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INTRO LOGICDAY 03
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Schedule for Unit 1
ü
ü IntroDay 1
Chapter 4Day 7
EXAM #1Day 8
Chapter 4Day 6
Chapter 4Day 5
Chapter 3Day 4
Chapter 2Day 3
Chapter 1Day 2
40% of Exam 1
60% of Exam 1
warm-up
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Chapter 2Sentential Logic
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Review
An argument is valid or invalidpurely 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 Logic
Logical terms Example Arguments
allsomenoarenot
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
In sentential logicthe logical terms are
statement connectives
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What is a Statement Connective?A statement connective (or simply, a connective)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 1
AND S2S1
AND grass is greensnow is white
AND it is sleetingit is raining
AND 3+3 = 62+2 = 4
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1-Place, 2-Place, …
a 1-place connective has 1 blanka 2-place connective has 2 blanksa 3-place connective has 3 blanksetc.
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Examples – 1-place
SIT IS FALSE THAT
SIT IS POSSIBLE THAT
SJay BELIEVES THAT
SKay HOPES THAT
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Examples – 2-place
AND S2S1
OR S2S1
IF S2S1
ONLY IF S2S1
IF THEN S2S1
UNLESS S2S1
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Examples – 3-place
S2 OTHERWISEIF S3S1
S2 IN WHICH CASEUNLESS S3S1
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Atoms and Molecules
A compound (molecular) statement isone that is constructed from one or more smaller statements by the application of a statement connective.
A simple (atomic) statement isone that is not constructed out of smaller statements by the application of a statement connective.
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A Simplification
Intro Logic is not concernedwith all connectives,
but only special ones – namely…
truth-functional connectives
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Truth-Values
the truth-value of a true statement is T
the truth-value of a false statement is F
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Truth-Functional
To say that a connective istruth-functional is to say thatthe truth-value of any compound statementproduced by that connectiveis a function of the truth-valuesof its immediate parts.
the whole is merely the sum of its parts
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Abbreviation 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 – Conjunction
( R & S )it is raining and it is sleeting
&and
Sit is sleeting
Rit is raining
abbreviationexpression
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Terminology
The symbol ‘&’ is called ampersand,
which is a stylized way of writingthe Latin word ‘et’,
which means “and”.
& & & & & &
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Terminology (cont)
R&S is called the conjunction of R and S.
R and S are individually called conjuncts.
the word ‘ampersand’ is a children’s pronunciation of the original word
and per se and
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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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Slogan
A conjunction A&B is trueif and only if
both conjuncts A and B are true.
A conjunction A&B is true if both conjuncts A and B are true; otherwise, it is false.
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Example 2 – Disjunction (‘or’)
( R ∨ S )it is raining or it is sleeting
∨or
Sit is sleeting
Rit is raining
abbreviationexpression
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Terminology
The symbol ‘∨’ is called wedge, which is a stylized way of writing the letter ‘v’,which initializes the Latin word ‘vel’,which means “or”.
R∨S is called the disjunction of R and S.
R and S are individually called disjuncts.
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Exclusive 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’ soup OR saladinclusive ‘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
F
F
T
T
R
F
T
F
T
S
case 4
case 3
case 2
case 1
R∨S
F
T
T
T
inclusive ‘or’
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Slogan
A disjunction A∨B is trueif and only if
at least one disjunct A or B is true.
A disjunction A∨B is false if both disjuncts A and B are false;
otherwise, it is true.
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a Connective that is not 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’)
~Rit is not raining
~not
Rit is raining
abbreviationexpression
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Terminology
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-functional
if R is true, then ∼R is falseif R is false, then ∼R is true
R and ∼R have opposite truth-values
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Example 4 – ‘if...then...’
( R → S )if my car runs out of gas, then my car stops
( S → R )if my car stops,then my car runs out of gas
→if… then…
Smy car stops
Rmy car runs out of gas
R→S is not equivalent to S→R.
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Terminology
A→C is called a conditional (of A and C).
A is called the antecedent.
if antecedent, then consequent
C is called the consequent.
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Aside
the prefix ‘ante’ means ‘before’
other words that contain ‘ante’
anteantechamberantediluvianantebellumante meridian (a.m.)antipasto (Italian form)
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Non-Truth-Functional ‘If-Then’
L → C if I lived in L.A., then I would live in CAL
CI live in California
N → Cif I lived in NYC, then I would live in CAL
NI live in New York City
LI live in Los Angeles
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NOT TRUTH-FUNCTIONAL!
T
L→C
F
I live in LA
F
I live in Cal
F
N→C
F
I live in NYC
F
I live in Cal
in one case "adding" F and F produces F
in one case "adding" F and F produces T
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Truth-Functional ‘If-Then’
R → Sif it rains,then I (will) shut the windows
SI shut the windows
Rit rains
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Truth-Functional version of ‘if-then’
F
F
T
T
R
F
T
F
T
S
case 4
case 3
case 2
case 1
R→S
T
T
F
T
true by “default”
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The 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".
