Unit 1-Lecture 5：Quantifiers & Predicate Logic

1 Predicate

Definition: Propositions with variables.

2 Quantifier

2.1 Definition

• An assertion that a predicate is always true is called a universal quantification. It’s upside-down A, for all, acts like an AND.
• An assertion that a predicate is sometimes true is an existential quantification. It’s backwards E– there exists– acts like an OR.

2.2 Mixing and Order of Quantifier

• Problems with translating A E quantifiers and E A quantifiers into English
• Translation from English into math is not something that can be done in a mindless mechanical way. Sometimes, the quantifiers really are meant to go the other way from the way that they literally appear.

2.3 Variables Over One Domain

• When all the variables in a formula are understood to take values from the same nonempty set, the unnamed nonempty set that x and y range over is called the domain of discourse, or just plain domain, of the formula.

2.4 Negating Quantifier

NOT(∀x.P(x)) IFF ∃x.NOT(P(x))

NOT(∃x.P(x)) IFF ∀x.NOT(P(x))

2.5 Validity and Satisfiable

• A predicate calculus formula is valid when it’s true for all possible domains of discourse that the variables range over.
• ∃x∀y.P(x,y) IMPLIES ∀y∃x.P(x,y)
• But this is not true:
  
  ∀x∃y.P(x,y) IMPLIES ∃y∀x.P(x,y)

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Reference

[1] Lehman E, Leighton F H, Meyer A R. Mathematics for Computer Science[J]. 2015.