How to say "regardless of " in predicate logic.

Question

Let's try to convert this statement to predicate logic:

"Every worker in this establishment have the same rights,granted by law, regardless of their position or sex"

W: set of workers    W(x): x is a worker
R: set of rights granted by law 
HR(x,y): x have right y
∀x:W.W(x) ⇒ ∀y:R.HR(x,y) 

I am not sure how to translate the "regardless of" part into predicate logic.

Also I don't know what to do with "position" property as it can mean so many different things. Let's say for the sake of simplicity, sex is binary (don't take transgender people or intersex people into account as that would complicate things a bit). I can formulate the sex property, but I can't formulate the position property since I wouldn't know the possible positions in that firm.

W: set of workers    W(x): x is a worker
R: set of rights granted by law 
HR(x,y): x have right y
M(x): x is a man
W(x): x is a woman
( (∀x:W.W(x).M(x)) ∨ (∀x:W.W(x).W(x)) ) ⇒ ∀y:R.HR(x,y)

I am not sure if "∨" conveys the meaning that I want or should (∀x:W.W(x).M(x)) be (∀x:W.W(x)∧M(x)).

Answer 1

Your

$( (∀x:W.W(x).M(x)) ∨ (∀x:W.W(x).W(x)) ) ⇒ ∀y:R.HR(x,y)$

has some problems. Yes, syntactically that should be changed to:

$( (∀x:W.W(x) \land M(x)) ∨ (∀x:W.W(x) \land W(x)) ) ⇒ ∀y:R.HR(x,y)$

but a bigger problem is with the $( (∀x:W.W(x) \land M(x)) ∨ (∀x:W.W(x) \land W(x)) )$:

First, the $(∀x:W.W(x) \land W(x))$ part unnecessarily repeats the $W(x)$ predicate. So, let's make that just $∀x:W.W(x)$

Second, the $(∀x:W.W(x) \land M(x))$ part is talking about all workers who are, at the same time, both a man and a woman. No, that's clearly not what you want. And notice, we already have a $∀x:W.W(x)$, so let's make this part $∀x:W.M(x)$

OK, so we have:

$( (∀x:W.M(x) ∨ (∀x:W.W(x)) ) ⇒ ∀y:R.HR(x,y)$

This gets us the third problem: You end up saying: "If all workers are men, or if all workers are women, then all rights ....". But note, that doesn't cover the case where you have some workers who are women, and some who are men.

To fix this, we can try to do:

$( (∀x:W.(M(x) ∨ W(x)) ⇒ ∀y:R.HR(x,y)$

But that is still not right! Indeed, the fourth problem with your translation is that the $x$ in $HR(x,y)$ is outside the scope of any of thew $\forall x$ quantifiers, and is indeed not quantified at all.

So, here's what you need to do. You need to say: "for anyone, whether they are man or woman, bla bla bla holds true for $x$"

This translates to:

$∀x:W.((M(x) ∨ W(x)) ⇒ ∀y:R.HR(x,y))$

OK, now it makes sense!

But, like you say, this only works if sex is binary ... which it is not. And, it doesn't solve the problem of the 'position'.

So, the best solution is to just ignore the 'sex' and 'position' thing, and just do:

$∀x:W.∀y:R.HR(x,y)$

Answer 2

The "regardless of" part of the statement is totally redundant. It emphasizes the fact that no distinctions should be made. In predicate logic, it is hence ignored. It is enough to state : $$\underbrace{\forall x_1 \, \forall x_2 \, \forall y \ HR(x_1,y) \leftrightarrow HR(x_2,y)}_{\textrm{all workers have the same rights}} \bigwedge \underbrace{\forall y \, \forall x \ R(y) \to HR(x,y)}_{\textrm{all workers have all rights granted by law}}$$

Logic doesn't need the emphasis since logic isn't sexist. ;-)