Logic: Lecture 5, 18 August 2015
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FOL syntax
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First order language L

  Sets R,F,C of relation, function, constant symbols
  Each symbol in R and F has an "arity"

  Variables:  Var = [v1,v2,...}

  Terms
    "Names" of elements
    c in C, x in Var are terms
    f(t1,...,tn) is a term

  Closed terms: no variables

Atomic formulas

  r(t1,...,tn)
  t1 = t2

Formulas of L

  Atomic formulas
  ~phi, phi or psi
  Ex phi

Notation

  Ax phi is ~Ex ~phi

FOL semantics
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L-structures and interpretations

  L-structure  M = (S,i)
    S is a set
    i maps R, F, C to relations, functions, constants over S

  Intepretation  I = (M,sigma)
    sigma : Var -> S     assignment to variables

  Notation about assignments
    sigma' = sigma[x1 |-> v1, ... , xn |-> vn]
    sigma' is sigma updated with new values for {x1,..,xn}

    I = (M,sigma)
    I[x1 |-> v1, ... , xn |-> vn] = (M,sigma[x1 |-> v1, ... , xn |-> vn])

  Interpretation of terms in I = (M,sigma)
    I(c) = i(c)
    I(x) = sigma(x)
    I(f(t1,..,tn)) = i(f)(I(t1),...,I(tn))

Satisfaction relation

  I |= phi
  I |= t1 = t2     if I(t1) = I(t2)
  I |= r(t1,..,tm) if (I(t1),...,I(tm)) is in i(r)
  I |= ~phi        if not(I |= phi)
  I |= phi or psi  if I |= phi or I |= psi
  I |= Ex phi      if there is a in S such that I[x|->a] |= phi

Check that

  I |= Ax phi      if for every a in S, I[x|->a] |= phi

Free and bound variables

  x is bound if it is within the scope of a quantifier
  x is free if it is not bound

  Define FV(phi) inductively.

  Note that the two sets are not complementary.  For instance, in
  (x or Ex x), x is both bound and free.  We can always rename
  bound variables to avoid this --- e.g. (x or Ey y).

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