So they're functionally the same but actually different as one is pointing out a group A are in group B and all of C is in group A therefore C is in group B. The other is pointing out consequence. Rule: an action will result if X happens. X happens therefore X will be executed as a consequence.
Can someone explain to me how the cat and restaurant arguments are formally equivalent? It seems to me like the cat argument has three variables: cats, mammals, and Garfield, (G>C>M) while the restaurant has only two variables: new restaurants and living standards. (NRO>LSI … NRO is true so LSI is true) would that not catch you out on a match the structure question? I understand they’re both considered conditional logic and we’re not going into the terminology details but ‘formally equivalent’ seems like a strong claim.
Cat: G is part of C, C is part of M, because G is part of C, G is "triggered" to be in M.
Rest: NRO is part of LSI, because NRO happens, LSI is triggered.
They are equal in the sense that the sufficient condition triggers the necessary condition simply by existing; in the cat argument, the member is just a way to represent the superset triggering because of the subset's existence. In the restaurant, there is no member, but the subset still happens, and thus the superset is triggered.
@MarisolSanchez the way i think of it is the sufficient condition is one that result in the necessary condition. However, the necessary condition does not always result in the sufficient condition.
If new restaurants open, then living standards will improve.
new restaraunts open > living standards improve
However, living standards could improve in several other ways too, so new restaruants opening is not needed for the living standards to improve.
@MarisolSanchez I think about it in terms of ‘if then’ statements. If (sufficient condition) then (necessary condition). The ‘if’ part is on the left, the ‘then’ part on the right. If Jedi, then force. If cat, then mammal. They will sometimes put the ‘then’ part before the ‘if’ part so look out for that. ‘You can use the force if you are a Jedi’ is still ‘if Jedi, then force.’ I believe he’ll address exceptions later, but this becomes harder when they use terms like ‘only if’ which proceeds (sufficient condition) only if (necessary condition) or ‘unless’. He’ll probably explain ‘unless’ better than I can though, it’s tricky.
@GraysonCogswell Garfield is not unique for being a cat, but he is an instance of a member of the set of all cats.
So for any generic thing, x, if x is a member of the set of all cats, it is a cat, which implies it is also a member of the set of all mammals in this case.
The criteria for being a member of the set of all cats then necessitates membership in the set of all mammals.
Equivalently then, this logically implies that if a generic thing x is not a member of the set of all mammals then it must also not be a member of the set of all cats.
Why does "all cats are mammals" not get translated in lawgic to C^M since it is stating that all Cats are members of Mammals and not "If cat, therefore mammals"?
@bbcream Just to add here after trying to think through this: Perhaps because "all cats" is not a singular but a category? Perhaps, members are individuals and therefore all cats cannot be a member but Garfield can as it is not a category but singular.
When visualizing it, "all cats" could technically be a dot within "Mammals" but it is actually a circle which would lead me to believe that C->M is the correct notation without reading the words.
Is it necessary (no pun) to diagram as a subset versus an outright sufficient - necessary arrow for uniformity purposes? To be clear, instead of writing out gC, can I write this out as G---C, similar to how the first premise is written out as C---M?
I think this makes more sense after mapping out my own examples - someone correct me if I'm wrong here.
If rent goes down, then living conditions will improve for residents of 7Sage Apartments. Rent is going down for 7Sage Apartments, therefore living standards will improve for their residents.
This is not an argument that will require sets, as there are many different ways living conditions can improve. Saying in order for living conditions to improve, rent must go down doesn't work. It is fine to say that rent going down will improve living conditions, but that's not the whole story.
Rent is going down in 7Sage Apartments, which means residents' living standards will improve. JY is a resident of 7Sage Apartments. Therefore, his rent is going down and his living standards will improve.
This argument implies that there are sets being used, as it establishes JY as a member of a set (resident of 7Sage Apartments). The first argument doesn't establish membership for anything specific, it only contains a conditional relationship.
One thing I noticed about this example is that the claim "All cats are mammals. Garfield is a cat." doesn't appear on its surface to be a conditional relationship the way the second example with downtown restaurants does. Just a note for consistency since this example of "X is Y." has been used elsewhere to demonstrate when something ISN'T a conditional relationship. I guess that's the point J.Y. is trying to make here but it honestly just makes me more confused because intuitively you can tell these arguments are different in form.
@bbcream I think it's translated to C->M because they are sets - Cats being the subset and mammals being the superset. Another way to think of it is since all cats are mammals: if one is a cat, then one is also a mammal, which mimics the Jedi form.
@Livandthecats but supersets/sets are denoted as C^M. The arrow is only used for conditional relationships, which as blosciale says, this doesn't appear to be that even if "if not mammal, then not cat" and "if cat, then mammal" makes logical sense here.
It is so interesting to see conditional logic explained this way! I majored in Math and we were taught set and conditional logic in one of our core math courses (MATH 220: Mathematical Proofs). While it was taught in a slightly different way, it seamlessly applies to the LSAT.
Came here after getting 149.4.23 wrong. Did not find the answer explanation as to why C over B helpful. Came here. Also not helpful, as the difference between subsets/supersets and conditional logic is key in that answer choice and I need more help understanding. Revamping and making this lesson more thorough would be helpful. #feedback
Could it be helpful, for modus ponens (and other arguments hinging on a certain set of circumstances then setting into motion more circumstances) consider the reality presented by the premises to be the individual we are assessing, just like how we assess whether or not Garfield is a member of the cat set.
For example:
- If new restaurants open downtown, then quality of life will improve for downtown residents
New restaurants open downtown, therefore quality of life improves
is the same as
Reality is a member of the set of instances where new restaurants opened downtown, therefore quality of life improves for downtown residents.
or to use a different argument
- All cats are mammals
Garfield belongs to the set cat, therefore Garfield is mammal
is the same as
Garfield satisfies the condition of being a cat, therefore Garfield is a mammal
Is this thought experiment just confusing? In assessing the arguments from this perspective, how are sets and conditions any different. I don't really see how modus ponens and a categorical syllogism can be substantially different, unless the content of the argument is a factor in classifying the type of logic rather than just the base form. I am not a logician, so I really don't know.
I did some research and it does appear that the distinction between the two does indeed lie in both the content, form and application of the argument.
A good post making the specific distinction for those interested (which I still don't understand all the particulars of): https://www.reddit.com/r/askphilosophy/comments/h9jglu/what_is_the_difference_between_a_hypothetical/
very interesting stuff, logicians and logic enthusiasts are crazy
Mmm... I disagree with this approach. I believe it is better to understand how these two arguments are different, since you rely on different methods to prove their validity. I use logical connectives for modus ponen/modus tollens and diagrams (circles and dots) for sets.
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77 comments
Is it just me or are the Lawgic written out examples just completely missing from this article?
So they're functionally the same but actually different as one is pointing out a group A are in group B and all of C is in group A therefore C is in group B. The other is pointing out consequence. Rule: an action will result if X happens. X happens therefore X will be executed as a consequence.
[This comment was deleted.]
@eborland precisely.
@eborland is it always true that the "if" is the sufficient condition and "then" is the necessary condition??
Can someone explain to me how the cat and restaurant arguments are formally equivalent? It seems to me like the cat argument has three variables: cats, mammals, and Garfield, (G>C>M) while the restaurant has only two variables: new restaurants and living standards. (NRO>LSI … NRO is true so LSI is true) would that not catch you out on a match the structure question? I understand they’re both considered conditional logic and we’re not going into the terminology details but ‘formally equivalent’ seems like a strong claim.
@MadeleineLoyd i understood it like this:
Cat: G is part of C, C is part of M, because G is part of C, G is "triggered" to be in M.
Rest: NRO is part of LSI, because NRO happens, LSI is triggered.
They are equal in the sense that the sufficient condition triggers the necessary condition simply by existing; in the cat argument, the member is just a way to represent the superset triggering because of the subset's existence. In the restaurant, there is no member, but the subset still happens, and thus the superset is triggered.
@MarisolSanchez the way i think of it is the sufficient condition is one that result in the necessary condition. However, the necessary condition does not always result in the sufficient condition.
If new restaurants open, then living standards will improve.
new restaraunts open > living standards improve
However, living standards could improve in several other ways too, so new restaruants opening is not needed for the living standards to improve.
How can we identify what the sufficient condition and what is the necessary one?
@MarisolSanchez (Someone please correct me if I'm wrong but) I've thought of it this way:
The sufficient condition is something that needs to be sufficient to satisfy what you're talking about.
The necessary condition is required, or something necessary you must have.
@MarisolSanchez I think about it in terms of ‘if then’ statements. If (sufficient condition) then (necessary condition). The ‘if’ part is on the left, the ‘then’ part on the right. If Jedi, then force. If cat, then mammal. They will sometimes put the ‘then’ part before the ‘if’ part so look out for that. ‘You can use the force if you are a Jedi’ is still ‘if Jedi, then force.’ I believe he’ll address exceptions later, but this becomes harder when they use terms like ‘only if’ which proceeds (sufficient condition) only if (necessary condition) or ‘unless’. He’ll probably explain ‘unless’ better than I can though, it’s tricky.
@MadeleineLoyd thank you!
is it me or does he sound like he's talking faster?
@LoganHjermstad You can slow down the playback speed to 0.8, thats what i did
All cats are mammals. Garfield is a cat. Therefore, Garfield is a mammal.
cats = C
mammals = M
Garfield = G
Cats --> Mammals
C --> M
Being a cat is sufficient to being a mammals but is not necessary.
Being a mammal is necessary to being a cat but is not sufficient.
Garfield is a cat, therefore he is also a mammal.
Premise 1: If the restaurant introduces chicken sandwiches, then sales will increase for the company.
Premise 2: The restaurant introduces chicken sandwiches to the menu.
Conclusion: Therefore, sales increase for the company.
I find the formal mathematics version of this more helpful to read for the Garfield example. It would be written as such:
For all generic things; if that thing is a cat then it is a mammal
Garfield is a cat
Therefore, Garfield is a mammal
@GraysonCogswell Garfield is not unique for being a cat, but he is an instance of a member of the set of all cats.
So for any generic thing, x, if x is a member of the set of all cats, it is a cat, which implies it is also a member of the set of all mammals in this case.
The criteria for being a member of the set of all cats then necessitates membership in the set of all mammals.
Equivalently then, this logically implies that if a generic thing x is not a member of the set of all mammals then it must also not be a member of the set of all cats.
Why does "all cats are mammals" not get translated in lawgic to C^M since it is stating that all Cats are members of Mammals and not "If cat, therefore mammals"?
@bbcream Just to add here after trying to think through this: Perhaps because "all cats" is not a singular but a category? Perhaps, members are individuals and therefore all cats cannot be a member but Garfield can as it is not a category but singular.
When visualizing it, "all cats" could technically be a dot within "Mammals" but it is actually a circle which would lead me to believe that C->M is the correct notation without reading the words.
Otherwise... not sure.
couldve said theyre "lawgically equivalent" too har har
Is it necessary (no pun) to diagram as a subset versus an outright sufficient - necessary arrow for uniformity purposes? To be clear, instead of writing out gC, can I write this out as G---C, similar to how the first premise is written out as C---M?
I find this easier to track.
I think this makes more sense after mapping out my own examples - someone correct me if I'm wrong here.
If rent goes down, then living conditions will improve for residents of 7Sage Apartments. Rent is going down for 7Sage Apartments, therefore living standards will improve for their residents.
This is not an argument that will require sets, as there are many different ways living conditions can improve. Saying in order for living conditions to improve, rent must go down doesn't work. It is fine to say that rent going down will improve living conditions, but that's not the whole story.
Rent is going down in 7Sage Apartments, which means residents' living standards will improve. JY is a resident of 7Sage Apartments. Therefore, his rent is going down and his living standards will improve.
This argument implies that there are sets being used, as it establishes JY as a member of a set (resident of 7Sage Apartments). The first argument doesn't establish membership for anything specific, it only contains a conditional relationship.
I really hope this makes sense, lol!
One thing I noticed about this example is that the claim "All cats are mammals. Garfield is a cat." doesn't appear on its surface to be a conditional relationship the way the second example with downtown restaurants does. Just a note for consistency since this example of "X is Y." has been used elsewhere to demonstrate when something ISN'T a conditional relationship. I guess that's the point J.Y. is trying to make here but it honestly just makes me more confused because intuitively you can tell these arguments are different in form.
@blosciale Agreed! That's why my question was why is "all cats are mammals" translated to C->M instead of C^M?
@bbcream I think it's translated to C->M because they are sets - Cats being the subset and mammals being the superset. Another way to think of it is since all cats are mammals: if one is a cat, then one is also a mammal, which mimics the Jedi form.
@Livandthecats but supersets/sets are denoted as C^M. The arrow is only used for conditional relationships, which as blosciale says, this doesn't appear to be that even if "if not mammal, then not cat" and "if cat, then mammal" makes logical sense here.
I am confused and would appreciate someone or a tutor explaining this/conditional arguments
Do you actually need to use this method?
Im glad i majored in philosophy, haha!
It is so interesting to see conditional logic explained this way! I majored in Math and we were taught set and conditional logic in one of our core math courses (MATH 220: Mathematical Proofs). While it was taught in a slightly different way, it seamlessly applies to the LSAT.
Until he explained how they were different. I assumed they were the same.
Came here after getting 149.4.23 wrong. Did not find the answer explanation as to why C over B helpful. Came here. Also not helpful, as the difference between subsets/supersets and conditional logic is key in that answer choice and I need more help understanding. Revamping and making this lesson more thorough would be helpful. #feedback
is this approach actually effective or not? How many people do this and how did it help you?
So far I am having extreme growing pains applying this method to solving LSAT questions ...
Could it be helpful, for modus ponens (and other arguments hinging on a certain set of circumstances then setting into motion more circumstances) consider the reality presented by the premises to be the individual we are assessing, just like how we assess whether or not Garfield is a member of the cat set.
For example:
- If new restaurants open downtown, then quality of life will improve for downtown residents
New restaurants open downtown, therefore quality of life improves
is the same as
Reality is a member of the set of instances where new restaurants opened downtown, therefore quality of life improves for downtown residents.
or to use a different argument
- All cats are mammals
Garfield belongs to the set cat, therefore Garfield is mammal
is the same as
Garfield satisfies the condition of being a cat, therefore Garfield is a mammal
Is this thought experiment just confusing? In assessing the arguments from this perspective, how are sets and conditions any different. I don't really see how modus ponens and a categorical syllogism can be substantially different, unless the content of the argument is a factor in classifying the type of logic rather than just the base form. I am not a logician, so I really don't know.
I did some research and it does appear that the distinction between the two does indeed lie in both the content, form and application of the argument.
A good post making the specific distinction for those interested (which I still don't understand all the particulars of): https://www.reddit.com/r/askphilosophy/comments/h9jglu/what_is_the_difference_between_a_hypothetical/
very interesting stuff, logicians and logic enthusiasts are crazy
is it okay to think of the sufficient condition as the "why?" for the necessary condition?
I don't think mentioning the difference was necessary in the first place. Now my brain will start marking them different sort of..
A --> B
A
Therefore, B
vs
A --> B
C --> A
Therefore, C --> B
Mmm... I disagree with this approach. I believe it is better to understand how these two arguments are different, since you rely on different methods to prove their validity. I use logical connectives for modus ponen/modus tollens and diagrams (circles and dots) for sets.