Today’s Lecture1/28/10

Chapter 7.1

! Symbolizing English Arguments

! 5 Important Logical Operators

!The Main Logical Operator

Quiz

State from memory (closed book and notes) thefive ‘famous’ valid forms and their names.

--be sure your name is on your paper

Announcements

Homework

-- Ex 7.1 pgs. 298-299 Part A and B (All)

Quiz on Tuesday (Feb 2nd)

--state from memory each logical operator, its translation,and its corresponding type of compound statement. Seethe table on p. 279. (Example: ~ ‘not’ negation)

Book Issues

Adding the Course

Validity v. Invalidity Again

An argument is valid if and only if it’s impossiblefor all of the premises to be true while theconclusion is false.

An argument is invalid if and only if it is possiblefor all of the premises to be true while theconclusion is false.

Validity v. Invalidity Again

--Our task shortly is to study a particularmechanical method for determining whether anargument is either valid or invalid.

--It will be the method of Truth Tables.

--But before we can employ this mechanicalmethod, it's essential that we be able to representEnglish arguments using a symbolic notation.

--To do this we need to learn all that is involvedwith translating English statements into symbols.

Atomic v. Compound Statements

An atomic statement is one that does not have anyother statement as a component.

Examples

!Grass is green.

!The library is adjacent to Sierra Tower.

!Tennis is challenging.

Atomic v. Compound Statements

A compound statement is one that has at least oneatomic statement as a component.

Examples

! It is false that grass is red.

!The library is adjacent to Sierra Tower and Iam hungry.

! If tennis is challenging, then tennis playershave a reason to practice.

Atomic Statements:

Symbolize AtomicStatements with a singleupper case letter.

B: Brass is a mixed metal.

C: Cathy called in sick.

N: Nadal wins trophies

T: Thought is mysterious.

Compound Statements

Symbolize Compound Statements by first symbolizingtheir Atomic Constituents, and then their logical words.

(logical words: “it’s not the case that”, “and”, “or”, “if,then”, “if and only if”, and their stylistic variants)

(We will just symbolize the Atomic Constituents for now)

Example:

It is false that grass is red.

It is false that G.

Compound Statements

More examples of partially translatedcompound Statements:

The library is adjacent to Sierra Tower andI am hungry -- L and I

If tennis is challenging, then tennis playershave a reason to practice -- If T, then P

Symbolizing the Logical Words

biconditionalIf and only ifdouble-arrow!

conditionalif, thenarrow"

disjunctionorveev

conjunctionanddot•

negationnottilde~

CompoundType

TranslatesNameOperator

~

NEGATIONS

Symbolizing Negations

Grass is not red (R: Grass is red)

is symbolized as this is our scheme of

~ R abbreviation

It is not the case that grass is red

~ R

It is false that grass is red

~ R

Negations of Compound Statements

It is false that Dallas wins and Phoenix wins (D: Dallas wins

is symbolized as P: Phoenix wins)

~ (D · P)

It’s not true that if Dallas wins, then Phoenix wins

~ (D " P)

The following is false: either Dallas wins or Phoenix wins.

~ (D # P)

Parentheses are Important

It is false that Dallas wins and Phoenix wins

~ (D · P) -- this a negation

(this says that it’s not the case that both of them win; it leaves usagnostic as to who actually wins; maybe both of them don’t win)

If we didn’t use parentheses we would get

~ D · P -- this is a conjunction, not a negation

(this says that Dallas does not win and Phoenix wins)

Parentheses are Important

It’s not the case that if Dallas wins then Phoenix wins.

~(D " P) -- this is a negation

(this says that Dallas winning does not entail Phoenix winning)

If we didn’t use parentheses we would get:

~D " P -- this is a conditional, not a negation

(this says if Dallas doesn’t win, then Phoenix wins)

Main Logical Operator

The most important step in knowing where toplace parentheses is finding the main logicaloperator (i.e. main connective) in the Englishstatement. This lets you know what kind ofcompound statement it is, be it a negation orconditional, or disjunction, etc.

Main Logical Operator

The main logical operator is, roughly, the operatorthat governs (or connects) the entire statement.

Finding the main operator depends upon yourability to see “what the statement is saying”. Thisjust takes practice.

More will be said concerning main operators andparentheses later.

Back to Negations

Again, all of these are examples of negations:

~ (D · P)

~(D "P)

~(D # P)

The ~ is the main logical operator

•

CONJUNCTIONS

Symbolizing Conjunctions

Grass is purple and life is good.

(G: grass is purple; L: life is good)

G • L

Stylistic Variants of ‘and’

! Grass is purple but life is good.

! Grass is purple; however life is good.

! Grass is purple yet life is good.

! Although grass is purple, life is good.

! While grass is purple, life is good.

! Grass is purple; nevertheless life is good.

Still,

G • L

Not all uses of 'and' are conjunctions

Not all uses of the English term 'and’ convey aconjunction. If they did, then we would be able totranslate the statements in question intoconjunctions and capture the essential meaning ofthe English statement. But we are unable to dothis in all cases.

For example…

Not all uses of 'and' are conjunctions

! Sometimes ‘and’indicates temporalorder

-Sarah cracked the safeand took the money.

-You made a joke and Ilaughed.

! Sometimes ‘and’indicates a relationship

-Dave and Val aremarried.

- Dave and Val movedthe couch.

These are all Conjunctions

P • Q

P • ~(Q # R)

(P " Q) • (Q " P)

The • is the mainoperator

~P •[Q # (R " S) ]

[Q " (P # R)] • S

(P # Q) •(R " S)

The • is the mainoperator

#

DISJUNCTIONS

Symbolizing Disjunctions

1. Grass is green or pizza is edible.

(G: Grass is green; P: pizza is edible)

G # P

2. Alfred will not pass tomorrow or Alfred will studytonight

(P: Alfred will pass tomorrow; S: Alfred will study tonight)

~P # S

Stylistic Variants of ‘or’

Alfred will not pass tomorrow and/or Alfred willstudy tonight.

Alfred will not pass tomorrow or Alfred will studytonight (or both).

Alfred will not pass tomorrow unless Alfred willstudy tonight.

Still ~P # S

Inclusive OR Exclusive OReither P or Q (or both) either P or Q (but not both)

Sometimes when peoplemake a disjunctive claim,they intend the ‘or’ to beread inclusively.

e.g.

If you want to live under myroof, either you get a jobor you go to college.

**The parent will not bebothered if you do both.

Sometimes when peoplemake a disjunctive claim,they intend the ‘or’ to beread exclusively.

e.g.

You may have the soup oryou may have the salad.

**The waitress will bebothered if you say ‘both’.

Logicians Treat ‘or’ as Inclusive

We will treat ‘or’ as inclusive in the absence of a contextthat suggests an exclusive reading.

There is, however, a way of translating an exclusive ‘or’which is, again, P or Q (but not both)

Consider:

You may have the soup or you may have the salad, but notboth.

Q: How would you translate this?

Exclusive OR

You may have soup or you may have salad, butnot both.

(S: you may have soup; L: you may have salad)

(S # L) • ~(S • L)

'Neither-Nor' is Not a Disjunction!

Neither Simon nor Garfunkel issad.

S: Simon is sad.

G: Garfunkel is sad.

Two Equivalent Readings

1. ~ (S # G)

2. ~S • ~G

These are all Disjunctions

P # Q

P # ~(Q • R)

(P " Q) # (Q • P)

The # is the mainoperator

~P #[Q • (R " S) ]

[Q " (P # R)] # S

(P # Q) #(R " S)

The # is the mainoperator

"

CONDITIONALS

Symbolizing Conditionals

If Lisa is identical to an immaterial soul, then Lisa is essentiallyinvisible.

L: Lisa is identical to an immaterial soul

I: Lisa is essentially invisible

L " I

If Lisa is identical to a material body, then Lisa is not essentiallyinvisible.

M: Lisa is identical to a material body

I: Lisa is essentially invisible

M " ~I

Some Stylistic variants of 'if-then'

If Gizmo is a cat, then Gizmo is a mammal

! Gizmo is a cat only if Gizmo is a mammal.

! Assuming that Gizmo is a cat, Gizmo is a mammal.

! Gizmo is a mammal if Gizmo is a cat

G: Gizmo is a cat; M: Gizmo is a mammal

G " M

(Note: there are other stylistic variants. See p. 286)

A note on ‘only if’

The term “Only if” (unlike “ if ”) introduces a consequent;the antecedent precedes the “only if”

Remember…

ANTECEDENT only if CONSEQUENT

The term “only if” intuitively (naturally) introduces anecessary condition (or a requirement). Since theconsequent of a conditional is a necessary condition for theantecedent, it’s a bit easier to see how “only if” introducesa consequent.

Sufficient and Necessary Conditions

Sufficient Conditions

" ‘P " Q’ is claiming thatthe occurrence of P issufficient condition for Q.

" A sufficient condition is acondition that guaranteesthat a statement is true (orthat a phenomenon willoccur).

Necessary Conditions

" ‘P " Q’ is also claimingthat the occurrence of Q isa necessary condition forP.

" A necessary condition is acondition that, if lacking,guarantees that astatement is false (or thata phenomenon will notoccur).

Some Examples

--If Alex knows he has hands,then Alex believes he hashands.

Knowing something issufficient for believing it.

--Alex knows he has hands onlyif Alex has good reason tobelieve he has hands.

Having good reason tobelieve something is anecessary condition onhaving knowledge of it.

--Given that one has aconscious pain, one is aware ofthe pain.

Being conscious of pain issufficient for being aware ofpain.

--You can legally drink only ifyou are at least 21.

Being at least 21 is anecessary condition onbeing able to legally drink.

These are all Conditionals

P " Q

P " ~(Q # R)

(P " Q) " (Q " P)

The " is the mainoperator

~P "[Q # (R " S) ]

[Q • (P # R)] " S

(P • Q) " (R " S)

The " is the mainoperator

“Unless” can be translated by the " aswell as the #

Depending on the intent of the speaker,

'Alfred will not pass tomorrow unless Alfred will studytonight' could read:

‘If Alfred will study tonight, then it’s false that he won’tpass tomorrow (i.e. he will pass)’ S" ~~P (or S " P).

This corresponds to the exclusive reading of 'or' in that theintent is not to say that Alfred could very well study andnot pass. For if he studies tonight, he will pass tomorrow.

“Unless” can be translated by the " aswell as the #

'Alfred will not pass tomorrow unless Alfred will studytonight' could read:

‘If Alfred will not study tonight, then Alfred will not passtomorrow’ ~S " ~P

This corresponds to the inclusive reading of 'or' in that itleaves open the possibility that he could study and notpass. The conditional just says that if he doesn't studytonight, he won't pass. It doesn't automatically followfrom this that if he does study, he will pass.

“Unless” can be translated by the " aswell as the #

Since we are sticking with an inclusive reading of ‘or’(and‘unless)’,

p unless q (where p and q stand for any statement,compound or atomic) should be symbolized as:

~q " p

!

Biconditionals

Symbolizing Biconditionals

Leslie is in her 30’s if and only if Leslie isbetween the ages of 30-39.

L: Leslie is in her 30’s

A: Leslie is between the ages of 30-39

L! A

Symbolizing Biconditionals

Jon is a bachelor if and only if Jon is not marriedand Jon is a male.

B: Jon is a bachelor

M: Jon is a male

R: Jon is married

Q: How would you symbolize this?

Symbolizing Biconditionals

Jon is a bachelor if and only if Jon is not marriedand Jon is a male.

B: Jon is a bachelor

M: Jon is a male

R: Jon is married

B ! (~R • M)

Stylistic variant of “if and only if”

Leslie is in her 30’s just in case Leslie is between theages of 30-39.

L: Leslie is in her 30’s

A: Leslie is between the ages of 30-39

M ! A

These are all Biconditionals

P ! Q

P ! ~(Q • R)

(P " Q) ! (Q • P)

~P ! [Q • (R " S) ]

[Q " (P # R)] ! S

(P # Q) ! (R " S)