Video summary
In this lecture, Professor Matthew Brown combines translation skills with proof strategies in quantified logic by working through three distinct arguments. The first example involves oranges being sweet and fragrant, which is translated into a universal statement asserting that for any object, if it is an orange, then it is both sweet and fragrant. To prove this, the professor demonstrates how to use substitution instances with a specific constant, apply universal elimination to derive individual properties, and then utilize conditional introduction within a subproof to show that assuming an object is an orange leads to the conclusion that it possesses both traits. Finally, universal introduction is applied to generalize the result back to all objects, effectively proving the original argument.
The second argument shifts focus to gardeners, industriousness, and respect, where the goal is to prove that Arthur and Katherine are respected because they are gardeners and all industrious people are respected. The proof strategy here relies heavily on universal elimination to break down the general rules about gardeners and industrious individuals into specific instances for Arthur and Katherine separately. Since the logical structure for proving that Arthur is respected is identical to proving that Katherine is respected, the professor illustrates an efficient method of copying and pasting the derivation steps while simply replacing the constant for Arthur with the constant for Katherine. The final step involves using conjunction introduction to combine these two individual conclusions into a single statement asserting that both individuals are respected.
The third and most complex argument deals with artichokes in the kitchen, their ripeness and flavor, and the resulting reactions of guests who will be surprised or pleased. This proof requires handling nested conditionals where the antecedents themselves contain universal quantifiers regarding all artichokes in the kitchen. The professor guides the viewer through a multi-layered process starting with universal elimination on the third premise to create a specific instance, followed by conditional introduction and elimination to derive the individual reactions of being surprised or pleased. By repeating this intricate sequence for both outcomes and then introducing a new constant to handle the final complex conclusion involving both surprise and pleasure, the proof successfully derives that all guests will be both surprised and pleased.
The lecture concludes by emphasizing that mastering these translation and proof techniques is essential for success on upcoming exams, as students must be able to perform this work independently. The professor notes that while the third example was lengthy and involved several layers of conditionals and quantifiers, it serves as a crucial exercise in managing complex logical structures. Looking ahead, the next session will integrate these proof strategies with semantic concepts from previous units, specifically exploring how to combine proofs with models to accomplish various logical tasks. Students are encouraged to practice similar problems to ensure they have fully grasped the material before moving forward to the more advanced topics involving semantics and truth conditions.
Read the full video transcript
hello and welcome back to Phil 320
deductive logic I'm professor Matthew
Brown and this is the second in our
series of lectures for unit 7 where
we're going to talk about translations
and proofs in ql so what we're doing
here is we're just combining the skills
we learned back in unit five chapter
four of the book where we did
translations to ql um with our proof
strategies right and and um so let's get
straight into it with some examples I
want you to take each of these three
English language arguments and I want
you to translate it into ql and provide
a proof I want you to do this pause the
video and do this for all three
arguments I've provided there uh a sort
of abbreviated symbolization key for the
different predicates and constants that
you'll use for each of these arguments
okay so pause the video give it a go
come back when you think you have it all
right welcome back let's see how you did
first um let's start with number one our
argument is Orange is our sweet we're
using o for orange and S is for sweet
also oranges are fragrant we're using f
for fragrant therefore oranges are sweet
and fragrant so this is how I would
translate this
argument um oranges are sweet um which
is uh unqualified it's not some oranges
are sweet it's all oranges are sweet
all oranges are
fragrant therefore all oranges are sweet
and fragrant right um for all X if it's
an orange then it is sweet and fragrant
right now let's head over to carap and
see if we can prove this argument so
here we are we've got the two premises
loaded up already let's see if we can
get our
conclusion here I am going to use the
conjunction roduction rule to to get us
what I what we want to get but before I
do that I am going to go ahead and get
some substitution instances of our first
pre and second premise let's use let's
use lowercase o as our constant so um oo
then O is a substitution instance of one
so we can apply our Universal
elimination rule there O then fo for
premise 2 okay now we're going to do a
conditional introduction we're going to
start by assuming o right and this
assumption we want to get to S so and fo
that's where we're trying to get so I
can get S so through conditional
elimination three and five I can get fo
the same way on four and five
I can get S so and fo through
conjunction introduction online six and
seven that's what I wanted to get so I
can close out my sub proof I have if oo
then s so and fo that is conditional
introduction on lines 5 through 8 but I
don't want just that I want the
universal and because I picked o
arbitrary at random when I start it out
on lines three and four I can get it
back by using the universal introduction
Ox then SX and
FX here that's Universal introduction
online n that gets me where I want to go
did you get to the same place I hope
that you did but let me know if you have
any questions let's move on to our
second argument which is this one all
gardeners are industrious furthermore
anyone industrious is respected Arthur
and Katherine are garders so they are
both respected I'm using G here for
gardeners I for industrious R for
respected and lowercase a for Arthur
lowercase C for Katherine so let's see
um all gardeners are industrious that's
fairly straightforward Universal so is
anyone industrious is respect Ed right
that is all industrious ones are
respected ones Arthur's a gardener and
Catherine's a gardener GA and
GC therefore Arthur is respected and
Katherine is
respected let's head over to carap and
try to prove this one so here we go I've
loaded the premises in already we want
to get to ra and RC I'm going to use
here my Universal elimination rules
again I need GA then I a right that's
Universal
elimination on line one and I need I a
then ra a that's Universal elimination
on line two okay next in order to apply
these conditionals I need the antecedent
I need GA right which I can get through
conjunction elimination on line three
now I can get IIA
through conditional elimination on line
four and six I can get ra through
conditional elimination on line five and
seven and I've got ra that's half the
battle now I'm going to have to do this
all over again to get RC but I know how
to do it
now I'm actually going to copy and paste
all those steps I'm going to replace the
a with a c so far so good to get RC I'm
going to have to change the N line
numbers here to 10 and 12 and my last
move here is to get R A and RC through
conjunction
introduction on line 8 where r a is and
line 13 where RC is and that is done
let's go ahead and look at our third
argument which is the longest one we
have so if all the artichokes in the
kitchen are ripe then the will be
surprised furthermore if the artichokes
in the kitchen are flavorful then all
the guests will be pleased all the
artichokes in the kitchen are ripe and
flavorful therefore the guests will be
surprised and pleased these are kind of
complicated because you have a universal
in that first line all the artichokes in
the kitchen are ripe but that's embedded
in a conditional so I would represent
that like this right we have for all
that x if x is in artichoke and X is in
the kitchen that's all the artichokes in
the kitchen then X is ripe right all the
artichokes in the kitchen are ripe if
that's true then all the guests will be
surprised okay so that's how I would
represent that first premise our second
premise is similar right all the
artichokes in the kitchen are flavorful
and if that's true then all the guests
will be pleased and finally all the
artichokes in the kitchen are ripe and
flavorful right that is for all x if x
is an arch choke in the kitchen then X
is ripe and flavorful therefore all the
guests will be surprised and
pleased so that's how I would represent
that argument let's see if we can head
over to carnap and prove
it so here we are we've got the premises
loaded up again and we're trying to get
to that final uh Universal all the
guests will be surprised and pleased
this is a complicated one but what I'm
seeing here is a series of conditionals
that I'm going to need to um break down
I'm going to start by trying to break
down this third
premise um which has the universal
quantifier in it with the universal
elimination rule so let's say um we're
going to
use um a AA and Ka a if AA and Ka a then
ra
a and Fa right I'm doing this Universal
elimination because I think it is going
to help me get the antecedence of my two
conditional arguments here that's my
guess so let's see the aned of the
condition add in one just has RA in it
so I need to get something that looks
like that I'm going to try to use the
conditional introduction
rule on AA and Ka and what I want is RA
a right so I can use this and the
conditional elimination rule to get ra
and Fa conditional elimination 4 five I
can get ra a right through conjunction
elimination on line six that's what I
wanted to
get so now I have AA and
Ka then ra a right that's conditional
introduction on my subproof 5 through 7
and that allows me because a was chosen
at random in line four uh and doesn't
appear in any premise or undischarged
assumption I can use the universal
introduction rule to get um for all X ax
and
KX then
RX through the universal introduction on
line
8 that is the antecedent of my line one
right um which will allow me to get for
all y gy then syy through additional
elimination 1 and N great now I need to
do all of those steps
again but uh just changing out the
relevant um pieces right so I'm going to
copy and paste here to make this go a
little faster of course it's going to
mess some things up so I have to think
through what exactly I'm getting a a and
Ka a yes ra a and Fa yes except this is
line 11 now right but what I want to get
now is not ra but fa right so I'm
changing this to line 12 I want um fa
here so my line numbers are changing
this is the sub proof of 11 through 13
so far so good here we're doing to FX
and change this to 14 good and here
we're changing this to
py which is what appears in the
conditional online
to and if I use 15 here I get that okay
we've come a long way um but we're not
there yet I've got these two universals
on line 10 and
16 but what I want remember is this more
complex one here so I know that I can do
it through a conditional introduction
but I'm going to need first to introduce
a substitution instance so let's call uh
let's use a different constant let's use
G GG then
SG that's Universal elimination on line
10 substituting the Y with a G I can
also get g g then PG Universal
elimination online
16 now I can do my conditional
introduction assume GG why because I
want to get SG and PG right that's my
goal here I get SG through conditional
elimination
1719 I get PG conditional elimination
1819 I get SG and PG through conjunction
introduction
2021 and that is what I wanted to get so
now I can close out my sub proof with
this conditional introduction
here conditional introduction 19 to 22
and that has the right form I just have
to apply my
Universal
introduction
to the conditional here get Sy and py
and that is universal introduction 23 I
can use the universal introduction here
again because G was chosen at random
doesn't appear in a premise and doesn't
appear in any undischarged
assumption up above well that was a big
one I hope it gives you uh a better
sense of how to do some more complex
proofs in ql all right um so uh give
those uh kinds of problems in the
practice exercises AG go you will have
to do some translation and proof work on
the exam so um you want to make sure
that you have that down pat um I will
see you next time where we put together
the ideas from this unit 7 with the
ideas on semantics from our last unit
six um to uh look at how to combine work
with proofs and models to accomplish a
variety of tasks good luck on the
practice exercises I will see you next
time