of paradoxes when dealing with self-reference. (Kripke, S., 1975. Outline of a Theory of Truth, The Journal of Philosophy, 72 690-716.). See also self-reference, paradoxes, liar paradox, truth theory._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
Kripke's Fixed Points/Read:
1. Separate truth and falsehood conditions (i.e. falsehood is not equal to non-truth).
2. Two sentence sets S1: true sentences, S2 false sentences.
3. Do evaluation on each level, therefore higher level - in this way, all sentences are "collected". Fixed point/(s): where evaluation is identical to input - Read: Success: then the extension fails - i.e. the meta language does not contain any further truth-attributions than the object language.
Kripke's Fixed Points/Kripke: the extension fails: meta language has no further truth-attributions - there is a paradox in the fixed point without truth value - falsehood does not equal non-truth!
Truth-predicate/Kripke's Fixed Points/Read: we separate the truth-predicates truth and falsehood - the truth-predicate is formed by the pair (S1, S2), whereby S1 contains the true sentences and S2 contains the wrong sentences. 1st level: here, a sentence has, e.g. ""Snow is white" is true" no truth value because evaluation at this stage is not possible - Solution: weak matrices for evaluating compound sentences, some of which are without truth value (without truth value) - (A v B) without truth value if one of A or B has no truth value (partial interpretation).
Fixed point/Kripke's Fixed Points/Kripke/Read: the fixed point is reached by transfinite induction - recursive or successive with partial evaluations - 1st transfinite level: all finite partial evaluations of S1 and S2 are collected separately - N.B.: at an early point (before adding all possible sentences), the reinterpretation of the truth-predicate no longer succeeds in adding something new. - Special case of the result about fixed points of normal functions over ordinal numbers. - Phi/f: represents the operation of expanding by allowing new interpretations. The fix point here is f(S1, S2) = (S1, S2).
Unfunded assertions: the separation of S1 and S2 leaves some statements without a truth value - e.g. "this statement is true" - it has no truth value at the minimum fixed point - A level higher we can give it an arbitrary value-but not to the liar.
Paradox/Kripke: follows Tarski: it cannot be expressed in one's own language - the entire discussion belongs to the meta language, as well as the predicates: "paradoxical" and "unfunded". They do not belong to the semantically terminated fixed point - Tarski's truth schema does not work here - (... + ...)._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
Me I Albert Menne Folgerichtig Denken Darmstadt 1988
HH II Hoyningen-Huene Formale Logik, Stuttgart 1998
Re III Stephen Read Philosophie der Logik Hamburg 1997
Sal IV Wesley C. Salmon Logik Stuttgart 1983
Sai V R.M.Sainsbury Paradoxien Stuttgart 2001