Why are paradoxes/contradictions (so) important? — Agent Smith

Not sure this is relevant but I generally accept that humans are clever animals who use language to help manage their environment. As a consequence, meanings and worldviews are riddled with inconstancies and subversions, some of them more striking than others. When I encounter a paradox it tends to remind me of the poetic, imprecise nature of language and the manufactured character of human understanding.

The question of all questions is "is the imprecision a bug in language or a feature of reality?"

My main concern is the existence/nonexistence of (true) paradoxes. If they exist then, classical logic is trivial unless it excludes some rule of natural deduction that prevents ex falso quodlibet. The rule that most logicians choose to exclude from natural deduction in order to prevent explosion is disjunction introduction/addition. Should we do that? It seems the right course of action assuming there are (true/real) paradoxes.

The bottom line is this: either resolve all paradoxes OR accept paraconsistent logic.

Classical logic has to use Occam's broom — Agent Smith

What about Occam's shaving gel? Makes the razor run smooth... Shave those needless hairs away! Smoooooth.....

The very definition of a paradox is that it can be resolved. The true paradox, that is.

:smile:

Negative self reference.

Just ask yourself, how many paradoxes involve this. Starting from Russell's paradox.

Why are paradoxes/contradictions (so) important?

Their significance to all (real) thinkers is that renders trivial the logical systems in which they arise. — Agent Smith

Perhaps I am not a (real) thinker, but all the excitement about paradoxes goes over my head. I just can't see how they have any practical meaning.

Not sure this is relevant but I generally accept that humans are clever animals who use language to help manage their environment. As a consequence, meanings and worldviews are riddled with inconstancies and subversions, some of them more striking than others. When I encounter a paradox it tends to remind me of the poetic, imprecise nature of language and the manufactured character of human understanding. — Tom Storm

It strikes me that many (most? all?) so-called paradoxes are really just playing with language. There was a discussion a month or so ago on the forum about whether the Liar's sentence/Russell's paradox undermine the validity of mathematics. Apparently Alan Turing actually believed that, because of those paradoxes, bridges designed with mathematics might fall down. I find that hard to grasp. [irony]Turing was somewhat smarter than I am.[/irony] I don't understand how he could believe that.

My main concern is the existence/nonexistence of (true) paradoxes. If they exist then, classical logic is trivial unless it excludes some rule of natural deduction that prevents ex falso quodlibet. The rule that most logicians choose to exclude from natural deduction in order to prevent explosion is disjunction introduction/addition. Should we do that? It seems the right course of action assuming there are (true/real) paradoxes. — Agent Smith

I wasn't familiar with the idea of logical explosion, so I opened my trusty Wikipedia. Here's the example used in that article:

As a demonstration of the principle, consider two contradictory statements—"All lemons are yellow" and "Not all lemons are yellow"—and suppose that both are true. If that is the case, anything can be proven, e.g., the assertion that "unicorns exist", by using the following argument:

[1] We know that "Not all lemons are yellow", as it has been assumed to be true.

[2] We know that "All lemons are yellow", as it has been assumed to be true.

[3] Therefore, the two-part statement "All lemons are yellow or unicorns exist" must also be true, since the first part "All lemons are yellow" of the two-part statement is true (as this has been assumed).

[4] However, since we know that "Not all lemons are yellow" (as this has been assumed), the first part is false, and hence the second part must be true to ensure the two-part statement to be true, i.e., unicorns exist.

To me, that example is based on a case of philosophical bait and switch. The two propositions in question "Not all lemons are yellow" and "All lemons are yellow," are concrete examples from the real world. Please show me a "paradox" like that. I can't think of any. The contradictory examples that tangle philosopher's and mathematician's shorts are all language paradoxes, e.g. "This sentence is false."

On an unrelated, or at least only semi-related subject, does the fact that light has both a wave and particle nature constitute a valid example of a real-life, concrete paradox, which I just denied the existence of?

:up:

t strikes me that many (most? all?) so-called paradoxes are really just playing with language — T Clark

:up:

It strikes me that many (most? all?) so-called paradoxes are really just playing with language. — T Clark

Escher's paradoxical ever up or down going stairs is about the angle of vision (that resolved the seeming contradiction). The twin paradox is about everyday experience and gravity, resolved by general relativity. "Contra-diction" is not always about diction.

Escher's paradoxical ever up or down going stairs is about the angle of vision (that resolved the seeming contradiction). The twin paradox is about everyday experience and gravity, resolved by general relativity. "Contra-diction" is not always about diction. — EugeneW

You're right. After I wrote that about language, I thought of Zeno's paradox. Now I'm trying to figure out how Zeno's and Russell's paradoxes are different from the liar's paradox, if they are. I think the twin paradox is only a paradox if you don't understand general relativity, which, of course, I don't. As for Escher, I would call the things he drew optical illusions. Is that the same thing as a paradox, just in a visual rather than a verbal medium? I'm not sure.

I don't think that changes my impression that the strain paradoxes supposedly put on philosophy is illusory. It seems pretty unlikely that I've got it right while some of the smartest people in history have it wrong, so I'm hoping to be enlightened.

Perhaps I am not a (real) thinker, but all the excitement about paradoxes goes over my head. I just can't see how they have any practical meaning. — T Clark

It's not that complicated. Given the natural deduction rules (there are 18 of 'em) of predicate logic, the existence of a true paradox means predicate logic is what logicians call trivial - it proves every statement conceivable is true via ex falso quodlibet (explosion).

The only way out is to adopt paraconsistent logic which accepts the existence of true contradictions, but prevents explosion by tweaking the rules of natural deduction e.g. it does away with the disjunction introduction/addition rule.

The yellow lemons, unicorn example argument of an ex falso quodlibet uses the disjunction introduction/addition rule in line [3].

Negative self reference.

Just ask yourself, how many paradoxes involve this. Starting from Russell's paradox. — ssu

I've seen at least two negative self-referential paradoxes: the liar sentence and Curry's paradox.

Your point?

It's not that complicated. — Agent Smith

I didn't say that the idea of paradoxes goes over my head, I said the excitement about them does. I just don't see why it's a big deal. They're not that hard to recognize. It's not like they can sneak up on you.

I didn't say that the idea of paradoxes goes over my head, I said the excitement about them does. I just don't see why it's a big deal. They're not that hard to recognize. It's not like they can sneak up on you — T Clark

Oh! Sorry, my bad. You didn't read my post thoroughly. I explain why paradoxes are a big deal.

Update

Paradoxes break (classical) logic.

Question: I'm told that any system of logic in which every sentence can be proven true is trivial. I know what "trivial" means, but in this case, it probably has a deeper meaning. Anyone knows what that is? Any help will be deeply appreciated. Thanks

Oh! Sorry, my bad. You didn't read my post thoroughly. I explain why paradoxes are a big deal. — Agent Smith

I did read your post thoroughly, Mr. Snooty. Agent Snooty. The explanation doesn't make sense to me. What difference does it make other than providing a bit of agita to some philosophers and mathematicians?

The zen masters loved to use paradox in their instructions to their students. For example “what is the sound of one hand clapping”. Language and words may only go so far, but truth lies beyond words. When you can talk without speaking, cry without weeping, scream without raising your voice, then you will understand.

Mr. Snooty. Agent Snooty — T Clark

:lol: I consider your actions an act of war!!

Read my last post, the post before you lost your mind. :smile:

Read my last post, the post before you lost your mind. — Agent Smith

I went back and reread your posts. I don't think there is any misunderstanding between us about the issue on the table. We just disagree on the implications. I have four answers to the question "What difference does it make that language paradoxes seem to undermine the value of logic?" Those answers are, in no particular order, none, zero, zilch, and nada.

Thanks for reminding me of Zen and its rather (unconventional) methods: the aim is to attain the so-called mushin no shin (mind without mind) state. The idea of Zen koans is to trigger a computer mind crash or Does not compute state.

On first hearing of this, I wondered if we were meant to experience what is it like to be a bat stone? but then no mushin no shin isn't no mind, it's mind without mind. Centrism/madhyamaka/the middle path.

I went back and reread your posts. I don't think there is any misunderstanding between us about the issue on the table. We just disagree on the implications. I have four answers to the question "What difference does it make that language paradoxes seem to undermine the value of logic?" Those answers are, in no particular order, none, zero, zilch, and nada. — T Clark

:ok: Great!

Three types of paradox

Falsidical – Logic based on a falsehood.
Veridical – Truthful.
Antinomy – A contradiction, real or apparent, between two principles or conclusions, both of which seem equally justified.

Maybe the optical illusion paradox should be included.

Paradoxes break (classical) logic — Agent Smith

Only when you use different types. The principle of explosion, based on two contradicting statements, was removed by the introduction of paraconsistent logìc, as its name suggests. It was used by Zermelo and Frankel to save set theory from disaster, Hilbert's paradox, with those strange sets including themselves, no longer paradoxed and the village barber could shave his own beard without repercussions...

Is an optical illusion a paradox? Can things move while standìng still? Can stairs go up while keeping you leveled? Can clocks run dìfferently for different observers? The last paradox is not an optical paradox. It's a veridical one.

What does it mean when you show a word you crossed out? Thought process?

A crashed computer can't be compared with a mind (is that why you crossed it out?). Zen means excluding all thoughts. Only experiencing the moment. But some structural organization remains. A mourning dove, a ticking clock, gas flowing in a pipe, an ache in the chest, a fly zooming in a glass, a door shutting, a fucking fridge, the whistle of a neighbor. It blends together in one... oh oooh...

Why is the twin paradox a paradox? Because people mostly can't imagine clocks to tick at different rates. But they do. It's why we fall down. I told this to someone and she said I am crazy...

I've seen at least two negative self-referential paradoxes: the liar sentence and Curry's paradox.

Your point? — Agent Smith

Usually the mathematical paradoxes/logical paradoxes are structured this way.

Do notice, with the same structure is also made theorems like Gödel's Incompleteness Theorem or Turings answer to the Halting Problem.

Above my pay grade, mate!

Self reference + Denial/Negation = Paradox.

I don't speak English! Oh, I just did! The only possibility, post elimination of the impossible, this (all I've written) is not English. :chin:

You noticed the point!

Give a reply to my comment that you won't never give in this forum.

Are there those comments that @Agent Smith doesn't give in this forum? Of course. Can you give them or utter them as @Agent Smith? Of course not! You are who you are.

The power of negative self reference.

You noticed the point! — ssu

Yaay! :smile:

I want to pick your brains on something. Why did you bring up negative self-reference? Do you have a specific reason for doing so? Are you, if I may ask, trying to say that all paradoxes can be reduced to a negative self-referential paradox?

I'd like to see you do that with Zeno's paradoxes if you don't mind that is. Can you?