There are ways to be clearer than either

> \[ \exists b. \phi(a,b) \wedge (\exists c. \psi(b,c) \wedge \upsilon(c,d)) \Longleftrightarrow \exists c. (\exists b. \phi(a,b) \wedge \psi(b,c)) \wedge \upsilon(c,d) \]

or

> There exists some common \\(b\\) such that the feasibility relation \\(\phi\\) between \\(a\\) and \\(b\\) holds, and there exists some common \\(c\\) such that the feasibility relation \\(\psi\\) between \\(b\\) and \\(c\\) holds, and the feasibility relation \\(\nu\\) between \\(c\\) and \\(d\\) holds, if, and only if, there exists some common \\(c\\) such that there exists some common \\(b\\) such that the feasibility relation \\(\phi\\) between \\(a\\) and \\(b\\) holds, the feasibility relations \\(\psi\\) between \\(b\\) and \\(c\\) holds, and the feasibility relation \\(\nu\\) between \\(c\\) and \\(d\\) holds.

Of course, Keith was deliberately trying to make the words as tiring as possible!

Look what happens when we replace things like "the feasibility relation \\(\phi\\) between \\(a\\) and \\(b\\) holds" by "\\(\phi(a,b)\\)" everywhere. I think this is getting better:

> There exists some common \\(b\\) such that \\(\phi(a,b)\\) and there exists some common \\(c\\) such that \\(\psi(b,c)\\) and \\(\nu(c,d)\\) if, and only if, there exists some common \\(c\\) such that there exists some common \\(b\\) such that \\(\phi\(a,b)\\), \\(\psi(b,c)\\), and \\(\nu(c,d)\\).

But it's still clunky. There are ways to improve the writing style to make it more digestible. For example:

> The following are equivalent:

> * There exists \\(b\\) such that \\(\phi(a,b)\\) and for some \\(c\\) we have \\(\psi(b,c)\\) and \\(\nu(c,d)\\).

> * There exists \\(c\\) such that for some \\(b\\) we have \\(\phi\(a,b)\\), \\(\psi(b,c)\\), and \\(\nu(c,d)\\).

I spend a few hours a day writing math, thinking about these things. It's frustrating but also fun.

Matthew's original formula is not bad if one is trying to understand exactly what rules of logic we're using to show the equivalence! We're using ['regular logic'](http://www.brics.dk/LS/98/2/BRICS-LS-98-2.pdf) , which is the fragment of logic governing \\(\exists\\) and \\(\wedge\\). This is the fragment of logic one needs to work with composition of relations. It holds in any topos, but even more generally, like in the category of vector spaces.

However, if one is not in the mood for logic, it's often less stressful to read text where the logical connectives are written as words.

> \[ \exists b. \phi(a,b) \wedge (\exists c. \psi(b,c) \wedge \upsilon(c,d)) \Longleftrightarrow \exists c. (\exists b. \phi(a,b) \wedge \psi(b,c)) \wedge \upsilon(c,d) \]

or

> There exists some common \\(b\\) such that the feasibility relation \\(\phi\\) between \\(a\\) and \\(b\\) holds, and there exists some common \\(c\\) such that the feasibility relation \\(\psi\\) between \\(b\\) and \\(c\\) holds, and the feasibility relation \\(\nu\\) between \\(c\\) and \\(d\\) holds, if, and only if, there exists some common \\(c\\) such that there exists some common \\(b\\) such that the feasibility relation \\(\phi\\) between \\(a\\) and \\(b\\) holds, the feasibility relations \\(\psi\\) between \\(b\\) and \\(c\\) holds, and the feasibility relation \\(\nu\\) between \\(c\\) and \\(d\\) holds.

Of course, Keith was deliberately trying to make the words as tiring as possible!

Look what happens when we replace things like "the feasibility relation \\(\phi\\) between \\(a\\) and \\(b\\) holds" by "\\(\phi(a,b)\\)" everywhere. I think this is getting better:

> There exists some common \\(b\\) such that \\(\phi(a,b)\\) and there exists some common \\(c\\) such that \\(\psi(b,c)\\) and \\(\nu(c,d)\\) if, and only if, there exists some common \\(c\\) such that there exists some common \\(b\\) such that \\(\phi\(a,b)\\), \\(\psi(b,c)\\), and \\(\nu(c,d)\\).

But it's still clunky. There are ways to improve the writing style to make it more digestible. For example:

> The following are equivalent:

> * There exists \\(b\\) such that \\(\phi(a,b)\\) and for some \\(c\\) we have \\(\psi(b,c)\\) and \\(\nu(c,d)\\).

> * There exists \\(c\\) such that for some \\(b\\) we have \\(\phi\(a,b)\\), \\(\psi(b,c)\\), and \\(\nu(c,d)\\).

I spend a few hours a day writing math, thinking about these things. It's frustrating but also fun.

Matthew's original formula is not bad if one is trying to understand exactly what rules of logic we're using to show the equivalence! We're using ['regular logic'](http://www.brics.dk/LS/98/2/BRICS-LS-98-2.pdf) , which is the fragment of logic governing \\(\exists\\) and \\(\wedge\\). This is the fragment of logic one needs to work with composition of relations. It holds in any topos, but even more generally, like in the category of vector spaces.

However, if one is not in the mood for logic, it's often less stressful to read text where the logical connectives are written as words.