Valid Arguments (Informative And Useful)

Video Series: Logic and Mathematical Proof (In-Depth Tutorials for Beginners)

(D4M) — Here is the video transcript for this video.

00:00
hi everyone welcome back i’m dave in this uh episode of valid arguments
informative and useful right so in this episode you’re going to learn about
valid arguments what they are and we’re going to practice uh constructing them
and so before we get started though i wanted to mention that this episode is
part of the series logic and mathematical proofs in-depth tutorials
for beginners and so um the link for the playlist is below in the description
so let’s see what we’re going to cover in today’s episode we’re going to talk
about first propositional uh consequence and then we’re going to talk about
testing for valid arguments and then we’re going to work out some exercises
and we’ll see how to do that and we’ll be able to do that pretty quickly
and i’ll show you how we can make it very simple
all right so let’s go ahead and get started [Music]

00:01
okay so up first we’re going to talk about uh what is a propositional
consequence so let’s put these together and here we go let’s get started so
first off we’re going to discuss valid arguments and then we’re going to talk
about inference rules um and the next episode is all about inference rules so
today’s about valid arguments um and then in the um
next video after inference rules i’m going to do a video over direct proofs
and then we’ll have a video over indirect proofs we’ll have a video over
proof by contradiction and the last episode will be over uh proof by cases or
um the episode after that one right there right so in the upcoming episodes
i just want to kind of uh you know show us exactly what we’re doing here um so
today’s this uh valid arguments uh part of the uh series here
all right so what is an argument so an argument first off is defined as a
statement right so we talked about what mathematical statements are in the first

00:02
episode it’s a sentence that is it’s a declarative sentence that has a true or
false value and it’s being a cue is being asserted
as a consequence of some list of statements so we have a list of
statements p1 p2 and how many we don’t know but we’ll enumerate them and
there’s finite number of them and so these statements are called the
premises or all of them to collect collectively we can call the hypothesis
of the argument and then the queue is called the conclusion um
of the argument all right so definition here um let’s see here
a statement is called a cons propositional consequence
of these statements right here so we’re going to make statement p1 p2 p3 p4 all
the way to pk and then we’ll draw a horizontal line and we’ll say q
and so q is called the propositional consequence
and we can write it out uh using connectives so we have of the hypothesis

00:03
here and then we have the consequence and we’re going to say that
this is a propositional consequence meaning that this right here is all a
tautology so again we talked about what tautology
is in a previous episode and we spent some time even talking about how to
figure out how to if something is tautology or not using some python so
we’ll look at that some more when we look at some exercises but anyways
here’s our first example of an argument so this is like a p1 here
and this is like a p2 here so we can say this argument would look something like
this p1 which is this statement here and then this would be like a p2
and this is our p2 statement here and this is our p3
and that’s that statement right there and then we have therefore
so this is like a substitution for the um
you know implication connective and then
we can say therefore here’s a symbol for therefore and then not p

00:04
so we could write all of this in this form right here also
and so it would look something like this let’s go right here so p or q
that’s the p1 and then an and so we’re going to end
all these statements here so not q implies r and and then we have not q or not r
and so then we have all of that implies the not p
and for readability we could perhaps include some brackets there
so for this argument if it’s valid we would need to check if this is a
tautology here or not so we could create
a truth table and determine if this is a tautology
now you might think oh we’re starting to get to mathematical proofs really
mathematical proofs are not about making truth tables so we’ll make the
transition when we talk about inference rules next time about working from truth

00:05
tables uh to making derivation proofs and so we’ll
talk about how to do that next time but today’s all about valid arguments
so here we go we have this uh statement right here and the question is is this a
tautology or not so we can phrase that and say now the question is whether or
not um whether or not it is a valid argument and we would check if it’s a valid
argument by checking if this is a tautology or not
so an argument is valid if the conclusion necessarily follows from the premises
and that’s exactly what this right here means we have to show that this is a
tautology for this p1 this p2 this p3 and this q right here this would be the
the q right here would be whatever is we’re concluding right here all right
perfect so let’s look at some examples we’re going to test for validity for the
following arguments so here’s the first argument here
so we have this statement right here p1 p2 p3 and then we have our conclusion

00:06
right here so how many variables do we have right so we have a p q and an r
so this is going to take eight rows and we can build out a um
truth table to to do that um so but before we do that though we
can actually go and let’s see here we can try to
you know work out some python let’s go over here real quick
let’s look at our setup here so we talked about this setup here last
in the last couple of videos and we’re going to use this logic uh file that
we’ve been using all along um in so we talked about that in how to
construct truth tables in the python episode
and so that will explain uh what the setup here is doing but basically we’re
importing this logic file we’re going to declare some variables
and then we can talk about whether or not we have some valid arguments or not
so this is um p if and only if q right here

00:07
in fact let’s go to this uh right here and so we’re looking at this example
right here example a and let’s see if we can make this a little bigger here
there we go and so now we have the p if and only if q and that parts right here
p if and only if q and then we have the qrr and so here’s
how we denote the qrr and we have an and right here also
and then we have a not not r and so that’s that’s the not r right there
and so then we have all of that with an implication
and then we have a not p there’s our conclusion right there so we can print
out the true table for that right there just hit shift enter to execute it and
you see that this right here are not all true so it’s not a tautology so this
right here is not a valid argument not a valid argument

00:08
so this a here is not a valid argument and we can just you know write out the
the statement right over here that we have um and so let’s look at another one
here’s b here so here’s another example of an argument
and the question is is it valid or not and so the way we would test that is we
would write out a um mathematical statement so it would be p or q and
and then we would do not q implies r and then an and and then not p or not r
and then we would take all those together that’s the hypothesis
and then we would um [Music] go here and we would say not p
and so we would need to check if this is a tautology or not
and so as we did in our previous videos we can make a truth table out or we can
just take a look at this in python real quick

00:09
so right here we have the um p or q and we have the and and we have not q
implies r and we have the and and then we have the not p or not r
and we have all of that with parentheses implies
um not p right we have the not p right here and then we have all that in
parenthesis and then we’re going to do dot because of the method we’re using is
to print a true table out all right so let’s shift enter execute that
and then we see that that is also not a tautology it’s a
tautology if the last column here is all trues so this is also not a valid
argument not a valid argument so both of these are not valid arguments
they may look valid to you well i don’t know that’s um
maybe maybe not they look valid right but we have a rigorous method to

00:10
determine if something is a tautology or not we’re just gonna go right here and
make the truth tables out and this is a lot faster way if you do this by machine
you know no one is going to write mathematical proofs using truth tables
so if you’re just trying to learn and get an understanding
i feel like it’s easier to just use a computer to help you out here all right
so now let’s um look at some [Music] let’s get some let’s get this erased here
and look at some [Music] um so here we go
the solution so part a is we’re going to take all the hypothesis here and we’re
going to look at this implication and the same thing for part b
and then we’re going to determine whether or not these are tautologies right
so you know that’s we just did using the
uh python code all right so the question is is the argument valid and so now
let’s look at some exercises so let’s bring this back over here and let’s look

00:11
at some exercises and so let’s look at exercise number one here
and so we have a p implies q so there we go right there p implies a q
and then we have a not r implies a not q
and then we have all that in parentheses and then we have implies and then we
have a not r implies and not p and we’ll print that true table out
there and when we do that we get all trues so this is a valid argument here
so this is a valid argument here now the problem with using truth tables
this is going to be something that we’re going to address in the next video the
next episode but the problem using truth tables right here is this is just a
brute force way of checking for tautology in other words it doesn’t
really give us any insight in terms of why this is a valid argument and we
start talking about inference rules then we get a better intuition as to why

00:12
something is true so we’ll talk about those here in the
next episode but i want to go through and make sure we can make sure that
something is a valid argument and so here we go here’s exercise number two
so maybe we can uh zoom in here a little bit more in case you’re having
a problem there we go um exercise number two so we have p if and only if q
so here we have p if and only if q in fact let me turn on my uh
let me turn on my mouse um highlighter all right so now maybe a little bit
easier to to see there we go all right so we have p if and only if q
and then we have p and all that is going to imply the queue
and now let’s go over here and hit shift enter and then we see that all of these

00:13
are true here so this one here is also valid so we’ll just say valid
all right very good so now for number three here um let’s go
down here and look at the third one here so for the third argument here we’re
gonna have p or q and then we’re gonna have not p we got right here
and then all those combined together with the and is going to imply a q
and so let’s print out that truth table there
and so that one gives us all trues here meaning it’s a tautology and so this is
a valid argument here this is also valid so i’ll put a check i’ll put a check
i’ll put a check so there’s three valid arguments there
all right so now let’s look at um number four here i gotta erase some of the
board here all right so let’s look at number four

00:14
and so here we have a p and q so here we got p and q
and then we have an and and then our next statement negation of q
implies uh sorry sorry negation of p implies q
and then all of that implies uh not q right so
we just simply type that up and then shift enter execute and we see we have a
false right here so the false right there means it’s not a valid argument so
i’ll just put an x here meaning not a valid argument and now let’s look at
number five and so now we have p implies a q and then we have a p
and then all those premises combined together
implies the conclusion let’s see if this is a valid argument or not
shift enter all of them are true this is a valid argument here p implies q and p
therefore we can conclude q so this is definitely a valid argument here

00:15
all right so then the last one here number six
and number six here we have p implies a q got that right here
and then we have an and and then a not q and then our conclusion right here
p implies r so let’s print that out and see what that true table looks like
and they’re all true and so we have another valid argument here so in fact
all of these i think were valid arguments except number four here
so you can’t just look at the p’s and q’s and guess right you have to actually
have some justification for it and so let’s go here now and
we have two more examples to look at here so let’s look at this a here
let’s go back here and look at a actually three more four more sorry
we got a here is q implies p and then we have a q
and then can we conclude a p from from that

00:16
let’s shift enter you see we get a false right here
and so this is not valid argument here so i’ll put an x right here not valid
now let’s look at b and so we can do a lot of these pretty quickly
now to be honest though it wouldn’t hurt you do all these by true table i mean
making true tables quickly is a is a good skill but
just for the sake of time in this video i’m just doing them using some code here
so we have p imply cq and then we have a not p and this implies a not q
so let’s execute that and we can see we get a false right here
so this is another not valid argument let’s look at c
all right c is a little bit more complicated right if p implies q we got
that right here and then and and then not q implies not r we got that

00:17
right here and then all of those premises combined together with an and
implies the conclusion r implies p and let’s see if this is a valid argument
and as you can see just one false value right there is not a valid argument
so not a valid argument there all right so let’s look at d here last one
and we have p implies the q got it and then and in the next one not p
implies the not q and then the last um premise here is we have p and not r
and then we have all of those combined together to imply s
now to be honest if you look at d here d is pretty obvious to me that it’s not
a valid argument there’s no s’s up here how can you conclude s
when there’s no s’s up here right but let’s just uh
you know make out the truth table and see in fact it was only false in one place

00:18
just because of the form of all these right here
um so you know i’m not saying that no matter no matter what you have up here
you cannot conclude something in any case um
yeah maybe you have a contradiction up here then you can conclude anything
maybe right in any case for this argument right here
as we can see there’s a false right here and so this is not a valid argument
right here so there’s four examples of arguments that are not valid
all right so uh that does it for this video right here
i’m hoping that you’re getting value out of these videos if you are please like
and subscribe in the next video we’re going to talk
about inference rules and we’ll start making progress towards writing
mathematical proofs because the video after the next video will actually start
writing direct proofs and it’ll be a lot of fun but you got we got to learn and
understand what inference rules are first so i look forward to seeing you in

00:19
the next video and thank you for watching
if you like this video please press this
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tell everyone what you do to succeed in your studies either way let us know what
you think in the comments

About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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