Probability and conclusions

[Dancingbambarina]

What are the rules in concluding from a contrapositive pertaining to probability. I see examples where it's okay, and examples where it's not (where groups or numbers are usually involved) to conclude from the contrapositive the probability matching in the conclusion from the normal form. So just to be clear, the negated necessary condition goes to the sufficient but the probabiity stays in the negated sufficient - now the necessary. So one term in a sense "hogs" the probability factor.

Thank you

[Adam Tyson]

Generally speaking, I avoid diagramming statements about probability as conditional statements, because true conditional statements are about certainty, not probability. Instead, I treat them as Formal Logic statements, with "most" arrows. Here's an example of why I think that way:

If you live in North Dakota, you probably don't like pineapple on your pizza.

Contrapositive:

If you like pineapple on your pizza, you probably don't live in North Dakota.

But what if there are only a small number of people who do like pineapple on their pizza, and they mostly (or all) live in North Dakota? Liking pineapple on your pizza might guarantee that you live in ND, even if you are in the minority there! Hence, the "contrapositive" is nonsense in this case.

Instead, if i felt the need to diagram, I would diagram this:

Live in ND Like Pineapple on Pizza

Instead of a contrapositive, I make the only valid inference that I can: that at least some people who don't like pineapple on their pizza live in North Dakota. A "some" arrow going the other way.

Now, if the probability is in the Sufficient Condition, then maybe we can make something of it:

If rain is likely tomorrow, I won't plan on going camping.

If I am planning on camping, so rain is not likely tomorrow.

That one works, and the probability stays with the condition.

[Dancingbambarina]

Thanks very much. That makes sense! Quite a lot of steps going into one conditional. Do you by any chance recall the one LSAT question where this is tested? I believe it's in the 70s but I could be wrong.

Thanks very much

[Dancingbambarina]

I don't believe my prior post posted, but Is there a reaosn why the sufficient inherents the probabiity when contrapositing, but not the other way round?

Thank you

[Dave Killoran]

Adam wasn't making a point like that, just more about what is "easier" to use or interpret. Since the contrapositive both reverses and negates, if there was probability on a side, a contrapositive would transfer it to the other term. Here's an example where the probabilistic term moves from sufficient to necessary:

Statement: If it is likely to rain today, I will not go to the beach today.

CP: If I went to the beach today, then it was not likely to rain today.

Thanks!

[Dancingbambarina]

Adam wasn't making a point like that, just more about what is "easier" to use or interpret. Since the contrapositive both reverses and negates, if there was probability on a side, a contrapositive would transfer it to the other term. Here's an example where the probabilistic term moves from sufficient to necessary:

Statement: If it is likely to rain today, I will not go to the beach today.

CP: If I went to the beach today, then it was not likely to rain today.

Thanks!

Thank you so much Dave,

I understand your point completely. What brings confusion is when transferring the probability from the necessary to the sufficient.

For example, here:

When I wear jeans, most of the time I tie my hair.

Wear Jeans --> Tie hair most of time

"From a contrapositive standpoint, if you don't tie your hair most of the time, then you aren't wearing jeans. So, if you don't ever tie it, you aren't wearing jeans. But note that allows for you to not tie it some of the time; that's okay under what the necessary specifies."

I am not fully understanding how the probability is switched here from the necessray to the sufficient after contraposition, and the new necessary does NOT inherit probability, unlike the example in this thread where probability is maintained in the necessary.

Is this because we're NOT dealing with that discrepancy Adam pointed out regarding what if all pizza-eaters lived in Idaho? My understanding is that as you say, normally this switch occurs, but as in the IDAHO example, sometimes it doesn't make sense - is this correct?

I understand using 'most' arrows allow us to accurately draw inferences. Is there a specific time you would recommend doing it the conditional way instead (perhaps the 'dirty way' as long as one is aware of the nuances)?

Sometimes the ''probably" or "likely" is placed awkwardly. Is there an eficient way to spot if the "most" (e.g.) is meant to be part of the sufficient or necessary? "Most As are Bs" typifies an easy examply, but sometimes it's more challenging "Mostly, men don't go to war at the turn of the century". It's sometimes very confusing to decide if there EVEN IS a most arrow, for example.

Thank you so very much. You are all helping me and I won't let you guys down.

[Dave Killoran]

Relooking at Adam's post, that's really a tough example and I'd likely remove the "probably" from the necessary and move it over to the sufficient. The reason he didn't is that it makes the sentence awkward and hard to use.

So, as that shows, it gets tough at times to keep all the "nots" and "mosts" and "somes" straight. Let's try the Jeans example using straight percentages and see if that helps:

• Statement: Wear Jeans --> Tie hair 51%-100% (most) of time
  
  Contrapositive: Tie hair 51%-100% (most) of time Wear Jeans
  This is the same as: Tie hair 0%-50% (not most) of time Wear Jeans

So, if you tie it 0% of the time (never), that meets the sufficient, or if you tie it, say, 25% of the time (which is some), that also meets the sufficient condition here. But regardless, you can see that the percentage is in fact moved from necessary to the sufficient, and while it may "sound" weird, it is logically solid.

As for using Mosts vs Arrows, usually with Some and Most I'll use the Formal Logic representation and not the Arrow. But, if I see a conditional term thrown in there like "if" or "when," I'll quickly consider whether the author is really trying to get across a conditional relationship in the traditional Arrow sense. Context matters in my evaluation, and I don't have a hard and fast rule. Either way will represent the relationship accurately, however, so it's matter of which is more comfortable to you.

Thanks!