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Syntax and semantics of propositional logic

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Lecture introducing propositional logic, Phil 57 section 3 ("Logic and Critical Reasoning"), San Jose State University, Fall 2010.

Lecture introducing propositional logic, Phil 57 section 3 ("Logic and Critical Reasoning"), San Jose State University, Fall 2010.

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  • 1. The Syntax and Semantics of Propositional Logic Phil 57 section 3 San Jose State University Fall 2010
  • 2. Valid arguments are truth-preserving
    • If premises are true, conclusion must be true.
    • Validity is a formal property – not a matter of content or context.
    • How to test whether an argument is valid?
    • Propositional Logic (aka Sentential Logic)
  • 3. PL as a formal system to test arguments:
    • Step 1: Identify argument “in the wild” (in a natural language, like English)
    • Step 2: Translate the argument into PL
    • Step 3: Use formal test procedure within PL to determine whether argument is valid
    • Note that good translation is crucial!
  • 4. Important features of PL
    • Symbols (to capture claims and logical connection between claims)
    • Syntax (the rules for how to take generate complex claims from simple ones)
    • Semantics (the meanings of the atomic units, and rules governing how meanings of atomic units are put together to form complex meanings)
  • 5. Syntax of PL
    • Using logical connectives and operators (which connect or operate on propositions)
    • Symbols:
    • Use letters (P, Q, R, … X, Y, Z) to stand for specific statements
    • Unary propositional operator: ~
    • Binary propositional connectives:  ,  ,  , 
    • Grouping symbols: ( ), [ ]
  • 6. Syntax of PL
    • Negation; not : ~ ~P
    • Conjunction; and :  P  Q
    • Disjunction; or :  P  Q
    • Material conditional; if … then .. :  P  Q
    • Biconditional: … if and only if … :  P  Q
  • 7. “ Good grammar” in PL: well-formed formula (wff)
    • Every statement letter P, … Z is a well-formed formula (wff)
    • If p and q are wffs, then so are:
    • (i) ~p
    • (ii) (p  q)
    • (iii) (p  q)
    • (iv) (p  q)
    • (v) (p  q)
    • (3) Nothing is a wff unless rules (1) and (2) imply that it is.
  • 8. Syntax of PL
    • Strings that are not wffs:
    • (P~Q)
    • (  QP)
    • (  R)
    • Strings that are wffs:
    • ((P  Q)  R)
    • ~(X  (Y  Z))
  • 9. Syntax of PL
    • In PL, every compound formula is one of the following:
    • negation
    • conjunction
    • disjunction
    • conditional
    • biconditional To determine which one, isolate main connective or operator.
  • 10. Syntax of PL
    • (P  Q) conjunction
    • ((P  Q)  R) biconditional
    • (Y  Z) conditional
    • (X  (Y  Z)) disjunction
    • ~(X  (Y  Z)) negation
  • 11. Syntax of PL
    • By convention, we can drop the outermost set of parentheses if the main connective is not unary (~)
    • (P  Q)  R
    • Y  Z
    • X  (Y  Z)
    • ~(X  (Y  Z))
  • 12. Syntax of PL
    • Important note:
    • ~(P  Q)
    • is not equivalent to
    • ~P  Q
  • 13. Semantics of PL
    • Semantic rules of PL tell us how the meaning of its constituent parts, and their mode of combination, determine the meaning of a compound statement.
    • Logical operators in PL determine what the truth-values of compound statements are depending on the truth-values of the formulae in the compound.
  • 14. Semantics of PL
    • Logical operators defined by truth-tables .
    • (T= true, F=false)
    • Negation:
    P ~P T F F T
  • 15. Semantics of PL
    • Conjunction:
    P Q P  Q T T T T F F F T F F F F
  • 16. Semantics of PL
    • Disjunction:
    P Q P  Q T T T T F T F T T F F F
  • 17. Semantics of PL
    • Material conditional:
    P Q P  Q T T T T F F F T T F F T
  • 18. Semantics of PL
    • Biconditional:
    P Q P  Q T T T T F F F T F F F T