Lecture_15_09 - Lecture 15 Lecture 15 Syntax Syntax...

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Unformatted text preview: Lecture 15 Lecture 15 Syntax Syntax Signature: constant symbols, relation symbols, e.g. H (unary), R (binary), c (constant). Terms: C (constants) and V (variables), and more if we allow function symbols. Language is defined inductively Base case: atoms built from terms and relation symbols, e.g. R(c,x) and H(x). How about R(H(x),y)? Syntax Syntax Inductive clauses (one for each connective): If ϕ ∈L, then ¬ ϕ ∈L If ϕ ,ψ∈L, then ϕ ∧ L ψ∈ If ϕ ∈L and x ∈V, then ∀x ϕ ∈L If ϕ ∈L and x ∈V, then ∃ x ϕ ∈L Bound and Free Variables Bound and Free Variables Scope Bound occurrences Free occurrences Sentences Definability Definability If M is an interpretation and ϕ is a formula with x as its only free variable, then ϕ defines the following subset of D: {a ∈ D : M ╞ ϕ (a)} Alert: a ∈ D need not be a symbol in our first­order language. So, if ‘ϕ (a)’ is not a wff, then what is meant by M ╞ ϕ (a)? This idea extends to subsets of Dn. Axiomatizability Axiomatizability If ϕ is a sentence in L, then ϕ partitions the class of all L­interpretations into those interpretations that satisfy ϕ and those that do not. If S is a collection of L­interpretations, then S is axiomatizable if there is a collection of sentences Γ ⊆ L such that Mod(Γ) = S. ...
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  • Spring '11
  • H
  • Syntax Syntax, Axiomatizability   Axiomatizability, Definability  Definability, relation symbols, Free Variables Bound

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