Where to start with mathematical logic?

[Dirac]

I'm writing a story right now in which there are machines being used to try automate the answering of questions, but the machines have difficulty answering questions about themselves. It's come to my attention that this seems to be related to mathematical logic, and Godel's incompleteness theorems. I've never studied this kind of maths before and wondered if anybody knows of a good place to start?

[notdavidlynch]

http://www.amazon.com/The-Logic-Book...the+logic+book

That's what I used IIRC.

I think the painting used as an illustration makes it worth all the money.

[Rhu]

I'm writing a story right now in which there are machines being used to try automate the answering of questions, but the machines have difficulty answering questions about themselves.

This immediately reminded me of the Halting Problem. Of course, this veers off a bit from the pure math logic, but may relate to your story if your machines fit the definition of Turing machines.

[Dirac]

http://www.amazon.com/The-Logic-Book...the+logic+book

That's what I used IIRC.

I think the painting used as an illustration makes it worth all the money.

Looks pretty decent, thanks. How maths-y is it though? Would prefer to get stuck in with that but I guess I've never actually studied logic so maybe I'm jumping the gun a bit.

This immediately reminded me of the Halting Problem. Of course, this veers off a bit from the pure math logic, but may relate to your story if your machines fit the definition of Turing machines.

The machines are a fairly recent addition to the story and the details aren't clear in my mind. I'm not sure how clear the details will actually need to be so who knows if they are Turing machines? Maybe I'll never find out.

[Perdix]

If it is a story and not a real life scenario I would just give them "strong" artificial intelligence by means of genetic algorithms...

You're essentially looking for a mathematical version of sentience -- no?

[notdavidlynch]

Don't try to get out of learning basic logic. It's essential for far more than your story.

[Dirac]

If it is a story and not a real life scenario I would just give them "strong" artificial intelligence by means of genetic algorithms...

You're essentially looking for a mathematical version of sentience -- no?

I don't really want them to be sentient, no. More like http://en.wikipedia.org/wiki/Automated_theorem_proving

Don't try to get out of learning basic logic. It's essential for far more than your story.

Yeah I sort of know that you're right though. Seems kinda dry to me without the maths though :P Also, if I'm going to buy an expensive book I'd like it to actually get to the stuff I'm primarily interested in. Maybe I'll be able to find a pdf of it though.

[pensive_pilgrim]

Also, if I'm going to buy an expensive book I'd like it to actually get to the stuff I'm primarily interested in.

You mean like digital logic? I used this for my digital logic design class, but it's pretty computer-focused. I used this for my discrete math class and it's more general and mathy. Both are available for free on tpb.

[Utisz]

I'm writing a story right now in which there are machines being used to try automate the answering of questions, but the machines have difficulty answering questions about themselves. It's come to my attention that this seems to be related to mathematical logic, and Godel's incompleteness theorems. I've never studied this kind of maths before and wondered if anybody knows of a good place to start?

If I understand your intent, you could always just ask the machine, right? ... "Is there any question you cannot answer?" ... or more correctly "Is there any question you cannot answer correctly and in finite time?"

If the machine is a deterministic Turing machine (the model of pretty much every [Turing-complete] machine we know), there are questions it certainly can't answer, proven by (as Rhu mentions) the "Halting Problem" and other undecidable problems.

If it is a clever machine (which it sounds like it is), it will be able to give you a question it cannot answer and then you can ask that question. Otherwise you win at step 1.


Would this work for your story?

[Johnny]

At one time I thought the "problem" with any sort of logic is that it starts with assumptions.

But then I heard someone say that philosophy is a game. I stopped thinking of problems and started playing after that.

Maybe losing oneself is a consequence?