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Presentation: "Truth Tables – Sentences". Introductory Logic PHI 120. Homework. Review WFFs Can 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

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Homework
Homework

  • Review

    • WFFs

      • Can 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”


  • In class
    In Class

    Have in hand

    Truth Tables HandoutSee especially “Building Truth Tables” section


    Review logical form

    Sentences (WFFs)

    Review – Logical Form


    Well formed formulas
    Well-formed Formulas

    • Simple WFFs

      • P, Q, R, S, ….

    • Complex WFFs

      • Negation~Φ

      • ConjunctionΦ&Ψ

      • DisjunctionΦvΨ

      • ConditionalΦ->Ψ

      • BiconditionalΦ<->Ψ

      • and nothing else

    Unary Structure

    Binary Structure


    The concept of truth value

    Truth Tables

    The Concept of Truth Value


    Theorem of the logic
    Theorem of the Logic

    Any statement (WFF) is either True or False

    T v ~T

    • This is a theorem of logic

      • Theorems are tautologies

      • Tautologies are necessarily true

    “A statement is true.” = T


    Theorem of the logic1
    Theorem of the Logic

    Any statement (WFF) is either True or False

    Φ v ~Φ

    • This is a theorem of logic

      • Theorems are tautologies

      • Tautologies are necessarily true


    Theorem of the logic2
    Theorem of the Logic

    Any statement (WFF) is either True or False

    P v ~P

    • This is a theorem of logic

      • Theorems are tautologies

      • Tautologies are necessarily true


    Theorem of the logic3
    Theorem of the Logic

    Any statement (WFF) is either True or False

    (P&~Q) v ~(P&~Q)

    • This is a theorem of logic

      • Theorems are tautologies

      • Tautologies are necessarily true


    The key to recognizing sentences
    The Key to Recognizing Sentences

    • Which connective is the weakest link in a sequence of symbols?

      (or as I like to ask)

    • Where can you most easily bend the sentence?

    Strongest

    ~

    &and/orv

    ->

    <->

    Weakest

    See

    page

    9


    What kind of sentence
    What kind of sentence?

    ~P

    ~P & ~Q

    P v Q -> R

    P v Q <-> R -> P

    negation: ~Φ

    conjunction: Φ&Ψ

    conditional: Φ->Ψ

    biconditional: Φ<->Ψ

    “the main connective”

    Metaphor of the Binding of a Book


    Building truth tables

    Sentences (WFFs)

    Building Truth Tables


    The simple
    The Simple

    • The truth-value of an atomic sentence


    The simple1
    The Simple

    • The truth-value of an atomic sentence


    Simple negation
    Simple Negation

    • The truth-value of a simple negation

    A negation (~) takes the opposite value of the statement being negated.


    Simple negation1
    Simple Negation

    • The truth-value of a simple negation

    A negation (~) takes the opposite value of the statement being negated.


    Building a truth table
    Building a Truth Table

    • Read the sentence

      P v ~P


    Building a truth table1
    Building a Truth Table

    • Read the sentence

      P v ~P

      The wedge is the main connective.

      Hence this is a disjunction.

    Φ v ~Φ

    P v ~P is an instance of our theorem


    Step 1 p v p
    Step 1P v ~ P

    • A Truth Table has two main columns

      • Left main column: ATOMIC SENTENCES

      • Right column: the WFF.

        • This row represents a header row.


    Step 2 p v p
    Step 2P v ~ P

    • Determine the number of rows for the WFF:

      • Rows = 2 (power of simple statements)


    Step 3 p v p
    Step 3P v ~ P

    • Fill in left main column first.


    Step 4 p v p
    Step 4P v ~ P

    • Right main column

      • assign truth-values for negation of simple statements.


    Step 4 p v p1
    Step 4P v ~ P

    • Right main column

      • assign truth-values for negation of simple statements.

    Notice that only one connective remains.


    Skip to last step p v p
    Skip to Last StepP v ~ P

    • Assign truth-values for the remaining wedge.

    See bottom of Truth Tables Handout


    Step 6b p v p
    Step 6bP v ~ P

    • Right main column

      • Main (or governing) connective

    A disjunction (a “v” statement) is FALSE only when both disjuncts are F.


    Step 5 6 p v p
    Step 5 & 6P v ~ P

    • Right main column

      • Main (or governing) connective

    A disjunction (a “v” statement) is FALSE only when both disjuncts are F.


    Step 5 6 p v p1
    Step 5 & 6P v ~ P

    • Right main column

      • Main (or governing) connective

    A disjunction (a “v” statement) is FALSE only when both disjuncts are F.


    Theorems are necessarily true
    Theorems are Necessarily True

    • This WFF is a Tautology.

      • regardless of whether P is true.

      • regardless of whether P is false.


    Homework1
    Homework

    • Review

      • WFFs

        • Can 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”


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