This lecture introduces modern symbolic logic as the abstract study of arguments, focusing specifically on deductive arguments and the concepts of validity and soundness. The core idea is to understand the formal structure that ensures truth is preserved from premises to conclusion.
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hi everyone
welcome to our first lecture video on
modern symbolic logic
fundamentally logic is a study of arguments
arguments
and we use logic all the time when we
argue and i don't really necessarily
mean like two people like yelling at
each other i just mean when we're trying
to convince someone of a point
now logic in the technical sense is an
abstract study of arguments so we're not
actually interested
in techniques of persuasion like
rhetoric we're actually
interested in the sort of logical
technical apparatus that makes arguments
work now in this sense we're going to focus
focus
on very sort of like clean arguments and
we're really only concerned that arguments
arguments
that are made up of statements so
statements are a very special type of
sentence they're a particular kind of sentence
sentence
and these are sentences that have a
truth value so they either true
or false so not all sentences have truth value
value
so it's really easy to come up with a
statement it's anything that's true or false
false
like it's raining outside or i like cheesecake
cheesecake
these are examples of statements they're
either true or false
now can you think of something that's
not a statement a sentence that is a non-statement
non-statement
so this would be something that actually
has no truth value to it at all
well you might sort of be struggling
right now because i've sort of primed
you to think about statements
but in fact there's lots any sort of
question is something that doesn't have
a truth value so how are you
that that's not true or false any sort
of command like clean your room or go away
away
these things also aren't statements
they're not true or false and any sort
of random expression that we use for
rhetorical sort of purposes like woo
or yay or whatever these things also
don't carry truth value
now we can sort of see that why we only
care about
statements in arguments in a technical
logic course because these non-statement
things don't really add any
genuine content to an argument they
might add rhetorical persuasiveness but that's
that's
not what we're really interested in in
this course
so what is an argument uh broadly
speaking an argument is a collection of statements
statements
uh where we have a set of statements
called the premises that are meant to
support the conclusion
and importantly a sk argument has a goal
the goal is for you or anyone i guess
to be convinced that the conclusion
follows from the premises
now what is follows from the premises
mean well actually
it depends there's lots of different
ways to understand that
so we're going to become very precise on
what it means to follow from the premises
premises
now there's actually a wide different
range of types of logic we can look at
with respect to arguments so this course
is going to focus on deductive arguments and
and
deductive arguments are like the gold
standard and the reason why is they have
this property of
certainty at the conclusion of a good
deductive argument
is certain and what that means is it
necessarily follows from the premises
so we're really going to sort of come to
terms with what that concept means
now another type of reasoning that's
really important is inductive reasoning
and there's also abductive reasoning and
not too many people know about abductive reasoning
reasoning
but this is sort of the reasoning style
that you see all the time in say science
and actually in our regular everyday lives
lives
so if you want to learn more about
inductive and abductive reasoning well you're
you're
probably going to have to take another
course we really want to be able to assess
assess
what a good deductive argument is the
thing is
it's too sort of vague to talk about
what is good there's lots of ways to
describe good that doesn't translate
from one thing to another
a good piece of poetry is not the same as
as
the same standards as a good piece of art
art
so for logic we have two senses
of goodness when we assess deductive arguments
arguments
and those senses are validity and
soundness whether an argument is valid
and whether an argument is sound
so we're going to focus on validity
here's the definition of validity now i don't
don't
want you to think that you have to sort
of memorize this definition immediately
eventually you will know this definition
very well but the way that we're sort of
going to understand and learn this
concept is by looking a lot of examples
and this is sort of going to be the way
that we're going to
proceed in the course regularly anyway
here's the definition
a deductive argument is valid if and
only if
whenever the premises are true the
conclusion must be true
or equivalently another way of phrasing validity
validity
is that it is impossible for the
premises to be true
and for the conclusion to be false at
the same time so those two things can
never happen
now if you think about it a little bit
you'll realize that these two conditions
are essentially the same
they're perfectly equivalent okay let's
look at an example of an argument
if it's raining then the sidewalk is wet
it's raining therefore the sidewalk is wet
wet
now this is a standard very simple
argument and it has two premises which
i've sort of tagged off there
and a conclusion the conclusion here is
marked off by the word therefore so that
we know that that's meant to follow
now the question we want to ask when we
have this type of argument and it's
in it's in a deductive form is whether
or not it is valid
or invalid and remember what we're
asking is if the premises are true
does the conclusion have to be true or equivalently
equivalently
is it possible for the premises to be
true and the conclusion to be false
so take a second think about it now a
lot of you will just be starting to
think of counter examples
is it possible that it's raining and the
sidewalk isn't wet
and of course it is possible lots of you
have seen examples of this because
you've seen stores like this
which has a little awning out front so
that if it's raining the sidewalk isn't wet
wet
so naturally it seems that because of this
this
we can invoke this awning that uh this
argument is invalid
it is actually possible for it to be
raining in the sidewalk to be dry we've
actually seen that all the time in our lives
lives
unfortunately that is not good reasoning
and it's actually
not invalid now the question is
why why is this on encounter example
not good why does it actually not prove
that this argument is invalid now the
reason why
comes from the definition of validity
validity says
if the premises are true then the conclusion
conclusion
must be true so we need to make sure is
that the premises are true
but we have a problem when we invoke the
awning the first
sentence in this argument the first
statement which is
premise one if it is raining then the
sidewalk is wet
essentially explicitly says hey you
cannot have things like awnings in this world
world
so this is sort of a weird example
because essentially what it's saying
is that for the premises to be true the
sidewalk always must get wet if it's raining
raining
so we can't say that there's something
stopping the sidewalk from getting wet
because that would actually make the
first premise false
so when we assess validity we need to
ensure the premises are true
so if i make the premises true if it's
raining then the sidewalk is wet
and it's raining then it must be the
case that the sidewalk is wet
there's no counter example that we can
think of so in fact this is a good argument
argument
in the sense that it is valid
here's another example if it's raining
then the sidewalk is wet
so the same first premise as before but
in the second premise we're going to say
it's not raining therefore the sidewalk
is not wet
is this valid or invalid assume that the
premises are true
does the conclusion follow or
equivalently is it possible for the
premises to be true
and the conclusion be false at the same
time if it is then we have a problem and
we can't say it's valid
well if we pause to think about it we
realize there's another good
counter example to this example as well
we've all seen reasons
why a sidewalk is wet even though it's
not raining
like this person who washes their
sidewalk that's a weird thing to do but
still lots of people like to do it
so if you wash your sidewalk then it
could be the case
that it's not raining and the sidewalk is
is
actually wet which means the premises
are true and the conclusion is false
so from this example we can conclude
that this argument
is invalid so you might be thinking
that i've pulled something fishy here
why is it that the washing the sidewalk
counter example
worked is okay and showed that the
argument is invalid
but the on encounter example is somehow
not okay
that one i said oh that doesn't show
that the first argument is invalid
well the answer is all about keeping the premises
premises
true when i invoked the washing of the sidewalk
sidewalk
that didn't render any of the premises
of the argument false
it's still the case that if it rains the
sidewalk gets wet and it's not raining
but the problem with the on encounter
example isn't
the awning itself it's the fact that by
invoking it
i made the premises false or at least
one of the premises false and you can't
do that
so the test for validity is always the same
same
make the premises true force them to be
true and then
see if the conclusion can be false if it
can be false
it means that the conclusion does not
follow from the truth of the premises
and validity requires that so a valid
argument essentially preserves truth
if the premises are truth it confers
that truth down towards the conclusion
and what we sort of see is that validity
is just about the form or the structure of
of
the argument it doesn't really matter so
much what the content is
it's the actual structure that confers
the truth forward to the conclusion
so here's a silly example this beetle
wind is nuzzling green or time
it is not nuzzling time therefore it is
nuzzling green
what does this mean who cares i actually
have no idea what it means
now maybe i was studying some other sort
of language or some other discipline
it might actually have meaning for all i
know but it doesn't matter
even if i don't know what it means i can
still study it in terms of logical
uh sort of technical nature and validity
and answer the question is this valid or
invalid so take a second
think about it and the answer is it's valid
valid
uh why well we'll actually come back to
this one a little later
so not only is validity only about form
or structure
that means it's not about actual truth
it's about possible truth so take a look
at this
argument if unicorns roam u of t campus
then i will give you all 100 unicorns do
roam the campus so you all get perfect
now is this valid yeah in fact this has
the exact same form as the very first
argument we looked at
if it rains then the sidewalk is wet
it's raining therefore the sidewalk is wet
wet
but what i'm trying to show you here is
that we can
say an argument is valid regardless of
whether or not the premises are true
you don't actually care if the premises
are actually true
you just make them true in some sort of
possible world or in your mind or
something like that essentially it's a
game of make-believe almost
here's the definition of invalidity i
haven't actually
explained what it means to be invalid
i've just invoked the word
but you can sort of imagine what it
means based off the fact that it's the
opposite of validity
so a defective argument is invalid if
it's not valid that's pretty straightforward
straightforward
but what that merely means is that it is
possible for premises to be true
and the conclusion be false at the same
time so a little tip is that it's often
easier to test for invalidity
and that's actually what we did by
trying to find those counter examples
but this is really sort of like a weird
way of phrasing it
testing for invalidity is the exact same
thing as testing for validity
so long as you know the definition of
validity you will always know how to test
test
arguments one thing you should have realized
realized
is because that the only thing that
matters is the form
and the structure and not the actual
content is that a really good way of
looking at
some arguments and trying to figure out
if it's valid or invalid
is to abstract the weird content away
so when i look at the beetlewind
argument how did i know it was valid
well i just sort of replaced it with general
general
placeholders so here's an example of a placeholder
placeholder
what this argument really says is circle or
or
square is true and then not
square so square is not true and the
conclusion says
therefore circle is true and what i'm
asking is
if the premises are true so if it is
genuinely the case that
circle or square must be right and not square
square
does it follow that circle is true and
so you can replace this with any sort of other
other
example so if i say i want fries or salad
salad
but i don't want salad today does that
mean i want fries
well yeah so it follows immediately when
you sort of abstract away
this type of abstraction is really
powerful and is essentially what we're
going to do
in this course after this first unit
we're not really going to look at arguments
arguments
like beetle winds or sidewalks reigning
we're going to look at arguments purely
in their abstract logical syntactical form
form
and there's lots of advantages to this
because then we don't get bogged down
in issues of real-world meaning we can
really just focus and zero in
on the abstract logical structure of statements
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