Kripke: Outline…, continued Till now:

1. Kripke: Outline…, continued Till now: L is a FOL with totally defined (presumably nonsemantical) predicates over some domain (that contains sentences codes). L*(S1,S2): L extended with a T(x) partially defined unary predicate (extension: S1, antiextension: S2). The operation  renders to any L*(S1, S2) another interpretation of the T(x) predicate where the extension of T(x) consists of the codes of true sentences and the antiextension of the codes of false sentences (plus the objects that are not sentence codes. The (possibly transfinitely) iterated application of  leads always to some fixed point. At a fixed point, T(x) is a truth predicate in the sense that T(x) is true resp. false exactly for the codes of true resp. false sentences. The (transfinite) sequence <L*n> starts with L*0=L*(, ) and continues by the iterated application of . I.e., L1 qualifies the sentences true resp. false by „matters of fact” only (in other words, no sentences is supposed to be true/false that is not made true/false by the interpretation of the other predicates), etc. The fixed point L of this sequence is the smallest fixed point, i.e. any other fixed point is an extension of it.

2. The level of a sentence is the smallest ordinal where it has a truth value in the above sequence. A sentence is grounded iff it has a truth value in L; ungrounded in the other case. There is an important difference between that a sentence is made true by an interpretation (i.e. it is true according to the interpretation, a metalanguage fact) and that it is true in an interpretation (i.e., its code falls in the extension of T(x)). If a sentence is made true first at some level, then it it is true first at the next level. At a fixed point, „is true” and „made true” falls together. The Nixon-Dean example (sentences (4) and (5) in the previous ppt): (4’) x(P(x)T(x)) (5’) x(Q(x)T(x)) Context: Dean makes some grounded and true statement; the smallest level of such a sentence be n. Then, at level n+1 (5’) proves to be grounded and false. Nothing prevents (4’) to be grounded and true. Its level is higher than n+1. It may be grounded and false, too. If all the other sentences of both person are grounded and false, then the siruation is paradoxical and both (4’) and(5’) become ungrounded.

3. The Truth-teller x(P(x)T(x)) (in short, Tt) (where P(x) is true only for the code of (3)) is ungrounded but not paradoxical. If the 0-level extension of T(x) contains this code (instead of beginning with the emptyset), then (3) will be true at every level - including the fixed point of this sequence. That means that we can assign consistently a truth-value to (3). But the assignment is arbitrary. A is paradoxicaliff it has no truth value at any fixed point. The Liar sentence is paradoxical. Is true (at a fixed point) -> is made false ->is false (contradiction). Is false -> is made true -> is true (contradiction again). A fixed point is a maximal fixed point if ith has no proper extension to another fixed point. By Zorn’s lemma, we can extend any fxed point to a maximal fixed point.

4. (12) Either (12) or its negation is true. (12) is ungrounded again (no truth value at the minimal fixed point – by induction). But it is true at every fixed point where it has a truth value. So its truth-value is not arbitrary. A fixed point is intrinsic iff it assigns to no sentence a truth value that contradicts to its truth value at any other fixed point. (The minimal fixed point is a such one.) A sentence has an intrinsic truth value iff it has a truth value at some intrinsic fixed point. E.g. ‘Tt Tt’ is a sentence that has the same truth values everywere where it has a truth value at all, but it doesn’t have an intrinsic truth value because it doesn’t have a truth value at any intrinsic fixed point. There is a largest intrinsic fixed point. This is the (unique) largest extension of the truth predicate which is consistent with our idea of truth and doesn’t involve arbitrary truth assignments.

5. Barwise-Etchemendy: The Liar Read the introductory considerations. First decision: the theory is about propositions. They are strictly two-valued. Main argument against „gappy” explanations: they don’t give a cause why sentences like the Liar sentence lack a truth value + the Strenghtened Liar. Important distinction: To assert a negative statement (mostly by a negated noun phrase, e. g. ‘Socrates is not running’). To deny something (mostly by sentence negation, e.g. ‘It is false that Socrates is running’). Next: „A budget of Liar-like paradoxes”