"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
You have been reading too much Wiki pages, and they can lead you to the wrong places unfortunately. — Corvus
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} — Corvus
Boolean values are applicable up to FOL, but FOL cannot express the full complexities in the world. Hence you are going up to HOL, which has the extended truth values, and can describe more complex states of the real world.A currently unknown Boolean value is still a Boolean value.
No such thing as a neutral Boolean value. — PL Olcott
In HOL, "What time is it?" would be translated into computable format, and can be processed for the proper truth values."What time is it?" has no Boolean value.
Contradiction proves False. — PL Olcott
Nonsense is not a logic world. It is an ordinary linguistic expression to mean False with added stupidity and foolishness connotation.{Nonsense} is reserved for expressions that cannot be true or false. — PL Olcott
Boolean values are applicable up to FOL, but FOL cannot express the full complexities in the world. Hence you are going up to HOL, which has the extended truth values, and can describe more complex states of the real world. — Corvus
In HOL, "What time is it?" would be translated into computable format, and can be processed for the proper truth values. — Corvus
Nonsense is not a logic world. It is an ordinary linguistic expression to mean False with added stupidity and foolishness connotation. — Corvus
Boolean values are properties of every Proposition
A proposition is a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. Propositions are also often characterized as being the kind of thing that declarative sentences denote. — PL Olcott
Nonsense !! Nonsense is just a colloquial expression saying, no you are bloody wrong mate.{Nonsense} is a stipulated term of the art of my formal three-valued formal system of logic
having values of {True, False, Nonsense} that only applies to expressions such as this: — PL Olcott
Many Wiki pages and Youtube videos are rubbish. Don't trust and worship them as if they are the bible. Think with your own mind, and if it doesn't make sense, then you should be able to say "Nonsense mate. This is what I think, because of this and that." As I said before, they may slag you for saying what you think is true, but at least you know you have been thinking with your own mind, rather than parroting what the Wiki pages and Youtubers said, or joined the herd of the inauthentic comedians seeking pleasure out of attacking the authentic self thinking man.The Strengthened Liar and Paradoxes of Incompleteness
https://www.youtube.com/watch?v=5LWQPGjAs3M — PL Olcott
A truck load of strawmen here. I didn't deny Boolean values, but I was simply saying that in FOL and HOL, you have the extended truth values including Boolean. — Corvus
I don't think that you can find any source that ever says anything like that for bivalent systems of logic. I don't think you can find any sources that say anything like that for (a) propositional logic (b) FOL, (c) SOL, (d) HOL.
https://en.wikipedia.org/wiki/Law_of_excluded_middle only works in bivalent logic. — PL Olcott
You should read some good Mathematical Logic books, not the Wiki pages.
But think about it even with your common sense. The world contains more problems, structures, events and objects than things that are just True or False. — Corvus
For simplest example, when you see a formula, X > 3, that is not true or false until you know the value of X. Until that moment, X > 3 remains unknown. — Corvus
If you say, "It is raining now." then it could be True in your town, but it could be false for someone living in some other part of the world, because it could be sunny. So your statement is contradictory when looking from both areas of the world. — Corvus
Some statements or formula depicting the real world structure, events or objects can be unknown, neutral or contradictory. You don't simply reject that as nonsense. You accept them as true, false, unknown, neutral or contradictory depending on the given formula, statements, and analysis. — Corvus
I have found that line of reasoning ineffective so I switched. We have to resolve my prior reply before you can begin to understand my updated reasoning.
— PL Olcott
Please reread my post above. — Corvus
Instead of a three values system with {True, False and Nonsense} I have bivalent systems of logic that derive a type mismatch error for any expression that is not a proposition. — PL Olcott
It is a Boolean valued system. When epistemological antinomies are involved they must be rejected
as a type mismatch error because that have no Boolean values. — PL Olcott
You are confusing between HOL and Computer Programming. In HOL, there is no such things as Boolean values. There are {Truth, False, Unknown, Contradiction, Neutral}, and they are the values of logical interpretation. — Corvus
For some unknown reasons, you changed the subject to Functions. There are differences in functions of math, and functions in computer programming. Can you explain the difference?They call functions of the Boolean type predicates in all orders of predicate logic. Functions of any other type are called functions. Predicate: >(5,2)==TRUE Function: +(5,2)==7 — PL Olcott
Could you please explain that in plain English? And how is it related to our discussion?Predicate: >(5,2)==TRUE Function: +(5,2)==7 — PL Olcott
Could you please explain that in plain English? And how is it related to our discussion? — Corvus
All bivalent systems of predicate logic only have (by definition of bivalent) two
Boolean values of True or False with nothing in between. What you have been
saying is the same as saying 2 == 5. — PL Olcott
Anyhow it shows you that bivalent logic is not useful and incapable for the real world uses in describing the complexities of the structures, events and objects. — Corvus
Not sure if your previous post was about the function call in Prolog, but it didn't look like the standard way of using function calls in the other PLs, hence I asked you about the difference between math functions and programming functions. — Corvus
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