# B Is there a symbol for "probably equals"?

#### John Bartle

Is there a symbol for "probably equals" ?

An example would be A probably equals B.

#### jedishrfu

Mentor
There’s one for approximately equals ie an equals with a wave mark over it and one for identically equals I think it’s an equals with a third horizontal Lin and one for not sure if they’re equal ie an equals sign with a question mark over it.

https://en.m.wikipedia.org/wiki/Equals_sign

https://en.m.wikipedia.org/wiki/List_of_mathematical_symbols

Here’s a discussion on Math Stack Exchange but it seems sadly there is not:

https://math.stackexchange.com/questions/1446010/symbol-for-probably-equal-to-barring-pathology

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#### John Bartle

@jedishrfu

Thanks. I'd rather have a probably equals symbol but I think I might just settle for the ≟ or the a.s. notation.

#### fresh_42

Mentor
2018 Award
Is there a symbol for "probably equals" ?

An example would be A probably equals B.
Depends on what probably means! If you do not know, then the question mark makes sense: $\stackrel{?}{=}$. If it is equal up to finitely many exceptions, or if the exceptions are otherwise a set of measure zero, then you can write ${=}_{a.e.}$ for "almost everywhere". Thus it all depends on what is "probably" to you. If you actually have a probability, then it is $P(x=a) = c\%$ with possibly an error margin $\pm e\%$ behind it.

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#### StoneTemplePython

Gold Member
@jedishrfu

Thanks. I'd rather have a probably equals symbol but I think I might just settle for the ≟ or the a.s. notation.
$X$ "probably equals $Y$ as meaning X equals Y in probability $X =_p Y$. Also saying $X =_{a.s} Y$ i.e. that X equal Y almost surely is a stronger criterion that X equals Y in probability.

This is tagged as B thread. If that tagging is correct, please don't use almost surely for anything as you'll misuse it
-- I'm pretty (almost?) sure on this.

#### John Bartle

@StoneTemplePython

I'm not real good with math in general, and I know even less concerning probability and statistics. What notation, for example, would you think makes sense for the statement:

cat very likely equals animal

I know cat is a subset of animal, but most people say a cat IS an animal also. I also want to point out the fact that it is only very likely an animal because nothing/not much is certain.

I would just go with cat =a.s. animal

#### StoneTemplePython

Gold Member
@StoneTemplePython

I'm not real good with math in general, and I know even less concerning probability and statistics. What notation, for example, would you think makes sense for the statement:

cat very likely equals animal

I know cat is a subset of animal, but most people say a cat IS an animal also. I also want to point out the fact that it is only very likely an animal because nothing/not much is certain.

I would just go with cat =a.s. animal
I appreciate the question but there are 2 big issues. One is that equalities run both ways so you need to be careful... I don't think people would claim an animal is a cat. You have the right idea with subsets. Why not $\text{cat} \subset \text{animal}$ with an asterisk saying that you are highly confident of this? Almost surely is a much stronger claim than this and I don't think that is what you are trying to say.

The other issue which really is a dead-end in philosophy (which is outside the scope of these forums) is a lot of people may define a cat to be a kind of animal -- once you start arguing about definitions and 'true meaning' it doesn't really go anywhere, and I don't think math (including probability) has much to say here.

#### John Bartle

One is that equalities run both ways so you need to be careful... I don't think people would claim an animal is a cat.
Oh yeah, I forgot about that.

You have the right idea with subsets. Why not $\text{cat} \subset \text{animal}$ with an asterisk saying that you are highly confident of this?
Uhh. Yeah, I might use the subset and an asterisk symbol. I'm not sure if that fits with my particular situation, but I'll consider it.

#### jedishrfu

Mentor
A better one might be a % over the equals

#### fresh_42

Mentor
2018 Award
If probably means likely in its linguistic sense, then it is not a mathematical expression and thus there is no sign for it. If it is quantifiable in any sense, then there are signs which depend on the context and the degree of quantification.

#### jedishrfu

Mentor
What signs are used for quantifiable items?

#### fresh_42

Mentor
2018 Award
Depending on context:

$\approx\quad \sim \quad\equiv\quad =\quad \stackrel{?}{=} \quad P(\ldots) \quad \pm \quad=_{a.e.}\quad \in \quad \subseteq \quad \triangleq\quad\circeq \quad \doteq \quad \cong \quad \fallingdotseq$

There is no sign for: "I have no idea but I guess probably ..."

#### jedishrfu

Mentor
Ahh okay. I didn’t think of those in a probalistic sense.

#### fresh_42

Mentor
2018 Award
If they are not otherwise used, authors will have some freedom to attach a certain meaning as all signs are context sensitive. In any case, there has to be a definition, although I would avoid regular signs as $=$ or $\cong$ etc. It always comes back to what probably means. I think I would write is as $=_{a.c.}$ for "almost certain".

#### John Bartle

A better one might be a % over the equals

I can't find that symbol. Can you tell me where I would find it?

#### John Bartle

Hey does anyone know if there is a symbol for "A necessarily equals B" ?

An example would be "ball necessarily equals sphere" or maybe "truth necessarily equals (not falsity)"

#### DaveC426913

Gold Member
Two different things here.
An example would be "ball necessarily equals sphere"
A ball is a subset of a sphere. They are not equivalent.

or maybe "truth necessarily equals (not falsity)"
This one is kind of a truism***.

To a programmer, it might be pedantically expressed as
T == !F or T==!!T

== means exactly the same as
!
means logical NOT
!!
means logical NOT NOT

So, T is exactly the same as NOT F
Or T is exactly the same as NOT NOT T.

Finally,
T === !F
means T is the very same thing as !F.
But that might be overstating the case.

***This assumes truth and falsehood are binary and there are no middle values. Not a good assumption.
Some things are not true, yet also not false. Saying "I bought five eggs" when in fact I bought six is not false, but it's also not the truth. It is a partial truth.

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#### nuuskur

A sphere is a subset of a (sufficiently large) ball or a boundary of a closed ball.

For modal logic we use the symbol $\diamond$ to mean "possibly". So if we have a statement $P$ asserting the equality of something, writing $\diamond P$ would mean the equality is possibly valid.

I am grossly over-simplifying, though.

I don't think there is any special notation for something as vague as "probably equals". We could write $\mathbb P (P) = 0.7$ or something tangible like that.

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#### BeedS

Is there a symbol for "probably equals" ?
A variable.
An example would be A probably equals B.
If A and B are not variables then A doesn't equal B. You cant compare two values without reading them, that is guessing and trying to compare. Your "probably" is selecting the right data type in software dev terms.

#### hilbert2

Gold Member
In some non-binary logic where the "truthlikeness" of a statement is a real number between 0 and 1, you could say that "truthlikeness of a=b is greater than 0.9" or similar.

"Is there a symbol for "probably equals"?"

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