Ordering and comma use in predicate logic

Question

Use of commas and ordering in notations of logic confuse the hell out of me especially when variables of different sets are concerned.

What I'm trying to formalize is that $G\left(a,b,r\right)=-G\left(b,a,r\right)$ for all possible combinations of these three arguments where $a\in P$ and $b\in P$ and $r\in R$.

To generalize this a bit let $K(a,b,r)$ be the statement "$G\left(a,b,r\right)=-G\left(b,a,r\right)$"

which of the following (if any) is the correct way to write this and most importantly, why?

$$\forall r\in R\ \forall a,b\in P,\ \ K(a,b,r)$$ $$\forall r\in R,\ \forall a,b\in P,\ \ K(a,b,r)$$ $$\forall r\in R,\ \forall a\in P,\ \forall b\in P,\ \ K(a,b,r)$$ $$\forall r\in R\forall a\in P\forall b\in P,\ \ K(a,b,r)$$ $$\forall \left(r\in R, a\in P, b\in P\right),\ \ K(a,b,r)$$ $$K(a,b,r),\ \forall r\in R\ \forall a,b\in P$$ $$K(a,b,r),\ \forall r\in R,\ \forall a,b\in P$$ $$K(a,b,r),\ \forall r\in R,\ \forall a\in P,\ \forall b\in P$$ $$K(a,b,r),\ \forall r\in R\forall a\in P\forall b\in P$$ $$K(a,b,r),\ \forall \left(r\in R, a\in P, b\in P\right)$$

Is there a simpler way to write this if $a$ and $b$ are treated as an ordered pair?

Answer 1

I don't think there's a correct answer. There is no universally agreed notation for predicate logic. There is the convention that $\implies$ means implication and $\forall$ means universal quanitfication and so on, but anyone is free to use different symbols than those and (as long as the structure of the system is the same) their logic will still be predicate logic.

Similarly, there is no convention about the commas. Some books use commas, others don't. Some put the $\forall$ in the end, others in the beginning. For formulas of predicate logic, the convention is that quantification comes to the left. However, you can find many mathematical books that make a statement and quantify ($\forall$) the variables on the right.

I don't get the question about the ordered pairs...

Answer 2

Maybe use $\forall (a, b, r) \in P×P×R$? ie treat them as an ordered triple chosen from the cartesian product of their sets? Then at least you're only using $\forall$ once and you'll end up with something more readable.

I'm new to this kind of formalism myself but it seems the easiest way to me.

Answer 3

The order in which you quantify is unimportant, as $\forall x\forall y~P(x,y)\iff \forall y\forall x~P(x,y)$ and so on.   (Note: It becomes important when mixing existential and universal quantifiers.)

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$\forall a{\in} P~\forall b{\in} P~\forall r{\in} R~K(a,b,r)$ is strictly correct.   The scope of the quantifiers is immediately to their right, so that's where the predicate always goes.

$\forall a{\in} P,\forall b{\in} P,\forall r{\in} R,K(a,b,r)$ is acceptable too; the punctuations marks are purely for association.   Some find it easier on their eyes than just using spacing.

$(\forall a{\in} P)(\forall b{\in} P)(\forall r{\in} R)~K(a,b,r)$ likewise uses brackets for this purpose.

$\forall a{\in} P~\forall b{\in} P~\forall r{\in} R:K(a,b,r)$ just uses a collon to separate the quantifiers from the predicates; you can mix this with one the above.

Now, sometimes when two or more terms are drawn from the same set, an abbreviation is accepted.   Thus you might see $\forall a{,}b{\in}P~\forall r{\in}R~K(a,b,r)$.   Mixing this with the above use of commas can be confusing though.

Alternatively, $\forall \langle a,b\rangle{\in}P^2~\forall r{\in}R~K(a,b,r)$, a Cartessian product notation might be preferred.   Perhaps even $\forall\langle a,b,r\rangle{\in}P{\times}P{\times}R~K(a,b,r)$ might be seen.

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Sometimes a statement might be made with the quantification tagged in the right margin.$$K(a,b,r)\tag{$\forall a{\in} P,\forall b{\in} P,\forall r{\in} R$}$$