self-referencing definition

[mr. vodka]

Is it allowed to define a proposition as follows:

Proposition B = "Proposition B is not true"

Probably not, but what is the ground reason for this? The fact that if you would be allowed to do that, that you'd come to a contradiction? That would imply that for defining any proposition, you first have to proof that it is possible to define a proposition as such? Does this sometimes get very tricky?

Thank you,
mr. vodka

[Hurkyl]

The unary predicate*
B \equiv \neg B
is certainly well-defined, but there is nothing that satisfies that relation.


Compare with the equation "x = x+1" where x is a real number valued variable.


*: A technical detail: if B is an n-th order predicate symbol, then that relation is of (n+1)-th order

[mr. vodka]

I see. But in the case of x = x + 1, x is undefined (--or maybe the right word is "undetermined", don't know if it matters--) itself (only the equality itself is defined). Isn't B itself defined/determined in my case? B = "B is not true" gives B a value, does it not? (namely, the fact that B is true, is the "content" of proposition B)

In other words, where do I cross the line in the following reasoning:

Define:
B = "B is not true"
If B were true, then "B is not true" is true, meaning B is not true. Contradiction.
If B were not true, then "B is not true" is false, meaning "B is true" would be true, implying B is true. Contradiction.

Thanks for your help.

[HallsofIvy]

Every proposition, by definition of the word "proposition" in logic, must be either true or false- even if we do not know which. Since, as you say, assuming either true or false leads to a contradiction, that is not a proposition.

[Martin Rattigan]

In general you can only define something in terms of the "undefined ideas" on which you base a subject, or previously defined things.

In your example proposition B is not previously defined, so the answer to the question, "Is it allowed ... ?", is, "no". Which is to say, of course you can do it, but don't expect it to make sense.

You'ld probably find "Vicious Circles" by Barwise & Moss an interesting read.

[Martin Rattigan]

Self reference, directly or indirectly, is not necessary for things to come adrift by the way.

If you try to define the propositions R_n for each n\in \mathbb{N} by

R_n="R_m is false for some m>n"

then each R_n would be both true and false.

In this example each R_n refers only to propositions for higher numbers so can't indirectly refer to itself.

[JSuarez]

To be a little more precise, it's perfectly possible to formulate a meaningful sentence like that one (here in the more common form):

"This statement is false"

But a sentence (roughly, a series of symbols in some media) is distinct from a proposition (again roughly, something that expresses a state of affairs); the current view is that meaningful sentences can be used to denote propositions, but not all (meaningful) sentences denote a proposition and the above is one of the best known examples.

As for ground reasons, there is the principle of bivalence that states that a proposition (not a sentence), must have a definite truth value.

[Hurkyl]

B = "B is not true" gives B a value, does it not?
No. If B were well-defined, and satisfied this equation, then you could manipulate B by substituting "not B" wherever B appears, and vice-versa.

x=x+1 is not a definition either. Why do you think "B = not B" is?


Incidentally, 2x=1 is not a definition either (x is a real number-valued variable). However, we can still use it to define x, because we can demonstrate that there is a unique real number that satisfies this equation.

[mr. vodka]

Hm, thank you all for the interesting replies.

I'm getting some mixed information, it seems (or I'm just interpreting it wrong). Let see if I'm getting it...

Hurkyl noted that the predicate (as you called it) "B = not B" is well-defined, but it implies that B is undetermined and the predicate is void because by definition of logic there is no B that satisfies it, meaning you can't build up a proof leading up to a contradiction, because there is no proposition to manipulate/use. So "B = not B" is a predicate, talking about a predicate which cannot exist? (I'm not saying Hurkyl said all this, I'm paraphrasing and adding some assumptions of my own, so of course any errors come onto my name).

On the other hand, HallsofIvy said "B = not B" is not a proposition at all, because it defies the definition of proposition. Maybe he meant the same as Hurkyl, but skipping the fact that it's still a definition, but one which will fit for no proposition.
But the fact that you say you cannot define a proposition if it's possibly non-false and non-true at the same time, that implies that with EVERY definition of a proposition you make, you have to proof that it can have a true/false-value before it can be seen as a real proposition? Why do we never do this?

One more question: why is it that "B = not B" is some sort of condition that must be satisfied and not a definition of B? What is the difference between "B = pi is larger than 4" and "B = not B"? I doubt it's the superficial fact that B appears twice in my statement, because then I'll show you "B = this proposition is false".

How do you know what you're dealing with?

PS: Martin, I'll check out the book, thank you

[JSuarez]

But the fact that you say you cannot define a proposition if it's possibly non-false and non-true at the same time, that implies that with EVERY definition of a proposition you make, you have to proof that it can have a true/false-value before it can be seen as a real proposition? Why do we never do this?


Because we don't define propositions: they, and their truth-values, exist independently of us; we state hypothesis and then attempt to prove that they correspond to some true proposition.

One more question: why is it that "B = not B" is some sort of condition that must be satisfied and not a definition of B? What is the difference between "B = pi is larger than 4" and "B = not B"? I doubt it's the superficial fact that B appears twice in my statement, because then I'll show you "B = this proposition is false".

In fact, "B = not B" does define something, if B is a predicate over a domain of objects: B is the (only) predicate that has an extension that is equal to the one of its complement; in other words it's the "always false" predicate or, if you are inh a context where sets are meaningful, the empty set.

The main differences between "B = pi is larger than 4" and "B = not B" is that in the former, you are defining the symbol B to be the (true) sentence "pi is larger than 4", where in the latter, you are defining B in terms of another expression that involves again B. In this case, if you interpret it as above, it's possible, but there are other cases where this may lead to improper definitions, like "B = the only natural number such that B is strictly smaller than itself".

[Martin Rattigan]

... in the former, you are defining the symbol B to be the (true) sentence "pi is larger than 4"

In what sense is this a true sentence?

[JSuarez]

The proposition referenced by the sentence "pi is larger than 4" is true: the number denoted by "pi" is larger than the number denoted by "4", given the usual model of the real numbers.

[Martin Rattigan]

Well, I'll have to think about that.

[mr. vodka]

Do you guys have a quote of the month?