How To Use Modus Ponens and the Substitution Rule

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 i show you how to use modest bonus and the substitution rule
did you know that people have been using modest opponents for thousands of years
it appears that a successor of aristotle was the first to use modest ponies but
what exactly is modest opponents and why does logic need a substitution rule
hi everyone welcome back i’m dave in this video how to use modest bonus and
the substitution rule is part of the series logic and mathematical proof
in-depth tutorials for beginners so let’s get started and see what we’re
going to do today so we’re going to talk about what is modest bonus
and then we’re going to talk about the substitution rule
and then we’re going to give some examples on using the substitution rule
so let’s go ahead and get started so what is modus ponens so

00:01
modest ponus is latin for the way that affirms by affirming
is is a simple is a valid simple argument and rule of inference
and so here’s the statement of modest bonus so we’re going to begin with two
statements p and q there any any mathematical statements
um so i talked about what mathematical statements are in the uh
in a previous uh video in this series uh make sure and check that out
um and p and q are tautologies uh we also talked about what tautologies are
in previous uh videos so the link to the playlist is below in the description
and so um so the conclusion is um q is is tautology as well
so if you know that p and p implies q are tautologies then q is a tautology as

00:02
well so let’s try to understand um you know why modest bonus is true
and to do that we’re going to look at two ways of looking at this theorem here
the first way is to not look at it as an inference rule but just as a
mathematical statement and see that it’s
a tautology so let’s do that first let’s go up here and write this as a
statement so we’re going to say p and p implies the q and all that’s going to
so um q is the conclusion here so implies q and i’m just using lowercase here
um but the statement over here says uses uppercase because these can be any
mathematical statements at all so i’m just going to use propositional
variables here and i’m going to say p and q so true false true false true true

00:03
and then false false and so then we have a p implies a q which is true false
true true and then we have a an and between these two right here
so p and p implies a q so we have true we’re doing an and between these two
columns so true false false false and then the last column here would be
all of this implies q so we could put parentheses here so p and p implies a q
all that implies q and so let’s put some columns here [Music]
so now we’re looking at all of this implies q
so all of this right here implies q so looking at true implies a true
which is true and fall and the rest of these are false

00:04
so they’ll they will imply whatever values are over here this is true now
so you can see this is a tautology and the name of this tautology is often
called modest bonus it corresponds to this rule of inference right here
um so you know i wanted to talk about this as a theorem though um so
you know this says that if p and p implies q are tautologies that’s not
what we have going on right here so this is just a logical statement over here
propositional statement over here and this is a tautology and
now let’s talk about this as a theorem though so p and q are statements
so i’ll put a column here um statements in queue is a statement here

00:05
and we may have some columns before these statements here
but these are tautology so this is true all the way down
all the way down to true and these are all trues
and then we may have some more columns in our true table
and then we have so is the statement q um sorry um
p implies eq so there should be a p implies a q here p implies q is a tautology
so we have this statement here whatever p is p is made up of some propositional
variables so i could have some other columns over here and this p and q are
mathematical statements and so i have this column right here because this is a
tautology my hypothesis here and then we may have some other columns

00:06
and then we have the statement q here and the conclusion is
q is a tautology also so this should be all trues down here
now let’s just play devil’s advocate here and say what if q here is not uh
a tautology what would be so wrong with that
so what what would happen if there was some false in here
so if there’s some false in here if q is not a tautology that means there’s at
least one false so what would happen in that row what would happen in that row
well these are trues in that row because this is a tautology and this is
the tautology and so then you have to ask yourself can this happen
can the q be false right here and the implication be true
and the answer is yes that could be false that’s the assumption that we’re under
is that there’s a false here so that could be false but the only way

00:07
the implication could be true if this is a false
is p would have to also be false but p is true so in fact this cannot be false
so this cannot happen right here again i’ll run through the argument
f is q is false under a hypothesis q is false so q is false here how can an
implication be true knowing the conclusion is false and that
would mean p has to be false false implies a false would be true
but p cannot be false so if q is false p has to be false but we’re under the
condition that p is a tautology if p is a tautology and the implication
is the statology then q has to be a tautology as well
so that’s the idea behind this theorem here
in terms of how it works and how we’re going to use it it’s a rule of inference

00:08
which we’re not really ready to talk about yet in this video
so in this series logic and mathematical proof we will be using modest opponents
a great deal when we start writing proofs
from from the perspective of this video though uh this is an introduction to
modest bonus and you can see that as a logical statement using logical
connectives and propositional variables we have a tautology and then i think you
can understand the basics of this theorem by looking at a case where it
fails and seeing that actually it cannot fail so in fact q must be a tautology
also okay so now let’s look at the substitution rule
okay actually before we look at the substitution rule i wanted to talk about
monas bonus a little bit more so um you know we’re going to talk about this
a great deal more in upcoming videos but let’s just talk a little bit
about it more because this may be a little perplexing to you because there’s

00:09
a difference between a valid argument and a sound argument and so let’s just
look briefly at what these things are but we’re going to talk a great deal
more about what arguments are so the form of a modest bonus argument has
two premises and a conclusion so it doesn’t matter which way you write
the premises you can say if p then q and p those are the two
premises there that’s the hypothesis or you could say p and then if p then q so
you could reorder these but then you have so you have two premises
and then you have a q therefore q so that’s the conclusion you’re concluding
that the q holds so this this argument right here is valid
but it has no bearing on whether any of the statements in the argument are
actually true that’s a different uh that’s a different
that’s a different idea because whether or not p is true or whether or

00:10
not this implication is true has no bearing on the structure
of modest bonus so for modest bonus to be a sound argument the premises must be
true for any in true instances of the conclusion
so let me give you in uh two examples here just just very briefly here
so here if we we can say here if x plus three is five then x equals two
and x plus five uh x plus three equals five that’s a statement therefore
x equals 2. so here’s an argument using modest bonus
and what i want to say is that this is a valid argument why is it valid because
we have the right structure of modest opponents we have an implication
we have the p right here so this is one statement here

00:11
and this is another statement here so we’re saying p and then now we’re
concluding the q so this is a valid argument here
now is the question is this sound right so this is a true implication so that’s
true and this is true right here and so this is a sound argument as well
so this is valid and a sound argument and so we can look at something like
if x plus three equals five then x equals three
and then we can say x plus three equals five and then we can say therefore
x equals three okay so now this argument here
two different arguments this argument here is valid
because the argument has the right structure to it

00:12
it’s an implication this is p and this is q and then the next statement is the p
and then therefore the q so this is the valid argument but this
is not a sound argument so this is unsound
and the reason why is because this first implication right here is not a true
implication so um it if they’re both true here so to be a sound argument
um the premises must be true for any true instances of the conclusion right
so that’s why this one is sound and this argument here is unsound so there’s a
difference between a valid argument both of them are valid
and and being sound and unsound so those two concepts are
uh often confused when you’re first starting out studying logic
so and because of this we’re going to study arguments a great deal
in some upcoming videos here okay so now let’s go back to the

00:13
propositional logic and we’re going to look at the substitution rule now
so what is the substitution rule let’s read that okay so the substitution rule
so we’re going to start off with p’s of tautology
and we’re going to suppose that p contains
statement variables p1 p2 through pn so it could have one or two or three all
the way there’s a finite number of them and there could be others as well
so there could be more but we just know that there’s these these right here
and then we’re going to suppose further that these cues are also statements
these could be any statements there’s no restriction upon the qs at all
now we’re going to use substitution so we’re going to replace
this p1 in this p here and we’re gonna substitute in capital q
one in here and q2 and q3 and all the way and then the statement that you get

00:14
the new statement that you get is also a tautology
that is the statement of substitution it’s a real
uh powerful i love i love this theorem here so let’s illustrate this by looking
at an example real quick so my statement here i’m going to look
at your capital p it needs to be a tautology so i’m going
to come up with the tautology here so we’re going to say p1 and p2
and that’s going to be by equivalent to p2 and p1 it’s going to be equivalent to
and that’s actually a tautology so far but i actually want to say perhaps
others as well so we’ll throw a little bit more onto
here and we’ll say if and only if r implies r r implies an r
okay so this is the tautology now it may not be clear to you that this

00:15
is a tautology so let me just uh briefly uh make out a truth table here and that
actually be important for what we’re going to see down here when we start
trying to understand [Music] why this after the substitution it’s
also a tautology so let’s look at a truth table right here
so we got p1 p2 and we got r and so let’s go here with true false true false
true false so one two three four five six seven eight and we got true true
false false true true false false and then true true true true false
also there’s all the possibilities here now let’s do this and here first
so we’re going to look at p1 and p2 so we’re looking at an and between the

00:16
first two columns and that’ll be true true false false false false false false
and then when we look at p two and p one to get this part right here
now we’re doing the uh and but in reverse so true and true true false false
and then the rest are false again [Music] so we can see these
are equivalent to each other so when we put equivalent right here
between these two right here they all match so this is going to be this is
going to be a tautology right here um and this right here is also going to
be a tautology so for all of this right here i’m going to use a capital p1
for all of this right up in here and we see that they’re all equivalent
so now i’m doing equivalent between these two columns so that you can see
their equivalent right so this is a tautology right here and then r implies r

00:17
r implies r that’s also a tautology true implies true
false applies to false true implies a true and so on so r r implies r
this is also a tautology and so then the last column would be the
implication here between p1 and r implies an r and that column is just p
so we can see that p is a tautology here and that’s we need in this problem p is
a tautology okay so let me draw some columns here alright so p is a tautology
and now we have p1 and p2 in our problem
right here and we have perhaps others as well so we have some other stuff over
here but we’re not going to use that in our substitution we’re going to

00:18
substitute q1 for p1 we’re going to substitute q2 for p2
now the q1 that i’m going to use is going to be so let’s say q1 is
let’s say q1 is i’m going to look at p or q and q 2 is p and s
where this p and this q and this s are other variables we haven’t used the
p yet that’s a lowercase p and a lowercase q and a lowercase s
and i’m going to substitute so so q one and q so all of these right here are
just statements they could be any statements at all you can introduce new
variables whatever q1 are just statements so here’s q1 and here’s q2
now when i substitute in q1 everywhere there’s a p1 and q2
everywhere there’s a q2 or sorry q2 wherever there’s a p2

00:19
then the new statement the resulting statement is also
tautology so let’s do that statement there so here’s p1 which is going to be
p or q so we got p or q and then and p2 which is p and s
so all of that right there is equivalent to p2 p and s and
p 1 which i’m substituting in p or q so that equivalence right there
and then if and only if and then it says perhaps others as well
but we’re not doing those substitutions so that’s just going to stay the same
there so r implies an r so here’s the new resulting statement
and this is the tautology by the substitution rule this is all

00:20
tautology i don’t need to go make a true table for this
and if if i did it would be longer because now we have four variables here
it would be 16 rows we would have a p q an s and an r
and you can see the table has grown doubled in size in fact from 8 to 16
because we introduced some new variables here but we don’t need to go do that
true table the substitution rule says this is a tautology
now what would happen if we were to go make this table anyways
what would happen if we were to make this table so we would have a p
uh for for little p we’d have a q we would have an s and we would have an r
and there would be 16 um 16 rows here to get all the possibilities of these
variables here but then we would start piecing it
together and we would look at p and q column which is the q1 column

00:21
and we would look at q2 right here that’s a q2 we would have a q2 column
and then we would have a q1 and a q2 column and then we would have a q2 and a q1
column right so we would keep making all these columns and then to the very last
thing we would get to this whole statement
here which i’m going to call because i haven’t used a capital q anywhere yet so
i’m going to call the whole thing here capital q
so the very last column would be capital
q now why is it that we don’t need to go make this uh truth table anymore
um the theorem says it’s already a tautology but why
why would this q here the whole thing here be a tautology well
it’s got four variables right so we got 16 rows we got all the possible
combinations right true false true false
true false and so on so out of all those possibilities there

00:22
we’re then still going to start building this column here right so q1
could have different uh you know it could be true false true
true they could all be trues or could be falses but it’s got some combinations
here we got some combinations here well what is q1 and q2 here q1 and q2
is taking the role of the p1 and p2 because q1 is getting substituted in for p1
but we already did all possible combinations for p1 and p2 there were
eight possible combinations so even though you’re doing 16
possibilities here there’s only going to
be so many possibilities for two columns here
so and we’ve already done all those possible combinations because remember q1
and q1 and p1 are the same so that those two all possible

00:23
combinations for those two have already been considered no matter what we’re
going to get uh combinations here we’ve already jumped to the end
where this whole statement here is um and we we’ve already gotten true so
this this will be all trues down here and the basic reasoning is that even
though we got more rows all the possible combinations right here have already
been verified in terms of the p1 and p2 and so since we’ve already verified all
these right here we’re going to get the same truth values
here you can go and check that you can see
this in action here but that’s the basic idea about why the substitution rule
is true and so let’s look at now some examples
and see how to how to use it to our advantage
so on this uh statement here we’re going to say
show this statement is a tautology and the goal is to not have to use of

00:24
truth table for this so and the reason why is because
you know it’s long so why do we want to make a truth table for it
and in general making a true table will take a lot of rows so if you have five
variables it’ll be 32 rows and so on so it’s exponential so is there a way to
show tautology without having to make a truth table or at least make a truth
table for a smaller simpler statement so when i’m looking at this right here
i’m going to say p capital p is going to be
the statement here and here it repeats doesn’t it so p and not q
so i’m going to say that’s p and not q and for capital q here i’m going to say
is this one right here so not p or not q now i notice that the

00:25
statement here has the form p implies q implies p
so that’s the form of the statement up here
so i’m going to substitute in capital p and capital q
this expression right here and right here and i’ll get this one right here
we’re trying to show this one as the tautology so i need to show this one as
the tautology first if i can show this one’s a tautology then i can substitute
these in and get this a statement and the substitution rule will say this is a
tautology as well so let’s go and check this so we have p and q
and then we have q implies a p and then we have p implies a q implies a p
so we have true false true false true true false false
and then we have q implies p so we have true false true
and then now we have p implies q implies p we have true implies true

00:26
true applies true false also implies a true so they’re all trues
here so this is a tautology so by the substitution rule if i
substitute in this p uh if i substitute in uh this right here for the for the p
and this right here for it right here and right here so by substitution rule
um the statement is a tautology the statement is a tautology
so we were able to get by showing a smaller true table
then perhaps this right here would have a little bit more work to do so
you know what would happen if we were to actually make this true table out to
just verify it directly it wouldn’t be much longer because there’s still just

00:27
you know p’s and q’s but in the end you’ll get a column where
you’ll have to consider the p’s and q’s and you know you may not even get all
four possibilities but you’ll have to consider those p’s and q’s so for
example if i start off with a p and a q and a not p and a not q
alright so true false true false true true false false
and then we have false false true true and then false true false true
right and so what will be to make this uh truth table here we would go make
this column next because we already have p and not q so we would do p and not q
but p not q is already the capital p so let’s just call that a capital p column
and then we would also make this this one right here

00:28
we would need somewhere we need to build this into our column
and that’s the capital q so when i do p here it’s going to be p
and not q so what i’m looking at here this is p and not q
so i’m looking at this column here and this column here with an and
so this so i’m looking at true and false so it’s false true and true
false and false false and false and then for this q right here we have
not p or not q so now we’re looking at these two columns here with an or
so we’re looking at false and false is false false and true is true
and then this is true and then this is true and so now look at the
possibilities for the p and q it’s false false but we’ve already
covered that case true true we’ve already covered that case
false true false true we’ve already covered that case and this is false true
but we’ve already covered that case so by making this true table we’ve already

00:29
covered all of the cases possible because we’re making this true table we
did all possible combinations and when we’re making this true table we
may not even need all those possible combinations so
for example in this this column down here we never even well we saw uh
we never saw true true implies a false down here so
we can continue on making this truth table we’ll we’ll uh we’ll we’ll need the q
implies a p next but we’ve already done that and we’ve
already done that column for all possible combinations
and then we’ll need the p implies a q implies a p column
so you see why this true table down here is really unnecessary because
the combinations the this is all possible combinations right here
and that may not even give rise to all possible combinations right here but in
case it does we’ve already done all possible combinations in the smaller

00:30
truth table here okay so by the substitution rule it’s a
tautology so we don’t even need this just to be clear we don’t even need this
down here i just wanted you to kind of see it to get a idea of how the uh
substitution rule is uh very important and it can really save you a lot of time
and just to make sure let’s do one more example here
so let’s look at this one right here now so this one is can you see the
the form of the statement here so these match right here right so
but we have a negation right here so i’m going to say capital p is
oops so i’m going to say capital p is p implies and not q
and then now what does this look like here this looks like p or not p

00:31
and is this a tautology if this is a tautology then i can just substitute in
this into it and then by the substitution rule that will also be a
tautology so let’s see if this is a tautology so we have p which could be
true or false and we have not p which is false true
and then we have the or between them and this is true and this is true and so
this is a tautology is a tautology so by the substitution rule
by the substitution rule this right here is a tautology p implies a not q
or the negation of p implies not q is also a tautology is also a tautology
all right so there’s a very nice uh usage for the substitution rule is to

00:32
generate more tautologies you know a very simple basic tautology you can plug
in new statements into that tautology and you still get the uh you still get a
tautology so it’s structured that’s important
not the actual specific variables it’s the structure between the statements and
the connectives that gives you the tautology
all right so i want to say thank you for watching if you have any questions or
ideas please let me know in the comment section below don’t forget that this
video is part of the series logic and mathematical proof
in-depth tutorials for beginners i want to say thank you for watching and
i’ll see you next time if you like this video please press this
button and subscribe to my channel now i want to turn it over to you math can be
difficult because it requires time and energy to become skills i want you to
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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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