The Double Arrow (p. 226)

[cmorris32]

Hello!

I have a question about The Double Arrow in the Conditional Reasoning chapter of the Logical Reasoning Bible. I am having a difficult time understanding the example given: "Ann will attend if and only if Basil attends." From my understanding, this example is split into two parts: (1) Ann will attend if Basil attends (2) Ann will attend only if Basil attends.

From these two sentences, we get
(1) Ann will attend if Basil attends: B --> A
(2) Ann will attend only if Basil attends: A --> B
Therefore, A <--> B

However, I don't understand the difference between those statements. I know 'if' is a sufficient condition indicator and 'only' is a necessary condition indicator, but it feels like the 2 sentences are saying the same exact thing. I feel like it will help me to understand the difference in the meaning of the sentences in order to understand the application of the double arrow. Could you please explain this?

Also, does it matter which condition goes on for the Double Arrow?

Thank you

[KelseyWoods]

Hi cmorris32!

The differences between sufficient and necessary conditions can seem subtle in terms of the wording of sentences but the meanings of statements can change drastically depending on which condition is necessary and which one is sufficient. Here, the difference between using a sufficient indicator like "if" and a necessary indicator like "only if" completely changes the meaning between those two sentences.

"Ann will attend if Basil attends" means that Basil's attendance is sufficient for Ann's attendance. If Basil is there, we can be certain that Ann will be there. If Ann is not there, it is certain that Basil will not be there. But if Ann is there, we are not sure whether or not Basil will be in attendance. Ann can be at the party without Basil. But Basil cannot be at the party without Ann.

"Ann will attend only if Basil attends" means that Basil's attendance is necessary for Ann's attendance. If Ann is there, we can be certain that Basil will be there. If Basil is not there, it is certain that Ann will not be there. But if Basil is there, we are not sure whether or not Ann will be in attendance. Basil could be at the party without Ann. But Ann cannot be at the party without Basil.

The relationship between Ann and Basil is reversed in the two sentences. In the first one, we can have Ann without Basil, but not Basil without Ann. In the second one, we can have Basil without Ann but not Ann without Basil. These differences are why, for the vast majority of conditional statements, it is so important to determine which condition is sufficient and which one is necessary and why when you diagram the statements the arrow always goes in only one direction, from the sufficient to the necessary. If you mix the conditions up, you totally change the relationship and end up making false inferences.

With the Double Arrow (indicated by phrases like "if and only if"), we have a rare relationship where the two conditions are both sufficient and necessary for one another. So you don't have to worry about which side of the arrow you place your conditions on because that arrow goes in both directions.

Hope this helps!

Best,
Kelsey