208 Logic as a Tool |= ∀x(x = f (x) → (P (f (f (x))) → P (x))) ∀x∀y (f (x) = y → g (y ) = x) |= ∀z (g (f (z )) = z ) ∀x∀y (f (x) = y → g (y ) = x) |= ∀z (f (g (z )) = z ) ∀x∀y (f (x) = y → g (y ) = x), ∀x∀y (g (x) = g(y ) → x = y ) |= ∀z (f (g (z )) = z ) For more exercises on derivations with equality, on sets, functions, and relations, see Section 5.2.7 (b) (c) (d) (e) 4.5.14 Consider the first-order theory of groups G, consisting of the following axioms, where e is the constant for group identity, ◦ is the (binary) group operation, and is the (unary) inverse operation, and all variables below are assumed universally quantified G1: (x ◦ y ) ◦ z = x ◦ (y ◦ z ) (associativity of ◦) G2: x ◦ e = x (e is a right identity) G3: x ◦ x = e ( is a right inverse operation) Using Resolution with Paramodulation, derive the following as logical consequences from the axioms G1–G3 Again, all variables in the formulae below are assumed universally quantified (a) x ◦ x = e ( is a left inverse operation) (b) e ◦ x = x (e is a left identity) (c) (x ) = x (d) (x ◦ y ) = y ◦ x (e) x =y◦z →y =x◦z ∧z =y ◦x (f) Adding to G the axiom x ◦ x = e, derive the commutativity law x ◦ y = y ◦ x (g∗∗) Adding to G the axiom x ◦ x ◦ x = e, derive the identity ((x, y ), y ) = e where (x, y ) = x ◦ y ◦ x ◦ y Jacques Herbrand (12.02.1908–27.07.1931) was a French mathematician and logician who made groundbreaking contributions to logic and automated reasoning, even though he died at the age of only 23 years old Herbrand worked in mathematical logic in particular and in what would become computability theory, where he essentially introduced recursive functions He also proved what is now called the Herbrand Theorem, informally stating that a first-order formula in a prenex form containing existential quantifiers is only valid (respectively, provable) in first-order logic if and only if some disjunction of ground substitution instances of the quantifier-free subformula of A is a tautology (respectively, derivable in propositional logic) Herbrand’s Theorem therefore reduces in a precise sense validity and theoremhood from first-order logic to propositional logic