Exam #3 will be given on MondayExam #3 will be given on Monday

Nongraded Homework: Now that we are Nongraded Homework: Now that we are familiar with the universal quantifier, try familiar with the universal quantifier, try http://www.poweroflogic.com/cgi/menu.cgi

(9.1, C, D, E and F – see (9.1, C, D, E and F – see ‘‘helphelp’’ link for link for symbol use; note about upside-down symbol use; note about upside-down ‘‘AA’’))

Exam #3Exam #3

One proof without SI or TI rules (10 pts.)One proof without SI or TI rules (10 pts.)

Two proofs using any of our rules (6 and Two proofs using any of our rules (6 and 10 pts.)10 pts.)

Four symbolizations in LMPL (three, Four symbolizations in LMPL (three, three, four, and four points)three, four, and four points)

The Universal QuantifierThe Universal Quantifier

((x), (x), (y), (y), (z), using other small-case z), using other small-case letters after letters after ‘‘ss’’, if needed., if needed.

It means, It means, ““the following is true of every single the following is true of every single thing in the universething in the universe”” or or ““for every value of x, for every value of x, the following comes out truethe following comes out true””

Like the existential quantifier and the tilde, Like the existential quantifier and the tilde, the universal quantifier applies to (or has in the universal quantifier applies to (or has in its scope) whatever immediately follows it.its scope) whatever immediately follows it.

Guidelines for symbolization in LMPL:Guidelines for symbolization in LMPL:

1. When using the universal quantifier in 1. When using the universal quantifier in translation, use an arrow as the main translation, use an arrow as the main operator within the scope of the quantifier operator within the scope of the quantifier (almost always).(almost always).

All philosophers are mortal: (All philosophers are mortal: (x)(Px → Mx)x)(Px → Mx)

In other words, for every single thing in the In other words, for every single thing in the universe, if ituniverse, if it’’s a philosopher, then its a philosopher, then it’’s mortal. 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.

ExceptionsExceptions

What about, What about, ““Everything has mass and Everything has mass and chargecharge””? ?

((x)(Mx & Cx)x)(Mx & Cx) is the right translation, but this is a is the right translation, but this is a very rare very rare casecase where the speaker wants to say that where the speaker wants to say that both predicates apply to every single thing in both predicates apply to every single thing in the universe.the universe.

B. All people are happy.B. All people are happy.Looks like it should beLooks like it should be

((x)(Px → Hx)x)(Px → Hx)

But since Forbes allows persons to be a domain of But since Forbes allows persons to be a domain of discourse, we dondiscourse, we don’’t actually need t actually need ‘‘PxPx’’ or the arrow or the arrow

((x)Hxx)Hx

is fine. Let the dictionary be your guide on HW and is fine. Let the dictionary be your guide on HW and exams. If exams. If ‘‘P_: _ is a personP_: _ is a person’’ appears in the appears in the dictionary, use it in your symbolization.dictionary, use it in your symbolization.

2. When using the existential quantifier, an 2. When using the existential quantifier, an ampersand should be the main connective ampersand should be the main connective within the scope of the quantifier (almost within the scope of the quantifier (almost always).always).

Some philosophers are happy. (P_: _ is a Some philosophers are happy. (P_: _ is a philosopher; H_: _ is happy)philosopher; H_: _ is happy)

((x)(Px & Hx) x)(Px & Hx)

Why not (Why not (x)(Px → Hx)?x)(Px → Hx)?

ItIt’’s well formed, but it doesns well formed, but it doesn’’t have the right t have the right truth-conditions. It says, truth-conditions. It says, ““for at least one for at least one thing P_ → H_ is truething P_ → H_ is true”” (with the same thing (with the same thing put in both blanks). put in both blanks).

But itBut it’’s too easy to make P_ → H_ true. Just s too easy to make P_ → H_ true. Just find something that isnfind something that isn’’t a philosopher (use a t a philosopher (use a table, for example). F → [any truth-value] is table, for example). F → [any truth-value] is true.true.

Exceptions?Exceptions?

What about, What about, ““Something is either rotten or deadSomething is either rotten or dead””??

((x)(Rx v Dx) is the right answer, but this is a rare x)(Rx v Dx) is the right answer, but this is a rare case.case.

Even here, the person speaking would probably Even here, the person speaking would probably mean mean ““Something in here is either rotten or dead.Something in here is either rotten or dead.”” How would you translate that?How would you translate that?

3. When using the universal quantifier in 3. When using the universal quantifier in translation, put the group you wish to talk translation, put the group you wish to talk about in its entirety to the left of the arrow about in its entirety to the left of the arrow that is the m.c. inside the scope of the that is the m.c. inside the scope of the universal quantifier.universal quantifier.

All frogs are green.All frogs are green.((x)(Fx → Gx) x)(Fx → Gx) ‘‘FxFx’’ is the antecedent because we want to is the antecedent because we want to say something about all frogs, not about all say something about all frogs, not about all green things.green things.

What about this one?What about this one?

Only U.S. citizens are allowed to be Only U.S. citizens are allowed to be president. (Cx: x is a U.S. citizen; Ax: x is president. (Cx: x is a U.S. citizen; Ax: x is allowed to be president)allowed to be president)

((x)(Ax → Cx)x)(Ax → Cx)

We want to say something about all people We want to say something about all people allowed to be president, not about all U.S. allowed to be president, not about all U.S. citizens.citizens.

4. When translating, group a noun and its 4. When translating, group a noun and its modifier together around an ampersand.modifier together around an ampersand.

All green frogs are poisonous.All green frogs are poisonous.

((x)[(Gx & Fx) → Px] x)[(Gx & Fx) → Px]

All frogs from Brazil are poisonous.All frogs from Brazil are poisonous.

((x)[(Fx & Bx) → Px] x)[(Fx & Bx) → Px]

Every philosopher who lives in Brazil speaks Every philosopher who lives in Brazil speaks German.German.

((x)[(Px & Bx) → Sx]x)[(Px & Bx) → Sx]

All frogs are slimy amphibians.All frogs are slimy amphibians.

((x)[Fx → (Sx & Ax)]x)[Fx → (Sx & Ax)]

Some rich philosophers are humble.Some rich philosophers are humble.

((x)[(Rx & Px) & Hx] x)[(Rx & Px) & Hx]

When should a quantifier When should a quantifier notnot be the main be the main operator of a predicate logic symbolization?operator of a predicate logic symbolization?

Some philosophers are good, and some Some philosophers are good, and some philosophers are not good.philosophers are not good.

((x)(Px & Gx) & (x)(Px & Gx) & (x)(Px & ~ Gx)x)(Px & ~ Gx)

There are two quantifiers, but neither is the There are two quantifiers, but neither is the main connective.main connective.

Guidelines for deciding whether the initial Guidelines for deciding whether the initial quantifier should be the main operator:quantifier should be the main operator:

1. Apply the sandwich principle: Everything 1. Apply the sandwich principle: Everything between an associated between an associated ‘‘eithereither’’ and and ‘‘oror’’ or or ‘‘ifif’’ and and ‘‘thenthen’’ should be grouped together. should be grouped together.

If any witness told the truth, then George is If any witness told the truth, then George is guilty. (W_: _ is a witness; T_: _ told the truth; guilty. (W_: _ is a witness; T_: _ told the truth; G_: _ is guilty; g: George)G_: _ is guilty; g: George)

((x)(Wx & Tx) → Ggx)(Wx & Tx) → Gg

Either all of the witnesses told the truth, or Either all of the witnesses told the truth, or George is guilty. (G_: _ is guilty; T_: _ told George is guilty. (G_: _ is guilty; T_: _ told the truth; W_:the truth; W_: _ is a witness; g: George)_ is a witness; g: George)

((x)(Wx → Tx) v Ggx)(Wx → Tx) v Gg

2. BUT, if, each time you pick a value for 2. BUT, if, each time you pick a value for xx, , you want to talk about the same thing you want to talk about the same thing throughout the entire instance, the initial throughout the entire instance, the initial quantifier should be the m.c.quantifier should be the m.c.

If any witness told the truth, then he or she is If any witness told the truth, then he or she is honest. (T_: _ told the truth; W_: _ is a honest. (T_: _ told the truth; W_: _ is a witness; H_: _ is honest) witness; H_: _ is honest)

((x)[(Wx & Tx) → Hx]x)[(Wx & Tx) → Hx]

Speakers have their names listed in the Speakers have their names listed in the program only if they are famous. (Sx: program only if they are famous. (Sx: xx is a is a speaker; Px: speaker; Px: xx’’s name is listed in the s name is listed in the program; Fx: program; Fx: xx is famous) is famous)

((x)[(Sx & Px) → Fx]x)[(Sx & Px) → Fx]

Some experienced mechanics are well paid Some experienced mechanics are well paid only if all the inexperienced ones are lazy. only if all the inexperienced ones are lazy. (E_: _ is experienced; M_: _(E_: _ is experienced; M_: _ is a mechanic; is a mechanic; W_: _ is well paid; L_: _W_: _ is well paid; L_: _ is lazy)is lazy)

((x)[(Ex & Mx) & Wx] → (x)[(Ex & Mx) & Wx] → (x)[(Mx & ~Ex) → x)[(Mx & ~Ex) → Lx ]Lx ]