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in this video we delve into understanding tautologies contradictions

and contingencies did you know that during the 1930s

propositional logic was formalized in terms of truth assignments

from this development we have our modern understanding of tautologies

contradictions and contingencies so let’s see what they are hi everyone

welcome back i’m dave and this video understanding tautology’s

contradictions and contingencies is part of the series logic and

mathematical proof in-depth tutorials for beginners so i recommend you check

out that series the link is below in the description and let’s get started

so what are tautologies contradictions and um

contingencies i’m going to cover that first and then we’re going to see some

examples of some contingencies and then we’re going to see some

examples of some tautologies and contradictions

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and then we’re going to see some useful tautologies at the end by far

tautologies is my favorite and you’re going to love love this and so let’s get

started so what are tautologies contradictions and contingencies

so we can classify statements as according to whether they’re true or not

always true always false or otherwise so if you have something that’s always

true regardless of the values the propositional variables have it’s called

the tautology so for example if you have some

let’s say you have some p’s and a q’s and some r maybe you have lots of

variables in here and so here you have all the possible

combinations you got some trues and falses and you got all the possible

combinations in here and who knows how many variables you have

but if your last column here is your statement right here let’s

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call it capital p if your last column is always true then you have a tautology

if your last column is always false no matter how many variables you have no

matter what combination of the variables

you have if you’re getting all falses in your last column for your for your

compound statement then this is then this compound statement is called a

contradiction now if you don’t have all trues or all

falses then you have some kind of combination of trues and falses and then

that’s called a contingency so there’s three types that we’re going

to talk about tautologies contradictions

and contingencies and well let’s just go ahead and see an example

so in this example right here we’re going to um you know look first at what a

contingency is and so here’s uh here’s the example

we’re going to work on right here and so let’s get some let’s get a true table

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going for this right here so we’re going to need here a um p

and a q so this this only has two variables in it p’s and q’s right and so

um let’s go here to uh look looking at only p’s and q’s

right here so p’s and q’s and then we need an and q and p

and then we have a p implies a q and p and then finally we have an or

so p implies a q and p all of that or q and so let’s see the true table for this

sorry so we got true false true false true true now we have an n

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now i’d like to just say this uh at this moment briefly

that this video is part of a series and in the two videos that came before this

one the first one we talked about logical connectives and in the second

one we talked about how to make how to set up true tables

so i showed you in the previous video how to set this true table up

now remember an and is so here we’re going to have a true a false true a false

and so here we’re going to have um false true false true false oops sorry i

made my p’s wrong there true true false false so actually this is what false

and then false right so now we have implication right here

so we’re going to be looking at these two columns right here

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because we’re going to say p implies the q and p so we have p implies q and p

so i’m looking at an implication now so true implies true that’s true

true implies a false that’s false and then these are both falses

so it doesn’t matter what’s here but these are both falses so these are true

okay so now what we’ve done here is in our ores we’ve done we made a column for

that we already have a column for this so now i’m doing the or between

these two columns here so i’m going to be looking at uh this column right here

and or q so i’m looking at this column right here also so i’m looking at the or

between these two right here so i have true or true that’s true

and i have false or false that’s false and i’d be true or true so that’s true

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i have a true or false so that’s true so because of this row right here

we’re seeing that [Music] right so here we have um

definitely this is a contingency contingency

all right so because if they were all truths then we would say it’s a

tautology if they were all falses we would say it’s a contradiction but

because we got some truths and some falses it’s a contingency

all right very good so now let’s look at our next example

and so up now we’re going to be looking at some examples of scientologies and

some contradictions and so for our first example right here

we’re going to ask the question is this statement a tautology or not

and so let’s look at that so in this statement right here we also

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have two variables so i’m going to go with here um a p and a q

and we’re going to need to look at q and p let’s put that column right here

and then we’re going to need to look at p implies that q and p

all right and we have an or and we have all this stuff over here also um

and we have a column for p and q but that’s actually the same as this

column right here q and p is the same as p and q um so i’m going to look at q

implies p and q next and then i’m going to be looking at the

or between these two right here so this is here

and then we have or this one right here so that’ll be my last column p implies q

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and p or q implies p and q all right very good so here we got true false

true false true true false false and let’s go ahead and put this here

we got these columns right here to work so q and p so that’ll be true and true

so this will be true and false so that’s false

false and true is false and this is false now i said earlier a minute ago

that that’s the same thing as p and q and you can see that that’s true because

true and true is true and then these have one false in them so

the end is the false and then these are both falses so that’s a false so these

columns are the same actually so just to save a little space i didn’t

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put another column here you could do that if you want but anyways

the next column is an implication i’m looking at p implies

the q and p so i’m looking at an implication between these two columns here

so i have a implication so true implies true

true true implies false false and these are both falses so this is true and true

here okay so now let’s look at these two columns here so i have a q implies

p and q which is the same as this one right here

so to get this column i’m looking at an implication between these two so true

implies a true false implies that false is true um true implies false is false

and then false implies a true is true very very good

so last thing is to do the or now so i need to do the or between these two

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right here so true and true is true true and false and true is true

true or false is true and then true or true is true

so you can see that this right here is a tautology simply because no matter what

the propositional variables whether you have true or true

you know whatever combination you have here the proposition the compound prop

compound statement is always true so there’s an example of a tautology

so let’s look at maybe one more is a statement right here at tautology

so this is one that you would seem to be convinced it is a tautology so

you know it seems that it should be a double negation right

so we have p which can be true or false and then we have the negation which is

false and true and then we have the negation of the negation

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which the negation here would be true and false and you can see that these are

the same so then the last one would be to simply make the equivalents

so these i’m so now i’m looking at these two columns here

and i’m doing the equivalence so true and true they’re the same so this is true

and then false and false they’re the same they match so this is true

so this is in fact the tautology here so what we can say from this

is pretty much what you would already guess is that anytime you have a tautology

so for example say p is a tautology and we looked at one a minute ago

so the one we looked at was p implies a q and p right or q implies a p and q

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so we looked at this a minute ago and we said that that was a tautology

so capital p a compound statement anytime you take any tautology not just

this one any any tautology if you do the negation of this then this will be a

contradiction because if you’re going to negate all the truths

then you’re going to end up with all falses

so anytime you have a tautology you also have a corresponding negation of that

tautology which is also a contradiction so we we’re going to spend a lot of time

looking at tautologies but just don’t forget that anytime you

see a tautology there’s also a negation just by taking

there’s a contradiction just by taking the negation okay so now if we

look at this question here in general there’s a proposition may contain in

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propositional variables so we’ve looked at the case where there’s p’s and q’s or

there’s p’s q’s and r’s but in fact you could have any uh number of them as long

as it’s finite so you could have two three four twenty

right but then your propositional your but then your truth table

uh grows extremely fast because if you have five propositional variables

then you’re going to need 2 to the fifth or 32 rows and that’s going to be a lot

of work what if you have 10 propositional variables 2 to the 10 rows

and so that’s going to grow very fast so the question is

you know this is called the tautology problem is there a better way to solve the

tautology problem than brute force method of constructing a true table

so in other words to determine if something is a tautology you could say

oh the truth table says yeah the last column is all trues but that may take a

tremendous amount of work to find the whole table and to know the last

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column is all true is there a better way than to simply make a truth table to

know if we have a tautology or not seriously this is a hard question

can anyone solve this all right so now let’s look at some useful tautologies

these are tautologies used every day and let’s see what some examples are

so the following statements are tautologies so let’s look at that right here

so the first one here i’m going to look at is p implies p

that’s the tautology let’s verify that so we have p which could be true or false

and then we have p implies a p so true implies a true it’s true

the false implies a false is true and so this is the tautology right here

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um the second one here is uh p implies q and q implies r

all that taken together implies p implies an r

so that kind of reminds you of the transitive property if you’re familiar

with the transitive property is let’s look at a true table real quick

so we have three variables here and so i’m going to say p q and r

and we’re going to go with p implies the q next

and then we’re also going to need a q implies an r

and then we’re also going to need the end of that so p implies a q and

u implies an r and then we’re going to need the uh p implies an r p implies an r

and then we’ll be taking the implication between these two right here

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so i’ll do that separately right here so here we go true false true false

yeah true false true false through false and then true false so we got true true

false false true true false false true true true true true oops false

false false so there’s how we set up for

a three variable now we’re looking at an

implication here between these two right here p and q so i’m just going to be

looking at the first two columns and i’m going to do an implication true implies

a true true implies a true and then these two are false and then

the last ones here are all true because this is false hypothesis here

all right so next column now i’m going to be looking at these two

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right here so i got true false true false true and true

okay now we’re going to look at um the and between these two right here so

now we’re going to be looking at these two columns here with an and

so here is true false true false true and true now the last one here

is the p implies the r so here we need to go back over here and

look at the p implies the r so where are we getting falses we’re

getting a false on the second and the fourth here we go true false true false

and then where else so we’re doing p implies q

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so all the rest of the p’s are false here so

you know just the second and the fourth were false all these are trues here

and then now the last one here if we look at the problem over here is we have

all of this implies that one right there so let’s look at that

we’re looking at the implication between these two right here

true true true true true true true truth so that second one right

there is a tautology i guess we could try to put a

problem in here it would just be all true

but i ran out of space on my board here all right so now we can do the um

[Music] equivalence right here so to do that let’s go to the just the

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market board here and let’s go look at this one right here

so this is going to be p and q and r and we’re going to be looking at p if

and only if q and we’re going to be looking at q if and only if r

and then we’re going to be looking at the and between these two

and then we’re going to be looking at the p if and only if r

and then the last column would be the whole thing p if and only if q and

q if and only if r if and only if i’ll call the whole statement q capital q okay

so we got true false true false true false true false true true false false

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true false true false wait true true false false true true false false

and now we got four trues and four falses

so looking at the equivalence between p and q whenever the same we get a true

so true true false false and then false false and then true true

now i’m looking at the equivalence between q and r

q and r here so i’m looking at true false true false false and then true

and now i’m looking at the and between these last two columns here

looking at the and between this one and this one so the end is true false true

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false and true now looking at the equivalence between p and r

so just look at p and r here so we got true false true false true false and true

okay and then remember the q was the equivalence between this one and this one

the q was the equivalence between this one and this one

so actually i could call this one here u1 and i can call this here q2

and so q here is just the equivalence between b1 and q2

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so the equivalence is true true oh true so they all match

except in this row right here and so what’s going on let’s check this out here

so p and q are equivalent is false q and r are equivalent is false

and then the and so false and false is false and then between p and r is true

and so um the equivalence right here okay so this right here are not equivalent

so in fact we don’t have equivalence right here we have implication

so if also implies a true that’s true so that is the

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statement is q is q1 implies q2 now we can get back here

and see that was number three there so p if and only if q and q if and only if r

all that implies and then you just saw how that’s not going to be an equivalent

um implies p if and only if r okay so that work right there was the missing

part right there in terms of why this is true or an implication

but it’s not equivalent there okay so very good now that is going to be um

one more thing to talk about here is wow there are a lot of lot more useful

tautologies so these are a lot of tautologies here that are used in

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proofs and in everyday uh basic logic that we use and so many of them have names

so there’s two things to do to go through each of these ten

and prove that it’s a tautology and then also to remember the name

so number one is called the excluded middle sometimes it’s called the law of

excluded middle p or not p then there’s the simplification p and q implies p

so it’s simple it simplifies because you know you got p and q but really all you

need is p so that’s simplification then you have construction

which is for the dual of the and of the simplification

now out of all of these here probably the

most important ones well that’s hard to say you can’t really say most important

ones but we’re gonna do a video um over the one two three four fifth one there

modest bonus we’re gonna do a video over that one so if you’re if you’re

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wondering what is that make sure and we’re also going to do a

video in fact that’s going to be the very next video over the contra positive

now look at the last two the proof by cases in the indirect proof

so proof by cases p implies r and q implies an r all of that is equivalent to

p or q implies r so that’s proof by cases and then we have something called an

indirect proof so i would definitely try to make a truth table out for those

uh tautologies there and make sure that not only can you make those tautologies

make those truth tables but you also learn those names

all right so when you do that you’ll be in great shape

all right so if you have any questions or ideas uh leave a comment below in the

in the comments um don’t forget this series is uh logic and mathematical

proof in-depth tutorials for beginners the link is below in the description i

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hope you check it out thank you for watching and i’ll see you next time

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