A Note on Conditional Sequences and their Rules

[Jon Denning]

A PowerScore student working through our Logic Games Bible recently wrote to me and inquired about an often-confusing idea he encountered in a discussion on sequencing and conditionality. More specifically, he wondered how to handle rules that have both sequencing elements like "A earlier than B," as well as conditional elements like "If A, then B"?

The rule in question was as follows:

“If A is earlier than B, then B is earlier than C.”

Diagrammatically we can show this as:

A > B B > C, and the contrapositive (assuming no ties) as C > B B > A

The problem for a lot of people is that they assume this means only one of two chains must happen:

1. A > B > C
2. C > B > A

And that’s true, to an extent. That is, those two orders are both entirely acceptable from the rule, and either could occur. But because this rule is conditional in nature, those orders only apply when the initial condition of either the original rule (A > B) or the contrapositive (C > B) happens!

If the original condition does not occur (not A > B, so B > A), and the contrapositive's first condition does not occur (not C > B, so B > C) then we can’t know anything and suddenly two other orders are allowed:

If B > A, then you could have C either earlier than B (#2 above), or

3. After A: B > A > C
4. Between B and A: B > C > A

So in fact we have four potential orders from that rule, not just two!

Fortunately, if that’s unclear there are a couple of other ways to think about this idea. For one, imagine the six possible orders of A, B, and C if no rules constrain their placement, and we don't have ties.

1. A > B > C
2. A > C > B
3. B > A > C
4. B > C > A
5. C > A > B
6. C > B > A

Now think about which of those six orders would violate the conditional rule A > B B > C, or its contrapositive C > B B > A. Begin by isolating each instance where a sufficient condition (A > B, or C > B) occurs:

1. A > B > C
2. A > C > B
5. C > A > B
6. C > B > A

That happens in options 1, 2, 5, and 6. Options 3 and 4 don’t contain either sufficient condition, so can be ignored.

Now see which of the four options with a sufficient (A > B, or C > B) violate(s) the necessary (B > C, and B > A, respectively):

1. A > B > C
2. A > C > B
5. C > A > B
6. C > B > A

Options 2 and 5 are both violations of either the initial rule (#2) or the contrapositive (#5), so those two sequences, and ONLY those two sequences, are not allowed here. The other four (including numbers 3 and 4 above) are all fine. Essentially the lesson is that when dealing with conditional rules, you must pay attention to not only the scenarios that would trigger the rule (sufficient condition met), but also those that do not (sufficient not met).

These can be tricky ideas to get your head around at first, but with continued practice I promise they become much, much easier. Keep at it!

Jon