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## Philosophy 57 lecture

**Philosophy 57 — Day 20**
• Quiz Today (3-Circle Venn Diagram Technique)
• New Versions of LogicCoach Available Online. See:

**– **LogicCoach is very useful for chapter 6 problems

• Extra-Credit Problems to be Posted Within 10 Days

**– **Will Be Problems from Chapter 6

**– **May Use Computer Programs (or any other tools!)

**– **Review of Basic Terminology and Set-up

**– **Translation from English into Propositional Logic

**Chapter 6 — Propositional Logic Review I**
• Capital letters denote simple or atomic statements. These are combined with
connectives (and, or, if, not, iff) to form compound or molecular statements.

**Simple Statements (symbolic):**
**Compound Statements (pseudo-symbolic):**
• Either it’s raining or it’s snowing.

• If Dell introduces a new line, then Apple will also.

• James Joyce wrote

*Ulysses*.

• Snow is white and the sky is blue.

• is not the case that Emily Bronte wrote

*Jane Eyre*.

→ It is not the case that

*E*.

• John is a bachelor if and only if he is unmarried.

→

*I *if and only if

*U*.

• Symbolic Connectives: “∼” for “not”, “∨” for “or”, “•” for “and”, “⊃” for
“if”, and “≡” for “if and only if”. Here’s a table of the five PL connectives:

**Operator**
**Logical Function**
**Used to translate**
• Using these symbols, we can put our compound examples in symbolic form:

**Pseudo-Symbolic:**
**Full Symbolic:**
• It is not the case that

*E*.

•

*I *if and only if

*U*.

• Our first topic is

*translation *from English into PL.

**Chapter 6 — Propositional Logic Review II**
**Statement Form:**
**Statement Name:**
**Statement Components:**
*p *called antecedent,

*q *called consequent
• NOTE: I use lower-case letters

*p*,

*q*, etc. as

*variables*, ranging over sentences
in PL. Lower-case letters are

*not *part of the PL language!
• PL sentences can be as complex as you like. But, they are always constructed
out of upper-case letters, and connectives (or operators).

• The main operator in a (compound) PL sentence is the operator that governs
the largest component(s) in the statement. What are the following?
“∼(

*G *⊃

*H*) ⊃ (

*I *∨

*J*)”
“(

*A *≡

*F*) • (

*C *≡

*G*)”
“

*A *⊃

*B *∨

*C *•

*D*”

**Chapter 6 — Propositional Logic Translations I**
• The following are all examples of negations:

**– **Rolex does not make computers. → ∼

*R*
**– **It is not the case that Rolex makes computers. → ∼

*R*
**– **It is false that Rolex makes computers. → ∼

*R*
• We always place the tilde

*in front *of the proposition it negates. The following
are also negations, since their main connectives are tildes:
∼[(

*A *≡

*F*) • (

*C *≡

*G*)]
• The following are

*not *negations. Watch the parentheses! What

*are *these?

**Chapter 6 — Propositional Logic Translations II**
• The following are all examples of conjunctions:

**– **Tiffany sells jewelry and Gucci sells cologne. →

*T *•

*G*
**– **Tiffany sells jewelry, but Gucci sells cologne. →

*T *•

*G*
**– **Tiffany sells jewelry, however, Gucci sells cologne. →

*T *•

*G*
**– **Tiffany and Ben Bridge sell jewelry. →

*T *•

*B*
• The following are all conjunctions, since their main connectives are dots:
(

*E *∨

*F*) • (

*G *∨

*H*)
[(

*R *⊃

*T *) ∨ (

*S *⊃

*U*)] • [(

*W *≡

*X*) ∨ (

*Y *≡

*Z*)]
• The following are

*not *conjunctions. Watch the parentheses! What

*are *these?
(

*A *• (

*B *≡

*C*)) ∨ (

*A *•

*D*)

**Chapter 6 — Propositional Logic Translations III**
• The following are all examples of disjunctions:

**– **Cigna expands operations or Aetna does. →

*C *∨

*A*
**– **Either Cigna or Aetna expands operations. →

*C *∨

*A*
**– **Cigna expands operations unless Aetna does. →

*C *∨

*A*
**– **Unless Cigna expands operations, Aetna does. →

*C *∨

*A*
• The following are all disjunctions, since their main connectives are wedges:
(

*F *•

*H*) ∨ (∼

*K *• ∼

*L*)
[

*S *• (

*T *⊃

*U*)] ∨ [

*X *• (

*Y *≡

*Z*)]
• The following are

*not *disjunctions. Watch the parentheses! What

*are *these?
(

*A *∨ (

*B *≡

*C*)) • (

*A *∨

*D*)

**Chapter 6 — Propositional Logic Translations IV**
• The following are all examples of conditionals:

**– **If Intel raises prices, then so does Compaq. →

*I *⊃

*C*
**– **Compaq raises prices if Intel does. →

*I *⊃

*C*
**– **Intel raises prices only if Compaq does. →

*I *⊃

*C*
**– **Delta lowers fares provided that United does. →

*U *⊃

*D*
**– **Delta lowers fares on condition that United does. →

*U *⊃

*D*
**– **United’s lowering fares implies that Delta does. →

*U *⊃

*D*
**– **Hilton’s opening a new hotel is a sufficient condition for Marriott’s doing

**– **Hilton’s opening a new hotel is a necessary condition for Marriott’s doing

• The following are all conditionals — their main connectives are horseshoes:
(

*A *∨

*C*) ⊃ (

*D *•

*E*)
[

*K *∨ (

*S *• ∼

*T *)] ⊃ [∼

*F *∨ (

*M *•

*O*)]

**Chapter 6 — Propositional Logic Translations V**
• The following are examples of biconditionals:

**– **Kodak introduces a new film if and only if Fuji does. →

*K *≡

*F*
**– **Kodak’s introducing a new film is a necessary and sufficient condition for

Fuji’s doing so. →

*K *≡

*F*
• Sidebar: Use the mnemonic device “SUN” to remember the important
distinction between sufficient

*versus *necessary conditions.

**– **If you rotate the “U” in “SUN” to the left, then you get “

*S *⊃

*N*,” which

means that sufficient conditions always imply necessary conditions.

**– **Sufficient conditions are antecedents. Necessary conditions are consequents.

• The following are biconditionals — their main connectives are triple bars:
(

*B *∨

*D*) ≡ (

*A *•

*C*)
[

*K *∨ (

*F *⊃

*I*)] ≡ [∼

*L *• (

*G *∨

*H*)]

**Chapter 6 — Propositional Logic Translations VI**
• Whenever 3 or more letters are appear, parentheses (or brackets or braces)
must be used carefully to indicate the proper range of the connectives.

• For instance, the string of symbols “

*A *•

*B *∨

*C*” is

*ambiguous*. It could
represent either “(

*A *•

*B*) ∨

*C*” or “

*A *• (

*B *∨

*C*)”. These have

*di*ff

*erent meanings*!
• A well-formed formula (WFF, for short) is a grammatical PL sentence. In
English, “Porch on the is cat a there” is ungrammatical. And, in PL, the
following strings of symbols are not WFFs, because they are ungrammatical:
• Here are some examples, to illustrate the importance of proper grouping.

1. Prozac relieves depression and Allegra combats allergies, or Zocor lowers
cholesterol. → (

*P *•

*A*) ∨

*Z*
2. Prozac relieves depression, and Allegra combats allergies or Zocor lowers

3. Either Prozac relieves depression and Allegra combats allergies or Zocor
4. Prozac relieves depression and either Allegra combats allergies or Zocor
5. Prozac relieves depression or both Allegra combats allergies and Zocor
6. If Merck changes its logo, then if Pfizer increases sales, then Lilly will
7. If Merck’s changing its logo implies that Pfizer increases sales, then Lilly
8. If Schering and Pfizer lower prices or Novartis downsizes, then Warner
• Do not confuse the following three statement forms:
“

*A *if

*B*” → “

*B *⊃

*A*”
“

*A *only if

*B*” → “

*A *⊃

*B*”
“

*A *if and only if

*B*” → “

*A *≡

*B*”

**Chapter 6 — Propositional Logic Translations VII**
• The tilde “∼” operates

*only *on the unit that immediately follows it. In
“∼

*K *∨

*M*,” ∼ affects only “

*K*”; in “∼(

*K *∨

*M*),” ∼ affects the entire “

*K *∨

*M*”.

• “It is not the case that

*K *or

*M*” is

*ambiguous *between “∼

*K *∨

*M*,” and
“∼(

*K *∨

*M*).” Convention: “It is not the case that

*K *or

*M*” → “∼

*K *∨

*M*”.

• “Not both

*S *and

*T *” → “∼(

*S *•

*T *)”. As we will see later (DeMorgan rule),
“∼(

*S *•

*T *)” ≈ “∼

*S *∨ ∼

*T *”. But, “∼(

*S *•

*T *)”
• Similarly, “Not either

*S *or

*T *” → “∼(

*S *∨

*T *)”. And, (DeMorgan rule again)
“∼(

*S *∨

*T *)” ≈ “∼

*S *• ∼

*T *”, but “∼(

*S *∨

*T *)”
• Here are some examples involving∼, •, and ∨ (not, and, or):
1. Shell is not a polluter, but Exxon is. → ??
2. Not both Shell and Exxon are polluters. → ??
3. Both Shell and Exxon are not polluters. → ??

4. Not either Shell or Exxon is a polluter. → ??
5. Neither Shell nor Exxon is a polluter. → ??
6. Either Shell or Exxon is not a polluter. → ??
• Summary of translations involving ∼, •, and ∨ (not, and, or):

**Pseudo-Symbolic**
**Propositional Logic (PL)**
• DeMorgan rules (we will

*prove *these rules later in the chapter):
∼(

*p *∨

*q*) ≈ ∼

*p *• ∼

*q*
∼(

*p *•

*q*) ≈ ∼

*p *∨ ∼

*q*
∼

*p *∨ ∼

*q *and ∼(

*p *•

*q*)

**Chapter 6 — Propositional Logic Translations VIII**
**English Expression**
**PL Operator**
not, it is not the case that, it is false that
and, yet, but, however, moreover, nevertheless, still, also,
although, both, additionally, furthermore
if . . . then . . . , only if, given that, in case, provided that, on
condition that, sufficient condition for, necessary condition for
(Note: do not confuse antecedents and consequents!)
if and only if (iff), is equivalent to, sufficient and necessary con-
dition for, necessary and sufficient condition for

**Chapter 6 — Propositional Logic Translations IX**
1. California does not allow smoking in restaurants.

2. Jennifer Lopez becomes a superstar given that

*I’m Real *goes platinum.

3. Mary-Kate Olsen does not appear in a movie unless Ashley does.

4. Either the President supports campaign reform and the House adopts
universal healthcare or the Senate approves missile defense.

5. Neither Mylanta nor Pepcid cures headaches.

6. If Canada subsidizes exports, then if Mexico opens new factories, then the
7. If Iraq launches terrorist attacks, then either Peter Jennings or Tom
8. Tom Cruise goes to the premiere provided that Penelope Cruz does, but

9. It is not the case that either Bart and Lisa do their chores or Lenny and
10. N’sync winning a grammy is a sufficient condition for the Backstreet Boys
to be jealous, only if Destiny’s Child getting booed is a necessary
condition for TLC’s being asked to sing the anthem.

11. Dominos’ delivers for free if Pizza Hut adds new toppings, provided that
12. If evolutionary biology is correct, then higher life forms arose by chance,
and if that is so, then it is not the case that there is any design in nature and
13. Kathie Lee’s retiring is a necessary condition for Regis’s getting a new
co-host; moreover, Jay Leno’s buying a motorcycle and David Letterman’s
telling more jokes imply that NBC’s airing more talk shows is a sufficientcondition for CBS’s changing its image.

Source: http://fitelson.org/57/lecture20.pdf

Guía clínica para el diagnóstico y el tratamiento de la enfermedad pulmonar obstructiva crónica J.A. Barberàa, G. Peces-Barbab, A.G.N. Agustíc, J.L. Izquierdod, E. Monsóe, T. Montemayorf y J.L. ViejogServei de Neumologia. aHospital Clínic. Barcelona. bFundación Jiménez Díaz. Madrid. cHospital Son Dureta. Palma de Mallorca. dHospital General. Guadalajara. eHospital Germans Trias i P

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