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In logic, we can use quantifiers, like $\forall$ or $\exists$, or binders, like $\sum_x, \min_x, \prod_x$, to bind variables. At least the quantifiers seem to also be called constants, while there are other constants like numbers like 2.

But sometimes I want to write formulae that have letters (which values I do not know), that are not bound by binders or quantifiers, but are assumed to be defined or "fixed" somewhere, such that my formula becomes truth-apt.

For example, I would like to say that (for the functions $M$ and $f$)

$$ \forall x ( M(x, \theta) = f(x)) $$

is not truth-apt, because $\theta$ is not bound. But, next, I find that

$$ \exists \theta, \forall x ( M(x, \theta) = f(x)) $$

and I also find this awesome $\theta$, which I shall call $\hat\theta$. Now I want to be able to say that

$$ \forall x ( M(x, \hat\theta) = f(x)), $$

but I am still missing the pre-definition of $\hat\theta$. Actually, $M$ and $f$ are also free variables.

However, I do not want to those down: $M$ and $f$ cannot be expressed with a mathematical expression (their values for a certain input can only be calculated via a complex computer algorithm and the details change all the time) and $\hat\theta$ is only conceptually available.

I am not even sure how to call these not-variables. Obviously they are no quantifiers or constant, but I do not want them to be variables either. I do not know the notation to make them bound.

Honestly, even if I could write $\theta$ down, I would not know how to express this.

$$ \forall x ( M(x, \theta=2) = f(x)), $$

or

$$ \theta=2, \forall x ( M(x, \theta) = f(x)), $$

or something else entirely?

Make42
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  • Obviously, if you know the value of $\theta$, there is no need to write: $∀x(M(x,θ=2))$; it is enough to write $∀xM(x,2)$. Thus, in a more general context, you can write something like: "let $\theta_0$ the known value of parameter $\theta$; then $∀x(M(x,θ_0)=f(x))$..." – Mauro ALLEGRANZA May 16 '22 at 14:55
  • @MauroALLEGRANZA: Regarding your first sentence: Sure, but (notationaly) I want to keep the "named" variable, just bind it to 2. Regarding you second sentence: Then, I am writing prose. I was looking for a logical expression. I expected that there was a common way of binding variables in a formal way without prose. – Make42 May 16 '22 at 14:58

1 Answers1

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It should be emphasised that an open formula, a formula with unbound variable(s), is not truth-apt is a misconception, perhaps, due to a textbook oversimplification. See this answer and my post.

So far as I understand the question, the notion of parameter can be employed (for more concrete examples of parametric definitions, see the article).

Tankut Beygu
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