Discrete Structures

Lecture Notes, 20 Sep 2010

Last Update: 20 September 2010

Note: NEW or UPDATED material is highlighted

Note: A username and password may be required to access certain documents. Please contact Bill Rapaport.

Index to all lecture notes
…Previous lecture

§1.3: Predicates & Quantifiers (cont'd)

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


    1. Base Cases:

      1. Atomic propositions (p, q, r, …) of propositional logic are well-formed (atomic) propositions of FOL.

      2. If t1,…,tn are terms (NPs)
        & if R is an n-place predicate,
        then R(t1,…,tn) 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,

        1. ¬A
        2. (AB)
        3. (AB)
        4. (AB)
        5. (AB)
        6. (AB)
        7. v[A]
        8. v[A]

        are WF (molecular) propositions of FOL

  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

  3. Syntax & Semantics of the Quantifiers:

    where v is any variable
    & where A is any proposition,
      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 →!

    all objects in the domain are such that tval(A)=T

    e.g., tval(∀xR(x))=T
    all objects in the domain have the property named by R

    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 ∧!!

    at least one object in the domain is such that tval(A)=T

    e.g., tval(∃xR(x))=T
    at least one obj in the domain has the property named by R

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

      W={1,2,3,…} N={0,1,2,3,…} Z={…–3,–2,–1,0,1,2,3,…}
    x[x > 0] TF
    (because ¬(0>0))
    x[x ≥ 0] TTF


  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)

  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)

Next lecture…

Text copyright © 2010 by William J. Rapaport (rapaport@buffalo.edu)
Cartoon links and screen-captures appear here for your enjoyment. They are not meant to infringe on any copyrights held by the creators. For more information on any cartoon, click on it, or contact me.