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I am currently working on describing a predicate logic for which the objects are mathematical variables. Thus I can say stuff like:

$\forall x: R(x) \implies \text{operator}(x)=1$

Here $x$ is a logical variable ranging over all mathematical variables in the domain and the equality operator $=$ is the mathematical equality, not the logical equality and $\text{operator}$ is the operator I am looking for (see below).

On the other hand I can say stuff like:

$\forall x,y: (R(x) \land R(y)) \implies x \not\equiv y$

whereby I mean that $x$ and $y$ denote different mathematical variables, not that the variables are different in value.

However, now I want to introduce a mathematical function which works on a mathematical value, given a mathematical variable. Therefore I need some kind of operator which, given a variable, returns the value.

$\text{operator}(x)$

What would be a logical name or symbol for such an operator?

Angelorf
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  • By $x=1.0$ I mean that the mathematical variable, which the logical variable $x$ denotes, has a real value of $1.0$. It should therefore actually say $\text{operator}(x) = 1.0$. – Angelorf May 27 '14 at 13:40
  • don't understand "whereby I mean that $x$ and $y$ denote the same mathematical variable, not that the variables are equal in value" if they are the same variable they are equal in value (because if they are the same they are equal in everything), maybe you mean the opposite: operator(x) = operator(y) (having the same value) does not imply x = y, like two brothers aren't the same person because they have the same father. – Willemien May 27 '14 at 19:21
  • @Willemien I don't think meaning is the same as implication. It is true that equality in value is implied, but it doesn't have to be the case that that is what the statement means. But all the same - I changed the part in question just for clarity. – Angelorf May 27 '14 at 20:17

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It seems to me that you are after some sort of valuation; although this concept is used in model theory to describe semantic truth for a given theory and logic.

It seems to me you want to talk about the values of the variables at the same time than the variables themselves. This will lead you to deal with expresions meaning different things: x will be a "variable", whereas operator(x) will be a "value of a variable". Thus, in your logic an expresion such as x = operator(y) should not make sense as it is compering two different kinds; or put other way, the syntax for "=" only makes sense for the same kind of expressions on both sides.

To address this then you need to introduce typed logic (if I remember correctly), but I should stop here as further discussion is only possible if my understanding is aligned with what you are trying to achieve.

Does the above make sense?

Edit: It's been a while since I haven't touch any Logic-related topics. But digging today I just found this freely available presentation about "Typed first-order logic" in academia.edu (registration is free.) I bet from there you can find lot more; I barely remember anything :)

carlosayam
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  • This absolutely makes sense! That is also why I used $\equiv$, to distinguish the logical sense from the mathematical $=$. Please tell me about typed logic. – Angelorf May 28 '14 at 12:59
  • @Angelorf, I updated my pseudo-answer; good luck! – carlosayam May 28 '14 at 20:48
  • Also, the concept of valuation is from model theory. But it seems you want to "internalise" valuation for the variable type. So, sorry about the confusion; although you will need some sort of semantic for your logic anyway... – carlosayam May 29 '14 at 00:56
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Some logic Texts ("Language Proof and Logic "Barwicse c.s. for example) call it function symbols, others just call it functions or operators.

Be aware that most logics assume that there is one and only one term where the operator refers to.

If there is no referent you get reference failures see http://en.wikipedia.org/wiki/Definite_description and http://en.wikipedia.org/wiki/Failure_to_refer

If there is more than one then it is not a function and you also can get into trouble.

If one of these options possibly exist then is better to replace the function symbol by a predicate: (this is always possible)

$ R(x,y)$ is true $ =_{df}$ (if and only if) "some condition" is true. For example

$R(4,2)$ and $R(4,-2)$ are both true for $R(x,y) =_{df} (x = y^2) $

While you can get into trouble if you use $ y = \sqrt x $ because in $Q$ ) there is no $ \sqrt{-1} $.

but against this logical better formulation there is the problem that functions are easier to read.

Good Luck

Willemien
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  • You seem to have totally misunderstood my question. I am not looking for what to call any operator, but I am looking for what to call this specific operator, or what symbol is used for this specific operator. In my logic it is assumed that any object in the domain of the logic refers to exactly one existing variable. Therefore I can use an operator, or as you call it, a function symbol. – Angelorf May 28 '14 at 13:02
  • ahh, sorry,

    I don't think there is a specific name for your operator,

    • it would be binary operator if $\forall x (operator(x)=1 \lor operator(x)=0 ) $ (so operator(x) has only 2 possible values).
    • a witness if $\forall x (operator(x) =1 \rightarrow R(x))$ (so in the other direction see http://en.wikipedia.org/wiki/Witness_%28mathematics%29 ),
    • a numerical truthvalue (informal my name) if $\forall x (R(x) \leftrightarrow operator(x) = 1) $ (a biconditional not a conditional)
    • a constant if $ \forall x (operator(x)=1 ) $

    But none of these descriptions seem to fit your operator.

     – Willemien May 28 '14 at 15:37