On the Costs of Nonclassical Logic

Abstract

Solutions to semantic paradoxes often involve restrictions of classical logic for semantic vocabulary. In the paper we investigate the costs of these restrictions in a model case. In particular, we fix two systems of truth capturing the same conception of truth: (a variant) of the system KF of Feferman (The Journal of Symbolic Logic, 56, 1–49, 1991) formulated in classical logic, and (a variant of) the system PKF of Halbach and Horsten (The Journal of Symbolic Logic, 71, 677–712, 2006), formulated in basic De Morgan logic. The classical system is known to be much stronger than the nonclassical one. We assess the reasons for this asymmetry by showing that the truth theoretic principles of PKF cannot be blamed: PKF with induction restricted to non-semantic vocabulary coincides in fact with what the restricted version of KF proves true.

Appendices

Appendix 1: The logic DM

The initial sequents of the system DM are:

$$\begin{array}{@{}rcl@{}} \begin{array}{lll} (\text{IN}) \quad \Gamma \Rightarrow \Delta, \text{if} ~\Gamma \cap\Delta\neq\varnothing & (\text{Cut})\quad\frac{\Gamma\Rightarrow\varphi,\Delta \qquad \Gamma,\varphi\Rightarrow\Delta}{\Gamma\Rightarrow \Delta}\\\\ (\text{W1}) \quad \frac{\Gamma\Rightarrow\Delta}{\Gamma,\varphi\Rightarrow\Delta} & (\text{W2}) \quad \frac{\Gamma\Rightarrow\Delta}{\Gamma\Rightarrow\varphi,\Delta}\\\\ (\text{Sub})\quad\frac{\Gamma\Rightarrow\Delta}{\Gamma(t/x)\Rightarrow\Delta(t/x)}& (\neg) \quad \frac{\Gamma\Rightarrow \Delta}{\neg \Delta\Rightarrow \neg\Gamma}\\\\ (\text{\textsf{d}}\neg 1) \quad \varphi \Rightarrow \neg \neg \varphi & ({\text{\textsf{d}}}\neg2) \quad \neg \neg\varphi\Rightarrow\varphi\\\\ (=1) \quad \Rightarrow t=t & (=2) \quad s=t,\varphi(s/x)\Rightarrow \varphi(t/x)\\\\ (\land 1) \quad \frac{\Gamma,\varphi,\psi\Rightarrow\Delta}{\Gamma,\varphi\land\psi\Rightarrow\Delta}& (\land 2) \quad \frac{\Gamma,\varphi\land\psi\Rightarrow\Delta} {\Gamma,\varphi,\psi\Rightarrow\Delta}\\\\ (\vee 1) \quad \frac{\Gamma\Rightarrow\varphi,\psi,\Delta}{\Gamma\Rightarrow\varphi\vee\psi,\Delta}& (\vee 2) \quad \frac{\Gamma\Rightarrow\varphi\vee\psi,\Delta}{\Gamma\Rightarrow\varphi,\psi,\Delta}\\\\ (\forall 1) \quad \frac{\Gamma\Rightarrow\varphi,\Delta}{\Gamma\Rightarrow\forall x\varphi,\Delta}& (\forall 2) \quad \frac{\Gamma\Rightarrow\forall x\varphi,\Delta}{\Gamma\Rightarrow \varphi(t/x),\Delta}\\\\ (\exists 1) \quad \frac{\Gamma,\exists x\varphi\Rightarrow\Delta}{\Gamma,\varphi(t/x)\Rightarrow \Delta}& (\exists 2) \quad \frac{\Gamma,\varphi\Rightarrow\Delta}{\Gamma,\exists x\varphi\Rightarrow\Delta} \end{array} \end{array} $$

In (¬), ¬Γ denotes the set of all negations of sentences in Γ. In ( ∀1) and ( ∃2), x is not free in the lower sequent.

Classical logic K is obtained by replacing, in DM, ( ¬) with the stronger

$$\begin{array}{@{}rcl@{}} \begin{array}{ll} (\neg\textsf{k}1) \quad \frac{\Gamma \Rightarrow \varphi,\Delta}{\Gamma,\neg\,\varphi\Rightarrow\Delta}& \qquad (\neg\textsf{k} 2) \quad \frac{\Gamma,\varphi\Rightarrow\Delta} {\Gamma\Rightarrow\neg\varphi,\Delta} \end{array} \end{array} $$

Appendix 2: Axioms of KF

PAT is the theory formulated in K (cf. Appendix 1) extended with initial sequents ⇒φ, for φ a basic axiom of PA, and the induction rule

$$ \frac{\Gamma,\varphi(x)\Rightarrow\varphi({S}x),\Delta}{\Gamma,\varphi(\overline 0)\Rightarrow\varphi(t),\Delta} $$

(IND)

with x not free in $\phantom {\dot {i}\!}\Gamma ,\Delta ,\varphi (\overline 0)$ and t arbitrary, for all formulas φ of $\phantom {\dot {i}\!}\mathcal {L}_{T}$. KF is obtained from PAT by adding the following initial sequents, where s,t range over (codes of) closed terms of $\phantom {\dot {i}\!}\mathcal {L}_{T}$:

KF1:

• 1. $\phantom {\dot {i}\!}{s^{\circ }}={t^{\circ }}\Rightarrow \textit {T}(\textit {s}\underset {\cdot }{=}\textit {t})$ (where (⋅)^∘ is the arithmetical evaluation function)

• 2. $\phantom {\dot {i}\!} \textit {T}(\textit {s}\underset {\cdot }{=}\textit {t})\Rightarrow {s^{\circ }}={t^{\circ }}$

KF2:

• 1. $\phantom {\dot {i}\!}{s^{\circ }}\neq {t^{\circ }}\Rightarrow \textit {T}(\underset {\cdot }{\neg }\, s= t)$

• 2. $\phantom {\dot {i}\!}\textit {T}(\underset {\cdot }{\neg }\, s= t)\Rightarrow {s^{\circ }}\neq {t^{\circ }}$

KF3:

• 1. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(x),\textit {T}\,\underset {\cdot }{\neg }\underset {\cdot }{\neg }\,x\Rightarrow \textit {T}x$

• 2. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(x),\textit {T}x\Rightarrow \textit {T}\,\underset {\cdot }{\neg }\underset {\cdot }{\neg }\,x$

KF4:

• 1. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(x\underset {\cdot }{\land }y), \textit {T}(x\underset {\cdot }{\land }y)\Rightarrow \textit {T}x\land \textit {T}y$

• 2. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(x\underset {\cdot }{\land }y), \textit {T}x\land \textit {T}y\Rightarrow \textit {T}(x\underset {\cdot }{\land }y)$

KF5 :

• 1. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(x\underset {\cdot }{\land }y),\textit {T}\underset {\cdot }{\neg }(x \underset {\cdot }{\land }y)\,\Rightarrow \,\textit {T}\,\underset {\cdot }{\neg } x\vee \textit {T}\,\underset {\cdot }{\neg } y$

• 2. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(x\underset {\cdot }{\land }y),\textit {T}\,\underset {\cdot }{\neg } x\vee \textit {T}\,\underset {\cdot }{\neg } y\,\Rightarrow \,\textit {T}\,\underset {\cdot }{\neg }\,(x\underset {\cdot }{\land }y)$

KF6:

• 1. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(x\underset {\cdot }{\vee }y), \textit {T}(x\underset {\cdot }{\vee }y)\Rightarrow \textit {T}x\vee \textit {T}y$

• 2. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(x\underset {\cdot }{\vee }y), \textit {T}x\vee \textit {T}y\Rightarrow \textit {T}(x\underset {\cdot }{\vee }y)$

KF7 :

• 1. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(x\underset {\cdot }{\vee }y),\textit {T}\,\underset {\cdot }{\neg }\,(x \underset {\cdot }{\vee }y)\,\Rightarrow \,\textit {T}\,\underset {\cdot }{\neg } x\land \textit {T}\,\underset {\cdot }{\neg } y$

• 2. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(x\underset {\cdot }{\vee }y),\textit {T}\,\underset {\cdot }{\neg } x\land \textit {T}\,\underset {\cdot }{\neg } y\,\Rightarrow \,\textit {T}\,\underset {\cdot }{\neg }\,(x\underset {\cdot }{\vee }y)$

KF8:

• 1. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(\underset {\cdot }{\forall } vx),\forall t\,\textit {T}(x(t/v))\Rightarrow \textit {T}(\underset {\cdot }{\forall } vx)$

• 2. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(\underset {\cdot }{\forall } vx),\textit {T}(\underset {\cdot }{\forall } vx)\Rightarrow \forall t\,\textit {T}(x(t/v))$

KF9:

• 1. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(\underset {\cdot }{\forall } vx),\exists t\,\textit {T}(\underset {\cdot }{\neg } x(t/v))\Rightarrow \textit {T}(\underset {\cdot }{\neg }\,\underset {\cdot }{\forall } vx)$

• 2. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(\underset {\cdot }{\forall } vx),\textit {T}(\underset {\cdot }{\neg }\,\underset {\cdot }{\forall } vx)\Rightarrow \exists t\,\textit {T}(\underset {\cdot }{\neg } x(t/v))$

KF10:

• 1. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(\underset {\cdot }{\exists } vx),\exists t\,\textit {T}(x(t/v))\Rightarrow \textit {T}(\underset {\cdot }{\exists } vx)$

• 2. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(\underset {\cdot }{\exists } vx),\textit {T}(\underset {\cdot }{\exists } vx)\Rightarrow \exists t\,\textit {T}(x(t/v))$

KF11:

• 1. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(\underset {\cdot }{\exists } vx),\forall t\,(\textit {T}\,\underset {\cdot }{\neg } x(t/v))\Rightarrow \textit {T}(\underset {\cdot }{\neg }\,\underset {\cdot }{\exists } vx)$

• 2. $\phantom {\dot {i}\!}{\text {Sent}_{\mathcal {L}_{T}}}(\underset {\cdot }{\exists } vx),\textit {T}(\underset {\cdot }{\neg }\,\underset {\cdot }{\exists } vx)\Rightarrow \forall t\,(\textit {T}\,\underset {\cdot }{\neg } x(t/v))$

KF12:

• T t ^∘⇔TṬ t

KF13:

• $\phantom {\dot {i}\!}\textit {T}\underset {\cdot }{\neg }\,\textit {\d {T}}t\Leftrightarrow \textit {T}\underset {\cdot }{\neg }\,{t^{\circ }}\vee \neg {\text {Sent}_{\mathcal {L}_{T}}}({t^{\circ }})$

KF14:

• $\phantom {\dot {i}\!}\textit {T}x\Rightarrow {\text {Sent}_{\mathcal {L}_{T}}}(x)$

KF ↾ is obtained by restricting (IND) to formulas of $\phantom {\dot {i}\!}\mathcal {L}$ and adding the sequents (Reg1) and (Reg2). We consider the extension of KF – that we call KF^∗– KF^∗ – via the disjunction of the following sentences, forcing complete or consistent extensions of the truth predicate respectively:

$$ \forall x({\text{Sent}_{\mathcal{L}_{T}}}(x)\land \neg T x\rightarrow T\underset{\cdot}{\neg} x) $$

(COMP)

$$ \forall x({\text{Sent}_{\mathcal{L}_{T}}}(x)\land T\underset{\cdot}{\neg}\,x\rightarrow \neg Tx) $$

(CONS)

Appendix 3: Completeness of DM

The bulk of this appendix is to prove Proposition 4 below. We recall that we write $\Gamma \vDash _{\mathcal S} \Delta$ to mean that for all four-valued models $\mathcal M$ of a set of sequents and rules $\mathcal S\supseteq \textsf {DM}$, the following are jointly satisfied:¹⁵

• if $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}} \varphi$ for all φ∈Γ, then $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}} \psi$ for at least one ψ∈Δ;

• if $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}}\neg \psi$ for all ψ∈Δ, then $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}}\neg \varphi$ for at least one φ∈Γ.

For notational simplicity, we do not deal with variable assignments.

The informal sketch of the strategy is as follows: we start with a pair (Γ,Δ) such that Γ⇒Δ is not derivable in $\phantom {\dot {i}\!}\mathcal S$, and build ‘maximal’ sets Γ⁺ and Δ⁺ such that Γ⁺⇒Δ⁺ is still not $\phantom {\dot {i}\!}\mathcal S$-derivable. We can in fact extend our notion of derivability (and underivability) in DM to sequents Λ⇒Θ where Λ,Θ are infinite: $\mathcal S\vdash {\Lambda }\Rightarrow {\Theta }$ iff for some finite Λ^′⊆Λ, Θ^′⊆Θ, we have $\phantom {\dot {i}\!}\mathcal S\vdash {\Lambda }^{\prime }\Rightarrow {\Theta }^{\prime }$. From Γ⁺,Δ⁺ a four-valued model $\phantom {\dot {i}\!}\mathcal M$ of $\phantom {\dot {i}\!}\mathcal S$ can be then read off, one that satisfies at least one of the following:

• if $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}}\varphi$ for all φ∈Γ⁺, then $\phantom {\dot {i}\!}\mathcal M\nvDash _{\textsf {DM}} \psi$ for ψ∈Δ⁺;

• if $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}}\neg \psi$ for all ψ∈Δ⁺, then $\phantom {\dot {i}\!}\mathcal M\nvDash _{\textsf {DM}} \neg \varphi$ for φ∈Γ⁺.

Henkin-style arguments of the kind given below can be found in [11] for a different sequent calculus and in [13] for a system of natural deduction. [1] hints at a more general strategy.

Proposition 4

Let $\phantom {\dot {i}\!}\mathcal S$ be a set of sequents in a language $\phantom {\dot {i}\!}\mathcal L$ and Γ,Δ finite sets of $\phantom {\dot {i}\!}\mathcal L$ -formulas. Then:

$$\text{if} ~\Gamma\vDash_{\mathcal S} \Delta, \text{then }\vdash_{\mathcal S} \Gamma \Rightarrow\Delta. $$

We prove the counterpositive. If $\phantom {\dot {i}\!}\nvdash _{\mathcal S} \Gamma \Rightarrow \Delta$, by (Cut), there cannot be a φ such that

$$\vdash_{\mathcal S} \Gamma\Rightarrow \Delta,\varphi ~ \text{and} ~ \vdash_{\mathcal S} \Gamma,\varphi\Rightarrow\Delta.$$

Let $\phantom {\dot {i}\!}\mathcal L^{+}:=\mathcal L\cup C$ the language $\phantom {\dot {i}\!}\mathcal L$ expanded with a countable set of new constants. Let {φ _n | n∈ω} and {c _i | i∈ω} enumerations of the formulae of $\phantom {\dot {i}\!}\mathcal L^{+}$ and constants in C respectively.

Next we inductively define Γ⁺ and Δ⁺ as follows:

• Γ₀:=ΓandΔ₀:=Δ

• if $\phantom {\dot {i}\!}\nvdash _{\mathcal S}\Gamma _{n},\varphi _{n}\Rightarrow \Delta _{n}$, then Δ _n+1=Δ _n and

  $$\Gamma_{n+1}:= \left\{\begin{array}{ll} &\Gamma_{n}\cup\{\varphi_{n},\chi(c_{i}/x)\},\hspace{10pt}\text{if }\varphi_{n}\text{ is }\exists x\chi\text{ and }c_{i}\text{ is the least constant in }C\\ &\hspace{100pt} \text{not occurring in }\Gamma_{n},\Delta_{n},\varphi_{n}.\\ &\Gamma_{n}\cup\{\varphi_{n},\neg\chi(c_{i}/x)\},\hspace{10pt}\text{if }\varphi_{n}\text{ is }\neg\forall x\chi\text{ and }c_{i}\text{ the least constant in }C\text{ not}\\ &\hspace{100pt}\text{ occurring in }\Gamma_{n},\Delta_{n},\varphi_{n}.\\ &\Gamma_{n}\cup\{\varphi_{n}\},\hspace{50pt}\text{otherwise} \end{array}\right. $$

• if $\phantom {\dot {i}\!}\vdash _{\mathcal S}\Gamma _{n},\varphi _{n}\Rightarrow \Delta _{n}$, we have Γ _n+1=Γ _n and

  $$\Delta_{n+1}= \left\{\begin{array}{ll} &\Delta_{n}\cup\{\varphi_{n},\chi(c_{i}/x)\},\hspace{10pt}\text{if }\varphi_{n}\text{ is }\forall x\chi\text{ and }c_{i}\text{ is the least constant in }C\text{ not}\\ &\hspace{100pt}\text{ occurring in }\Gamma_{n},\Delta_{n},\varphi_{n}.\\ &\Delta_{n}\cup\{\varphi_{n},\neg\chi(c_{i}/x)\},\hspace{10pt}\text{if }\varphi_{n}\text{ is }\neg\exists x\chi\text{ and }c_{i}\text{ the least constant in }C\text{ not}\\ &\hspace{100pt}\text{ occurring in }\Gamma_{n},\Delta_{n},\varphi_{n}.\\ &\Delta_{n}\cup\{\varphi_{n}\},\hspace{60pt}\text{otherwise} \end{array}\right. $$

Let $\phantom {\dot {i}\!}\Gamma ^{+}=\bigcup _{n\in \omega }\Gamma _{n}$ and $\phantom {\dot {i}\!}\Delta ^{+}=\bigcup _{n\in \omega }\Delta _{n}$. Obviously we have Γ⊂Γ⁺ and Δ⊂Δ⁺.

Next we state some properties of the pair (Γ⁺,Δ⁺).

Lemma 9

• 1. If Γ ^′ ⊂Γ ⁺ , Δ ^′ ⊂Δ ⁺ , with Γ ^′ ,Δ ^′ finite, and $\phantom {\dot {i}\!}\mathcal S\vdash \Gamma ^{\prime }\Rightarrow \Delta ^{\prime }$ , then there is an n∈ω such that

  $$\mathcal S\vdash \Gamma_{n}\Rightarrow \Delta_{n}$$

• 2. $\phantom {\dot {i}\!}\nvdash _{\mathcal S}\Gamma _{n}\Rightarrow \Delta _{n}$ for all n;

• 3. $\phantom {\dot {i}\!}\Gamma ^{+}\cap \Delta ^{+}=\varnothing$

• 4. t=t∈Γ ⁺ and t≠t∈Δ ⁺;

• 5. If φ∉Γ ⁺ , then $\phantom {\dot {i}\!}\vdash _{\mathcal S}\Gamma ^{+}\cup \{\varphi \}\Rightarrow \Delta ^{+}$ (symmetrically for φ∉Δ ⁺);

• 6. s=t∈Γ ⁺ and φ(s/x)∈Γ ⁺ , then φ(t/x)∈Γ ⁺;

• 7. s≠t and φ(s/x)∈Δ ⁺ , then φ(t/x)∈Δ ⁺;

• 8. s=t∈Γ ⁺ if and only if s≠t∈Δ ⁺;

• 9. there is no $\phantom {\dot {i}\!}\varphi \in \mathcal L^{+}$ such that

  $$\Gamma^{+}\Rightarrow\Delta^{+},\varphi\text{ and }\varphi,\Gamma^{+}\Rightarrow\Delta^{+}. $$

A four-valued model $\phantom {\dot {i}\!}\mathcal M$ can now be defined as follows:

$$M:=\{[\![ t]\!]\;|\;t\in \textit{Term}_{\mathcal L^{+}}\} $$

where

$$[\![t]\!]:=\{ s\in \textit{Term}_{\mathcal L^{+}}\;|\;s=t\in \Gamma^{+}\} $$

that is the equivalence class of terms of $\phantom {\dot {i}\!}\mathcal L^{+}$ determined by s = t∈Γ⁺. Moreover,

$$\begin{array}{@{}rcl@{}} &P_{\mathcal M}^{+}:=\{([\![t_{1}]\!],...,[\![t_{n}]\!])\;|\;P(t_{1},...,t_{n})\in \Gamma^{+}\}\\ &P_{\mathcal M}^{-}:=\{([\![t_{1}]\!],...,[\![t_{n}]\!])\;|\;\neg P(t_{1},...,t_{n})\in \Gamma^{+}\} \end{array} $$

The term model $\phantom {\dot {i}\!}\mathcal M$ so-defined enables us to prove the following lemma, which will in turn yields the desired result:¹⁶

Lemma 10

For all $\phantom {\dot {i}\!}\varphi \in \mathcal L^{+}$,

• 1. φ∈Γ ⁺ if and only if $\phantom {\dot {i}\!}\mathcal M\vDash _{\textnormal {\textsf {DM}}}\varphi$;

• 2. if φ∈Δ ⁺ , then $\phantom {\dot {i}\!}\mathcal M\nvDash _{\textnormal {\textsf {DM}}}\varphi$.

Proof

By induction on the positive complexity of φ.¹⁷ We mostly focus on the base and on the existential quantifier cases.

Case 1::

φ:= (s = t). (s = t)∈Γ⁺ iff [ [s] ]=[ [t] ] iff $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}} s=t$. If (s = t)∈Δ⁺ and $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}} s=t$, then [ [s] ]=[ [t] ] and s = t∈Γ⁺, which would imply the $\phantom {\dot {i}\!}\mathcal S$-derivability of Γ⁺⇒Δ⁺, against Lemma 9.

Case 2::

φ:= (s≠t). If (s≠t)∈Γ⁺, by Lemma 9(8), s = t∈Δ⁺. By Lemma 9(2), s = t∉Γ⁺, therefore [ [s] ]≠[ [t] ] and $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}} s\neq t$. If s≠t∈Δ⁺, again by Lemma 9(8), s = t∈Γ⁺ and so $\phantom {\dot {i}\!}\mathcal M\nvDash _{\textsf {DM}} s\neq t$.

Case 3::

φ: = P(t ₁,...,t _n ). By construction of $\phantom {\dot {i}\!}\mathcal M$, φ∈Γ⁺ iff $\phantom {\dot {i}\!}([\![t_{1}]\!],...,[\![t_{n}]\!])\in P_{\mathcal M}^{+}$, that is $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}} P(t_{1},\ldots ,t_{n})$. If P(t ₁,...,t _n )∈Δ⁺, then by Lemma 9(3), P(t ₁,...,t _n )∉Γ⁺; therefore $\phantom {\dot {i}\!}([\![t_{1}]\!],...,[\![t_{n}]\!])\notin P_{\mathcal M}^{+}$.

Case 4::

φ:=¬ P(t ₁,…,t _n ). Similar to the previous one.

Case 5::

φ:=¬¬χ. It’s easy to check that φ∈Γ⁺ iff χ∈Γ⁺ (otherwise Γ⁺,χ⇒Δ⁺ or Γ⁺,¬¬χ⇒Δ⁺ by Lemma 9, thus Γ⁺⇒Δ⁺ by the DM-rules). We can then apply the induction hypothesis.

If φ∈Δ⁺, then χ∈Δ⁺ (otherwise Γ⁺⇒χ,Δ⁺, and thus Γ⁺⇒Δ,φ). Again we apply the induction hypothesis.

Case 6::

φ:=∃v _i χ.

By construction of Γ⁺, φ∈Γ⁺ if and only if χ(c)∈Γ⁺ for suitable c. This is then equivalent to $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}}\chi (c)$ and $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}} \exists v_{i}\chi$ by induction hypothesis and definition of $\phantom {\dot {i}\!}\vDash _{\textsf {DM}}$.

If φ∈Δ⁺, then χ(t/v _i )∈Δ⁺ for any t (otherwise $\vdash _{\mathcal S}\Gamma ^{+}\Rightarrow \Delta ^{+},\chi (t/v_{i})$ , by Lemma 9 and $\vdash _{\mathcal S} \Gamma ^{+}\Rightarrow \Delta ^{+}$ since $\mathcal S\vdash \chi (t)\Rightarrow \exists v\chi$ ). By induction hypothesis, $\mathcal M\nvDash _{\textsf {DM}} \chi (t/v_{i})$.

Case 7::

φ:=¬ ∃v _i χ. If φ∈Γ⁺, then ¬χ(t/v _i )∈Γ⁺ for any t (as ¬χ(t)∉Γ⁺ for some t entails $\phantom {\dot {i}\!}\mathcal S\vdash \Gamma ^{+},\varphi \Rightarrow \Delta ^{+}$, thus Γ⁺⇒Δ⁺, quod non). Also, if ¬χ(t/v _i )∈Γ⁺ for any t, then ¬∃v _i χ∈Γ⁺ (as if φ∉Γ⁺, then φ∈Δ⁺, thus ¬χ(c)∈Δ⁺ for suitable c∈C). By induction hypothesis, this is equivalent to $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}} \neg \chi (t)$ for any t and $\phantom {\dot {i}\!}\mathcal M\vDash _{\textsf {DM}}\varphi$.

By definition of Δ⁺, if φ∈Δ⁺, then ¬χ(c/v _i )∈Δ⁺ for suitable c∈C. By induction hypothesis, $\phantom {\dot {i}\!}\mathcal M\nvDash _{\textsf {DM}}\neg \chi (c/v_{i})$. Thus $\phantom {\dot {i}\!}\mathcal M\nvDash _{\textsf {DM}} \neg \,\exists v_{i}\chi$.

□

With Lemma 10 at hand one can check that all initial sequents in $\phantom {\dot {i}\!}\mathcal S$ are satisfied by $\phantom {\dot {i}\!}\mathcal M$.

This conclude the proof of Proposition 4. From the assumption $\phantom {\dot {i}\!}\nvdash _{\mathcal S} \Gamma \Rightarrow \Delta$ we constructed (Γ⁺,Δ⁺) and the term model $\phantom {\dot {i}\!}\mathcal M$. The reduct $\phantom {\dot {i}\!}\mathcal M\restriction \mathcal {L}$ of $\phantom {\dot {i}\!}\mathcal M$ to $\phantom {\dot {i}\!}\mathcal {L}$ is such that $\phantom {\dot {i}\!}\mathcal M\restriction \mathcal {L}$ is a model of $\phantom {\dot {i}\!}\mathcal S$ and

for all φ∈Γ, $\phantom {\dot {i}\!}\mathcal M\mathpunct \upharpoonright \mathcal L\vDash _{\textsf {DM}}\varphi$ and $\phantom {\dot {i}\!} \mathcal M\mathpunct \upharpoonright \mathcal L\nvDash _{\textsf {DM}} \psi$ for all ψ∈Δ.

Thus $\phantom {\dot {i}\!}\Gamma \nvDash _{\mathcal S} \Delta$. This completes the proof of Proposition 4.