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A truth table is a tool. You can do lots of logical work with it. "Frinstance"... you
can plug in an argument and test whether it's valid, or you can write in a single
sentence and see whether it's a contradiction or a tautology. But we
won't do any of that in this video. We'll focus like a laser beam on just one task:
How to construct a truth table. A truth table is built out of rows and
columns. The columns aren't hard. Usually the question is: How many rows will you
need? Let's start with a simple case. Let's have this P stand for, "The
polar bear hunts on ice." Now P is either true or false,
so we've already answered our question... We'll need two rows because P is either
true or false. You probably know that P is actually true— the polar bear does
hunt on ice, but our table needs to show all the possible truth values and there
are two. Let's go to another simple case: Not-P.
Here we have added a tilde. But look— we don't need more rows. P is either true or
false. When P is true not-P is false and when P is false not-P is true.
Now this isn't a terribly interesting truth table, but it does do something—it
captures the negation function. That is, it illustrates when a negated statement
is true. A negation is true when the statement it
is negating is false. In this case it's true that the polar bear hunts on ice, so
it's false to say the polar bear does not hunt on ice. But we're focusing on
table construction, so the real lesson here is that the negation operator did
not require a new row. So what we've seen is: one letter, two rows.
But look what happens when we add a second letter.
"Frinstance"... "If the polar bear hunts on ice, then seals better watch out."
Now P and W are standing in for the propositions, or statements, and let's
imagine that we don't know whether they are true or false. So we have to cover
all the possibilities: maybe they're both true, maybe one is true and one is not,
and vice versa. Finally, maybe they're both false. Now we've got all the
possibilities covered and we can fill out the circumstances where the sentence
as a whole is true and where it is false. This sentence is a conditional, and the
only time a conditional is false is when the antecedent is true and the
consequent false. So once again our truth table has done some work here. It nicely
shows the truth function for conditional. But that's not the topic of this video.
We're more interested in these patterns right here. Look how we alternated T's
and F's singly in this column and we alternated doubly in this column. I don't
know if "alternated doubly" is the correct English, but you see what I mean.
Here's the point: when constructing a table with two different letters, you'll
need four rows, alternating singly under one letter and doubly under the other. That's
how you capture all the possibilities: they might both be true, one or the other
false, or both false. So one letter, two rows and
two different letters, four rows. Can you guess how many rows we'll need
if there are three different letters? Well, let's find out. Here we have another
conditional, but it's more complicated than the previous one. Maybe it says, "If
the polar bear hunts on ice and eats seals, then seals better watch out. And
again, let's say we don't know the truth values of P or S or W. So we need the
table to cover all the possibilities: maybe each of them is true, maybe each of
them is true except for W, maybe each of them is true except for S, and so on. Take
a look. Hit pause for a moment, if it helps. See how we've covered all the
possibilities? And check it out... our pattern is back: alternating singly,
alternating doubly, and alternating... What? Quadruply? I don't know what to call it,
but you see what I mean. You can also see that it requires eight rows. I hope you
like patterns because we have another important one here. One letter, two rows;
two different letters, four rows; three different letters, eight rows. In other
words, the number of rows needed increases exponentially with each new
letter. That means four different letters, 16
rows. Check it out. Hit pause so that you can really take a
look. And I hate to break it to you, but five different letters will require 32
rows! So full truth tables get to be a pain pretty quickly
That's exactly why other techniques are important, like partial tables or natural
deduction. I won't go into those right now because we've got one last loose end
to tie up. Pop quiz: How many rows do we need for our truth table?
The answer is... just four. We only have two different letters here: P and E. When
a letter appears more than once, it must have the same distribution of truth
values underneath it. That's why some people like to make truth tables like
this: they make a little key off to the side. All the different sentence letters
go in the key. In this case, we only have two: P and E. Two different letters, so
four rows, using the pattern we've learned.
Now you can work the truth table row by row. In this row, P is true. That's what
the key tells us. E is also true in this row. In this row
P is true but E is false. Again it's the key that guides us. This is no different
from what we've been doing through the whole video, only we're setting up the
truth values off to the side first. So a truth table is a tool for doing
things in logic. We didn't really discuss how to use that tool, but now at least
you know how to get that tool set up... in proper working condition.
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