Lol. My background is in philosophy so I was always putting these 2 as different types of arguments so I'm assuming that was holding be back in making unnecessary steps.
The only thing that confuses me is when we are supposed to use our real world knowledge and not? I feel like in multiple lessons its been used and others not so much.
@LogicallyUnreasonable Like for some of these, the answers are incorrect because alternative world examples exist to disprove them, even though they aren't existing in the stimulus universe.
@BrieV Premises aren't valid or invalid; arguments are. The question is whether the premise is true. In this example, 'Garfield is a cat' is treated as true and because all cats are mammals, the conclusion follows validly!
if the two relationships are considered the same for our purposes, why is is not okay for me to use the visualization of the circles for both. I know it is wonky to do it for things that are not really objects, but it helps me a lot. Am I shooting myself in the foot?
@Hashibiro, if it helps you visualize it, go for it. It just gets wonky because the modus ponens doesn't have members and the sufficient condition is more of a true or false, so it might cost you time when it comes to test taking, but while we practice, who cares?
Hey guys! Wondering if anyone has another perspective to explain why only the second premise gets translated into lawgic and not the first. I wish I could think about this more deeply, but unfortunately my brain does not like math like representative explanations. Thanks in advance!
The first is a set and grouping question. Think of it as groups inside groups. Banana is inside (an element) of the fruit set. So are grapes, apples, and tomatoes.
Set of fruit: {banana, grapes, apples, and tomatoes}
Now, in an argument, we can now make rules for those items (elements) in the set of fruit.
For example, Rule: “all Fruit are from seeds.” This rule is placed on the entire collection of fruit inside the set of fruit. It applies to apples since apple was stated as an element of our fruit set.
Therefore since all fruit are from seeds, and apples is a fruit, then apples must be from seeds.
In fact, by saying “all Fruit”, we’re saying, for something to be a fruit, it’s necessary to have came from a seed. If something has not come from a seed it cannot be a fruit since all fruits come from seeds.
however, if something came from a seed that does not mean it must be a fruit.
the first argument is saying if x is apart of this set or group, then x must be “this”.
The second argument is saying if x occurs, x creates y, and if y exists, Z must follow. (It’s a casual inference chain argument, not an argument involving groups).
Both have the same exact argument structure.
Argument 1
where A = cat, B = mammal
all “cats” are “mammals” (A -> B)
X is a cat (x is A, and remember all As are Bs).
Therefore, x is a mammal. (Since x is A and all A->B, x must also be B)
Argument 2
Let A = more restaurants open, B = improved living standard.
If more restaurants open then improved living standards. (A -> B).
More restaurants opened. (x = A, A = True -> B must also be True).
Living standards improve. (therefore B must have resulted, since A -> B).
Notice how argument 1 uses groups and rules/qualities about those groups (If under18 -> minor. X is under18, therefore X is minor).
While argument 2 uses causal reasoning. (If it’s winter time there’s less food, and it’s winter time, so therefore there’s less food.)
“Less food” is an amount, not a clear grouping like “fruit”. This is the reason why the second argument does not get translated like the first argument. “Increases, decreases, improves.” These are verbs of unspecified change, not nouns “Apple” or “fruit”.
what’s most important is that these two arguments produce the same logical structure despite one containing sets and groups, while the other contains A casual inference chain .
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.
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Lol. My background is in philosophy so I was always putting these 2 as different types of arguments so I'm assuming that was holding be back in making unnecessary steps.
sigh
Thank you for adding this lesson! This has confused me for some time.
The only thing that confuses me is when we are supposed to use our real world knowledge and not? I feel like in multiple lessons its been used and others not so much.
@LogicallyUnreasonable GLAD IM NOT ALONE. I just asked this same question a few videos back lmfaoo.
@LogicallyUnreasonable Like for some of these, the answers are incorrect because alternative world examples exist to disprove them, even though they aren't existing in the stimulus universe.
I'm confused on how you know if the premise is valid. Like is it because Garfield has both the C and the M meaning it follows the C - M?
@BrieV Premises aren't valid or invalid; arguments are. The question is whether the premise is true. In this example, 'Garfield is a cat' is treated as true and because all cats are mammals, the conclusion follows validly!
if the two relationships are considered the same for our purposes, why is is not okay for me to use the visualization of the circles for both. I know it is wonky to do it for things that are not really objects, but it helps me a lot. Am I shooting myself in the foot?
@Hashibiro No, I don't think you are. It's a useful visual for If-then too.
@Kevin_Lin Thanks Kevin!
@Hashibiro I do the reverse, where I prefer representing everything with arrows and exclamation points, e.g. A ⟹ B and !B ⟹ !A
I think both of our methods are fine :)
@Hashibiro, if it helps you visualize it, go for it. It just gets wonky because the modus ponens doesn't have members and the sufficient condition is more of a true or false, so it might cost you time when it comes to test taking, but while we practice, who cares?
my predicate logic professor would lose his mind if i told him i was conflating categorical syllogism and modus ponens
Hey guys! Wondering if anyone has another perspective to explain why only the second premise gets translated into lawgic and not the first. I wish I could think about this more deeply, but unfortunately my brain does not like math like representative explanations. Thanks in advance!
@juliairelandd
The first is a set and grouping question. Think of it as groups inside groups. Banana is inside (an element) of the fruit set. So are grapes, apples, and tomatoes.
Set of fruit: {banana, grapes, apples, and tomatoes}
Now, in an argument, we can now make rules for those items (elements) in the set of fruit.
For example, Rule: “all Fruit are from seeds.” This rule is placed on the entire collection of fruit inside the set of fruit. It applies to apples since apple was stated as an element of our fruit set.
Therefore since all fruit are from seeds, and apples is a fruit, then apples must be from seeds.
In fact, by saying “all Fruit”, we’re saying, for something to be a fruit, it’s necessary to have came from a seed. If something has not come from a seed it cannot be a fruit since all fruits come from seeds.
however, if something came from a seed that does not mean it must be a fruit.
the first argument is saying if x is apart of this set or group, then x must be “this”.
The second argument is saying if x occurs, x creates y, and if y exists, Z must follow. (It’s a casual inference chain argument, not an argument involving groups).
Both have the same exact argument structure.
Argument 1
where A = cat, B = mammal
all “cats” are “mammals” (A -> B)
X is a cat (x is A, and remember all As are Bs).
Therefore, x is a mammal. (Since x is A and all A->B, x must also be B)
Argument 2
Let A = more restaurants open, B = improved living standard.
If more restaurants open then improved living standards. (A -> B).
More restaurants opened. (x = A, A = True -> B must also be True).
Living standards improve. (therefore B must have resulted, since A -> B).
Notice how argument 1 uses groups and rules/qualities about those groups (If under18 -> minor. X is under18, therefore X is minor).
While argument 2 uses causal reasoning. (If it’s winter time there’s less food, and it’s winter time, so therefore there’s less food.)
“Less food” is an amount, not a clear grouping like “fruit”. This is the reason why the second argument does not get translated like the first argument. “Increases, decreases, improves.” These are verbs of unspecified change, not nouns “Apple” or “fruit”.
what’s most important is that these two arguments produce the same logical structure despite one containing sets and groups, while the other contains A casual inference chain .
If A -> B
A
Therefore, B. Same structure.
Hope that helped.
Is it just me or are the Lawgic written out examples just completely missing from this article?
@SavanahHoffstein I see it in the diagrams of argument 1 and 2
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.
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@eborland precisely.
@eborland is it always true that the "if" is the sufficient condition and "then" is the necessary condition??
@GennaBishop Yes!
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!