PHI 120 Presentation Truth Tables Sentences Homework Review WFFs Can you read sentences correctly Print Truth Tables handout Building TTs Sentences and Sequents Connectives when are they false ID: 419850
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Slide1
Introductory LogicPHI 120
Presentation: "Truth Tables – Sentences"Slide2
HomeworkReviewWFFsCan you read sentences correctly?
Print:
Truth Tables
handout
"Building TTs: Sentences and Sequents"
"Connectives – when are they false"
Allen/Hand
Section 2.1, esp. pages 40-41
p. 47-8: “tautology,” “inconsistency &
contingent sentence”Slide3
In ClassHave in hand
Truth Tables Handout
See especially “Building Truth Tables” sectionSlide4
Review – Logical FormSentences (WFFs)Slide5
Well-formed FormulasSimple WFFsP, Q,
R
,
S
, ….
Complex
WFFs
Negation
~
Φ Conjunction Φ & Ψ Disjunction Φ v Ψ Conditional Φ -> Ψ Biconditional Φ <-> Ψ and nothing else
Binary Structure
Unary StructureSlide6
The Concept of Truth ValueTruth TablesSlide7
Theorem of the LogicAny statement (WFF) is either True or FalseT v ~TThis is a theorem of logic
Theorems are
tautologies
Tautologies
are necessarily true
“A statement is true.” =
TSlide8
Theorem of the LogicAny statement (WFF) is either True or FalseΦ v ~Φ
This is a theorem of logic
Theorems are
tautologies
Tautologies
are necessarily trueSlide9
Theorem of the LogicAny statement (WFF) is either True or FalseP v ~PThis is a theorem of logic
Theorems are
tautologies
Tautologies
are necessarily trueSlide10
Theorem of the LogicAny statement (WFF) is either True or False(P&~Q) v ~(P&~Q)This is a theorem of logic
Theorems are
tautologies
Tautologies
are necessarily trueSlide11
The Key to Recognizing SentencesWhich connective is the weakest link in a sequence of symbols? (or as I like to ask
)
Where can you
most easily
bend
the sentence?
See
page
9
Strongest
~& and/or v-><->WeakestSlide12
What kind of sentence?~P~P & ~Q
P v Q -> R
P v Q <-> R -> P
negation:
~
Φ
conjunction:
Φ
& Ψconditional: Φ -> Ψbiconditional: Φ <-> Ψ
“the main connective”
Metaphor of the Binding of a BookSlide13
Building Truth TablesSentences (WFFs)Slide14
The SimpleThe truth-value of an atomic sentence
PSlide15
The SimpleThe truth-value of an atomic sentence
P
1
T
2
F
1Slide16
Simple NegationThe truth-value of a simple negation
P
~
P
1
T
2
F
1
2
3
A negation (~) takes the opposite value of the statement being negated.Slide17
Simple NegationThe truth-value of a simple negation
P
~
P
1
T
F
2
F
T
1
2
3
A negation (~) takes the opposite value of the statement being negated.Slide18
Building a Truth TableRead the sentenceP v ~PSlide19
Building a Truth TableRead the sentenceP v ~PThe wedge is the main connective.Hence this is a disjunction.
Φ v ~Φ
P v ~P is an instance of our theoremSlide20
Step 1P v ~ PA Truth Table has two main columns
Left
main column: ATOMIC SENTENCES
Right
column: the WFF.
This row represents a header row.
P
P
v
~PSlide21
Step 2P v ~ PDetermine the number of rows for the WFF:Rows = 2 (power of simple statements)
P
P
v
~
P
1
2Slide22
Step 3P v ~ PFill in left main column first.
P
P
v
~
P
1
T
2
F
1
2
3
4
5Slide23
Step 4P v ~ PRight main columnassign truth-values for
negation of simple statements
.
P
P
v
~
P
1
T
2
F
1
2
3
4
5Slide24
Step 4P v ~ PRight main columnassign truth-values for
negation of simple statements
.
P
P
v
~
P
1
T
F
2
F
T
1
2
3
4
5
Notice that only one connective remains.Slide25
Skip to Last StepP v ~ PAssign truth-values for the remaining wedge.
CONNECTIVES – when they are false
~Φ
A negation is false
if
the statement being negated (Φ) is true
Φ & Ψ
A conjunction is false
if
one or both
of the conjuncts is false
Φ v Ψ
A disjunction is false
only if
both disjuncts
are false
Φ -> Ψ
An conditional is false
only if
antecedent
(Φ)
true
and
consequent
(Ψ) false
Φ <-> Ψ
A biconditional is false
only if
the
two conditions
have a
different
truth value
See bottom of
Truth Tables HandoutSlide26
Step 6bP v ~ PRight main columnMain (or governing) connective
A disjunction (a “v” statement) is FALSE only when both
disjuncts
are F.
P
P
v
~
P
1
T
F
2
F
T
1
2
3
4
5Slide27
Step 5 & 6P v ~ PRight main columnMain (or governing) connective
A disjunction (a “v” statement) is FALSE only when both
disjuncts
are F.
P
P
v
~
P
1
T
T
F
2
F
T
1
2
3
4
5Slide28
Step 5 & 6P v ~ PRight main columnMain (or governing) connective
A disjunction (a “v” statement) is FALSE only when both
disjuncts
are F.
P
P
v
~
P
1
T
T
F
2
F
T
T
1
2
3
4
5Slide29
Theorems are Necessarily TrueThis WFF is a Tautology.
regardless of whether P is true.
regardless of whether P is false.
P
P
v
~
P
1
T
T
F
2
F
T
T
1
2
3
4
5Slide30
HomeworkReviewWFFsCan you read sentences correctly?
Print:
Truth Tables
handout
"Building TTs: Sentences and Sequents"
"Connectives – when are they false"
Allen/Hand
Section 2.1, esp. pages 40-41
p. 47-8: “tautology,” “inconsistency & contingent sentence”