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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]

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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the next video and thank you for watching

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