Chapter 5 Quantifiers

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Chapter 5 Quantifiers

• Chapter 6 Quantified truth trees
• Chapter 7 Quantified natural deduction
• Chapter 8 Identity and function symbols

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Quantifiers

• 5.1 Constants and quantifiers
• 5.2 Categorical sentence forms
• 5.3 Polyadic predicate
• 5.4 The language Q
• 5.5 Symbolization

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Aristotles Syllogism

• All F are G
• No F are G
• Some F are G
• Some F are not G
• Where F and G are general terms.
• An argument form consists of two premises and a
  conclusion

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Examples

• All F are G
• Some H are F
• Therefore, some H are G
• No F are G
• All H are F
• Therefore, no H are G

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History, and Modern Logic

• Aristotle Sentential reasoning vs. Syllogism
• PL Sentences lacking logical connectives is not
  analyzable.
• Syllogism deal with general terms.
• Modern logic Unified formal treatment of two
  levels of logical analyses.
• Frege and Peirce introduced symbolic quantifiers
  into the representation.

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Modern logical analysis

• Syllogism To be is a monadic predicate
• Proposition logic as a subsystem of quantified
  predicate logic
• Dyadic predicates and polyadic predicates
• Compound and complex formulas

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Examples

• All the beads are either red or blue.
• All the children found either red beads or blue
  beads.
• Some of the boys played with Karen.
• Logical issues
• How many logical components we have here?
• How to make formal representations of them?

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5.1 Constants and Quantifiers

• Constants Proper names (Karen, Tom, Carnegie
  Hall, White house)
• Universal Quantifier ?
• Existential Quantifier ?
• Definitions ?xAx df ?xAx

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Predicate-Argument Structure

• Monadic predicates Properties
• Dyadic predicate Binary relations
• Polyadic Predicates Relations with more then two
  arguments
• Arguments Individual variables
• Predicate-argument structures are open, need to
  be quantified to become statements

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5.2 Categorical sentence forms

• Objects and general domain for arguments
• All F are G For all x, if Fx, then Gx
• Some F are G There is some x, Fx and Gx
• The vs. Truth conditions