In a nutshell, if you try to define the multiplication rules any other way, arithmetic stops working in a big hurry. When you multiply X by a positive integer Y you’re adding it to itself Y times. So it makes sense that a positive times a positive is positive. If the “negative times a negative” quadrant on the lower left were all negative instead of positive (e.g., “ “), then the rows and columns that go through it will suddenly have to switch patterns (e.g., “increasing by 3’s” to “decreasing by 3’s”) when they pass zero. On a case-by-case basis, it’s not obvious that a negative times a negative should be positive.