>> Today's lesson's going
to work on the Venn diagrams that we learned a while back. But rather than just looking at
single statements we're going to use Venn diagrams to
analyze entire arguments. So if we remember the old
Venn diagram system we had the subject term and
the predicate term. And this, if you wanted to
say all S or P shade the left. If you wanted to say no S
or P shade the middle space. If you wanted to say some S
or P, put an x in the middle. And if you wanted to say some S or not P you put
an x on the left. That's going to hold still. In this case the S is going to
stand for the subject position at the conclusion
or the minor term. The P is going to
stand for the predicate of the conclusion
or the major term. And we're going to add one
more term and that's going to intersect the two circles
and that's going to be, we're going to use an M
for the variable for that. And that's going to
stand for the middle term or the term that's
repeated in both premises. So to see how this would
look in an argument form, I have an argument example to the top left here that's
a IAI figure 3 argument. So I is some blank are blank. A is all blank are blank. And I is some blank
are blank again. The figure 3 argument
tells that we need to put at the middle term in the subject position
of both premises. And then whatever goes
in the subject position of the conclusion needs to
go on the bottom premise. And whatever goes in
the predicate position of the conclusion needs
to go on the top premise. Okay, so here's our
completed argument. And now the question
is is how do we go about evaluating this argument
using three part Venn diagrams. What I'm going to do is
go ahead and talk through and solve this argument using
the three part Venn diagrams. And on the next slide I'll walk
you through each of the steps for doing this for
constructing the Venn diagrams for evaluating them. And then on the later slides
we'll actually look at step by step a couple of arguments
and see how that's done after we've seen
the formal steps. So this is just kind of
warm you up, get you ready for what we need to be doing. So step one we need to take
and set up three circles for our Venn diagrams. I'm going to have
circle one, circle two and then circle three
intersecting the two. The bottom left circle is
going to be your S term, bottom right circle is going
to going to be your P term and the top circle is
going to be the M term. Now the first step for
constructing your Venn diagram or for evaluation your Venn
diagram is you're going to plot out the premises. And the first thing you're going to do is find your universal
premises and plot those first. So in this case the universal
premise is all M are S. And in order to do that if
we numbered these spaces so we're all sure what I'm
talking about, one, two, three, four, five, six,
seven spaces total. The top space is an
M that is not an S. So space one is an
M that's not an S. And space four is also
part of the M circle but it's not part
of the S circle. So if we looked at these two
circles without the P circle in the way, you would
see M S, if we wanted to say all M's are S, we
need to shade out the part of the M circle that is
not an S because we want to say all M's are S. And now
we have all M's are S. Now if we took those, these two
circles and put them right on top of the three part Venn
diagram you'd see the shaded part overlaps that
space one and four. So you'd shade in the
space one and four. And don't worry as
we go we're going to practice this quite a bit. Okay then the next step is to
plot out your remaining premise, in this case some M are
P. And here some M are P, if we look at our old
Venn diagram system. So we're used to seeing
the M on the left side, the P on the right side. Some M are P, you'd
put an x in the middle. But now we're going to have
to rotate the circles down so that P is on the
bottom and M is up top. The x still stays in the middle. Now if we slid this rotated
two part Venn diagram over the three part
Venn diagram, you'd see that x can either go
in space three or space four. Both of those are M's
that are also P's and both of those say some M are P. But because space four
is already shaded out, the only space the x can go in, the only space the x can
go in is space three. Okay, so that's part
one of this evaluation of arguments using
Venn diagrams. Part two, you're
going to put your pen down and ask a question. The question is can
I read the conclusion in what I've plotted out here? And so the conclusion
says some S are P, the old school Venn diagram. In this case the old school Venn
diagram looks the exact same as the new three part one
except that M circle's kind of in the way distracting us. But some S are P, what you need to see is an x somewhere
in the middle. There's two spaces in the
three part Venn diagram that would have an
x in the middle. Either space three or space six. If you see an x in either
of those spaces it would say that some S's are P's. And sure enough space
three has an x in it. And so it does say that there
is an S that's also a P, some S are P. And so
what this tells us then, the Venn diagram tells us that if these premises
are true we do know with certainty this conclusion. And so we would say
this argument is valid. So again what you're doing
is you're going to plot out your premises and then see
if you can find your conclusion in what you've already
plotted out. If you can find it,
the argument's valid. If you can't find it,
the argument's invalid. Okay, let's go ahead
and look through step for step the way we
would analyze three part Venn diagrams. Part 1, setting up the premises. So you're going to put the
minor term on the bottom left, the major term on
the bottom right and the middle term goes up top. Another way of saying that,
you're going to put the subject of your conclusion on the
bottom left, the predicate of your conclusion
on the bottom right and the middle term
is going to go up top. That's where you're going to
put the terms for the circles. Step two, you're going to plot
your universal premises first. Universal premises
always take up two spaces. And there's a little hint
here, you're either going to be looking for a football or like a crescent
moon or a Pacman thing. And what that means is this. So if you have your three
circles and you're trying to say no S's are M's. You'd be shading in two spaces. It's the spaces between the
S and the M. And if you look at those two spaces that
got shaded it looks kind of like a football. There is logic teacher
Mark Thorsby who has on YouTube posted
his entire lectures for Patrick J. Hurley's, A Concise Introduction
to Logic Textbook. They're really good lectures
if you want to use those as supplements, by
all means use them. And his example is
looking for a football, that's where I got
this idea from. It's pretty clever. And then the other thing to
look at, or I think it's clever at least, the other
thing to look at is the Pacman or
the crescent moon. This is [inaudible] ideas well. So if you want to say all
M's are P's, you're going to have two spaces that you'd
have to get rid of to leave us with all M's being P's. And it's going to be like that. Now the reason this is like
a Pacman or a crescent moon, you have the circle
for the M term. You have the circle
for the P term. Then if you look right here,
that part that's shaded, that part looks kind of like
a crescent moon or a Pacman. If you had a Pacman it would be, the ghost would be here
running away from Pacman. Because Pacman's
trying to gobble him up. So you're either going to look
for a crescent moon or a Pacman. The important part is
you're always going to be shading two spaces when you're doing your
universal premises. Okay, the next thing is is
if you have a x on the line of an argument that's going
to indicate uncertainty as to where the particular
premise goes. Particular premises
always go in one of two possible spaces
or on the line. So if you can't put it in
one of two spaces and it goes in the other space,
if you can put it in both spaces then the x
would have to go on the line. For example. [ Pause ] If you have some S's are
M's, space one, two, three, four, five, six, seven. If you look at space two that's
an S that's an M. If you look at space three that's also
an S that's an M. And so that could go in
either of those spaces. And so this is what that
number three says on the left. An x on the line
indicates uncertainty as to where the particular
premise goes. And so in this case because it
could go in either space two or three and we're not sure
where it goes, we'd put the x on the line between
two and three. Now what if we had already
plotted the premise no M's or P's before that and shaded
out space three and four? Now if you say some S's are
M's that could either go in space two or three but
three doesn't exist anymore. So the only space left is
for the x to go in space two. And this is actually
rule number two applies, we plot our universal
premises first. Because a lot of times what
will happen is we'll plot our universal premises and then
we move on to our particulars that plotting of the universals
kind of kicks the particulars out of those awkward
positions of being on the line. And it says okay, you
can only be in one space. So an x on the line
indicates uncertainty is where the particular
premises go. The particular premises are
either going to go in one of two possible spaces. If one of the spaces is shaded
out, it goes in the other space. If no space is shaded
out then it goes on the line between two spaces. And so that's setting
up the premises. Once you've set up your premises
the next phase you're going to move onto is evaluating
your argument. So what you do is
one, stop writing. You're not going
to write anything down for your conclusion. Once you've written everything out for your premises
you're done. Then you ask yourself
a question. Can you read the conclusion
in what you've plotted? If you can, the argument's
valid. If you can't, the
argument's invalid. As we go through this we'll
talk about some tricks for helping you see if you
can identify the conclusion and what you've plotted. But for now let me
give you an example. So M S P, let's say the first
premise was no S's are M. And the second premise
was all P's are M's. [ Pause ] And the conclusion
was no S's are P's. So let's focus on
our conclusion. No S are P. With that
conclusion, no S are P, the question is can you read that conclusion what
you've plotted? Well the old school Venn diagram
system you have S on the left, P on the right, shaded
in the middle. On this three part Venn
diagram you do have the middle between S and P is shaded. And so this example you
can read the conclusion in what you've plotted. And so this argument is valid. There's a couple hints. The first hint is more
information is okay, less is not. Before I erase this so
you can read everything, let me explain what that means. In the example drawn
here the space between S and the M is shaded
in two spots. And then you have the space
where the P is is shaded out over where the P is. And this looks like a
little bit more is done than the final conclusion
Venn diagram. So for instance what
it'd really look like is if you have the top
part of the S shaded and then you had a big chunk
of this P shaded as well. Now that's kind of what the
old school Venn diagram would look like. This is more information than
is needed and that's okay. So in this case the
argument's still valid because it still does say if you're an S you are
not a P, no S's are P's. Where you get into a problem is if you had less information
there. So for instance if you had. [ Pause ] The space between P and
M, it said no P are M. And then you had the space
between S and M. It said no S or M. And your conclusion was
still no S are P. Here you have less than your conclusion
written. Because the bottom section
in the middle between the S and the P, space one, two,
three, four, five, six, seven, space six, needs to be shaded
in as well and it's not. So that's less information. If that was your, if
this was your argument, this would be an
invalid argument. So this is what we mean by
more information is okay, less is not. Again as we go through the
specific examples you'll see this worked out in
much more detail. And I have some practice
exercises that really spend some
time dealing with this. So if you're kind of
unsure as to what was going on right there, don't
worry we're going to go through it again. Here's the hint, more information is
okay, less is not. Second hint, an x on the line
means the argument is invalid. So if you're ever unsure as
to where the x should go, it could go equally
between two spaces, the shading doesn't force you
into one space or the other and you put that x on the line, you know right away the
argument is invalid. And a final hint is a space
that has been shaded twice, that also means the
argument's invalid. It just happens to
work out that way. Same with the x on the line, it just happens to
work out that way. So those are some extra
tips that'll help you out. Everything before the
hint is what you really, really need to know. But if you need some extra
help, the hint helps as well. Okay, let's go ahead
and practice with some specific arguments. This first argument is
no felines are dogs. All cats are felines
therefore no cats are dogs. So we'll take this
one step at a time. The subject of the conclusion or the minor term goes
on the bottom left. And so in this case that's cats. The predicate of the conclusion or the major term goes
on the bottom right. And in this case that's dogs. And then finally
the middle term, in this case felines
goes up top. Okay, so we'll put
all that together now. So you have felines, cats and dogs are the three
terms we're dealing with. Okay, now we'll move onto
plotting our premises. So we have spaces
one, two, three, four, five, six, and seven. The first move you're
going to make is to fill in your universal premises. In this case both
premises are universal. So it doesn't matter
which one you do first. Really if you're ambidextrous
you can do them both at the same time. I'm not, so I always
just pick the, if they're both universal I
just do the top premise first. So we're going to do
no felines are dogs. In this case we're going to
shade out the part where felines and dogs overlap because we want
to say that no felines are dogs. The old school Venn
diagram you'd have felines on the left side,
dogs on the right side and you'd shade out the middle. But to match this to the three
part Venn diagram we're going to have to rotate this up
so that felines is up top and dogs is going
to be bottom right. So you have felines up
top, dogs bottom right, keep your shading in the middle. Now if I had transparency paper
I'd take the transparency paper and make overlapping circles. So I'd do the three
part Venn diagram. And then do two circles
on the transparency paper. And then you can slide the
two circles over the top of the three part Venn diagram and see exactly where
things fit. In this case just pretend that we're sliding this
rotated old school Venn diagram over the three part
Venn diagram. And as you slide it over
you see that it's going to overlap the green
circle and the red circle. And specifically the shaded
part is going to be right where spaces three
and four are found. And that says no
felines are dogs. Now we're going to move
onto the next premise. So remember the universals are
always shading in two spaces. We're either looking for
footballs or crescent moons. For the no felines are dogs
we did a football shading. For the A form we're going to
do a crescent moon shading. So in this case we're
doing all cats are felines. So the old school Venn diagram
you have cats on the left and felines on the right. You're going to shade
in the left side. But now felines is up
top, cats on the bottom. For the three part Venn diagram,
so we need to rotate the circle so that felines is up top,
cats is on the bottom. And then we're going
to go ahead and shade in that cat section again. Now as we slide the
old school Venn diagram over the three part Venn
diagram we would see that space, spaces five and six are
going to get shaded here. Spaces five and six get shaded. And there you have,
remember we're looking for either the crescent moon
or the Pacman, spaces five and six make that crescent moon. So now the only space
left for cats to be in is also in the feline circle. And so that does say that
all cats are felines. And finally we move into
the conclusion phase. At this point we
put down our pen. We're done plotting out
anything on the Venn diagram. So we don't tough the
Venn diagrams again. And the questions that's asked
is can you read your conclusion in what you've already plotted? More information is
okay, less is not. The space where dogs and
cats overlap is shaded out. So there's no way to be
both a cat and a dog. In other words, it does
say no cats are dogs. Specifically two, three, four,
five, six, seven, spaces three and six are the spaces
where cats and dogs overlap. And that would be where
you'd have a cat/dog hybrid. And spaces three and six
are overlap or shaded out which means nothing
can exist there. And so that does say
that no cats are dogs. So this argument is valid. Now when we go back to that
more information is okay, less is not. Space five and space
four is more information. And that's okay. Because space, really what's
going on is between spaces two and seven, that's, space two
is where all your cats occur. And space seven is where
all of your dogs occur. And so it still does say no cats
are dogs even though there's more information like
space five discussion and space four discussion. What wouldn't be acceptable is
if say space six wasn't shaded. Because space six if it
wasn't shaded would be a space where you are allowing a cat
to be a dog which violates that statement, no
cats are dogs. So that's what we mean by
more information is okay, less is not. But once again because
in this argument, spaces three and six are shaded. The argument no cats
are dogs is valid. Really quick, to look at
the old school Venn diagram, if you had cats on the
left, dogs on the right, the middle space
would be shaded. And if you carry that
over and look, yep, the middle space is shaded. But once again we
didn't write anything down for the conclusion. We only wrote stuff
down for the premises. And then the question is is
can we read the conclusion in the premises. So what happened is if
these premises are true, we do know this conclusion
with absolute certainty. And that tells us then
that the argument is valid. Let's try another one. So this argument says no
cats are nice animals. Some pets are cats. Therefore some pets
are not nice animals. Step one, we're going to plot out our universal
premises first. In this case the
universal is the top one, no cats are nice animals. So we'll go the old
school Venn diagram. We have cats on the left,
nice animals on the right, and we want to say no
cats are nice animals. But then we have to
rotate the Venn diagram. So we're going to have cats up
top, nice animals bottom right and we're going to
shade in the middle. Let me write the
numbers on this, three, four, five, six, seven. So as we slide the old
school Venn diagram over the three part Venn diagram
we'd see that it goes right over the top of spaces
three and four, cats being the green circle, nice animals being
the red circle. So spaces three and four
are what we overlap. And that's how you plot
no cats are nice animals. Okay, now for the
second premise. We're going to do
some pets are cats. The old school Venn
diagram, you have pets and you have cats,
some pets are cats. You're going to put an
x in the middle there. We're going to rotate
the C up to the top and keep the pets
down on the bottom. So we'll have cats up top,
pets down on the bottom, we put an x in the middle there. Now as slid this Venn diagram,
old school one, over the top of the three part Venn
diagram we see spaces two and three are the two spots
where that x would need to go. But because we already
shaded out space three and shading means that
that space doesn't exist, the only space left for that
x to go is in space two. And so that is the second
premise for the argument. The x could have gone in two
spaces but one has been shaded out so now it can
only go in one space. So now we move into
the evaluation phase. You put down your pen. And the question is do
you see the conclusion in what you've already charted? In this case, yes. There's an x in the
pet circle but not in the nice animal circle. And so that argument does say
some pets are not nice animals. So it's valid. Let me show you with the
old school Venn diagram what that looks like. You have pets, nice animals, if you were saying some
pets are not nice animals, you'd put an x in the far left. In this case there is
an x on the far left. It's in the top of the far
left but it is on the far left. That would be the equivalent
of just putting an x on the top part of your
old school Venn diagram where just the P part of the
circle, so the far left space. It does say that some
pets are not nice animals. And so as you write out your
premises the conclusion is already there. You didn't have to
write anything new for that conclusion. The two spaces that would
have worked is space two where you did see the x. The
other space that you could have, if you saw an x it would have
been valid is in space five. Now at this point the
question might come up, what if you had an x on the
line between space two and five? This argument, that's
not the case but let's say we had an argument
where you had an x between two and five like that on the line. And the conclusion was some
pets are not nice animals. Would that be a valid argument? For whatever reason, so
go back to that hint, an x on the line means the
argument is always invalid. For whatever reason this
situation will never arise. Likewise if you have
a double shaded space and the question comes up, what
if it's a double shaded space but it gives us the conclusion? For whatever reason double
shaded space will never give you the conclusion. If you see a double shaded space
the argument is always going to be invalid. So it's kind of a nice way
of double checking your work. So that's it for these
practice arguments. Let's go ahead and practice
getting the component parts figured out so as we go
through arguments we're going to be a little more
fluid with them. We're going to use
this Venn diagram where the spaces are one, two,
three, four, five, six, seven. You have S on the bottom left, P on the bottom right
and M up top. And we're going to
answer some questions. So question one, how would
you show N M's are P's? As I go through these I'm
going to go at a natural pace. But I encourage you to
hit pause and try it on your own before you
see me solve this through. So in this case no M's are P's. We're going to shade in
the space three and four between the M and the P circles. And so there's the football. Next question, how would
you show all S's are P's? In this case we're going to
shade out spaces two and five and there's the Pacman
or the crescent moon. As I'm going through this if
you happen to get one or two of them wrong, just go ahead
and rewind it, to finish it all out rewind it and try it
again in about ten minutes . You'll probably forget all of
this and then you're just going to be re-practicing
from scratch. You could probably get through
about four runs of, you know, waiting ten minutes, rewind, start again before you
have this stuff memorized. So you'll be able to practice
it again and again and again as though it's brand new to you. Okay, so that one we
shaded in two and five. The next one, how would
you show no S are P? In this case we're going to
shade in the space between the S and the P, that's three and six. That says no S are P. How
about some M's are S's. Here an M that's
an S is space two, an M that's an S is
also space three. And so because we're not
sure if it goes in space two or three we're going to put it
on the line between the two. So there are x on the
line of two and three. Some P are not M. So seven
is a P that's not an M, six is a P that's not an M. It's
not clear which of these it is, so we're going to put an x on
the line between six and seven that says some P are not
M. This time we're going to have spaces three and four
already shaded in and we go through another series
of questions. So how would you show
some P are not S? So spaces four and seven
are P's that are not S's. But space four is
already shaded in. So you can't go there. The only space left to
go is in space seven. How about some S's are not M,
some S's are not M. Let's see, so space five is an
S that's not an M, space six is an S that's
not an M. It's not clear which space it goes it so we're
going to put that x on the line, so x goes on the
line of five and six. All right, what about
some S's are P's? So S on the bottom left,
P on the bottom right. Space three is an S that's a P,
space six is an S that's a P. But space six or space
3 is an S that's a P, space six is an S that's a P.
But space three doesn't exist so the x has to go in space six,
that's the only place it can go. Okay and finally
all M's are P's. Let's see, so space one is an
M that's not a P and we want to say all M's are P's. Space two is an M that's
not a P and we want to say all M's are P's. So space one and two have to go. If this was a real
argument it'd be very odd because it's basically
saying that M doesn't exist because you already
shaded out three and four. So nothing exists anywhere for
space M which is kind of weird. Anyways, so all M's are P's,
space one and two get shaded in. And there's your Pacman shape
or your crescent moon shape. All right, next set
of questions. The, what we're trying to figure
out now is what you would need to see to get the
following conclusions. So if we want, if we had a
conclusion in an argument that said some S's are P's. The question is, what would you
need to see happen where for that conclusion to occur? In this case some S's are P's. You would need to see an x
in either space three or six. If you saw an x in either three or six the argument
would be valid. What about for some S are not P? In that case an x in
either space is two or five would be what
you'd need to see to make the argument valid. But again because these are your
conclusions you're not going to write those x's in. It's only if you can read them. If the x is actually there,
the argument is valid. If the x is not there,
the argument is invalid. So in this case because there's
no x anywhere the argument would be invalid if that
was your conclusion. What if you had no S are P? You'd have to have spaces
three and six shaded in. And what about all S are P? You'd have to have space
two and five shaded in. Both of those spaces would
need to be shaded in. Okay, bonus round. I'm going to draw this so you can see exactly
what's being talked about. If two, three and
five were shaded, so there's two, three and five. If that was shaded when an
argument with the conclusion of all S are P be valid? The answer to that question
is yes, it would be valid. It has more than
the conclusion says. But remember that hint
in the very beginning, more information is
okay, less is not. And if you look visually
space six is the only space where your S's can exist. And space six are
part of the P circle. And so every single S
really is a P which is what that conclusion says,
all S's are P's. So this argument would be valid. So let me back up a
minute and walk you through one more time the
reading these conclusions. I'm going to set up the old
school Venn diagrams right beside it so you can be, we can
be sure that you're really clear on what you're looking for. So the old school Venn
diagram, if you have your S on your left side, your P on the
right, and what's really going on here is space two is
up there, it's cut off, space three, space four, space
five, space six, space seven and then this is up
where your M circle. We're going to get
rid of that M circle. We don't need it. And that's what we're
looking at. So space two and five, three
and six, four and seven, that's where the
conclusion exists. So if you saw some S are
P, an x in space three or six are both spaces
that do say some S are P for the old school Venn diagram. If you said some S are not P,
an x in space, either space two or five is what it would take to make the old school
Venn diagram work. And so space, x in space two
or five is what it would take to make the three part
Venn diagram work. If you were saying no S are
P then you'd need to shade in both three and six. And if you said all S are
P then you'd need to shade in both space two and five. Okay, so that's it for the
three part Venn diagrams for analyzing arguments. It's a lot to take in. I really recommend going through
the practice section a few times before you really start
working through exercises. But if you've got a
grip of it go ahead and get going on it now. I think the tricky thing
is is that because there's that third circle in there a lot
of times it confuses students. And the trick is is if you
go back to those color slides that I had in the beginning,
let me back up to that part. Here as you're plotting out
your premises it helps to think in terms of these three colors. And so when you're going
no felines are dogs, that's the top premise. You're pretending the blue
cat circle doesn't exist. And so when you say
no felines are dogs. If the blue cat circle
doesn't exist and you're saying no F are D,
well that space is the space that needs to be shaded. And then when you say
all cats are felines. That's between the blue
and the green circle. You need to pretend the
red circle doesn't exist. And so all C are F. If the red
circle didn't exist you'd shade in this space here. Remember the red circle
doesn't exist right now. And so the trick is is
you're always focusing only on the two circles that matter. Then when you're reading your
conclusion, no cats are dogs, pretend the green circle
doesn't exist, the felines. And is the space between
cats and dogs shaded? In this case it is. So I think if you think in terms of like those are
three separate circles and you are only focusing on
two circles at any given time, the thing that helps to
practice then is to figure out what circle do I need
to pretend isn't there. And that's kind of a tricky
thing to get the hang of. But as you get there
it comes with practice, just keep working on it. Okay, that's it for today. Take care and see you next time.