Ah okay. Sorry for being an a-hole, but some of the comments here are just... You asked a question in good faith and I mistook it.
So, it’s simple:
Imagine you’re playing with LEGO blocks.
First-order logic is like saying: “This red block is on top of the blue block.” You’re talking about specific things (blocks), and how they relate. It’s very rule-based and clear.
Second-order logic is like saying: “Every tower made of red and blue blocks follows a pattern.” Now you’re talking about patterns of blocks, not just the blocks. You’re making rules about rules.
Why can’t machines fully “do” second-order logic? Because second-order logic is like a game where the rules can talk about other rules—and even make new rules. Machines (like computers or AIs) are really good at following fixed rules (like in first-order logic), but they struggle when:
The rules are about rules themselves, and You can’t list or check all the possibilities, ever—even in theory. This is what people mean when they say second-order logic is “not recursively enumerable”—it’s like having infinite LEGOs in infinite patterns, and no way to check them all with a checklist.
Maybe I should have asked: In what sense are machines “fully doing” first-order logic? I think I understand the part where first logic formulas are recursively enumerable, in theory, but isn’t that intractable to the point of being useless and irrelevant in practice?
Ah okay. Sorry for being an a-hole, but some of the comments here are just...
You asked a question in good faith and I mistook it.
So, it’s simple:
Imagine you’re playing with LEGO blocks.
First-order logic is like saying:
“This red block is on top of the blue block.”
You’re talking about specific things (blocks), and how they relate. It’s very rule-based and clear.
Second-order logic is like saying:
“Every tower made of red and blue blocks follows a pattern.”
Now you’re talking about patterns of blocks, not just the blocks. You’re making rules about rules.
Why can’t machines fully “do” second-order logic?
Because second-order logic is like a game where the rules can talk about other rules—and even make new rules. Machines (like computers or AIs) are really good at following fixed rules (like in first-order logic), but they struggle when:
The rules are about rules themselves, and
You can’t list or check all the possibilities, ever—even in theory.
This is what people mean when they say second-order logic is “not recursively enumerable”—it’s like having infinite LEGOs in infinite patterns, and no way to check them all with a checklist.
Maybe I should have asked: In what sense are machines “fully doing” first-order logic? I think I understand the part where first logic formulas are recursively enumerable, in theory, but isn’t that intractable to the point of being useless and irrelevant in practice?