this post was submitted on 31 Jul 2024
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It's not pedantic. You can mathematically prove math.
You can't mathematically/algorithmically prove an event happened or did not happen.
Adding "mathematically/algorithmically" in front of the word "prove" as if it were always implicitly there, and suggesting that it's the only way we should be using the word "prove" seems pretty darned pedantic to me.
We're describing the behavior of software. It must be implicitly included. Software cannot do anything that isn't algorithmic.
You can prove mathematical logic and you can (not 1-to-1) tie that to symbolic logic, but since it's not 1-to-1, because of ambiguity of symbols, there will be much more complexity. I personally think that the future of various machine assistants lies there, and what LLM's now do is going to be used in auxiliary roles for that.
The problem is that mathematical proofs rely on the basic premise that the underlying assumptions are rock solid, and that the rules of the math are rock solid. It's rigorous logic rules, applied mathematically.
The real world is Bayesian. Even our hard sciences like physics are only "mostly" true, which is why stuff like relativity could throw a wrench in it. There's inherent uncertainty for everything, because it's all measurement based, with errors, and more importantly, the relationships all have uncertainty. There is no "we know a^2 and b^2, so c^2 must be this". It's "we think this news source is generally reliable and we think the sentiment of the article is that this crime was committed, so our logical assumption is that the crime was probably committed". But no link in the chain is 100%. "Rock solid" sources get corrupted, generally with a time lag before it's recognizable. Your interpretation of a simple article may be damn near 100%, but someone is still going to misread it, and a computer definitely can.
Uncertainty is central to reality, down to the fact that even quantum phenomena have to be talked about probabilistically because uncertainty is built in all the way down.
This is why many philosophers came to criticize metaphysical logic in the 1800s, viewing it as dealing with absolutes when reality does not actually exist in absolutes, stating that we need some other logical system which could deal with the "fuzziness" of reality more accurately. That was the origin of the notion of dialectical logic from philosophers like Hegel and Engels, which caught on with some popularity in the east but then was mostly forgotten in the west outside of some fringe sections of academia. Even long prior to Bell's theorem, the physicist Dmitry Blokhintsev, who adhered to this dialectical materialist mode of thought, wrote a whole book on quantum mechanics where the first part he discusses the need to abandon the false illusion of the rigidity and concreteness of reality and shows how this is an illusion even in the classical sciences where everything has uncertainty, all predictions eventually break down, nothing is never possible to actually fully separate something from its environment. These kinds of views heavily influenced the contemporary physicist Carlo Rovelli as well.
You are describing LLMs, yes. But not what I'm describing.
I'm talking about machine finding syllogisms and checking their correctness. This can't be rock solid because of interpretation of the statement in natural language with its fuzzy semantics, but everything after that can be made rock solid. While in LLMs even it isn't.
That's what I'm talking about.
Humans make mistakes, but not such as LLM-generated texts contain.
I mean that one can build a reasoning machine which an LLM isn't.
I'm not describing LLMs. LLMs are completely irrelevant, and my examples had nothing to do with LLMs.
Formal logic requires propositions be Boolean in nature. They're true, or they're false.
That's not the real world. There are no booleans in the real world. In the real world, everything, down to the fundamental particles, is inherently probabilistic.
Our "certainty" is at most 99. a lot of 9s. It's never 100%. You can't say "the New York Times said X", and "the New York Times is perfectly reliable", so "X must be true". It's "given that the NYT said X and the NYT has a history of reporting facts with reasonably high accuracy, the probability X is true is...". If they get caught being shady, the estimates of previous information learned from them is retroactively changed. But there is no "proof", because there is no certainty anywhere in the chain. The world and human understanding of it has to be Bayesian. Again, down to the Uncertainty Principle about low level particles. Uncertainty is fundamental to reality. There is no certainty.
Why are you writing this to me?
Do you know what a syllogism is?
It doesn't require being certain of the information we're building it on. Only of existence of such categories.
Naturally people in Antiquity and Middle Ages who used symbolic logic were even less certain of the actual truths and lies in the world than we are.
It allows the truth to be subjective, but not the logical constructions. This is a very important trait both then and now.
The difference between the filter and the data going through it.
Of course you can't just feed all the data of all the PoVs and similar cases on something, integrate it into a model and expect your PoV to not clash with its output.
It's philosophically the same as why using dialectics is bad for science.
A syllogism is a tool for theoretical reasoning that doesn't actually apply in the real world, because it relies on Boolean possibility spaces. There is never an "all articles by X are correct", and there is no theoretical possibility that "all articles by X are correct" in the real world. The connections in the real world are literally always probabilistic. In every case. Every time.
You can't use formal logic for any real world use case because there are no valid starting assumptions. The only thing logic can ever prove is internal consistency, not fact.
Yes, and being able to build structures with internal consistency would be an advantage.
Nobody says you can prevent any "AI" oracle from saying things that aren't true.
But a tool which would generate a tree of possible logical conclusions from something given in language and then divided into statements on objects with statistical dependencies could be useful.