Lecture Notes, 20 Sep 2010

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§1.3: Predicates & Quantifiers (cont'd)

1. (Recursive) Definition of Well-Formed Proposition of FOL:

   Remember:

   • "terms" (or NPs) name or describe objects in the domain;
   • "predicate" (or VPs) name properties of objects or relations among ≥2 objects
   • "variables" (or pronouns) are like variables in programming languages
   1. Base Cases:
      1. Atomic propositions (p, q, r, …) of propositional logic are well-formed (atomic) propositions of FOL.

      2. If t₁,…,t_n are terms (NPs)
         & if R is an n-place predicate,
         then R(t₁,…,t_n) is a WF (atomic) proposition of FOL (a "subatomic" proposition)

   2. Recursive Cases:
      • If A, B are WF (atomic or molecular) propositions of FOL,
        & if v is a variable,
        then:
        1. ¬A
        2. (A ∧ B)
        3. (A ∨ B)
        4. (A ⊕ B)
        5. (A → B)
        6. (A ↔ B)
        7. ∀v[A]
        8. ∃v[A]

        are WF (molecular) propositions of FOL

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2. Definition of Term of FOL:
   1. If v is a variable, then v is a term of FOL.
      E.g., x, y, z, … are terms

   2. If c is a constant, then c is a term of FOL.
      E.g., "fred", "evelyn", 1, √2, … are terms

   3. (another clause to be presented later, maybe…)
   4. Nothing else is a term.
      • In particular, no predicate is a term!
        I.e., if R is a predicate, then R is not a term.

        So: predicates can never appear in term-position;
        i.e., in FOL, predicates cannot be terms of other predicates;
        i.e., predicates cannot appear inside other predicates

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3. Syntax & Semantics of the Quantifiers:

   NAME	SYNTAX	ENGLISH	E.G.	SEMANTICS
   universal quantifier	∀v[A], where v is any variable & where A is any proposition, including: all of the "old" ones & any of these "new" ones; if A is atomic, the brackets can be omitted; e.g., ∀xR(x)	Everything in the domain satisfies A. For all objects v in the domain, A is the case.	All humans are mortal ≡ ∀x[Human(x) → Mortal(x)] Note the use of →!	tval(∀vA)=T iff all objects in the domain are such that tval(A)=T e.g., tval(∀xR(x))=T iff all objects in the domain have the property named by R
   existential quantifier	∃v[A], as above e.g., ∃xR(x)	Something satisfies A There is (or: there exists) an object v in the domain for which A is the case.	Some human is mortal ≡ ∃x[Human(x) ∧ Mortal(x)] Note the use of ∧!!	tval(∃vA)=T iff at least one object in the domain is such that tval(A)=T e.g., tval(∃xR(x))=T iff at least one obj in the domain has the property named by R

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4. The truth value of a quantified proposition depends on the domain:

   proposition	Domain:
   	W={1,2,3,…}	N={0,1,2,3,…}	Z={…–3,–2,–1,0,1,2,3,…}
   ∀x[x > 0]	T	F (because ¬(0>0))	F
   ∀x[x ≥ 0]	T	T	F

   Notes:

   "∀x[x > 0]" can also be written (slightly more grammatically) as: ∀x[>(x, 0)]

   "∀x[x ≥ 0]" can also be written (slightly more grammatically) as: ∀x[>(x, 0) ∨ =(x, 0)]

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5. Finite domains:

   Consider the domain be {0,1,2,3}.
   Then tval(∀xR(x)) = T iff, for all values of x in the domain, tval(R(x)) = T
                                         iff tval(R(0))=T & tval(R(1))=T & tval(R(2))=T & tval(R(3))=T

   ∴ ∀xR(x) ≡ R(0) ∧ R(1) ∧ R(2) ∧ R(3)

   Similarly, we can show that: ∃xR(x) ≡ R(0) ∨ R(1) ∨ R(2) ∨ R(3)

   • ∀xR(x) is like a for-loop!
   • ∃xR(x) is like a search procedure!

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6. De Morgan's Laws for Quantifiers (or: How to Negate a Quantifier):

   In a finite domain, say {0,1}:

   ∀xP(x) ≡ P(0) ∧ P(1).

   ∴ ¬∀xP(x) ≡ ¬(P(0) ∧ P(1))
                       ≡ (¬P(0) ∨ ¬P(1)), by DeMorgan for the negation of ∧
                       ≡ ∃x[¬P(x)]

   Not only is this also true in an infinite domain, but so is this:

   ¬∃xP(x) ≡ ∀x[¬P(x]

   and these:

   ∃xP(x) ≡ ¬∀x¬P(x)
   ∀xP(x) ≡ ¬∃x¬P(x)

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