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Aren't there definitions of Truth that are not the negation of Falsehood? Can't there be a function True(x) that is not equal to !False(x)? Can't there be a thi

by TZubiri 3mo ago

Aren't there definitions of Truth that are not the negation of Falsehood? Can't there be a function True(x) that is not equal to !False(x)? Can't there be a third function Paradox(x) such that these counterexamples can be considered paradoxes and therefore outside of the truth? I'm admittedly not a logician and don't formally study paradoxes, but I never quite understood the whole category of "this sentence is false" paradoxes, I feel that the sentence is paradox and therefore it's neither false nor true. I do agree that it's a truth vector sounds like a silly panacea fantasy, though. But more logical formality is not the counter argument that would convince me of it, rather I believe that there's less formal and rigorous ways to get closer to truth.

UltraSane 3mo ago
There are a LOT of different logical system. Paraconsistent Logic Intuitionistic Logic Dialetheism
galaxyLogic 3mo ago
Interesting point about "self-referential sentences". I tend to agree. In my view a sentence saying something like "This sentence ..." does not have valid semantic meaning. It says nothing because, what "This" in "This sentence" means is ill-defined. If terms we use are not well-defined, then sentences using such terms can not have meaning. But for the sake of argument let's explore, what could the "this" in (so called) "self-referential" sentences refer to? Do they refer to a specific encoding of the sentence you are reading, as some bits in computer memory perhaps? That would require that those bits somehow have a unique "identity" and the "this" in a self-referential sentence would have to refer to those bits in specific addresses of a specific memory-chip. But of course the "this" does not specify which memory chip, which specific (concrete) encoding of its (purported) meaning it is referring to. And if it did, then it would be talking about that specific set of bits in that specific memory-chip, not of "itself". The fallacy is that what we perceive as a "sentence we read" is somehow "speaking" of something. But no, the sentence is not a subject, a sentence can not speak, and THEREFORE it specifically can not speak of itself. A sentence can not speak, only actors, only subjects, like humans and AI, can "speak". And their speech must be encoded in some physical medium. A written sentence like "This sentence is ..." gives us the false impressions that somehow the SENTENCE IS SPEAKING of itself! But speech can not speak, speech is the product of speaking. Hence, in my view, "self-referential sentences" do not have any meaning and whatever paradoxes they might seem to create are results of confusion between ontological levels of "Subject" vs. "Speech".
- erehweb 3mo ago
  As the article notes, the sentence "This sentence is written in English" is well understood, and true. "This sentence is written in French" is also well understood and false.
  - TZubiri 3mo ago
    Right, my original comment wasn't about self-reference, but paradoxes, and not all self-references result in paradoxes. Not sure what the difference between the language of a sentence and the truthiness of a sentence is. Maybe it has to do with the fact that the language refers to syntactical features that can be determined at 'compile' time, while truthiness refers to a semantic quality feature that can only be determined at 'runtime'. (See en.wikipedia.org/wiki/colorless_green_ideas_sleep_furiously for at the very least a funny canonical example of a syntactically valid, but semantically nonsensical sentence.) The fact that the language function can also be mapped to reduced parts of the sentence, might also be relevant, you can deduce that "This-" and "This sentence is-" are English, while you cannot partially compute that "This sentence is-" is true. A single negation bit being enough to change the result of the operation.
  - galaxyLogic 3mo ago
    Note, that "This sentence is written in English" is not about the meaning of a sentence but about its encoding. Statements about Truth are about the meaning of something. What does "This" in the above sentence refer to? It refers to the expression of an idea, how the idea is written, in what language. It does not refer to the idea, or the meaning of the sentence, what we think of as the "sentence" (vs. its encoding). Therefore I would say it is not self-referential sentence, or if it is in a very different way than "This" in "This sentence is false".
- aesthesia 3mo ago
  Quines produce similar issues to self-referential sentences without being directly self-referential. e.g. "'Yields falsehood when preceded by its quotation' yields falsehood when preceded by its quotation."
roywiggins 3mo ago
There is no lack of more elaborate logics: https://en.wikipedia.org/wiki/Non-classical_logic https://en.wikipedia.org/wiki/Non-classical_logic
cgio 3mo ago
Fuzzy logic?
codechicago277 3mo ago
I’m not a logician either but believe this is what Tarski’s definition of truth solves for. In order to make a statement about the statement itself, you have to introduce a new meta language. Then a statement in the meta language is only true if the underlying statement is true. Much more rigorous explanation: https://plato.stanford.edu/entries/tarski-truth/ https://plato.stanford.edu/entries/tarski-truth/
layer8 3mo ago
You run into the same problems when you have True(x) = NOT (False(x) OR Paradox(x)). You can define NotTrue(x) := False(x) OR Paradox(x), and then you have True(x) = NOT NotTrue(x), which one can build the usual diagonalizations on.
- HappMacDonald 3mo ago
  Well, I mean True(x) does = NOT (NOT True(x)), so..

by TZubiri 3mo ago

UltraSane 3mo ago

There are a LOT of different logical system. Paraconsistent Logic Intuitionistic Logic Dialetheism

galaxyLogic 3mo ago

Interesting point about "self-referential sentences". I tend to agree. In my view a sentence saying something like "This sentence ..." does not have valid semantic meaning. It says nothing because, what "This" in "This sentence" means is ill-defined. If terms we use are not well-defined, then sentences using such terms can not have meaning. But for the sake of argument let's explore, what could the "this" in (so called) "self-referential" sentences refer to? Do they refer to a specific encoding of the sentence you are reading, as some bits in computer memory perhaps? That would require that those bits somehow have a unique "identity" and the "this" in a self-referential sentence would have to refer to those bits in specific addresses of a specific memory-chip. But of course the "this" does not specify which memory chip, which specific (concrete) encoding of its (purported) meaning it is referring to. And if it did, then it would be talking about that specific set of bits in that specific memory-chip, not of "itself". The fallacy is that what we perceive as a "sentence we read" is somehow "speaking" of something. But no, the sentence is not a subject, a sentence can not speak, and THEREFORE it specifically can not speak of itself. A sentence can not speak, only actors, only subjects, like humans and AI, can "speak". And their speech must be encoded in some physical medium. A written sentence like "This sentence is ..." gives us the false impressions that somehow the SENTENCE IS SPEAKING of itself! But speech can not speak, speech is the product of speaking. Hence, in my view, "self-referential sentences" do not have any meaning and whatever paradoxes they might seem to create are results of confusion between ontological levels of "Subject" vs. "Speech".

erehweb 3mo ago
As the article notes, the sentence "This sentence is written in English" is well understood, and true. "This sentence is written in French" is also well understood and false.
- TZubiri 3mo ago
  Right, my original comment wasn't about self-reference, but paradoxes, and not all self-references result in paradoxes. Not sure what the difference between the language of a sentence and the truthiness of a sentence is. Maybe it has to do with the fact that the language refers to syntactical features that can be determined at 'compile' time, while truthiness refers to a semantic quality feature that can only be determined at 'runtime'. (See en.wikipedia.org/wiki/colorless_green_ideas_sleep_furiously for at the very least a funny canonical example of a syntactically valid, but semantically nonsensical sentence.) The fact that the language function can also be mapped to reduced parts of the sentence, might also be relevant, you can deduce that "This-" and "This sentence is-" are English, while you cannot partially compute that "This sentence is-" is true. A single negation bit being enough to change the result of the operation.
- galaxyLogic 3mo ago
  Note, that "This sentence is written in English" is not about the meaning of a sentence but about its encoding. Statements about Truth are about the meaning of something. What does "This" in the above sentence refer to? It refers to the expression of an idea, how the idea is written, in what language. It does not refer to the idea, or the meaning of the sentence, what we think of as the "sentence" (vs. its encoding). Therefore I would say it is not self-referential sentence, or if it is in a very different way than "This" in "This sentence is false".
aesthesia 3mo ago
Quines produce similar issues to self-referential sentences without being directly self-referential. e.g. "'Yields falsehood when preceded by its quotation' yields falsehood when preceded by its quotation."

roywiggins 3mo ago

There is no lack of more elaborate logics: https://en.wikipedia.org/wiki/Non-classical_logic https://en.wikipedia.org/wiki/Non-classical_logic

cgio 3mo ago

Fuzzy logic?

codechicago277 3mo ago

I’m not a logician either but believe this is what Tarski’s definition of truth solves for. In order to make a statement about the statement itself, you have to introduce a new meta language. Then a statement in the meta language is only true if the underlying statement is true. Much more rigorous explanation: https://plato.stanford.edu/entries/tarski-truth/ https://plato.stanford.edu/entries/tarski-truth/

layer8 3mo ago

You run into the same problems when you have True(x) = NOT (False(x) OR Paradox(x)). You can define NotTrue(x) := False(x) OR Paradox(x), and then you have True(x) = NOT NotTrue(x), which one can build the usual diagonalizations on.

HappMacDonald 3mo ago
Well, I mean True(x) does = NOT (NOT True(x)), so..