Hello,
the liars sentence is a famous antinomy / paradox of classical logic:
L:= „This statement is not true“
If we assume, that L is true, then because of the definition of L it has to be not true (=false). If we assume, that L is not true, then because of the definition of L it has to true
So in both cases L is true and false at the same time, and that is not allowed.
Most solutions of the problem do not allow sentences like L and especially no selfreferences.
But there is an other posiibility: A new logic called „layer logic“ or should I say „liar logic“?
The idea with layer logic is, that every determination of a truth value belongs to a layer (0,1,2,3,...), and different truth values are allowed in different layers (whatever layers are – see beyond). As layers are „blind“ to themselfes and to higher layers, we always have to use a higher layer if we „talk“ about a layer. In the rules of layer logic we have this formulation of the liar L:
„For all k=0,1,2,3,...: This proposition LL is true in layer k+1, if LL is not true in layer k and LL is false in layer k+1 else.“
In layer logic all propositions are „undefined“ in the lowest layer 0.
Therefore LL is undefined in layer 0.
Layer 1: This proposition LL is true in layer 0+1, if LL is not true in layer 0 and LL is false in layer 0+1 else..“ Therefore LL is true in layer 1.
Layer 2: This proposition LL is true in layer 1+1, if LL is not true in layer 1 and LL is false in layer 1+1 else..“ Therefore LL is false in layer 2.
Layer 3: This proposition LL is true in layer 2+1, if LL is not true in layer 2 and LL is false in layer 2+1 else..“ Therefore LL is true in layer 3.
Layer 4: This proposition LL is true in layer 3+1, if LL is not true in layer 3 and LL is false in layer 3+1 else..“ Therefore LL is false in layer 4.
We see that LL has an alternating truth value with increasing layers – there is no contradiction. As propositions belong to all layers, LL is self-referential but not within a layer.
And what is with liars that speak about all layers?
„LA:= This proposition is not true in all layers“
In layer logic meta propositions about layers and truth values have to be nearly classic: They can only be true or wrong and have to have the same truth value in all layers >=1.
So if LA is true in layer 1 it has to be true in all layers, so LA is false – in all layers, that would be a contradiction. If LA is not true in layer 1 it has to be not true in all layers, so LA is true – in all layers, that would be a contradiction.
Therefore LA is not an allowed meta proposition in layer logic.
Maybe there are better examples for extended liars?
The layers may look somehow strange at first glance.
The layers were first just a formal parameter to differentiate truth values (for example in the first part of a proof and the second). Meanwhile I see them as a kind of a new dimension, of meta levels or cause and effect order or a new part of time. Even without knowing exactly what a layer is, we can use layer logic and layer set theorie, as the rules for using them are mostly independent of this.
In a restricted way Professor Ulrich Blau in Munich invented a logic with layers
some 20 years before me, the reflection logic. He counted how often we reflected about the truth value of a proposition
and those meta levels were his layers.
Here links with more detailed information about layer logic and layer set theory: www.researchgate.net/post/Is_this_a_new_valid_logic_And_what_does_layer_logic_mean In German: www.ask1.org/threads/stufenlogik-trestone-reloaded-vortrag-apc.17951/#post-492741
About Professor Ulrich Blau: https://ivv5hpp.uni-muenster.de/u/rds/blau_review.pdf In German: https://link.springer.com/chapter/10.1007/978-94-017-1456-3_20 https://books.google.de/books?id=9x...kQAQ#v=onepage&q=reflexionslogik blau&f=false As it is unusual and bulky I can understand that not many are going to study layer logic, but in my eyes the possible results – a new look on logic and the world and a way out of many antinomies - it is worth the effort.
On the other hand I am interested to learn,more about the liar, extensions and the solutions?
Yours Trestone
For most people this is new and obscure, so added references and explanations would be a good idea.
to solve antinomies like yours, the tortoise antinomie or the kreta antinomie by different metalevels is not new and has been used long ago before blau. blau has nothing new to offer.
Hello Stanley Wilkin,
in the listed thread more details to layer logic are given:
www.researchgate.net/post/Is_this_a_new_valid_logic_And_what_does_layer_logic_mean In German: www.ask1.org/threads/stufenlogik-trestone-reloaded-vortrag-apc.17951/#post-492741
Yours
Trestone
Hello Albert Hinnerk, yes, meta levels have since long been used to solve antinomies with self references. But most times it was by eliminating and forbidding the self references between different languages, types or meta levels.
In the reflection logic of Prof. Blau and in my layer logic selfrefential propositions are allowed, and there are different truth values for the same proposition in different meta levels or layers. In layer logic the layers are extended for all propositions – so it is not easy to find an universal illustration or interpretation for them – they seem to be more than meta levels, something „new“.
And I never have heard of an other logic (or meta levels theory) that does not only give a solution to antinomies like the liar but also relativates Cantors Diagonalization and probaply also the Uncompleteness Sentences of Gödel and the Halting Problem of Informatics.
So there should be something new in layer logic – or a fundamental failure / problem.
Yours Trestone
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