I wrote it to see whether, if we explain our logic clearly, we can come to an agreement ....
The parenthetical phrase if we explain our logic clearly should be set off with a comma pair.
Also, if you want to be pedantic (and I do), the if in the main clause would more properly be a whether. To quote Grammar Girl,
Although in informal writing and speech the two words are often used interchangeably, in formal writing, such as in technical writing at work, it's a good idea to make a distinction between them because the meaning can sometimes be different depending on which word you use. The formal rule is to use if when you have a conditional sentence and whether when you are showing that two alternatives are possible.