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PROFESSOR: In this final
segment on predicate logic,

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there are two issues that
I'm going to talk about.

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The first is some problems with
translating A E quantifiers

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and E A quantifiers
into English-- or rather

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from English into logic.

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We've seen examples in class
that English is ambiguous.

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And I want to show you
two that are I think

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interesting and provocative.

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Just as a warning that
the translation is not

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always routine.

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And then the second
topic is an optional one,

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just to kind of
make some comments

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about the amazing results
in metamathematics,

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the mathematics of
mathematics, or more

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precisely, the mathematics
of mathematical logic.

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And two fundamental
theorems about properties

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of predicate calculus,
which go beyond this class

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and are optional,
I would suggest

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it's worth listening to
the A E in English example.

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And if you want to skip the
short discussion of the meta

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theorems, that's fine,
because it's never going

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to come up again in this class.

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So let's look at this phrase in
English, where the poet says,

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"all that glitters is not gold."

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Well, a literal
translation of that

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would be that, if we
let G be glitters,

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and I can't use G again,
so we'll say Au is gold,

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then this translated literally
would say for every x,

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G of x, if x is gold
implies that not gold of x.

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So is that a
sensible translation?

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Well, it's clearly
false, because gold

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glitters like gold.

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And you can't say
that gold is not gold.

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So this is not what's meant.

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It's not a good translation.

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It doesn't make sense.

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Well, what is meant,
well, when the poet says,

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"all that glitters
is not gold," he's

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really leaving out a key word
to be understood from context.

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All that glitters is
not necessarily gold.

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He was using poetic license.

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You're supposed to fill in
and understand its meaning.

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And the proper
translation would be

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that it is not true that
everything that glitters

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is gold.

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It is not the case that
for all x, if x glitters,

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then x is gold.

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So it's just an example
where a literal translation

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without thinking about
what the sentence means

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and what the poet who
articulated this sentence

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intended will get you
something that's nonsense.

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It's one of the problems
with machine translation

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from natural language into
precise formal language.

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Let's look at another
example of the same kind.

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The poet says this
time, "there is a season

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to every purpose under heaven."

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This is a variant of
a biblical phrase.

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So what does it mean?

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Well, the literal
translation would

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be, there exists an
s that's a season,

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such that for every p that's
a purpose, s is for p.

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Well, that from the way
the quantifiers work

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means that there's some season,
say summer, that's supposed

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to be good for all purposes.

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Well, that's not right,
because summer is not

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good for snow shoveling.

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And if your purpose is to
shovel snow, then summer

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will not do for you as a season.

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So even though
it's phrased, there

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is a season to every
purpose under heaven,

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it's not the case that the
intended translation is there

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is a season for every purpose.

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In fact, the poet really means
to flip the quantifiers, which

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is what's shown here.

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We're going to switch them
around so that we are really

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saying, for every purpose,
there is a season,

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such that s is for p.

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For snow shoveling,
winter is good.

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For planting, spring is good.

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For leaf watching, fall is good.

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And that is, in fact,
the intended translation

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here, although I
remind you that there's

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a famous historical man, Sir
Thomas More, who was described

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as a man for all seasons.

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That would be a case where
there was one man who

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was good for all seasons.

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He was a polymath--
a writer, a cleric,

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and the chancellor of
England for many years

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until he had a falling
out with Henry VIII, which

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served him ill.

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That's the end of those two
examples, whose point is just

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to warn you that translation
from English into math

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is not something
that can be done

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in a mindless mechanical way.

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Sometimes, the
quantifiers really

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are meant to go the
other way from the way

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that they literally appear.

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Now, we're going to shift to
another topic, which is just

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two profound theorems
from mathematical logic

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about the properties
of predicate calculus

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that are worth
knowing about and that

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describes sort of the
power and limits of logic.

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So these are called
meta-theorems,

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because they're
theorems about theorems.

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They're theorems about
systems for proving theorems.

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And that phrase, meta, means
going up above a level.

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So the first theorem
is a good news theorem.

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It says that, if you
want to be able to prove

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every valid assertion
of predicate calculus,

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there really is only
a few axioms and rules

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that will do the job.

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As a matter of fact,
the rules that you need

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are just rules that
we've seen already,

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namely modus ponens and
universal generalization

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and a few valid axioms
which we've seen already.

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So let's go back a little bit.

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So there's a remark here
that says that, in practice,

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if you're really going to try
to do automatic theorem proving,

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you need much more than
this minimal system.

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But it's intellectually
interesting and satisfying

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that a fairly economical set
of axioms and inference rules

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are in theory
sufficient to prove

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every logically valid sentence.

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And this is known as Godel's
completeness theorem.

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Godel was the great
mathematician,

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German mathematician, who spent
the latter part of his life

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at the Institute for
Advanced Study in Princeton

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as an emigre.

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And he has two major
theorems, at least,

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that are results of his.

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One is the completeness
theorem-- this one.

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Then there's an incompleteness
theorem, which maybe we'll

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talk about in a few lectures.

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But for now, the good news is
you can prove everything that's

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valid using a few simple rules.

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Now, the bad news
is that there's

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no way to tell
whether you're attempt

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to find a proof for
something that you think

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is valid is going to succeed.

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There's no way to
test whether or not

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a quantified formula is valid.

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This is in contrast to the
case of propositional formulas,

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where you can do it
with a truth table.

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A truth table may
blow up, so it becomes

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pragmatically infeasible.

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But at least,
theoretically, there's

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an exhaustive search
that will enable

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you to figure out whether a
propositional formula is valid.

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That's not the case
with predicate calculus.

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Predicate calculus
is undecidable,

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meaning that it's logically
impossible to write a computer

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program that will take an
arbitrary predicate calculus

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formula in and print out
true or false depending on

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whether or not it's valid.

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Can't be done.

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Now, as I said, we're
not going to go further

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into these theorems.

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These are the kind
of basic results

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that would be proved in an
introductory course in logic.

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Usually, they take about
a half a term to do,

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maybe a little less.

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And it goes beyond our course.

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You can look over in
the math department

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for introductory
courses in logic.

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And you will learn about these
two profound meta-theorems

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about logic and math.