A Simple and Practical Valuation Tree Calculus for First-Order Logic

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8 Ján Kľuka and Paul J. Voda2.15 Lemma (Completeness). If T ⇒ p S, then D ⊢p T ⇒ p S for a free cutfree D.Proof. If T ⇒ p S, then by completeness of the calculus G3cp and by compactnessthere is a derivation E of a sequent Γ ⇒ ∆ for some Γ = C 1 , . . . , C k ⊆ Tand ∆ = D 1 , . . . , D n ⊆ S. Cuts are eliminable in G3cp [TS00, page 92], so wecan assume that E is cut free. By Lemma 2.14, there is a free cut free E ∗ s.t.E ∗ ⊢p Γ ⇒ p ∆. By Lemma 2.12, E ∗ ⊢p T ; Γ ⇒ p S; ∆ holds. We construct a freecut free D ∗ s.t. D ∗ ⊢p T ⇒ p S as follows:◦D 1◦ E ∗D n.C k◦. ◦ ⊢ T ⇒p SC 1Note that all indicated leaves witness a weakening of the identity axiom (Ax).⊓⊔2.16 Corollary (Soundness and completeness of ⊢p). We have T ⇒ p Siff D ⊢p T ⇒ p S for a free cut free D.⊓⊔3 Quantification LogicIn this section, we extend our calculus to the first-order quantification logicwithout equality.3.1 Semantics of first-order quantification logic. We assume the standardnotion of a structure M for a first-order language L not containing the equalitysymbol =. A structure has a non-empty domain M, assigns meaning to functionand predicate symbols of L, and extends the meaning in the usual way to termsand formulas. We write M A when M satisfies the sentence A.We define M T (M is a model of T ), T S, M ⇒ T , and T ⇒ Sanalogously to their propositional counterparts with the additional conditionthat if T S or T ⇒ S, then T and S are theories in the same language.3.2 Logical consequence in expansions of structures. We denote by L[ ⃗ F ]a language which is just like L, but contains new predicate or functions symbols⃗F = F 1 , . . . , F n . Let M be a structure for a language L. As usual, a structure M ′is an expansion of M to L[ ⃗ F ] if M ′ is a structure for L[ ⃗ F ] and coincides with Mon all symbols of L. This implies that the domains of M and M ′ are identical.We write T ⇒ L, ⃗ F S if all of the following conditions are satisfied: (a) L is alanguage, and ⃗ F are new symbols, (b) T is a theory in L, (c) S is a theory in L[ ⃗ F ],and (d) for each structure M for L such that M T , there is an expansion M ′of M to L[ ⃗ F ] such that M ′ ⇒ S.
A Simple and Practical Valuation Tree Calculus for First-Order Logic 93.3 Lemma.(a) If T ⇒ p S, and T , S are in the same language, then T ⇒ S.(b) T ⇒ S with T and S in L iff T ⇒ L,∅ S,(c) (Weakening) If T ⇒ L, ⃗ F S, then T, T ′ ⇒ L, ⃗ F , ⃗ G S, S ′ for any new ⃗ G, T ′ in L,and S ′ in L[ ⃗ F , ⃗ G].Proof. Straightforward.⊓⊔3.4 Initial quantification properties. The initial quantification propertiesare (i) the initial properties from Fig. 2 with ⇒ p replaced by ⇒ L,∅ provided Aand B are from L; (ii) the following initial properties of quantifiers:(R∃) A[x/t] ⇒ L,∅ ∃x A[x], (L∀) ∀x A[x] ⇒ L,∅ A[x/t],(L∃) ∃x A[x] ⇒ L,c A[x/c], (R∀) ∅ ⇒ L,c ∀x A[x], ¬A[x/c]provided x is the only free variable in A[x] from L, t is a closed term from L,and c /∈ L.Note that by using the notion of logical consequence in expansions of structureswe never have to deal with formulas containing free variables. We thusescape the annoying problem of a variable capture by a quantifier when substitutinga term with free variables for a variable.3.5 Lemma (Initial quantification properties). Initial quantification propertiesare true.Proof. Propositional properties are true by Lemma 3.3(a)(b). Quantifier instantiationproperties (R∃) and (L∀) hold trivially with , thus also with ⇒, andhence with ⇒ L,∅ by Lemma 3.3(b). For the Henkin witnessing property (L∃) itsuffices to expand any M for L s.t. M ∃x A[x] by interpreting c as a witnessto satisfy A[x/c]. The Henkin counterexample property (R∀) is made true byinterpreting c in an expansion of M as a counterexample ¬A[x/c] wheneverM ∀x A[x].⊓⊔3.6 Theorem (Expansion cuts).T ⇒ L,∅ S iff (i) A[ ⃗ F ], T ⇒ L[ ⃗ F ],∅ S and (ii) T ⇒ L, ⃗ F A[ ⃗ F ], S.Proof. In the direction (→), (i) take any M such that M A[ F ⃗ ], T . Contractit to M ′ for L. We have M ′ T , and we get M ′ ⇒ S, and hence M ⇒ S, fromthe assumption. (ii) Take any M T , we get M ⇒ S from the assumption.Arbitrary expansion of M to M ′ for L[ F ⃗ ] will satisfy one of S.In the direction (←), take any M for L satisfying T . From the assumption (ii),we get an expansion M ′ satisfying one of A[ F ⃗ ], S. If it is one of S, we also haveM ⇒ S. Otherwise M ′ A[ F ⃗ ]. Now the assumption (i) applies, and we getM ′ ⇒ S, and hence M ⇒ S.⊓⊔
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