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## Formal Logic

General questions relating to LSAT Logical Reasoning.
Foti
• Posts: 14
• Joined: Mar 31, 2020
#74591
When dealing with formal logic, I understand that things that are mutually exclusive cannot occur together. A positive trigger leading to a negative result always renders a mutually exclusive situation, correct? That being said when diagramming these mutually exclusive relationships according to the LR Bible, we are taught to use the double-not arrow, <--I-->.

What is the difference between A <--I--> B and A <---> ~B? Is there any? If you were to translate the second example it would imply that ~B --> A and that is not what is necessarily being said. If it is ~B it can also be ~A, they can both not occur. So my question is what is the difference between A <- I--> B and A <---> ~B? It seems like the second example is just the wrong way to illustrate the relationship but that it is intended to mean the same thing.

Is it safe to say that if there is any relationship that has a positive trigger leading to a negative result, can I ALWAYS illustrate this with double-not arrow?

I understand that "all except/all but" is the opposite of "if and only if". "All except/all but" means that it has to be one or the other, and that it can't be neither. This makes it different from a positive trigger leading to a negative trigger, such as A --> ~B. In this situation, it does not have to be either, it can be that they both do not occur. So this is where I get confused. How would you diagram problems with "all/except/all but"? Are there any LSAT examples from the past that I can look at to help me practice with "all except/all but"?
Dave Killoran
• PowerScore Staff
• Posts: 5808
• Joined: Mar 25, 2011
#74594
Foti wrote:What is the difference between A <--I--> B and A <---> ~B? Is there any? If you were to translate the second example it would imply that ~B --> A and that is not what is necessarily being said. If it is ~B it can also be ~A, they can both not occur. So my question is what is the difference between A <- I--> B and A <---> ~B? It seems like the second example is just the wrong way to illustrate the relationship but that it is intended to mean the same thing.
The short answer is that A B allows for neither A nor B to occur whereas the other diagram does not. So, the second diagram is too restrictive, and eliminates and outcome that exists under the double-not arrow. This is why we use that symbol. If we could get away with not doing it, we definitely would!

Foti wrote:Is it safe to say that if there is any relationship that has a positive trigger leading to a negative result, can I ALWAYS illustrate this with double-not arrow?
Yes, but it's not limited to positive sufficient conditions. In our material and discussion you will see the discussion of the negative sufficient condition as well. Keep reading on!

Foti wrote:I understand that "all except/all but" is the opposite of "if and only if". "All except/all but" means that it has to be one or the other, and that it can't be neither. This makes it different from a positive trigger leading to a negative trigger, such as A --> ~B. In this situation, it does not have to be either, it can be that they both do not occur. So this is where I get confused. How would you diagram problems with "all/except/all but"? Are there any LSAT examples from the past that I can look at to help me practice with "all except/all but"?

Phrases like this are infrequently used on the LSAT in stimuli and answers, and almost never does getting the problem right revolve around you making the right diagram (I don't recall an example). Here are a few usage examples (and not necessarily as controlling points in the argument):

February 1995 LR1 Section II, #17
December 2004 LR1 Section I, #10
September 2014 LR2 Section IV, #18

"All except," I don't recall that on a test but it's the same as all but.

Thanks!
Katya W
• Posts: 42
• Joined: Dec 03, 2019
#74926
Hello PS! I was hoping I could get your help on my dilemma below regarding the conditional phrasing "either/or". Apologies ahead of time if this Topic thread was not the correct one.

I am struggling to understand why the below two hypothetical conditional statements containing "either/or" are translated differently (I made these up based on two separate examples I saw):

1. If the bar does not have a karaoke machine, then it will have either a pool table or a dart board.
Translation: /Karaoke Machine Pool table OR Dart board

2. Unless the world burns, we will go to either the baseball game or the renaissance festival.
Translation: /Baseball game AND /Renaissance festival world burns.

My confusion is why in the first example the "either/or" translates to an OR between the two terms, but in the second example the "either/or" translates to AND.

Can someone help me with this? My brain is melting! (Hence world burning, and my public, non-social distancing examples).

Thank you! ~ Katya
Dave Killoran
• PowerScore Staff
• Posts: 5808
• Joined: Mar 25, 2011
#74929
Hi Katya,

Thanks for the message! I'll give you a hint here: the second one contains "Unless" whereas the first one does not. That changes everything! So, apply the Unless Formula, and that means the either/or gets negated, which isn't happening in the first one.

Please let me know if that helps stop the brain melt If not we'll come back and explain it in detail.

Thanks!
Katya W
• Posts: 42
• Joined: Dec 03, 2019
#74930
Ah...hmm.... I had a feeling that's what it was. That sorts things out entirely! I was indeed missing that the OR and AND will flip too from what the statement says when you are converting the Unless statements into shorthand. Thank you, Dave!! Brain melt aborted. Recovery in progress.
taylorudtner
• Posts: 3
• Joined: Aug 13, 2022
#97357
Hello!

I have a few questions about this so I am sorry for the length of this post in advance.

I am confused about how formal logic appears in LR, specifically with concern about the double-not arrow. For the purposed of LG I'm solid, but when applied to LR I get much more confused. I was under the impression that a double not arrow (A<--|-->B) was the same as A-->~B and it's contrapositive B-->~A because it also implies that if A or B do not occur, we cannot assume anything else (meaning that neither can occur). Weirdly, understanding the double negative not arrow (like ~A<--|-->~B) for conditional statements with a negated sufficient condition helped me understand this because it means that a situation with neither cannot exist while a regular double not arrow means that a situation with both cannot exist, right?

However, when we go to LR is that still the case? I get confused trying to think about how a stimulus with a double-not arrow would be phrased.

For example, if the stimulus said "No ducks in the lake are purple. All animals that eat a specific type of berries are dyed purple.' That would be diagrammed as:

D-->~P / P-->~D and P-->AB
so
D<--|-->P-->AB
right?

So we cannot infer from that that no ducks in the lake at the berries. Instead we have to go against the arrow and turn the P-->AB into a 'some' relationship (D<--|-->P<--(some)-->AB) and from that we infer that some animals that did not eat the berries are ducks and some ducks did not eat the berries (~D<--(some)-->AB)? Can you negate either variable in the latter?

Are there any other conditional statement where you know that you will have to revert to 'some' to make an inference?

But then what if the situation was D<--|-->P<--AB? Would that just be a complicated way of saying D-->~P-->~AB? No duck in the lack is purple, if something is not purple it did not eat the purple-dying berries?

Also what if it was an unless statement with 'some'? Like some ducks in the lake are not purple unless they eat the berries (D<--(some)-->P-->AB)? But we cannot infer anything from that because to do so it would end up being a double-some train. What if it wasn't some ducks but most ducks? Same thing-no inference?

I'm thinking myself in a circle here. In writing this I both clarified things for me and confused myself--every bit of phrasing is so particular. I would appreciate some clarification and help in this regard and perhaps a real-life example (I apologize for the horrible one I provided, I hope it made sense).

Thank you!
Rachael Wilkenfeld
• PowerScore Staff
• Posts: 1358
• Joined: Dec 15, 2011
#97385
Hi Taylor,

I think your diagrams are a bit twisted around, which is probably causing some of your confusion. Let's start with the first example you give:
For example, if the stimulus said "No ducks in the lake are purple. All animals that eat a specific type of berries are dyed purple.' That would be diagrammed as:

D-->~P / P-->~D and P-->AB
so
D<--|-->P-->AB
right?
You are right on with your duck purple relationship. You could write it either as the statement with contrapositive (as you do in the first line) or as a double not arrow (as you do in the second line). However, "all animals that eat a specific type of berries are purple" is the opposite of what you stated. Your diagram looks as if everything that is purple eats a certain type of berry, while the relationship you presented is that if something is an animal that eats the berries, then it's purple. So the correct diagram for the second sentence would be animal who eats specific berries (AB) Purple (P).

Now we can look at the two rules together:

D P
AB P

Here's the key: the double not arrow is something that includes a negation as part of the relationship. As you wrote above, it includes two conditionals D P and P D. We can actually make a chain here:

AB P D which yields the inference AB D and D AB (or you could write the double not arrow AB D).

In terms of reversals, all reverses to some, most also reverses to some. The specific examples you gave were not consistent with your original statements so we wouldn't be able to diagram them here in a way that makes sense. You couldn't have ANY ducks that are purple. But your unless statement would look something like this:

If ducks in the lake are purple, then those ducks are animals that eat the specific berries. PD AB.

I hope that helps clear things up for you!

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