564 Lecture 7 Sept. 14 1999

1. ~Âx(Student(x)-->Happy(x)) given

~Âx(~Student(x)vHappy(x)) Conditional law

~Âx(~Student(x)v~~Happy(x)) Double negation

~Âx~(Student(x)&~Happy(x)) DeMorgan's law (b)

¤x(Student(x)&~Happy(x)) Quantifier negation (iv)

2. a. grandfather of:

~Symmetry:

#If John is the grandfather of Bill, then Bill is the grandfather of John.

~Reflexive:

#John is the grandfather of John.

~Transitive

#If John is the grandfather of Bill, and Bill is the grandfather of Joe, then John is the grandfather of Joe.

b. be the same color as:

Symmetry

If the book is the same color as the cloth, the cloth is the same color as the book.

Reflexive

The book is the same color as the book.

Transitive

If the book is the same color as the cloth, and the cloth is the same color as the ball, then the book is the same color as the ball.

c. sit next to:

Symmetry:

If Sue sits next to Mary, then Mary sits next to Sue.

~Reflexive:

#Sue sits next to Sue.

~Transitive:

#If Sue sits next to Mary, and Mary sits next to Felicia, Sue sits next to Felicia.

d. west of

~Symmetry

#If Tucson is west of the Rincons, then the Rincons are west of Tucson.

~Reflexive

#Tucson is west of Tucson

Transitive

If theTucsons are west of Tucson, and Tucson is west of the Rincons, then the Tucsons are west of the Rincons.

e. differ from

Symmetry

If Josephine differs from Alethia, then Alethia differs from Josephine.

~Reflexive

#Alethia differs from Alethia

~Transitive:

#If Josephine differs from Alethia, and Alethia differs from Arline, then Arline differs from Josephine.

f. earlier than

~Symmetry

#If John is earlier than Mary, then Mary is earlier than John.

~Reflexive

#Mary is earlier than Mary

Transitive

If John is earlier than Mary, and Mary is earlier than Kate, then John is earlier than Kate.

g. be married to

Symmetry

If Heidi is married to Art, then Art is married to Heidi.

~Reflexive

Art is married to Art.

?Transitive

??If Art is married to Heidi and Heidi is married to Bill, then Art is married to Bill.

h. guest/host

Converse

If John is the guest of Alethia, then Alethia is the host of John.

i. teacher-student

Converse

If John is the student of Alethia, then Alethia is the teacher of John.

4. a. ancestor of:

~Symmetry

#If Sue is the ancestor of Bill, then Bill is the ancestor of Sue.

~Reflexive

#Sue is the ancestor of Sue.

Transitive

If Sue is the ancestor of Bill, and Bill is the ancestor of Mabel, then Sue is the ancestor of Mabel.

mother of:

~Symmetry

#If Sue is the mother of Bill, then Bill is the mother of Sue.

~Reflexive

#Sue is the mother of Sue.

~Transitive

#If Sue is the mother of Mabel, and Mabel is the mother of Amy, then Sue is the mother of Amy.

b. be older than

~Symmetry

#If Sue is older than Bill, then Bill is older than Sue.

~Reflexive

#Sue is older than Sue.

Transitive

If Sue is older than Bill, and Bill is older than Felicia, then Sue is older than Felicia.

be one year older than.

~Symmetry

#If Sue is one year older than Bill, then Bill is one year older than Sue.

~Reflexive

#Sue is older than Sue.

~Transitive

#If Sue is one year older than Bill, and Bill is one year older than Felicia, then Sue is one year older than Felicia.

c. sister of

~Symmetry

#If Sue is the sister of Bill, then Bill is the sister of Sue.

~Reflexive

#Sue is the sister of Sue.

Transitive

If Sue is the sister of Mary, and Mary is the sister of June, then Sue is the sister of June. ("full" sister...)

sibling of

Symmetric

If Sue is the sibling of Bill, then Bill is the sibling of Sue.

~Reflexive

#Sue is the sibling of Sue.

Transitive

If Sue is the sibling of Mary, and Mary is the sibling of June, then Sue is the sibling of June.

d. be taller than

~Symmetric

#If Sue is taller than Alethia, Alethia is taller than Sue.

~Reflexive

#Sue is taller than Sue.

Transitive

If Sue is taller than Alethia, and Alethia is taller than Amy, Sue is taller than Amy.

be as tall as

Symmetric

If Sue is as tall as Alethia, Alethia is as tall as Sue.

Reflexive

Sue is as tall as Sue.

Transitive

If Sue is as tall as Alethia, and Alethia is as tall as Amy, Sue is as tall as Amy.

5. i. "Fido likes someone"

f = Fido

Like = like

¤x(Like(f,x))

ii. "People who live in NY love it"

(note: "it" = "NY", not "living in NY"... )

ny = NY

Livein = live in

Love = love

Âx(Livein(x, ny)-->Love(x, ny))

iii. "Although no one made any noise, John was annoyed."

j = John

Annoyed = annoyed

Make = make noise

Noise = noise

Âx~¤y(Noise(y) & Make(x,y)) & (Annoyed(j))

iv. "Everything is black or white".

Black = black

White = white

Âx(Black(x)vWhite(x))

v. "If someone is noisy, he annoys everybody."

("All noisy people annoy everybody")

Noisy = noisy

Annoy = annoy

Âx((Noisy(x))--> Ây(Annoy(x,y)))

vi. "All that glitters is not gold".

Glitter = glitter

Gold = gold

~Âx(Glitter(x)-->Gold(x)) (normal reading)

Âx(Glitter(x)-->~Gold(x)) (funny reading)

N.B. Âx~(Glitter(x)-->Gold(x)) is not a possible reading. What does it mean?

5. (Couldn't get the underlinings to translate to html easily, so I've left it out.

6. Muggedat = mugged at (a time)

3seconds = is 3 seconds long

Timeperiod = is a time period

Man = man

Âx((Timeperiod(x) & 3seconds(x))-->¤y(Man(y)&Muggedat(y, x)))

"For all x, if x is a time period and x is three seconds long, there exists a y such that y is a man and y is mugged at x"

¤yÂx((Timeperiod(x) & 3seconds(x))-->(Man(y)&Muggedat(y, x))

There exists a y such that, for all x, if x is a time period and x is three seconds long, y is a man and y is mugged at x.

Recall that for every statement in predicate logic, it must be evaluated as true or false with respect to a particular model (M).

(i) to each individual constant a member of U

(ii) to each one-place predicate a subset of U

(iii) to each two-place predicate a subset of UxU

(and in general:

(iv) to each n-place predicate a subset of UxUx...U

n

(that is, a set of ordered n-tuples of elements of D)

So, a statement in predicate calculus is true or false with respect to a particular model.

Now, when you have a quantified expression, all of whose variables are bound, it is a statement just like P(c) is a statement, and its evaluation with respect to a model should be a truth value. However, it's made up of a quantifier, and an open statement, like P(x), which is not a statement and doesn't have a truth value of its own. To evaluate the truth of the full, quantified, expression, though, we need to consider the truth value of every possible interpretation (or assignment) for the variable. In order to do that, the rule for semantic interpretation of the quantifier needs to be able to refer to and change the variable assignments, to do the "checking" that is needed to evaluate whether Âx(P(x)) is true.

Before getting into varying assignments, though, let's consider what an assignment function can do by itself. DeSwart gives this example:

(2) a. She invited him to the party.

b. Invite (x,y)

c. Ann likes Bill, so she invited him to the party.

d. Carol felt sorry for Dave, so she invited him to the party.

In the predicate calculus, the clause with pronouns by itself (2a) is represented by the open proposition Invite(x,y) (2b), where the variables can be different people in different contexts. In 2c, the context provides an assignment function which assigns particular values to the variables x and y, in particular, Ann and Bill. In 2d, the context provides an assignment function which assigns different particular values to x and y, in particular, Carol and Dave. In that case, the particular model and the two different assignment functions might be as represented in (3), e.g.

(3) Model:

[| c |] = Carol

[| d |] = Dave

[| a |] = Ann

[| b |] = Bill

[| s |] = Sue

[| j |] = John

g(x) = Ann

g(y) = Bill

g'(x) = Carol

g'(y) = Dave

g"(x) = Sue

g"(y) = John

So, now, we've got the equipment to consider how to compute the truth values of quantified statements. All interpretation is now done with respect to a model M and a particular assignment of values to variables, g: [| |]M, g. So, for instance, in context 2c, "Ann likes Bill", [| Invite(x, y) |]M, g = 1, because g assigns Ann to x and Bill to y, but

[| Invite(x, y) |]M, g" = 0, because g" assigns Sue to x and John to y, and they are not in the set of ordered pairs denoted by Invite.

The idea of checking the range of possible values for a variable, which is our intuitive understanding of quantification, is implemented by checking all the possible assignments for a variable. So, let's imagine that our interpretation function is working on a given formula with respect to an assignment to variables g, like that one given above,where g(x) is Ann:

(4) Interpretation with respect to an assignment

At the point where the interpretation function gets to the quantifier, it will have decided that (Mortal(x)) is true iff Ann is mortal (since it will have assigned the value Ann to x, according to the assignment function g). But then it discovers that that variable is quantified, when it gets to Âx. What does it then have to do? Holding everything else constant, it has to check all possible values for x, to be sure that no matter what value x is assigned, Mortal(x) turns out to be true. That is, it has to check the assignment, g[x/e] which is the same as g except that assigns the individual e to x... and all other possible assignments where the only thing that varies is the value of x, until all possible values for x are checked in the argument position. (If you're evaluating the truth of something like ¤x(Mortal(x)), then you only have to check until you get the value 1 for one particular assignement; to evaluate the truth of Âx(Mortal(x)), you only have to check until you get the value 0 for one particular assignment. Of course, if the appropriate verifying/falsifying value doesn't show up, you have to check every possible assignment in both cases). So, having said all that, here's the rules for semantically interpreting quantified statements.

(5) For g(x/e) a variable assignment which differs from g at most in that it assigns the variable e to x:

That completes our semantic interpretation for predicate logic.

Now, let's consider the interpretations we can assign to "All that glitters is not gold", given a particular model universe and particular assignment. Let's say we have a model with three constants, two predicates, and one variable, illustrated in (6).

(6) Model:

[| r |] = Ring

[| f |] = Foil

[| s |] = Stone

g(x) = Ring

Now, let's practice some with shuffling around quantified formulas. For example, let's change the universally quantified formula in 6b into an existentially quantified one:

(7)Manipulating the funny reading:

"It is not the case that there exists an x such that x glitters and x is gold.

Note that you can't just rearrange quantifiers and negation to suit yourself, so it's obvious, for instance that 8a and 8b don't mean the same thing:

But it might be less obvious that 9a and 9b don't mean the same thing, when you just read them aloud.

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

"For all x, it is not the case that if x glitters, then x is gold."

In fact, 9b is a very strange sentence that, if you manipulate it a bit, you can see clearly is not a possible meaning for the English sentence "All that glitters is not gold". Here's the manipulation:

"For all x, x glitters and is not gold".

This is false in any world in which something doesn't glitter, or in which something glitters and is gold.

These in (11) are sort of like the distributive laws of propositional logic, but the last two are only true in one direction (notice the arrow isn't a biconditional). Take 11c, e.g. If everything in the universe is either male or female, it doesn't mean that either everything is male or everything is female. Similarly for 11d: if there exists something that is male, and there exists something that is female, it is not necessarily the case that there exists something that is male and female.

(11) Laws of Quantifier Distribution:

In 12, we see that if all the quantifiers over a given formula are of the same type, it doesn't matter which order they're in. 12c, however, indicates that if there is a unique thing x such that for all y, there's some two-place predicate that's true P(x,y), then for all y there is at least one thing x such that the two-place predicate is true (x,y). The two sides (as in c and d above) are not equivalent, but there is a logical one-way implication. (If there's a particular journalist that every senator hates, then every senator hates at least one journalist).

(12) Laws of Quantifier (In)Dependence

These laws say when you can move quantifiers to the front of a conditional formula without changing truth conditions, when the conditions about free variables within the formula are met. They (in combination with the laws of quantifier distribution, above) let you find the prenex normal form of a formula, which is just a formula where all the quantifiers precede the quantifier-free matrix. This can be useful when you're trying to use the rules of inference; having all the quantifiers at the beginning make it much easier to keep track of your instantiations and generalizations.

(13) Laws of Quantifier Movement:

provided that x is not free in P

provided that x is not free in P

(14). "Some person likes every book".

A translation like ¤xÂy(Like(x,y)) doesn't work for this sentence, because this just says there's something that likes everything. There's got to be predicates for each N, person and book, Person(x) and Book(x). If we do this, the appropriate translation is (for the reading where there is at least one unique person who likes every book):

(15) ¤x(Person(x) & Ây(Book(y)-->Like(x,y))

But now, since the quantifiers were embedded in the NPs, we've had to embed the quantifiers next to their appropriate predicates. Let's convert it to prenex normal form (PNF):

(16) ¤x(Person(x) & Ây(Book(y)-->Like(x,y))

¤x~~(Person(x) & Ây(Book(y)-->Like(x,y)) Double negation

¤x~(~Person(x) v ~Ây(Book(y)-->Like(x,y)) DeMorgan's law

¤x~(~Person(x) v ¤y~(Book(y)-->Like(x,y)) Quantifier negation (i)

¤x~(Person(x) --> ¤y~(Book(y)-->Like(x,y)) Conditional law

¤x~ ¤y(Person(x) --> ~(Book(y)-->Like(x,y)) Quantifier movement (b)

¤xÂy~(Person(x) --> ~(Book(y)-->Like(x,y)) Quantifier negation (i)

¤xÂy~(~Person(x) v ~(Book(y)-->Like(x,y)) Conditional law

¤xÂy(~~Person(x) & ~~(Book(y)-->Like(x,y)) DeMorgan's law

¤xÂy(Person(x) & (Book(y)-->Like(x,y)) Double negation

All of that just to show that one can move a quantifier from the second half of a conjunct to the front of the formula! (as long as the quantified variable is not free in the first half of the conjunct, of course).

Next time: Natural deduction in predicate logic: Universal Instantiation, Universal Generalization, Existential Generalization, Existential Instantiation.