Understanding Tautologies, Contradictions, and Contingencies

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

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

00:00
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

00:01
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

00:02
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

00:03
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

00:04
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

00:05
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

00:06
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

00:07
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

00:08
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

00:09
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

00:10
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

00:11
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

00:12
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

00:13
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

00:14
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

00:15
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

00:16
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

00:17
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

00:18
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

00:19
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

00:20
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

00:21
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

00:22
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

00:23
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

00:24
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

00:25
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

00:26
hope you check it out thank you for watching and i’ll see you next time
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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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