"This sentence is not true" is not a truth bearer thus not a proposition thus cannot be included in any Boolean logic system. — PL Olcott
This seems your source of misunderstanding. In propositional logic, you would day "This sentence is not true." But in predicate logic, it can be translated into "Some sentence is not true."
In FOL it can be translated into "X is not true." which are all perfectly true or false depending on the truth criteria of the quantifiers and variables. — Corvus
The Variables of propositional logic and every other order of bivalent logic must have a Boolean value. — PL Olcott
When you widen the scope into predicate logic, FOL and HOL, the concept of truth and falsity has multifaceted nature. FOL enables you employ the variables for the individuals and subjects. HOL can deal with the variables for the relations, operators and properties within the sentence. — Corvus
None-the-less in every bivalent system of logic we must be able to reduce every variable to a Boolean value. Your reply did not seem to understand that. — PL Olcott
You are still under confusion, or don't want to see the real point. We have not been only talking about bivalent system of logic here. If you can recall the OP is about HOL. Not 2000 year old propositional logic. Hence it was necessary and relevant considering and looking into the multifaceted nature of truth, which are in the domains of FOL and HOL.Your reply merely stated that variables in higher orders of logic represent more complex things than in Propositional logic. — PL Olcott
You have been reading too much Wiki pages, and they can lead you to the wrong places unfortunately.This is contrasted with the more commonly known bivalent logics (such as classical sentential or Boolean logic) which provide only for true and false.
https://en.wikipedia.org/wiki/Three-valued_logic — PL Olcott
If some thing is Nonsense, then it is equivalent to False. In FOL HOL, truth values can be far more than just 3 above you listed. : {True, False, Unknown, Neutral, Contradiction}A three-valued logic system that can easily handle self-contradictory expressions would have the values of: {True, False, Nonsense}. — PL Olcott
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