Self-reference and Logic (2005) [pdf] | Hacker News Reader

Self-reference and Logic (2005) [pdf]

imm.dtu.dk

60 pointsidiliv21 comments

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.

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

"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.

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.

No, there are 3 sentences implied in the context, but only 1 sentence is visible in this specific example.

Natural language normally isn't a fully self-contained representation.

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

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?

What type of logic uses period and quote operators? What are their semantics?

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.

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!

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?

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

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.

P = P -> Q

P = !P v Q

Q |= _|_

P = !P v _|_

P = !P

Your reasoning only works for valid statements. That is, statements that have a truth value. As you’ve demonstrated P does not have one. For no admissible statement is P equal to not P.

1. Premise: P -> Q

2. Premise: ((P -> Q) -> P) -> P

3. (P -> Q) -> P, from 1

4. P, from 2 and 3

5. Q, from 1 and 4

The P, in your original post, is not a valid formula since it does not have a truth value. You can construct all the proofs and whatnot you want but if P is not a logical statement (one that is either true or false but not both) then the arguments are not valid.

Your Q was generic and could have been any statement. In particular one that has been provably shown to be false. If your arguments are correct then you have the ability to prove any false statement is true.

Do you really think you have found a way to demonstrate that the standard two-valued logic used in mathematics leads to a contradiction?

Just repeating Curry's paradox from Wikipedia.

I see. The difference in our views comes from me being a mathematician. Curry’s Paradox is not a paradox in mathematical logic due to my objections. In mathematical logic we don’t allow statements that don’t have a well defined truth value. The premise in the start of Curry’s Paradox is a statement without a well defined truth value.

Curry’s Paradox shows the limits if naive set theory. Thus mathematically we have to be OK with the idea that not all naive set constructions produce valid sets.

In informal setttings we can argue that

If this sentence is true then Santa Claus exists

can’t be false. But then one usually assumes it must then be true. I would argue this isn’t the case. There are sentences that have no truth value. For instance,

This sentence is false.

Thanks, this clarifies things for me.

Runtime Error: Stack Overflow in recursive function.

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