**Proof.**

1. Let Γ *φ* be derived in <sup>H</sup>*<sup>s</sup>*(**ND**PL).

> Then, Γ, ¬*φ* ⊥ is derived in <sup>H</sup>*<sup>s</sup>*(**ND**PL), by (HRAA).

> Hence, Γ, ¬*φ φ* is derived in <sup>H</sup>*<sup>s</sup>*(**ND**PL), by (¬HE2).


Since (*φ* → *ψ*) → ((*ψ* → *θ*) → (*φ* → *θ*)) is a classical tautology, Γ (*φ* → *ψ*) → ((*ψ* → *θ*) → (*φ* → *θ*)) is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL).

Hence, by Modus Ponens, Γ (*ψ* → *θ*) → (*φ* → *θ*) is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL). (\*)

Now, suppose that Γ, *φ θ* is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL).

Then, by item 2, Γ *φ* → *θ* is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL).

Therefore, Γ *ψ* → *θ* is derivable in H*s*(**ND**PL) by (→ HE2) applied to the latter and (\*). Then, finally, Γ, *ψ θ* is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL), by (→ HE1).

4. Let Γ *φ* ↔ *ψ* be derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL).

> Suppose that Γ, *θ φ* is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL).

Then, Γ, *θ* → *φ* is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL), by claim 2. (\*\*)

Since (*φ* ↔ *ψ*) → ((*θ* → *ψ*) → (*θ* → *φ*)) is a classical tautology, Γ (*φ* ↔ *ψ*) → ((*θ* → *ψ*) → (*θ* → *φ*)) is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL).

Therefore, Γ (*θ* → *ψ*) → (*θ* → *φ*) is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL).

Hence, Γ, *θ* → *ψ* is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL), by (→ HE2) applied to the latter and (\*\*).

Then, finally, Γ, *θ ψ* is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL), by (→ HE1).

5. It suffices to prove the claim when *k* = 2 and then apply a straightforward induction.

Suppose Γ, *ψ*1, *ψ*2 *θ* is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL).

Then, Γ (*ψ*1 → (*ψ*2 → *θ*)) is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL), by applying claim 2 twice.

Since (*ψ*1 → (*ψ*2 → *θ*)) ↔ ((*ψ*1 ∧ *ψ*2) → *θ*) is a classical tautology, Γ (*ψ*1 ∧ *ψ*2) → *θ* is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL), by claim 4.

Then, finally, Γ, *ψ*1 ∧ *ψ*2 *θ* is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL), by (→ HE1).

The converse direction is similar.

Given a truth assignment *δ* : Prop → {<sup>f</sup>,<sup>t</sup>}, for any propositional variable *p* ∈ Prop, let us define *pδ* := *p* if *<sup>δ</sup>*(*p*) = t, else *pδ* := <sup>¬</sup>*p*.

**Lemma 2.** *Let* Γ *be a finite set of propositional formulae and let* {*p*1, ..., *pn*} *contain all propositional variables occurring in formulae in* Γ*. Suppose δ is a truth assignment satisfying* Γ *and let* Γ*δ* = Γ ∪ {*pδ*1, ..., *<sup>p</sup>δn*}*. Then,* Γ*δ* ⊥ *is derivable in* <sup>H</sup>*<sup>s</sup>*(*ND*PL)*.*

**Proof.** By items 3 and 5 of Lemma 1, it suffices to prove the claim assuming that all formulae in Γ are transformed to equivalent ones in CNF and then replaced by the list of elementary disjunctions occurring as conjuncts in that CNF. Thus, without loss of generality, we can assume that Γ = {*<sup>γ</sup>*1, ..., *<sup>γ</sup>k*}, where all *γi* are elementary disjunctions.

Take the satisfying assignment *δ*. By definition, *δ* also satisfies all literals in {*pδ*1, ..., *<sup>p</sup>δn*}. Furthermore, *pδ*1, ..., *pδn* ⊥ is an atomic refutation axiom, hence derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL).

Now, select from each *γi* in Γ a literal disjunct *αi* that is satisfied by *δ*. Then, *αi* must be in {*pδ*1, ..., *<sup>p</sup>δn*}. Hence, {*pδ*1, ..., *<sup>p</sup>δn*, *α*1, ..., *<sup>α</sup>n*} = {*pδ*1, ..., *<sup>p</sup>δn*}.

Therefore, *pδ*1, ..., *<sup>p</sup>δn*, *α*1, ..., *αn* ⊥ is an atomic refutation axiom, hence derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL). (\*) In addition, *αi* → *γi* is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL), for each *i* = 1, ..., *n*. Therefore, by applying repeatedly item 3 of Lemma 1, we can replace successively each *αi* by *γi* in (\*), thereby eventually proving the claim.

By Anti-Monotonicity of , Lemma 2 immediately implies the following.

**Corollary 2.** *Let* Γ *be a finite satisfiable set of propositional formulae. Then,* Γ ⊥ *is derivable in* <sup>H</sup>*<sup>s</sup>*(*ND*PL)*.*

*5.2. Ł-Completeness and Ł-Adequacy of* H*s*(*ND*PL)

**Theorem 1.** *The hybrid derivation system* H*s*(*ND*PL) *is Ł-complete for the classical propositional logic* PL*.*

**Proof.** Due to the deductive completeness of **ND**PL, of which H*s*(**ND**PL) is a deductively conservative extension, it suffices to prove the R-completeness of <sup>H</sup>*<sup>s</sup>*(**ND**PL), i.e., that the refutation of every non-valid in PL sequent is derivable there. Let Γ |= *θ*. Then, there is a truth assignment *δ* satisfying Γ and falsifying *θ*. Therefore, *δ* satisfies Γ ∪ {¬*θ*}. By Corollary 2, it follows that Γ, ¬*θ* ⊥ is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL). Then, by Rule (¬HE2), Γ, ¬*θ θ* is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL). Finally, by the Anti-Monotonicity Rule Mon , we obtain that Γ *θ* is derivable in <sup>H</sup>*<sup>s</sup>*(**ND**PL). QED.

Proposition 3 and Theorem 1 together imply the following.

**Corollary 3.** *The hybrid derivation system* H*s*(*ND*PL) *is Ł-adequate for* PL *and, therefore, it provides a syntactic decision procedure for* PL*.*

The system H*s*(**ND**PL) and the ND-style refutation system developed in [2] are equivalent in terms of formal refutability, by virtue of the respective Ł-soundness and Ł-completeness results. Still, they are fairly different in style and it would be instructive to compare their proof-theoretic features, strengths and weaknesses, for the sake of possibly designing a better structured system of practical derivations based on <sup>H</sup>*<sup>s</sup>*(**ND**PL).

**Remark 8.** *Note that only some of the derived hybrid refutation rules were used in the proofs of Ł-soundness and Ł-completeness, hence the others must be derivable, or at least admissible, in the reduction of* H*s*(*ND*PL) *obtained by removing them. I leave the question of identifying a minimal Ł-complete subsystem of* H*s*(*ND*PL) *to future investigation. In particular, however, the rule* HRAA is *used in the proof of Lemma 1, hence that proof is not applicable to the system* H*s*(*ND*PL) *of Natural Deduction for the intuitionistic propositional logic* IPL*. Of course, it should not be applicable for* IPL*, e.g., because the refutation axiom* (*p* ∨ ¬*p*) *ought to be derivable there, while it is not in* <sup>H</sup>*<sup>s</sup>*(*ND*PL)*.*

#### **6. Towards a Meta-Proof Theory of Hybrid Derivation Systems**

Adding the relation for syntactic refutation and building systems of formal derivations that involve it together with the standard provability relation can be regarded as first steps towards internalising the notion of hybrid derivation into the logical language and then developing a theory for that notion that mirrors the proof theory of . In particular, derivability and refutability can now be treated on a par, as two related primitive concepts rather than as complementary ones where refutability is to be represented syntactically by non-provability. (Note, however, that, for any complete logic or theory, and applied to sequents of sentences are readily inter-reducible as complementary relations.) Thus, a proof theory of hybrid derivation systems emerges, extending and combining both the traditional proof theory and the theory of refutation systems.

Furthermore, the basic logical concepts of soundness, completeness, consistency, and satisfiability that relate syntax and semantics of a given logical system can now be all expressed and treated purely syntactically in terms of and . Thus, a "meta-proof theory" of hybrid derivation systems now emerges too, studying the meta-logic of these concepts respective to the given logical system L. Here, I will only set the stage for development of such meta-proof theory and will raise some generic questions, but I leave its systematic study to future work.

To begin with, let us add a new meta-symbol **F**, for "absurd", "falsum", or "contradiction", to the meta-language of hybrid derivation systems. Now, new hybrid derivation rules can be added to the thus extended framework, in order to reflect basic meta-properties of the given hybrid derivation system:

✄ **Cons**, stating consistency:

$$\frac{-\oint\_{\gamma} + \phi}{\mathbf{F}}$$

,

✄ "Ex (meta-)falso quodlibet", **EFQ**:

$$\frac{\mathbf{F}}{\!\!\!\!\!\!-\phi'} \qquad \frac{\mathbf{F}}{\!\!\!\!-\phi'} $$

✄ **Ł-Comp**: "Ł-completeness":

$$\begin{array}{c} \left[ \vdash \phi \right] \\ \vdots \\ \left[ \begin{array}{c} \text{F} \\ \hline \end{array} \right] \\ \end{array}$$

,

.

✄ Ł-**RAA**: "Ł-Reductio ad absurdum"

$$\begin{aligned} [\![\![\phi]\!] \!] \\ \vdots \\ \mathbf{F} \\ \hline \mathbf{\dot{}} \end{aligned} $$

Deductive completeness and Ł-completeness can now be *internalised* and stated as additional hybrid rules:

$$\begin{array}{ccccc} \left[\neg\Phi\right] & \left[\neg\Phi\right] & \left[\neg\Phi\right] & \left[\neg\Phi\right] \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ \left[\neg\Psi\right] & \left[\neg\Psi\right] & \left[\neg\Psi\right] & \left[\neg\Psi\right] & \left[\neg\Psi\right] \\ \end{array}$$

$$\begin{array}{ccccc} \left(\textbf{Ded}\right) & \begin{array}{ccc} \neg\Psi & \vdash\psi & \neg\Psi \\ \hline \vdash\psi & \neg\Psi & \end{array} \end{array}$$

Some natural questions arise:


A next natural step would be to strengthen the meta-language even further, to a full-fledged logical meta-language, involving meta-variables and quantification over derivable and refutable formulae (or, sequents). Then, for instance, the semantic relationship between and can be postulated in the meta-language as Γ *φ* ⇔ ∼ Γ *φ* (where ∼ is the meta-negation). (Some initial steps into studying propositional meta-theory of acceptance and rejection of formulae (sequents with empty lists of premises) in a similar spirit can be found in [14].) I leave the general study of the meta-proof theory of hybrid derivation system to future work.

**Remark 9.** *It should be noted that what I call here 'meta-proof theory' has essentially been studied in great depth for theories of the arithmetic in the context of Gödel's incompleteness theorems and, more generally, in the context of* axiomatic theories of truth*; see [24]. However, the general meta-proof theory proposed here makes no assumptions about the expressiveness of the object logic regarding definability of truth predicates in it, or in general, and consequently it has a much wider scope.*

## **7. Conclusions**

#### *7.1. Some Applications of Hybrid Derivation Systems*

Arguably, hybrid derivation systems have a number of potential applications, both conceptual and technical, including:


## *7.2. Current and Future Work*

Due to space and time limitations, this paper leaves many open ends and related questions, some of which have already been mentioned so far. In addition, here are some topics of current and follow-up work:


**Funding:** This work was partly supported by research gran<sup>t</sup> 2015-04388 of the Swedish Research Council.

**Acknowledgments:** I thank Sara Negri and Tom Skura, as well as the anonymous referees for careful reading, helpful suggestions, and some important corrections. I also thank the participants in the Refutation Symposium in Pozna ´n 2018 and in the CLLAM seminar at the Philosophy Department of Stockholm University for some useful comments on earlier versions of this work presented at these events.

**Conflicts of Interest:** The author declares no conflicts of interest.
