INTRO LOGIC
Chapter 2
DAY 03
Sentential Logic
1
Schedule for Unit 1
ü Day 1
Intro
ü 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
3
Review
An argument is valid or invalid
purely in virtue of its form.
warm-up
Form is a function of the arrangement
of the terms in the argument,
where the
LOGICAL TERMS play a primary role.
40% of
Exam 1
60% of
Exam 1
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Classical Syllogistic Logic
Logical terms
all
some
no
are
not
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.
Example Arguments
all X are Y
all Y are Z
/ all X are Z
all X are Y
no Y are Z
/ no X are Z
statement1
all X are Y
some X are not Z
/ some Y are not Z
connective
statement2
statement3
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Example 1
Sentential Logic
In sentential logic
the logical terms are
S1
AND
S2
statement connectives
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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Examples – 2-place
1-Place, 2-Place, …
a 1-place connective has 1 blank
a 2-place connective has 2 blanks
a 3-place connective has 3 blanks
etc.
IF
S1
AND
S2
S1
OR
S2
S1
IF
S2
S1
ONLY IF
S2
S1
UNLESS
S2
S1
THEN
S2
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Examples – 1-place
IT IS FALSE THAT
S
IT IS POSSIBLE THAT
S
Jay BELIEVES THAT
S
Kay HOPES THAT
S
Examples – 3-place
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S1
IF
S2
OTHERWISE
S3
S1
UNLESS
S2
IN WHICH CASE
S3
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3
Atoms and Molecules
Truth-Values
the truth-value of a true statement is T
A compound (molecular) statement is
one that is constructed from one or more smaller
statements by the application of a statement
connective.
the truth-value of a false statement is F
A simple (atomic) statement is
one that is not constructed out of smaller
statements by the application of a statement
connective.
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A Simplification
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Truth-Functional
To say that a connective is
truth-functional is to say that
the truth-value of any compound statement
produced by that connective
is a function of the truth-values
of its immediate parts.
Intro Logic is not concerned
with all connectives,
but only special ones – namely…
truth-functional
connectives
the whole is merely the sum of its parts
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Abbreviation Scheme
Terminology
The symbol ‘&’ is called
ampersand,
which is a stylized way of writing
the Latin word ‘et’,
which means “and”.
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
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Terminology (cont)
expression
abbreviation
it is raining
R
it is sleeting
S
the word ‘ampersand’ is a children’s pronunciation
of the original word
and
&
and per se and
it is raining and it is sleeting
(R&S)
R&S is called the conjunction of R and S.
R and S are individually called conjuncts.
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Conjunction is truth-functional
Example 2 – Disjunction (‘or’)
R
S
R&S
expression
abbreviation
case 1
T
T
T
it is raining
R
case 2
T
F
F
it is sleeting
S
case 3
F
T
F
or
∨
case 4
F
F
F
it is raining or it is sleeting
(R∨S)
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Slogan
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Terminology
A conjunction d&e is true
if and only if
both conjuncts d and e are true.
The symbol ‘∨’ is called wedge,
which is a stylized way of writing the letter ‘v’,
which initializes the Latin word ‘vel’,
which means “or”.
A conjunction d&e is true
if both conjuncts d and e are true; otherwise, it is false.
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
Disjunction is truth-functional
R
S
R´S
case 1
T
T
T
case 2
T
F
T
would you like a baked potato,
OR French fries?
case 3
F
T
T
would you like coffee or dessert?
case 4
F
F
F
would you like soup, OR salad?
would you like cream or sugar?
inclusive ‘or’
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Exclusive ‘or’ vs. Inclusive ‘or’
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Slogan
exclusive ‘or’ soup OR salad
inclusive ‘or’ cream or sugar
A disjunction d∨e is true
if and only if
at least one disjunct d or e is true.
Latin has two words:
‘aut’ is exclusive ‘or’
‘vel’ is inclusive ‘or’
A disjunction d∨e is false
if both disjuncts d and e are false;
otherwise, it is true.
Legalistic English has the word ‘and/or’
Logic concentrates on inclusive ‘or’.
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a Connective that is
not Truth-Functional
Terminology
R
S
R because S
S because R
T
T
???
???
T
F
F
F
F
T
F
F
F
F
F
F
The symbol ‘;’ is called “tilde”
(as in ‘matilda’);
which is a highly stylized way of
writing the letter ‘N’,
which is short for ‘not’.
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’)
expression
abbreviation
it is raining
R
not
~
it is not raining
~R
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Negation is truth-functional
if
if
R is true, then ;R is false
R is false, then ;R is true
R and ;R have
opposite truth-values
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Example 4 – ‘if...then...’
Aside
my car runs out of gas
R
the prefix ‘ante’ means ‘before’
my car stops
S
other words that contain ‘ante’
if… then…
²
if my car runs out of gas,
then my car stops
(R²S)
if my car stops,
then my car runs out of gas
(S²R)
ante
antechamber
antediluvian
antebellum
ante meridian (a.m.)
antipasto (Italian form)
R²S is not equivalent to S²R.
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Terminology
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Non-Truth-Functional ‘If-Then’
A²C is called a conditional (of A and C).
I live in Los Angeles
L
A is called the antecedent.
I live in New York City
N
I live in California
C
if I lived in L.A.,
then I would live in CAL
if I lived in NYC,
then I would live in CAL
L²C
C is called the consequent.
if antecedent, then consequent
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N²C
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NOT TRUTH-FUNCTIONAL!
I live in LA
I live in Cal
L²C
F
F
T
I live in NYC
I live in Cal
N²C
F
F
F
Truth-Functional version of ‘if-then’
R
S
R²S
case 1
T
T
T
case 2
T
F
F
case 3
F
T
T
case 4
F
F
T
in one case "adding" F and F produces T
in one case "adding" F and F produces F
true by “default”
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Truth-Functional ‘If-Then’
it rains
R
I shut the windows
S
if it rains,
then I (will) shut the windows
R²S
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The Oddness of Cases 3 and 4
If you promise to shut the windows
IF 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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