SYMBOLIC LOGIC UNIT 3: COMPUTING TRUTH VALUES Truth Values The - - PowerPoint PPT Presentation

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SYMBOLIC LOGIC UNIT 3: COMPUTING TRUTH VALUES Truth Values The - - PowerPoint PPT Presentation

Jan 19, 2023 249 likes •735 views

SYMBOLIC LOGIC UNIT 3: COMPUTING TRUTH VALUES Truth Values The truth value of a formula/sentence is always (at least in this class) either true (T) or false (F). Notes: Truth values has nothing to do with how valuable a sentence is.

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SYMBOLIC LOGIC UNIT 3: COMPUTING TRUTH VALUES

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Truth Values The truth value of a formula/sentence is always (at least in this class) either true (T) or false (F). Notes: Truth values has nothing to do with how valuable a sentence is. “Truth value” is a technical term, which derives from the mathematical concept of the value (output) of a function.

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Truth-Functional Operators If an operator is truth functional, then we can determine the truth value of any formula in which it is the major

  • perator just by knowing the truth values of its most

complex subcomponent formula(s). In particular: If all of the operators in a compound sentence are truth functional, then the truth or falsity of this sentence can always be determined just by knowing the truth or falsity

  • f its component parts.
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Truth-Tables If a sentential operator is truth-functional, then we can easily compute the truth value of any formula in which it is the major operator if we know the truth values of its parts.

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CONJUNCTION p q (p • q) T T T T F F F T F F F F DISJUNCTION p q (p ∨ q) T T T T F T F T T F F F CONDITIONAL p q (p ⊃ q) T T T T F F F T T F F T BICONDITIONAL p q (p ≡ q) T T T T F F F T F F F T NEGATION p ~p T F F T

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CONJUNCTION p q (p • q) T T T T F F F T F F F F

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DISJUNCTION p q (p ∨ q) T T T T F T F T T F F F

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BICONDITIONAL p q (p ≡ q) T T T T F F F T F F F T

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NEGATION p ~p T F F T

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CONDITIONAL p q (p ⊃ q) T T T T F F F T T F F T

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How to Compute Truth Values

(A • X) ∨ (B • Y)

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(A • X) ∨ (B • Y)

Step 1: write the truth values for the simple components above them. T T F F

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(A • X) ∨ (B • Y)

Step 2: write the truth values of the simplest subformulas beneath their major operators. Use arrows to represent dependencies. T T F F F F

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(A • X) ∨ (B • Y)

Step 3: Continue down the tree until you get to the major operator. T T F F F F F

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SYMBOLIC LOGIC UNIT 4: SYMBOLIZING ENGLISH SENTENCES

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“Since we have only the five operators and since English is an extremely rich and complex language, with any number of ways of forming compound sentences, there will necessarily be a fair bit of “squeezing” involved to get the English to fit our simple logical language. Something, sometimes a good bit, is bound to be lost in translation.”

—Klenk, pp.51–52

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Steps for Symbolizing in Sentential Logic 1. Identify the simple sentences and operators.

  • 2. Determine which of the operators are truth

functional.

  • 3. Identify the major operator.
  • If it is truth functional, replace it with the relevant

symbol.

  • If not, represent the whole sentence as a single

letter.

  • 4. Continue step 3 for subformulas.
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p ⊃ q

antecedent consequent sufficient for necessary for

CONDITIONAL p q (p ⊃ q) T T T T F F F T T F F T

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if

  • nly if

provided that unless Four versions of ⊃ in English

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  • 1. You’ll pass the course if you’ll learn logic.
  • 2. If you’ll pass the course, you’ll learn logic.

You’ll pass the course. ≡ P You’ll learn logic ≡ L

Four versions of ⊃ in English

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  • 1. You’ll pass the course only if you learn logic.
  • 2. Only if you’ll pass the course will you learn logic.

You’ll pass the course. ≡ P You learn logic ≡ L

Four versions of ⊃ in English

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You’ll pass the course provided that you learn logic. Provided that you pass the course, you’ll learn logic. You’ll pass the course. ≡ P You learn logic ≡ L

Four versions of ⊃ in English

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You’ll pass the course unless you learn logic. Unless you pass the course, you’ll learn logic. You’ll pass the course. ≡ P You learn logic ≡ L

Four versions of ⊃ in English

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