@misc{Lexicon of Arguments,
title = {Quotation from: Lexicon of Arguments – Concepts - Ed. Martin Schulz, 29 Mar 2024},
author = {Logic Texts},
subject = {Fixed Points},
note = {Read III 196
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 you have to choose a higher level. In this way, all sentences are "collected".
Fixed point/(s): where evaluation (output) 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(1).
>Meta language, >Object language, >Levels (Order).
III 197
Kripke's Fixed Points/Kripke: the extension fails: the 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 value.
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.
>Truth predicate.
1st level: here, a sentence has, e.g. ""Snow is white" is true" has no truth value because evaluation at this stage is not possible.
>Valuation.
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).
III 198
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 fixed point here is f(S1, S2) = (S1, S2).
III 200
Unfounded 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.
One level higher we can give it an arbitrary value-but not to the liar paradox.
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 "unfounded".
They do not belong to the semantically terminated fixed point.
Tarski's truth schema does not work here - (... + ...).
>Disquotation schema.
1. Saul Kripke Outline of a Theory of Truth (1975) in: R.L.Martin (Ed.) Recent Essays on Truth and the Liar Paradox Clarendon Oxf/NY 1984},
note = {
Re III St. Read Thinking About Logic: An Introduction to the Philosophy of Logic. 1995 Oxford University Press German Edition: Philosophie der Logik Hamburg 1997 },
file = {http://philosophy-science-humanities-controversies.com/listview-details.php?id=221715}
url = {http://philosophy-science-humanities-controversies.com/listview-details.php?id=221715}
}