Formal Methods 101: Second-Order (and Higher-Order) Logic

Introduction:

  • Second-Order Logic extends First-Order Logic to Quantify Sets. Whereas first-order logic may only quantify Variables in a domain of discourse (set), Second-Order logic may also quantify of sets themselves.
  • Example SOL Expression:
    • (∃T)(∀Q)(∀D)(∃A)((TQ) ∧ (A, D∈T) ∧ ((C(x,y) ∧ ¬D) → ((A ∧ E(t,z)) ∨ F)))
    • This is read there exists a set T, for all sets Q, for all D, there exists an A, such that set T is a proper subset of a set from Q, and A and D are elements of set T, and C and not D, implies A and E or F.
  • Second-Order Logics is even MORE expressive. This can be used to build even MORE extensive complex models of the domains of discourse.
  • Note: Higher-order logics (3rd, 4th, 5th, etc.) extend the quantification operators to quantifications over quantifications over quantifications…this is analogous to types of types of types… in type-theory. Hence the connection to programming languages types.

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