Self-reference and Logic (2005) [pdf]

http://www.imm.dtu.dk/~tobo/essay.pdf

by idiliv • 8 years ago

60 21 8 years ago

imm.dtu.dk

hs86• 8 years ago

This is great for understanding why simpler logics (Boolean) are computable but too expressive ones (FO) are not. As soon as the logic is able to make statements about itself, things get 'weird'. A not formal(!) but imho still intuitive example:

"This sentence has five words."

This is a true statement. And we know that the conjunction of two true statements should also be true.

"This sentence has five words and this sentence has five words."

With eleven words each part of this statement is obviously false and here we can't rely on the "and", which we are used to from the simpler logics, anymore.

code_duck• 8 years ago

This is also the primary topic explored in the well known book Godel, Escher, Bach.

gerdesj• 8 years ago

"This sentence has five words and this sentence has five words." In English "this" need not always be self referential, so I'll retort with "This sentence is imprecise".

The two "this"s could refer to two different things. However, the second this would normally become that to indicate comparison but grammatically the original is still sound [with notes]:

This sentence [indicates a sentence with five words in it] has five words and this sentence [indicates another sentence, also with five words in it] has five words.

jjnoakes• 8 years ago

Even if you try to separate what each 'this' refers to, there is still only one sentence, so the sentence becomes meaningless if you do that.

pretendscholar• 8 years ago

Simple, add a period after each statement: This sentence has five words. and. This sentence has five words. Fields medal please.

krackers• 8 years ago

Why isn't this a valid solution?

"This sentence has five words" AND "this sentence has five words" seems to be logically consistent. Doesn't trying to construct the logical proposition in natural english violate some sort of rule?

schoen• 8 years ago

I think this is more a problem of deixis or indexicality than of self-reference particularly.

https://en.wikipedia.org/wiki/Indexicality#In_linguistic_pra...

https://en.wikipedia.org/wiki/Deixis

For example, if I say "it's sunny out now" or "today is a warm day" or "Berkeley is east of here" I also run into problems related to the prospect that references may not have the same referent in different contexts. (For example, two speakers can respectively affirm and deny each of these propositions even though they may both agree in all of their beliefs!)

The example that you give can also be interpreted as a deictic problem because the referent of "this sentence" can change from sentence to sentence, even though we normally don't expect the referents of noun phrases to change this way.

CliffStoll• 8 years ago

An excellent paper, especially reviewing the connections of Peano, Tarski, and Godel.

I'm surprised that he did not mention Jim Propp's self=referential aptitude test. It starts with:

1. The first question whose answer is B is question (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

And, of course, Bolander's paper ends with an utterly delightful final sentence!

keithnz• 8 years ago

Runtime Error: Stack Overflow in recursive function.

yters5• 8 years ago

If this sentence is true, then Santa Claus exists. Therefore, Santa Claus exists.

skh4• 8 years ago

If P then Q is a true statement when P is false. (Assuming standard two valued logic.). The statement

If P then Q

being true does not mean Q has to be true. I don’t think you can conclude Santa Claus exists based on your argument.

yters5• 8 years ago

P = P -> Q

P = !P v Q

Q |= _|_

P = !P v _|_

P = !P

firethief• 8 years ago

Footnote 17 mentions a proposed approach of "extending logic to include contexts", but doesn't name any names. Anyone have a reference I can follow?

Self-reference and Logic (2005) [pdf]

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