By Lieger R., Vahan H., Rumatowski G.

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Let S be a set of universal closed formulas. Let ϕ be a closed formula such that ¬ϕ is universal. Then S is Herbrand satisfiable iff S is Tarski satisfiable and S |=Hb ϕ iff S |= ϕ. Corollary 89. Herbrand unsatisfiability and Herbrand entailment are not decidable, but they are semi-decidable for formulas meeting the conditions above. Theorem 90 ([88]). For formulas that do not meet the conditions above, Herbrand unsatisfiability and Herbrand entailment are not semi-decidable. Herbrand Model Theory for Finite Herbrand Base.

Xn ((x1 =x ˙ 1 ∧ . . ∧ xn =x ˙ n ∧ p(x1 , . . , xn )) ⇒ p(x1 , . . , xn )) – – – – Note that EQL may be infinite, depending on L. Actually, symmetry and transitivity of = ˙ follow from reflexivity of = ˙ and the substitution axiom for = ˙ and could be omitted. Theorem 71 (Equality axioms) – For each interpretation I, if I is normal then I |== EQL . – For each interpretation I with I |= EQL there is a normal interpretation I= such that for each formula ϕ: I |= ϕ iff I= |== ϕ. – For each set S of formulas and formula ϕ: EQL ∪ S |= ϕ iff S |== ϕ.

N ψ iff for each finite interpretation I: if I |=fin ϕ then I |=fin ψ. Finite interpretations are special Tarski interpretations, hence there are obvious inclusions between corresponding notions: each valid formula is finitely valid, each finitely satisfiable formula is satisfiable, etc. These inclusions are proper. For example, { ∀x ¬(x < x), ∀x∀y∀z(x < y ∧ y < z ⇒ x < z), ∀x∃y x < y } is a satisfiable, but finitely unsatisfiable set of formulas, and the single formula [∀x ¬(x < x) ∧ ∀x∀y∀z(x < y ∧ y < z ⇒ x < z)] ⇒ ∃x∀y ¬(x < y) is finitely valid, but not valid.

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