only if and contraposition
Just a quick question: Do you think that any of the apparent counterexamples to modus tollens/contraposition would work if the conditional were stated using "only if" as the conditional marker?
1. If it rained, it didn't rain hard.
1a. It rained only if it didn't rain hard.
Try out any of the other apparent problems for MT/contraposition with only if in place.
The logically equivalent paraphrase in (1a) doesn't seem to have the same plausibility as (1), that is, I can't give a good context for using 1a.
I'm curious if any of you have thoughts about this phenomenon.
--Ian
Labels: conditionals (general), contraposition, modus tollens, only if
posted by dovester @ 10:33 AM
4 Comments:
How about this:
1. If he lives in New England, then he doesn’t live in Boston. (And contraposition gives “If he lives in Boston, then he doesn’t live in New England.)
1a. He lives in New England only if he doesn’t live in Boston.
Suppose you tell me that you’ll live in New England only if you don’t have to move to Boston because you hate it. Also suppose you have possible plans to move, and suppose that I know this, but I don’t know if you for-sure moved or not.
Now someone asks me, “Does Ian live in New England?” I say, “Well, he lives in New England only if he doesn’t live in Boston.”
Is this a plausible context?
Jesse
Interesting. Do you think that the "only if" in (1a) could be paraphrased using "but only if"?
1a'. He lives in New England, but only if he doesn't live in Boston.
If this captures the meaning of your (1a), then your statement isn't really a conditional. Rather, it would be a biconditional. To see illustrative examples of "but only if" as biconditionals, see Pospesel's textbook on propositional logic.
I think it makes sense of your context to think of your 1a as a "but only if" biconditional.
--Ian
Hy guys,
Hope you don't mind me jumping in here. I haven't been to the blog in a while and thought I would see what the recent topics are.
I find this one very interesting. I have been trying to think of a situation that would be plausable to use (1a), but can't really come up with one.
However, consider this...
Suppose you are a scientist that is on an expedition deep in the jungle of Borneo, and you see a creature that has never been seen before. You want to determine if it is a mammal or not.
When you turn to your experienced guide (who knows that when you are that deep in the jungle, you kind of have to expect the unexpected ... or in other words that it's often the case that things look differently than what you would normally classify them as) he says:
1a. It's a mammal only if it doesn't look like a mammal.
which is to say:
1. If it's a mammal, it doesn't look like a mammal.
Ok so this may be a little far fetched, but I think that Ian is right that there is something about "only if" that makes counterexamples to contraposition seem a little off.
--Ryan
I don't find Ryan's only if example plausible, though the if version makes sense So I think it is an example of my analysis and not a counterexample.
To say:
That's a mammal only if it doesn't look like one,
suggests that not looking like a mammal is in some way a part of it's being a mammal.
One way to understand this is in terms of why we choose only if constructions generally: we do it to emphasize the necessary conditiion as a condition. When I say:
I'll give a make-up test only if you have a good excuse.
I want to emphasize the good excuse as a (necessary) condition for giving the exam. The same isn't true when we use the truth-functionally equivalent statement:
If I give you a make-up exam, then you will have a good excuse.
The necessary condition is buried in this statement.
--Ian
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