Unlike first-order logic, second-order logic is not recursively enumerable—less computationally tractable, more fluid, more human. It operates in a space that, for now, remains beyond the reach of machines still bound to the strict determinism of their logic gates.
In what sense is second-order logic “beyond the reach of machines”? Is it non-deterministic? Or what are you trying to say here? (Maybe some examples would help)
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?
Think of it like this: Why is Gödel’s attack on ZFC and Peano Arithmetic so powerful...
Gödel’s Incompleteness Theorems are powerful because they revealed inherent limitations USING ONLY first-order logic. He showed that any sufficiently expressive, consistent system cannot prove all truths about arithmetic within itself… but with only numbers.
First-order logic is often seen as more fundamental because it has desirable properties like completeness and compactness, and its semantics are well-understood. In contrast, second-order logic, while more expressive, lacks these properties and relies on stronger assumptions...
According to EN, this is also because second order logic is entirely human made.So what is second-order-logic? The question itself is a question of second-order-logic. If you ask me what first order logic is… The question STILL is a question of second-order-logic.
First order logic are things that are clear as night and day. 1+1, what is x in x+3=4… these type of things.
In what sense is second-order logic “beyond the reach of machines”? Is it non-deterministic? Or what are you trying to say here? (Maybe some examples would help)
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?
Think of it like this: Why is Gödel’s attack on ZFC and Peano Arithmetic so powerful...
Gödel’s Incompleteness Theorems are powerful because they revealed inherent limitations USING ONLY first-order logic. He showed that any sufficiently expressive, consistent system cannot prove all truths about arithmetic within itself… but with only numbers.
First-order logic is often seen as more fundamental because it has desirable properties like completeness and compactness, and its semantics are well-understood. In contrast, second-order logic, while more expressive, lacks these properties and relies on stronger assumptions...
According to EN, this is also because second order logic is entirely human made.So what is second-order-logic?
The question itself is a question of second-order-logic.
If you ask me what first order logic is… The question STILL is a question of second-order-logic.
First order logic are things that are clear as night and day. 1+1, what is x in x+3=4… these type of things.