I think of the is-ought gap in terms of arguments. If the conclusion of an argument has an “ought” or “should” in it, there has to be a premise that has an “ought” or “should.” If it has none, there must be one that is implicit and unarticulated, or the conclusion doesn’t follow. We can argue validly about “oughts,” as in “everyone ought to X, so I ought to X.” But the argument can only establish that the conclusion follows from the premises, which must include a prescriptive premise, not that some prescriptive statement follows from purely descriptive premises.
If I were to convince myself that the category of morality was actually a subset of prudence rather than a distinct category, would I count as a moral anti-realist?
"Because no ordinary meaning can be derived from the statement “This sentence is false” by a reasonable person (either through the speaker’s intention, listener’s perception, reference to an objective state of the world, or any other possible definition of meaning), the liar’s paradox can be dismissed as meaningless nonsense"
You're just taking your desired result ("that sentence is meaningless nonsense") for granted.
The fact that minds much larger than yours dedicated so much thought and research to it should rather give you pause, not in an "argument from authority" way, but in a "maybe I'm not the biggest genius and they're all stupid" humble moment.
For starters, the Liar's Paradox has been a key insight into creating type theory, and understanding meta-statements.
It has also spawned variations like Russel's Paradox, Godel's incompletness proving technique, and Turing's halting theorem, that not only proved it fruitful, but gave some of the most important results and insights into set theory, algebra, and computer science.
"1=2" and your other "nonsense" examples never did anything of the sort. That alone should hint you that there's a substantial difference between the Liar's Paradox statement and your examples.
"We don’t need overly complex semantics of meaning that butcher our ordinary understanding of language to make sense of a nonsense statement"
Sure, if we just hand-wave the issue away, and insist loudly enough that it's a non-issue.
"The fact that you can’t divide by 0 isn’t a challenge posed for some math hero to solve, but a logical fact that should be accepted and incorporated into our model of mathematics"
Actually it's just an artifact of specific axioms chosen. There are valid mathematics where division by 0 is totally fine (a category of algebras called "wheel algebras").
Thanks for the review. As a general point, the article is only discussing the "meaning" of the liar's paradox - it makes no claims as to its alleged contributions. As mentioned, there is value in understanding why meaningless nonsense "is" meaningless nonsense. (And many others have attributed this to the liar's paradox as well, I'm not stumbling on anything new). This article only discusses meaninglessness in discourse.
Just because 1=2 hasn't participated in the same contributions as the liar's paradox doesn't mean the two aren't equivantely self-contradictory. All propositions assume their truth (see deflationary accounts of truth). So when a proposition says that it is also false, it becomes self-contradictory. You can't proclaim truth and falsehood the same way you can't proclaim a shape both has 3 sides and 4 sides—not at least without being self-contradictory. You might as well be saying that 1=2. This is why I argue that the liar's paradox is meaningless, the speaker couldn't be conveying any information and the listener couldn't have gained anything.
You either have to either accept that the sentences which I provided above "aren't" self-contradictions or the liar's paradox is as well.
I think of the is-ought gap in terms of arguments. If the conclusion of an argument has an “ought” or “should” in it, there has to be a premise that has an “ought” or “should.” If it has none, there must be one that is implicit and unarticulated, or the conclusion doesn’t follow. We can argue validly about “oughts,” as in “everyone ought to X, so I ought to X.” But the argument can only establish that the conclusion follows from the premises, which must include a prescriptive premise, not that some prescriptive statement follows from purely descriptive premises.
If I were to convince myself that the category of morality was actually a subset of prudence rather than a distinct category, would I count as a moral anti-realist?
"Because no ordinary meaning can be derived from the statement “This sentence is false” by a reasonable person (either through the speaker’s intention, listener’s perception, reference to an objective state of the world, or any other possible definition of meaning), the liar’s paradox can be dismissed as meaningless nonsense"
You're just taking your desired result ("that sentence is meaningless nonsense") for granted.
The fact that minds much larger than yours dedicated so much thought and research to it should rather give you pause, not in an "argument from authority" way, but in a "maybe I'm not the biggest genius and they're all stupid" humble moment.
For starters, the Liar's Paradox has been a key insight into creating type theory, and understanding meta-statements.
It has also spawned variations like Russel's Paradox, Godel's incompletness proving technique, and Turing's halting theorem, that not only proved it fruitful, but gave some of the most important results and insights into set theory, algebra, and computer science.
"1=2" and your other "nonsense" examples never did anything of the sort. That alone should hint you that there's a substantial difference between the Liar's Paradox statement and your examples.
"We don’t need overly complex semantics of meaning that butcher our ordinary understanding of language to make sense of a nonsense statement"
Sure, if we just hand-wave the issue away, and insist loudly enough that it's a non-issue.
"The fact that you can’t divide by 0 isn’t a challenge posed for some math hero to solve, but a logical fact that should be accepted and incorporated into our model of mathematics"
Actually it's just an artifact of specific axioms chosen. There are valid mathematics where division by 0 is totally fine (a category of algebras called "wheel algebras").
Thanks for the review. As a general point, the article is only discussing the "meaning" of the liar's paradox - it makes no claims as to its alleged contributions. As mentioned, there is value in understanding why meaningless nonsense "is" meaningless nonsense. (And many others have attributed this to the liar's paradox as well, I'm not stumbling on anything new). This article only discusses meaninglessness in discourse.
Just because 1=2 hasn't participated in the same contributions as the liar's paradox doesn't mean the two aren't equivantely self-contradictory. All propositions assume their truth (see deflationary accounts of truth). So when a proposition says that it is also false, it becomes self-contradictory. You can't proclaim truth and falsehood the same way you can't proclaim a shape both has 3 sides and 4 sides—not at least without being self-contradictory. You might as well be saying that 1=2. This is why I argue that the liar's paradox is meaningless, the speaker couldn't be conveying any information and the listener couldn't have gained anything.
You either have to either accept that the sentences which I provided above "aren't" self-contradictions or the liar's paradox is as well.