Transcript Chapter 5 Quantifiers

Chapter 5
Quantifiers
Chapter 6 Quantified truth trees
Chapter 7 Quantified natural deduction
Chapter 8 Identity and function symbols
Quantifiers
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5.1
5.2
5.3
5.4
5.5
Constants and quantifiers
Categorical sentence forms
Polyadic predicate
The language Q
Symbolization
Aristotle’s 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
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
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.
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
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?
5.1 Constants and Quantifiers
• Constants: Proper names (Karen, Tom,
Carnegie Hall, White house)
• Universal Quantifier: 
• Existential Quantifier: 
• Definitions xAx =df ¬ x¬Ax
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
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