Transcript Slides from 10/22/14

Exam #3 will be given on Monday
Nongraded Homework: Now that we are
familiar with the universal quantifier, try
http://www.poweroflogic.com/cgi/menu.cgi
(9.1, C, D, E and F – see ‘help’ link for
symbol use; note about upside-down ‘A’)
Exam #3
One proof without SI or TI rules (10 pts.)
Two proofs using any of our rules (6 and
10 pts.)
Four symbolizations in LMPL (three,
three, four, and four points)
The Universal Quantifier
("x), ("y), ("z), using other small-case
letters after ‘s’, if needed.
It means, “the following is true of every single
thing in the universe” or “for every value of x,
the following comes out true”
Like the existential quantifier and the tilde,
the universal quantifier applies to (or has in
its scope) whatever immediately follows it.
Guidelines for symbolization in LMPL:
1. When using the universal quantifier in
translation, use an arrow as the main
operator within the scope of the quantifier
(almost always).
All philosophers are mortal: ("x)(Px → Mx)
In other words, for every single thing in the
universe, if it’s a philosopher, then it’s
mortal.
No trees are animals.
("x)(Tx → ~ Ax)
The group you’re talking about in its entirety is the group of
trees.
What does ‘("x)(Tx & ~ Ax)’ say?
For anything in the universe, it is a tree and not an animal.
In other words, everything in the universe is a tree and
not an animal.
This formula is well formed and it has truth-conditions, but
it’s unlikely to appear in a symbolization, because it’s
unlikely anyone would ever say something with that
formula’s truth-conditions.
Exceptions
What about, “Everything has mass and
charge”?
("x)(Mx & Cx)
is the right translation, but this is a very rare
case where the speaker wants to say that
both predicates apply to every single thing in
the universe.
B. All people are happy.
Looks like it should be
("x)(Px → Hx)
But since Forbes allows persons to be a domain of
discourse, we don’t actually need ‘Px’ or the
arrow
("x)Hx
is fine. Let the dictionary be your guide on HW and
exams. If ‘P_: _ is a person’ appears in the
dictionary, use it in your symbolization.
2. When using the existential quantifier, an
ampersand should be the main connective
within the scope of the quantifier (almost
always).
Some philosophers are happy. (P_: _ is a
philosopher; H_: _ is happy)
($x)(Px & Hx)
Why not ($x)(Px → Hx)?
It’s well formed, but it doesn’t have the right
truth-conditions. It says, “for at least one
thing P_ → H_ is true” (with the same thing
put in both blanks).
But it’s too easy to make P_ → H_ true. Just
find something that isn’t a philosopher (use a
table, for example). F → [any truth-value] is
true.
Exceptions?
What about, “Something is either rotten or dead”?
($x)(Rx v Dx) is the right answer, but this is a rare
case.
Even here, the person speaking would probably
mean “Something in here is either rotten or dead.”
How would you translate that?
3. When using the universal quantifier in
translation, put the group you wish to talk
about in its entirety to the left of the arrow
that is the m.c. inside the scope of the
universal quantifier.
All frogs are green.
("x)(Fx → Gx)
‘Fx’ is the antecedent because we want to
say something about all frogs, not about all
green things.
What about this one?
Only U.S. citizens are allowed to be
president. (Cx: x is a U.S. citizen; Ax: x is
allowed to be president)
("x)(Ax → Cx)
We want to say something about all people
allowed to be president, not about all U.S.
citizens.
4. When translating, group a noun and its
modifier together around an ampersand.
All green frogs are poisonous.
("x)[(Gx & Fx) → Px]
All frogs from Brazil are poisonous.
("x)[(Fx & Bx) → Px]
Every philosopher who lives in Brazil speaks
German.
("x)[(Px & Bx) → Sx]
All frogs are slimy amphibians.
("x)[Fx → (Sx & Ax)]
Some rich philosophers are humble.
($x)[(Rx & Px) & Hx]
When should a quantifier not be the main
operator of a predicate logic symbolization?
Some philosophers are good, and some
philosophers are not good.
($x)(Px & Gx) & ($x)(Px & ~ Gx)
There are two quantifiers, but neither is the
main connective.
Guidelines for deciding whether the initial
quantifier should be the main operator:
1. Apply the sandwich principle: Everything
between an associated ‘either’ and ‘or’ or
‘if’ and ‘then’ should be grouped together.
If any witness told the truth, then George is
guilty. (W_: _ is a witness; T_: _ told the
truth; G_: _ is guilty; g: George)
($x)(Wx & Tx) → Gg
Either all of the witnesses told the truth, or
George is guilty. (G_: _ is guilty; T_: _ told
the truth; W_: _ is a witness; g: George)
("x)(Wx → Tx) v Gg
2. BUT, if, each time you pick a value for x,
you want to talk about the same thing
throughout the entire instance, the initial
quantifier should be the m.c.
If any witness told the truth, then he or she is
honest. (T_: _ told the truth; W_: _ is a
witness; H_: _ is honest)
("x)[(Wx & Tx) → Hx]
Speakers have their names listed in the
program only if they are famous. (Sx: x is a
speaker; Px: x’s name is listed in the
program; Fx: x is famous)
("x)[(Sx & Px) → Fx]
Some experienced mechanics are well paid
only if all the inexperienced ones are lazy.
(E_: _ is experienced; M_: _ is a mechanic;
W_: _ is well paid; L_: _ is lazy)
($x)[(Ex & Mx) & Wx] → ("x)[(Mx & ~Ex) →
Lx ]