INTRO LOGIC

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INTRO LOGICINTRO LOGICDAY 03 DAY 03

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Schedule for Unit 1Schedule for Unit 1

Day 1 Intro

Day 2 Chapter 1

Day 3 Chapter 2

Day 4 Chapter 3

Day 5 Chapter 4

Day 6 Chapter 4

Day 7 Chapter 4

Day 8 EXAM #1

40% of Exam 1

60% of Exam 1

warm-up

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Chapter 2Chapter 2Sentential Logic Sentential Logic

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ReviewReview

An argument is valid or invalid purely in virtue of its form.

Form is a function of the arrangement

of the terms in the argument, where theLOGICAL TERMS play a primary role.

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Classical Syllogistic LogicClassical Syllogistic Logic

Logical terms Example Arguments

all

some

no

are

not

all X are Yall Y are Z/ all X are Z

all X are Yno Y are Z/ no X are Z

all X are Ysome X are not Z/ some Y are not Z

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Sentential Logic Sentential Logic

In sentential logic

the logical terms are

statement connectivesstatement connectives

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What is a Statement Connective?What is a Statement Connective?

A statement connectivestatement connective (or simply, a connectiveconnective)

is an "incomplete" expression –

i.e., an expression with one or more blanks –

such that,

whenever the blanks are filled by statements,

the resulting expression is also a statement.

connective statement2statement1

statement3

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Example 1Example 1

S1 AND S2

snow is white AND grass is green

it is raining AND it is sleeting

2+2 = 4 AND 3+3 = 6

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1-Place, 2-Place, …1-Place, 2-Place, …

a 1-place1-place connective connective has 1 blank

a 2-place2-place connective connective has 2 blanks

a 3-place3-place connective connective has 3 blanks

etc.

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Examples – 1-placeExamples – 1-place

IT IS FALSE THAT S

IT IS POSSIBLE THAT S

Jay BELIEVES THAT S

Kay HOPES THAT S

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S1 AND S2

S1 OR S2

S1 IF S2

S1 ONLY IF S2

IF S1 THEN S2

S1 UNLESS S2

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S1 IF S2 OTHERWISE S3

S1 UNLESS S2 IN WHICH CASE S3

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Atoms and MoleculesAtoms and Molecules

A compoundcompound (molecularmolecular) statement isone that is constructed from one or more smaller statements by the application of a statement connective.

A simplesimple (atomicatomic) statement isone that is not constructed out of smaller statements by the application of a statement connective.

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A SimplificationA Simplification

Intro Logic is not concerned

with all connectives,

but only special ones – namely…

truth-functional connectives

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Truth-ValuesTruth-Values

the truth-value of a true statement is T

the truth-value of a false statement is F

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Truth-FunctionalTruth-Functional

To say that a connective istruth-functionaltruth-functional is to say that

the truth-value of any compound statementproduced by that connective

is a function of the truth-values

of its immediate parts.

the whole is merely the sum of its parts

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Abbreviation SchemeAbbreviation Scheme

1. atomic sentences are abbreviated by upper-case letters (of the Roman alphabet)

2. connectives are abbreviated by special symbols (logograms)

3. compound sentences are abbreviated by algebraic-combinations of 1 and 2

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Example 1 – ConjunctionExample 1 – Conjunction

( R & S )it is raining and it is sleeting

&and

Sit is sleeting

Rit is raining

abbreviationexpression

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TerminologyTerminology

The symbol ‘&’ is called ampersandampersand,

which is a stylized way of writing

the Latin word ‘etet’,

which means “and”.

& & & & & &

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Terminology (cont)Terminology (cont)

R&S is called the conjunctionconjunction of R and S.

R and S are individually called conjunctsconjuncts.

the word ‘ampersandampersand’ is a children’s pronunciation of the original word

and per se and

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Conjunction is truth-functional Conjunction is truth-functional

F

F

T

T

R

F

T

F

T

S

case 4

case 3

case 2

case 1

R&&S

F

F

F

T

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SloganSlogan

A conjunction & is trueif and only if

both conjuncts and are true.

A conjunction & is true if both conjuncts and are true; otherwise, it is false.

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Example 2 – Disjunction (‘or’) Example 2 – Disjunction (‘or’)

( R S )it is raining oror it is sleeting

oror

Sit is sleeting

Rit is raining


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The symbol ‘’ is called wedgewedge,

which is a stylized way of writing the letter ‘v’,

which initializes the Latin word ‘vel’,

which means “or”.

RS is called the disjunctiondisjunction of R and S.

R and S are individually called disjunctsdisjuncts.

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Exclusive Sense vs. Inclusive SenseExclusive Sense vs. Inclusive Sense

would you like soup, OR salad?

would you like coffee or dessert?

would you like a baked potato, OR French fries?

would you like cream or sugar?

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Exclusive ‘or’ vs. Inclusive ‘or’Exclusive ‘or’ vs. Inclusive ‘or’

exclusive ‘or’ soup OR salad

inclusive ‘or’ cream or sugar

Logic concentrates on inclusive ‘or’.

Latin has two words:

‘aut’ is exclusive ‘or’

‘vel’ is inclusive ‘or’

Legalistic English has the word ‘and/or’

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Disjunction is truth-functional Disjunction is truth-functional

F

F

T

T

R

F

T

F

T

S

case 4

case 3

case 2

case 1

RS

F

T

T

T

inclusive ‘or’

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SloganSlogan

A disjunction is trueif and only if

at least one disjunct or is true.

A disjunction is false if both disjuncts and are false;

otherwise, it is true.

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a Connective that is a Connective that is notnot Truth-Functional Truth-Functional

R because S

F

F

T

T

R

F

T

F

T

S S because R

F

F

F

???

F

F

F

???

merely knowing that R and S are both true tells us nothing about whether one is responsible for the other

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Example 3 – Negation (‘not’)Example 3 – Negation (‘not’)

Rit is not raining

not

Rit is raining


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The symbol ‘’ is called “tilde”

(as in ‘matilda’);

which is a highly stylized way of writing the letter ‘N’,

which is short for ‘not’.

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Negation is truth-functionalNegation is truth-functional

if R is true, then R is false

if R is false, then R is true

R and R have opposite truth-values

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Example 4 – ‘if...then...’Example 4 – ‘if...then...’

( R S )ifif my car runs out of gas, thenthen my car stops

( S R )ifif my car stops, thenthen my car runs out of gas

if… then…if… then…

Smy car stops

Rmy car runs out of gas

RS is not equivalent to SR.

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AC is called a conditionalconditional (of A and C).

A is called the antecedentantecedent.

ifif antecedentantecedent, thenthen consequentconsequent

C is called the consequentconsequent.

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AsideAside

the prefix ‘ante’ means ‘before’

other words that contain ‘ante’

ante

antechamber

antediluvian

antebellum

ante meridian (a.m.)

antipasto (Italian form)

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NonNon-Truth-Functional ‘If-Then’-Truth-Functional ‘If-Then’

I live in Los Angeles L

I live in New York City N

I live in California C

if I lived in L.A., then I wouldwould live in CAL

L C

if I lived in NYC, then I wouldwould live in CAL

N C

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NOT TRUTH-FUNCTIONAL!NOT TRUTH-FUNCTIONAL!

I live in LA I live in Cal LC

F F T

I live in NYC I live in Cal NCF F F

in one case "adding" F and F produces T

in one case "adding" F and F produces F

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Truth-Functional ‘If-Then’Truth-Functional ‘If-Then’

it rains R

I shut the windows S

if it rains, then I (will) shut the windows R S

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Truth-Functional version of ‘if-then’Truth-Functional version of ‘if-then’

F

F

T

T

R

F

T

F

T

S

case 4

case 3

case 2

case 1

RS

T

T

F

T

true by “default”

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The Oddness of Cases 3 and 4The Oddness of Cases 3 and 4

If you promise to shut the windowsIF it rains, then only one scenario (case) constitutes breaking your promise –

the scenario in which it rains but you don’t shut the windows.

In case 3 and case 4, you keep your promise "by default".

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THE ENDTHE END

INTRO LOGIC

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