<p>Specifically relating to consistency:</p>
<p>Consistency is a property of logical systems (First order logic, type theory, simply typed lambda calculus) etc. It's the property that there are no proofs of False, which in turn implies that not every proposition has a proof.</p>
<p>A Turing machine is not a logical system. It's a specific syntax for defining a set of words over some alphabet, that happens to be either equivalent or more powerful than every other model of computation we can think of.</p>
<p>You could relate a Turing Machine to a logic by e.g. making a Turing machine that accepts a specific encoding of theorems in a logic, or to accept well formed proof trees in a given logic. But without specifying such a relationship the phrase &quot;inconsistent Turing machine&quot; is simply nonsense.</p>

CC BY-SA 4.0
2024-05-25T19:09:49.243Z
broadly speaking, an *inconsistent Turing machine == a Turing machine "implementing" an inconsistent logical system* is something I can live with.
Barney
CC BY-SA 4.0
2024-05-25T23:12:09.750Z
@Barney but what does implementing mean? Are you deciding the system? Semi-deciding it? Enumerating theorems? Checking proof terms? Checking typing deprivations?
Joey Eremondi
CC BY-SA 4.0
2024-05-26T18:23:09.190Z
Well, depends, that's why I quoted it. In different contexts, that can mean any of that. What I want to say, the relation between the system and the machine must be explicitly specified to get any sense of it.
Barney
CC BY-SA 4.0
2024-05-25T18:19:12.490Z
2024-05-25T18:19:12.490Z
Joey Eremondi
<p>The source you pointed to is not worth worrying about. Not definining the notion of &quot;inconsistent Turing machine&quot; is the least of its problems. One can derive arbitrary conclusions by taking a mathematical model of computation (Turing machine), vaguely map it onto a non-mathematical phenomenon (human mind), mix that with speculation that has not been supported by anything (quantum effects might matter in the functioning of a human mind), and pretend that everything is logically crisp so that &quot;logical&quot; conclusions about the situation having any standing.</p>
<p>Look, I can do it too:</p>
<ol>
<li>The grammar of untyped <span class="math-container">$\lambda$</span>-calculus is context free.</li>
<li>Humans are capable of parsing non-context-free grammars.</li>
<li>Therefore, humans are more powerful than the <span class="math-container">$\lambda$</span>-calculus.</li>
</ol>
<p>Of course, what I wrote is worthless, but I bet with a little bit of obfuscation I could generate a tiresome discussion.</p>

CC BY-SA 4.0
2024-05-25T08:36:28.370Z
yes, this is fine, but the mind-machine discussion was just where I encountered the term. I am still wondering if an *inconsistent Turing machine* is a thing? Or is it purely context-dependent?
Barney
CC BY-SA 4.0
2024-05-25T08:48:27.527Z
But why do you worry about some unexplained concept used in a discussion that is riddled with problems? Consistency is a property of formal systems. Anyone who says "inconsistent Turing machine" (in a context where they discuss consistency of fomral systems) has to explain themselves. It's not up to everyine else to try to make sense of whatever they're saying.
Andrej Bauer
CC BY-SA 4.0
2024-05-25T09:18:34.323Z
Fair enough, this is exactly what I wanted to know - if this was an established term I didn't understand, or some made-up, context-dependent, further unspecified one. Thanks!
Barney
CC BY-SA 4.0
2024-05-25T08:27:27.707Z
2024-05-25T09:36:24.303Z
2024-05-25T09:36:24.303Z
Andrej Bauer
<p>I am reading <a href="https://iep.utm.edu/lp-argue/" rel="nofollow noreferrer">responses to the Lucas-Penrose argument</a> and many make sense to me. Some of them employ inconsistent Turing machines as a model of mind to escape Gödel’s first incompleteness theorem (as it applies only to consistent systems).</p>
<p>I wonder how a Turing machine can be inconsistent? By definition, it MUST consistently follow the rules (the transition function).</p>
<p>Maybe not inconsistency but rather correctness is meant? Or better to say, consistency of the theory the Turing machine generates.</p>
<p>For instance, we want a Turing machine which, say, shall print 1 if the input is 1, and 0 otherwise. If we construct a Turing machine that always prints 1 regardless of the input, can we call it inconsistent in terms of those responses?</p>
<p>Another possibility could be the inconsistency of an actual machine caused by its physical limitations, such as a slow CPU or running out of memory. However, this is more of an error than inconsistency. Moreover, Gödel’s theorem operates on a more platonic level.</p>
<p>If not, what makes a Turing machine inconsistent, resp. consistent? What is a simple example of such an inconsistent Turing machine?</p>

<turing-machines>
Inconsistent Turing machine
Barney
# Inconsistent Turing machine

(Asked by: username_0 on 2024-05-25)

I am reading [responses to the Lucas-Penrose argument](https://iep.utm.edu/lp-argue/) and many make sense to me. Some of them employ inconsistent Turing machines as a model of mind to escape Gödel’s first incompleteness theorem (as it applies only to consistent systems).

I wonder how a Turing machine can be inconsistent? By definition, it MUST consistently follow the rules (the transition function).

Maybe not inconsistency but rather correctness is meant? Or better to say, consistency of the theory the Turing machine generates.

For instance, we want a Turing machine which, say, shall print 1 if the input is 1, and 0 otherwise. If we construct a Turing machine that always prints 1 regardless of the input, can we call it inconsistent in terms of those responses?

Another possibility could be the inconsistency of an actual machine caused by its physical limitations, such as a slow CPU or running out of memory. However, this is more of an error than inconsistency. Moreover, Gödel’s theorem operates on a more platonic level.

If not, what makes a Turing machine inconsistent, resp. consistent? What is a simple example of such an inconsistent Turing machine?

**Tags:** turing-machines

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## 2 answers

### Answer by username_1 on 2024-05-25. Score: 6 | Accepted answer

Specifically relating to consistency:

Consistency is a property of logical systems (First order logic, type theory, simply typed lambda calculus) etc. It's the property that there are no proofs of False, which in turn implies that not every proposition has a proof.

A Turing machine is not a logical system. It's a specific syntax for defining a set of words over some alphabet, that happens to be either equivalent or more powerful than every other model of computation we can think of.

You could relate a Turing Machine to a logic by e.g. making a Turing machine that accepts a specific encoding of theorems in a logic, or to accept well formed proof trees in a given logic. But without specifying such a relationship the phrase "inconsistent Turing machine" is simply nonsense.

> **Comments:**
> **username_0**: broadly speaking, an *inconsistent Turing machine == a Turing machine "implementing" an inconsistent logical system* is something I can live with.  
> **username_1**: @Barney but what does implementing mean? Are you deciding the system? Semi-deciding it? Enumerating theorems? Checking proof terms? Checking typing deprivations?  
> **username_0**: Well, depends, that's why I quoted it. In different contexts, that can mean any of that. What I want to say, the relation between the system and the machine must be explicitly specified to get any sense of it.  

---

### Answer by username_2 on 2024-05-25. Score: 9

The source you pointed to is not worth worrying about. Not definining the notion of "inconsistent Turing machine" is the least of its problems. One can derive arbitrary conclusions by taking a mathematical model of computation (Turing machine), vaguely map it onto a non-mathematical phenomenon (human mind), mix that with speculation that has not been supported by anything (quantum effects might matter in the functioning of a human mind), and pretend that everything is logically crisp so that "logical" conclusions about the situation having any standing.

Look, I can do it too:

1. The grammar of untyped $\lambda$-calculus is context free.
2. Humans are capable of parsing non-context-free grammars.
3. Therefore, humans are more powerful than the $\lambda$-calculus.

Of course, what I wrote is worthless, but I bet with a little bit of obfuscation I could generate a tiresome discussion.

> **Comments:**
> **username_0**: yes, this is fine, but the mind-machine discussion was just where I encountered the term. I am still wondering if an *inconsistent Turing machine* is a thing? Or is it purely context-dependent?  
> **username_2**: But why do you worry about some unexplained concept used in a discussion that is riddled with problems? Consistency is a property of formal systems. Anyone who says "inconsistent Turing machine" (in a context where they discuss consistency of fomral systems) has to explain themselves. It's not up to everyine else to try to make sense of whatever they're saying.  
> **username_0**: Fair enough, this is exactly what I wanted to know - if this was an established term I didn't understand, or some made-up, context-dependent, further unspecified one. Thanks!  

---

