## Does Tarski Undefinability apply to HOL ?

145678Next
• 1.7k
"This sentence is not true" is not a truth bearer thus not a proposition thus cannot be included in any Boolean logic system.

A physical analog would be a digital logic inverter (NOT gate) with its output connected back to its input. Such a circuit forms an oscillator, with the output continually swinging back and forth between 0 and 1.
• 516
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.

The Variables of propositional logic and every other order of bivalent logic must have a Boolean value. Any variables that cannot possibly be true or false must be excluded from every bivalent logic system. https://en.wikipedia.org/wiki/Three-valued_logic can have the values {True, False, Nonsense}.
"This sentence is not true" has the semantic value of {nonsense}.

The predicate Is_Not_True(X) summarize by the operator "~" is fine unless
X is defined as X := ~True(X). Then it is the oscillator that wonderer1 referred to.
• 516
A physical analog would be a digital logic inverter (NOT gate) with its output connected back to its input. Such a circuit forms an oscillator, with the output continually swinging back and forth between 0 and 1.

Good job that is a perfect analogy!
• 3k
The Variables of propositional logic and every other order of bivalent logic must have a Boolean value.

Variables in propositional logic is for the whole sentence, not the elements in the sentence, hence its limitation. You are still talking under the propositional logic domain only.

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.
• 516
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.

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. Your reply merely stated that variables in higher orders of logic represent more complex things than in Propositional logic.

It seems that you are simply failing to understand this:
In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean,[1] sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false, and some third value. 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

A three-valued logic system that can easily handle self-contradictory expressions would have the values of: {True, False, Nonsense}.
• 3k
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.
Your reply merely stated that variables in higher orders of logic represent more complex things than in Propositional logic.
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.

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
You have been reading too much Wiki pages, and they can lead you to the wrong places unfortunately.

A three-valued logic system that can easily handle self-contradictory expressions would have the values of: {True, False, Nonsense}.
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}
145678Next
bold
italic
underline
strike
code
quote
ulist
image
url
mention
reveal