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Natural languages like English or Spanish are dynamic and fluid,
full of nuance and ambiguity. That's great for poetry...
not so great for logic. For the clarity and precision logic demands,
we must turn to a formal language. That is,
an artificial symbolic language built with one purpose in mind:
to express complex logical ideas simply. Here's an example. This might look
confusing now, but with a little practice, it's a huge improvement over the same
sentence written in English. Now maybe you've had a bad experience in
math and your heart rate goes up when you see a symbolic formula like
this, but stick with me. Almost anyone can master the basics of a
formal logical language. In fact, there are only three elements
that make up a sentence of truth functional symbolic logic.
we use a single capital letter to stand for a sentence of English.
So the English sentence, "Koalas have fingerprints,"
becomes K in our formal language. You get to decide what constant stands
for what sentence, so you don't *have* to use the letter K—
you could use F instead, or even Z—but the point is that in our
formal language, the English sentence "Koalas have
fingerprints," is replaced by a constant—a capital
letter.
The operators represent the five truth functions.
Let's go through each one. This is a tilde.
Your textbook may use this symbol instead or some other notation—
that's okay. The point is that it stands for the negation operator.
The way to read this operator is "not" or "it is not the case." So, for instance, this
constant standing alone reads, "The narwhal's horn is actually a
tooth," but now it reads "It is not the case that
the narwhal's horn is actually a tooth," or more simply, "The narwhal's horn is not
actually a tooth." This is an ampersand—it stands for the
conjunction operator. Depending on your textbook, this symbol
or even this one might stand for a conjunction operator.
It really doesn't matter. Whatever the exact symbol, it stands for
the English "and". Thus, we can take the constant K and the
constant S and put them together with the
ampersand to get "Koalas have fingerprints and sloths are
skilled swimmers." Next we have the disjunction operator.
Its symbol is usually the wedge. This operator reads as the English "or"
as in, "Either that is an octopus or that is a squid."
The horseshoe stands for the conditional operator. Your textbook
might simply use an arrow. The conditional operator forms
"if, then" sentences. Let's say E stands for "That animal is an elephant," and J stands
for, "It can jump." So ~J reads
"It cannot jump." Put them together with the horseshoe and we get
"If that animal is an elephant, then it cannot jump."
Lastly, we have the biconditional operator ,typically represented by triple
bar or double-headed arrow. It reads "if and
only if" as in, "That animal is a Grizzly Bear if
and only if that animal is Ursus Horribilis."
Lastly our formal language has parentheses, braces,
and brackets. Students sometimes worry about the differences between them, but
it doesn't really matter— these are all just options for grouping
when you need to do so. More on grouping later. So that's it. Our
formal language is made up of constants, operators, and grouping indicators. Pretty
simple, right? Now we just need to know how to put them
all together properly. Let's call a correctly constructed
sentence of our formal language a "well-formed formula" or wff for short.
There are only three rules of grammar for our language and they're pretty easy.
This is a wff. It says "Giraffes tongues are purple."
And this is a wff. It says "Octopuses have three hearts."
So the first rule of our language is simple: if you have a constant
by itself, it's a wff. This is a wff— it says "Pandas only eat
bamboo." So this is a wff, too. It says "It is not
the case that pandas only eat bamboo." Or if you prefer, "Pandas do not only eat
bamboo."
This is a wff because it's a sentence letter standing alone.
This is a wff because it's a sentence letter tagged with a negation operator.
So this is a wff. It reads "Butterflies taste with their feet
and do not taste with their mouths." Here's another: "If it's a Honey Badger,
then it's a tough creature and immune to venom." This is a wff,
this is a wff, so this grouping is a wff.
And this wff is connected to this wff with the conditional function.
So it's a wff. And just like—that we're done.
Our formal logical language is pretty simple. It's made up of constants,
operators, and grouping indicators—and we put them together according to three
simple rules of grammar. Let's close with a few examples to test
your understanding. Ready? Is this a wff?
No. The conjunction operator can join two wffs
to create a new wff, but it can't just be tagged on the front of a wff.
So this is not a wff. How about this one?
Is it a wff? No. You can tag a wff with the negation
function and the result will be a wff— for instance, this is a wff—but you
cannot join two different wffs with the negation
function to create a new wff. Here's another: wff or not a wff?
It's a wff. Because this is a wff and so tagging it with the negation operator
is a wff. Same goes here. So these are wffs.
And when you join a wff to another wff using the conditional operator...
the result is a wff. Now here's a tough one: is this a wff?
Yes! We just said this is a wff, and according to our rules you can tag a
wff with the negation function and the result will always be a wff.
The parentheses show that the negation operator is tagged onto
this wff, but what if we didn't use parentheses
and just tagged on another negation operator like this? Is it a wff?
Yes. This is a wff because it's a constant standing alone.
This is a wff because a wff tagged with a negation operator
is a wff. And since we just said this is a wff...
this is a wff, too. Of course, this is a wff over here,
and the two of them are joined with the conditional operator...
so the end result is a wff. Last one: is this a wff? No.
Our rules of grammar say that you can connect a wff to a wff
using disjunction, but the rules don't allow us to connect a wff to a wff to
a wff! This is where grouping indicators come
in handy. Check it out. Now this is a wff because we have a
wff connected to a wff. And this wff is connected to this
wff using the disjunction operator. So we're all set. It's now a wff. And
since I've now said "wff" 68 times... it's time to end this video.
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