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♪ Paul: Today, Mark and I are back
at Golden Gardens Park
in Seattle, Washington, upper level in the woods--
in the woods above Puget Sound.
And a topic for this episode
is how to make a truth table for a single formula
and then read information off of that table.
So...we're gonna take a formula
that has just one letter in it or one component.
And let's say... let's take the formula
A and tilde A, quantity negated.
So this would read, it's not the case that...
and then, the quantity A and not A.
And we already know how
to calculate the truth-value of that
if we assign truth-values directly to A.
But now, we're gonna do this on a truth table,
so I'm gonna draw the table underneath the formula,
like this, and then have Mark calculate it
after I give a couple remarks.
So we put the formula here on the right...
upper-right quadrant.
And over here, we write all the letters that are in the formula.
In this case, the formula has one letter,
but it's repeated twice.
And then, underneath, we write all the possible truth-values
that that letter could be assigned.
And of course, a single letter can either be true or false,
so there's only two truth-value assignments.
It's either true or false.
And then, a couple of definitions.
We say that the scope of a connective
is the connective plus whatever it applies to.
So the scope of this tilde here would be itself plus the A.
So the scope of this tilde would stretch from here to here.
And then, the scope of the ampersand
would be itself plus whatever the ampersand applies to,
so its scope starts here and goes to here.
So that's the scope of the ampersand.
And then, the tilde here applies to the parenthesis,
which pairs with this parenthesis.
And so, the scope of this tilde, of course,
is everything to the right.
It's itself plus everything to the right.
And that second definition will say
that the main connective of a formula
is the connective of greatest scope,
and so, therefore, that's the smallest scope connective.
That's the next biggest scope connective.
And then, the connective of greatest scope
is this tilde applied to the whole.
Therefore, that's the main connective.
Mark: Size matters.
Paul: Size matters when it comes
to operators or connectives, yes.
Although, I don't know if the bigger the better.
Mark: That's true.
Paul: Yeah...so now, what we do
when we calculate the value of...when we fill in the table
and calculate the value of a formula
for every possible truth-value
that its components could be assigned,
we start with the smallest scope connective and calculate that.
If there's a tie, the order doesn't matter.
If there's two connectives of equal scope,
the order doesn't matter.
We start with the smallest.
We go to the next biggest connective,
and we end up at the end doing the main connective.
And so, this is a, uh...inside-out procedure.
You go inside the formula and work your way out
to the greatest scope connective.
Uh...and then, another definition.
The values that are underneath the main connective
form what we call the final column.
So the final column of a table
will be the values under the main.
So now, Mark is gonna fill the table in.
Mark: Oh, okay, I can do this.
I'll stand here, 'cause I'm right-handed.
Paul: Okay.
Mark: Working...you can work horizontally or vertically.
I'll just work horizontally here.
On this first row, A's true,
so I'll just be consistent and make all the As true.
And as Paul said, I wanna start off
with the connectives with the smallest scope,
so that would be this tilde.
If the A's true... that tilde's gonna be false.
Next, we have the ampersand in size in terms of scope.
Uh...the left conjunct is true.
The right conjunct is false, so we know, then,
that the ampersand's gonna be false.
Then, finally, the main connective.
This tilde is negating this entire conjunction,
so if the conjunction's main operator...
main connective is false, then this tilde must be true.
So it looks like, given A's true,
this thing as a whole would be true.
And let's just complete the table now.
Enter Row 2, the As are false.
Again, if that A is false,
the tilde next to it's gonna be true.
♪ And if we have an ampersand
with a false left side and a true right side,
we know the ampersand's false.
And once again, this tilde is negating the conjunction,
and this is the main operator of the conjunction.
So if that's false, the tilde's true.
The final column has nothing but truths--all truths--
which would make it a tautology.
If it was all Fs, it would be a contradiction.
If it was a mixture of Ts and Fs,
it would be a contingency.
Those are the three choices for single statements.
Since we have all Ts in the final column,
we have a tautology.