Vtyjkyuk

I'm trying to figure out how to best formalize the following interrelationship in first order logic:

A material has (electric) resistance $r$ and conductance $g$, and the two are related as $r cdot g = 1$.

The idea is that, if I know either $r$ or $g$ for some material, I should be able to infer the other.

Let the relation $mathrmres(m,r)$ mean that material $m$ has resistance $r$, and similarly $mathrmcon(m,g)$ for conductance, and let $mathrmmul(x,y,z)$ denote the multiplication relation $x cdot y = z$. The following formulas express the semantics of resistance/conductance:

1. $mathrmres(m,r) land mathrmcon(m,g) rightarrow mathrmmul(r,g,1)$


2. $mathrmres(m,r) land mathrmmul(r,g,1) rightarrow mathrmcon(m,g)$


3. $mathrmcon(m,g) land mathrmmul(r,g,1) rightarrow mathrmres(m,r)$

I think this works as expected: for example, if I know $mathrmres(X,10)$ for some specific material $X$ (a constant), then rule (2) together with the fact $mathrmmul(10,0.1,1)$ gives $mathrmcon(X,0.1)$. However, it feels like we should be able to express the relation between $r$ and $g$ more compactly$-$do we really need three formulas for this?

Intuitively, I thought there should be a single formula describing the relation, but I can't find one that works. Am I missing something, or is there something lurking here that's beyond the expressivity of first order logic?

Are you somehow not allowing your logical language to contain function symbols?

I would write
$$ forall x(operatornamematerial(x) to operatornameres(x)cdotoperatornamecon(x) = 1)$$
and be done with it.

(Even though first-order logic can be formulated with no function symbols without losing expressivity, the standard presentations allow any combination of predicate symbols, function symbols and constant symbols).

If I understood you correctly, this relationship could be written just as
beginequation
res(m,r) leftrightarrow con(m,frac1r)
endequation

EDIT (after discussion in the comments):
beginequation
(res(m,r) land con(m,g)) leftrightarrow mul(r,g,1)
endequation
For a specific material (say $m=X$), we have the following;

• If $res(X,r)$ holds, $con(X,g)$ must hold for $g$ such that $mul(r,g,1)$ holds.


• If $con(X,g)$ holds, $res(X,r)$ must hold for $r$ such that $mul(r,g,1)$ holds.

I am aware that this lacks the restriction on $m$, but I figured that this formula is not to be used without knowing either $r$ or $g$ of a specific material $m$.