# Hybrid Deduction–Refutation Systems

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

#### 1.1. Semantic vs. Deductive Refutability

#### 1.2. Related Work and Main Contributions

#### Related Work

- The idea of ’complementary systems’ for sentential logic, suggested by Bonatti and Varzi in [8] is related in spirit, though technically different from the idea of hybrid refutation systems, as it considers the complementary systems, for deductions and for refutations, acting separately.
- Similarly, in [9], Skura studies ‘symmetric inference systems’, that is, pairs of essentially non-interacting inference systems, and shows how they can be used for characterizing maximal non-classical logic with certain properties. In particular, the method is applied there to paraconsistent logic.
- In [10], Wybraniec-Skardowska and Waldmajer explore the general theory of deductive systems employing the two dual consequence operators, the standard logical consequence, inferring validities, and the refutation consequence, inferring non-validities. Again, no interaction of these consequence operators is considered there.
- In [11], Caferra and Peltier, motivated by potential applications to automated reasoning, take a unifying perspective on deriving accepting or rejecting propositions from other, already accepted or rejected, propositions, thus considering separately each of the four consequence relations arising as combinations.
- In [12], Goré and Postniece combine derivations and refutations to obtain cut-free complete systems for bi-intuitionistic logic.
- In [13], Negri explores the duality of proofs and countermodels in labelled sequent calculi and develops a method for unifying proof search and countermodel construction for some modal and intuitionistic propositional logic over classes of Kripke frames with suitable frame conditions. In particular, for some of this logic, the method provides a decision procedure.
- In [14], Citkin considers essentially multiple-conclusion generalisations of hybrid inference rules studied here. Citkin discusses consequence relations and inference systems employing such rules and proposes a meta-logic for formalising propositional reasoning about such systems. Even though with different motivation and agenda, and with no technical results of the type pursued here, this work appears to be the closest in spirit to the idea of hybrid deduction–refutation systems studied in the present work.
- In [17], Rumfitt considers “reversals” of the rules of propositional Natural Deduction, to formalise derivations between “accepted” and “rejected” sentences. While the motivation is different from the one related to refutation systems, most (but not all!) resulting rules are essentially the same as the “hybrid refutation rules” obtained by contrapositive inversion of the rules of propositional Natural Deduction considered in Section 4. See Remark 7 on the distinction between the two types of rules.

#### Contributions and Structure of the Paper

## 2. Preliminaries

#### 2.1. Refutation Rules and Systems: Basic Concepts

**pure rule of refutation inference**is a rule scheme of the type

**Disjunction rule**:

**Reverse substitution rule scheme**:

**mixed refutation rules**, which are relativised to a given underlying deductive system $\mathbf{D}$ for the logical system $\mathrm{L}$, as follows:

**D**(hence, assuming soundness of

**D**, proved valid in $\mathrm{L}$) and each $\sigma ({\psi}_{1}),\dots ,\sigma ({\psi}_{n})$ has been derived as non-valid in $\mathrm{L}$, then $\sigma (\gamma )$ is derived as non-valid in $\mathrm{L}$ too. A typical example is Łukasiewicz’s rule

**Reverse modus ponens**(aka,

**Modus Tollens**):

**refutation system (associated with a given underlying deductive system $\mathbf{D}$**) is a set $\mathcal{R}$ of (generally, mixed) refutation rules (where ⊢ is indexed with $\mathbf{D}$). Refutation rules with no premises are called

**structural refutation axioms**, and I will write them simply as $\u22a3\theta $.

**Remark**

**1.**

**D**are done separately, in advance or “on demand”, whenever needed for the derivation of the target refutation, and as part of that derivation. In either case, the deductive system

**D**is assumed to play only an auxiliary role for the functioning of the refutation system $\mathcal{R}$. Formally, a

**refutation derivation in $\mathcal{R}$**, or just an

**$\mathcal{R}$-derivation**, for a formula $\theta $ is a sequence ${S}_{1},...,{S}_{t}$, where ${S}_{t}$ is $\u22a3\theta $ and every ${S}_{i}$ is either a refutation axiom, or is of the form ${\u22a2}_{\mathbf{D}}\psi $ or is obtained from some already listed items in the sequence by applying a refutation rule from $\mathcal{R}$, by deriving the conclusion from suitable substitution instances of the premises. We now say that a formula $\theta $ is

**refutable in $\mathcal{R}$**(or, just

**$\mathcal{R}$-refutable**) iff there is a refutation derivation for $\theta $ in $\mathcal{R}$.

**refutation-sound**, or**Ł-sound, for $\mathrm{L}$**, if only non-valid in $\mathrm{L}$-formulae (more generally, logical consequences in $\mathrm{L}$) are $\mathcal{R}$-refutable,**refutation-complete**, or**Ł-complete, for $\mathrm{L}$**, if all non-valid in $\mathrm{L}$-formulae (more generally, logical consequences in $\mathrm{L}$) are $\mathcal{R}$-refutable.

#### 2.2. Basic Refutation Systems for Classical Logic

**Refutation axiom:**$\u22a3\perp $.

**Refutation rules:**

**RS**:

**MT**:

**Remark**

**2.**

**MT**is replaced by one for the respective modal logic.

## 3. Hybrid Derivation Systems: Basic Theory

#### 3.1. Hybrid Deduction–Refutation Rules and Systems

**(single-conclusion) sequent**, we mean an expression of the type $\Gamma \bowtie \theta $, where $\Gamma $ is a list (treated as a set) of formulae in $\mathrm{L}$, $\theta $ is a formula in $\mathrm{L}$, and $\bowtie \in \{\u22a2,\u22a3\}$. Sequents of the type $\Gamma \u22a2\theta $ will be called

**deductions**, while those of the type $\Gamma \u22a3\theta $ will be called

**refutations**. (From a general perspective, both deductions and refutations in our sense are treated syntactically as logical deductions, but we need a more differentiating and unambiguous terminology here.)

**sound**in $\mathrm{L}$ if $\Gamma \vDash \theta $ is a valid logical consequence in $\mathrm{L}$. Respectively, a sequent $\Gamma \u22a3\theta $ is

**sound**in $\mathrm{L}$ if $\Gamma \vDash \theta $ is a non-valid logical consequence in $\mathrm{L}$.

**Remark**

**3.**

**hybrid deduction rule of inference**(based on a given deductive system

**D**) is a rule of the type:

**hybrid refutation rule of inference**(based on a given deductive system

**D**) is a rule of the type:

**hybrid rules of inference**. Hybrid rules with no premises will be called respectively

**deduction axioms**and

**structural refutation axioms**, and we write them simply as sequents $\Gamma \u22a2\theta $, respectively $\Delta \u22a3\theta $.

**HDR**, resp. $\sigma (\Delta )\u22a3\sigma (\theta )$ in the case of

**HRR**. The respective semantic interpretation of the hybrid rules above in the case of propositional logical systems can be given as follows: for any uniform substitution $\sigma $, if each of the logical consequences $\sigma ({\Gamma}_{1})\vDash \sigma ({\phi}_{1}),\dots ,\sigma ({\Gamma}_{m})\vDash \sigma ({\phi}_{m})$ is derived as valid and each of $\sigma ({\Delta}_{1})\vDash \sigma ({\psi}_{1}),\dots ,\sigma ({\Delta}_{n})\vDash \sigma ({\psi}_{n})$ has been derived as non-valid, then $\sigma (\Gamma )\vDash \sigma (\theta )$ is derived as valid in the case of

**HDR**, respectively $\sigma (\Delta )\vDash \sigma (\theta )$ derived as non-valid in the case of

**HRR**, as defined above.

**refutation axiom schemes**of the type $\Gamma \u22a3\theta $, where closure under substitution is not assumed, but syntactic constraints are imposed on $\Gamma $ and $\theta $. A simplest example is a scheme $p\u22a3q$, where $p\ne q$. Clearly, allowing closure under substitution would produce unsound refutation sequents, such as $p\u22a3p$. Of course, structural refutation axioms are special kinds of refutation axiom schemes, but it would be helpful to consider both types separately. Structural refutation axioms and refutation axiom schemes will be called collectively just

**refutation axioms**. Note that, to make the general hybrid rules applicable, they must act in combination with some rules with no premises, i.e., deduction and refutation axioms, which provide an initial stock of derived sequents.

**Remark**

**4.**

**D**, which provides the initial stock of derived sequents $\Gamma \u22a2\theta $ only, but rather that they define the notions of deduction derivations and refutation derivations on a par, by a mutual induction defined as expected, which I combine in one notion of hybrid derivation, defined further.

**sound**for a given logical system $\mathrm{L}$ if it respects the intuitive interpretation above, i.e., whenever applied to sound premises in $\mathrm{L}$, it produces a sound conclusion in $\mathrm{L}$.

- All standard deduction rules (in particular, axioms) are particular cases of hybrid deduction rules. In particular, such are all rules of sequent calculi and systems of natural deduction.
- The refutation rules defined in Section 2.1 are particular cases of hybrid refutation rules.
- In addition, suitable meta-properties of the given logical system $\mathrm{L}$ can be used to extract and justify specific new hybrid inference rules for it. An important example is the
**Deductive consistency rule**$$(\mathbf{Cons})\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{\u22a2\phi}{\u22a3\neg \phi},$$

**hybrid deduction–refutation system**, or (for shorter) a

**hybrid derivation system**, is a set $\mathcal{H}$ of hybrid rules of inference for a given logical language. A

**hybrid derivation in $\mathcal{H}$**, or just an

**$\mathcal{H}$-derivation**, for a sequent $\Gamma \bowtie \theta $ is a sequence of sequents ${S}_{1},...,{S}_{t}$, where ${S}_{t}$ is $\Gamma \bowtie \theta $ and every ${S}_{i}$ is either a deduction axiom or a refutation axiom, or is obtained from some already listed sequents in the sequence by applying a hybrid rule of inference from $\mathcal{H}$. Then, we say that the sequent $\Gamma \bowtie \theta $ is

**derivable in $\mathcal{H}$**. Furthermore, we say that the logical consequence $\Gamma \vDash \theta $ is

**deduced/deducible in $\mathcal{H}$**if $\Gamma \u22a2\theta $ is derivable in $\mathcal{H}$, and that $\Gamma \vDash \theta $ is

**refuted/refutable in $\mathcal{H}$**if $\Gamma \u22a3\theta $ is derivable in $\mathcal{H}$.

**D**(which can be an axiomatic system, a sequent calculus, a system of natural deduction, or a system of semantic tableaux). In such case, the derivations in $\mathcal{H}$ extend those in

**D**, by enabling not only derivations of refutations based on

**D**, but also possibly of some deductions not derivable in

**D**(esp. in case

**D**is incomplete).

**Remark**

**5.**

**deductively sound for $\mathrm{L}$**, or**D-sound for $\mathrm{L}$**, if only logical consequences that are valid in $\mathrm{L}$ are $\mathcal{H}$-deducible.**refutationally sound for $\mathrm{L}$**, or**R-sound for $\mathrm{L}$**, if only logical consequences that are non-valid in $\mathrm{L}$ are $\mathcal{H}$-refutable.**Ł-sound for $\mathrm{L}$**, if it is both D-sound and R-sound for $\mathrm{L}$.**Ł-consistent**, if there is no $\Gamma $ and $\theta $ such that both $\Gamma \u22a2\theta $ and $\Gamma \u22a3\theta $ are derivable in $\mathcal{H}$.**deductively complete for $\mathrm{L}$**, or**D-complete for $\mathrm{L}$**, if all logical consequences that are valid in $\mathrm{L}$ are $\mathcal{H}$-deducible.**refutationally complete for $\mathrm{L}$**, or**R-complete for $\mathrm{L}$**, if all logical consequences that are non-valid in $\mathrm{L}$ are $\mathcal{H}$-refutable.**Łukasiewicz-complete for $\mathrm{L}$**, or**Ł-complete for $\mathrm{L}$**, if it is both D-complete and R-complete for $\mathrm{L}$.**Ł-saturated**, if for all $\Gamma $ and $\theta $, either $\Gamma \u22a2\theta $ or $\Gamma \u22a3\theta $ (possibly both) is derivable in $\mathcal{H}$.**Ł-adequate for $\mathrm{L}$**, if it is both Ł-sound and Ł-complete for $\mathrm{L}$.**Ł-balanced**, if it is both Ł-consistent and Ł-saturated.

**Proposition**

**1.**

- 1.
- If $\mathcal{H}$ is Ł-sound for $\mathrm{L}$, then $\mathcal{H}$ is Ł-consistent.
- 2.
- If $\mathcal{H}$ is Ł-complete for $\mathrm{L}$, then $\mathcal{H}$ is Ł-saturated.
- 3.
- If $\mathcal{H}$ is Ł-adequate for $\mathrm{L}$, then $\mathcal{H}$ is Ł-balanced.
- 4.
- If $\mathcal{H}$ has a recursive set of rules and is Ł-adequate for $\mathrm{L}$, then it provides a decision procedure for the valid logical consequences in $\mathrm{L}$.

**Proof.**

**Remark**

**6.**

**R**-soundness and

**R**-completeness should be replaced by “falsifiable", without assuming that the latter implies the former. Still, claims 1 and 3 in Proposition 1 will no longer hold for such semantics. (Thanks to the reviewer who pointed that out.) Still, note that, even if the deduction fragment of a hybrid derivation system may be D-unsound, or D-incomplete, for the given logical system, its refutation fragment may still be R-sound, or R-complete, and vice versa. An interesting example is the simple Ł-complete refutation system for Medvedev’s logic of finite problems (for which no recursive axiomatization is known yet, but it has a co-r.e. set of validities) designed in [23], employing as the underlying deductive system the weaker Kreisel–Putnam’s logic KP. Thus, the resulting hybrid system is D-incomplete but R-complete for Medvedev’s logic.

#### 3.2. Inversion of Rules and Derivative Hybrid Rules

**derivative rules**from existing ones by using

**inversion**: swapping one premise with the conclusion of the given rule and swapping ⊢ with ⊣ in both sequents. (The use of the term ‘inversion’ here is different from ‘inversion principle’ widely used in proof theory, see [5], but related to the term ‘inversion’ used in [2], when applied to single-premise rules. In addition, the idea of inverting inference rules was essentially used in the design and proof of completeness of the sequential refutation system for $\mathsf{PL}$ in [3].) For example, applying inversion to the rule Modus Ponens

#### 3.2.1. Inversion of Deduction Rules

#### 3.2.2. Inversion of Refutation Rules

#### 3.2.3. Soundness of Derivative Rules

**Proposition**

**2.**

**Proof.**

#### 3.3. Canonical Hybrid Extensions of Deductive Systems

**D**, its

**canonical hybrid extension**$\mathcal{H}(\mathbf{D})$ is obtained by adding to

**D**the derivative rules of all deduction rules (incl. axioms) of

**D**.

**D**is D-sound for a given logical system $\mathrm{L}$, then $\mathcal{H}(\mathbf{D})$ is Ł-sound for $\mathrm{L}$. If

**D**is also D-complete for $\mathrm{L}$, then $\mathcal{H}(\mathbf{D})$ cannot add more derivable deduction sequents, so it is D-complete too. In this case, $\mathcal{H}(\mathbf{D})$ extends

**D**conservatively with respect to deductions, but it generally does add derivable refutation sequents. However, even then $\mathcal{H}(\mathbf{D})$ may generally not be R-complete, hence not Ł-complete, either. In particular, it cannot be R-complete if $\mathrm{L}$ is not decidable. The question of when $\mathcal{H}(\mathbf{D})$ is Ł-complete is one of the main questions of the general theory of hybrid derivation systems.

**canonical hybrid extension**$\mathcal{H}(\mathcal{R})$ is obtained by adding to $\mathcal{R}$ the derivative rules of all refutation rules (incl. axioms) of $\mathcal{R}$. Again, by Proposition 2, if $\mathcal{R}$ is R-sound for a logical system $\mathrm{L}$, then $\mathcal{H}(\mathcal{R})$ is Ł-sound for $\mathrm{L}$. The question of Ł-completeness of $\mathcal{H}(\mathcal{R})$ is, again, generally open.

**Corollary**

**1.**

## 4. Hybrid Extensions of the System of Natural Deduction for $\mathsf{PL}$

#### 4.1. Hybrid Derivatives of the Rules for Natural Deduction for $\mathsf{PL}$

#### 4.2. Hybrid Derivatives of the Introduction Rules of ${\mathit{ND}}^{\mathsf{PL}}$

#### 4.3. Hybrid Derivatives of the Elimination Rules of ${\mathit{ND}}^{\mathsf{PL}}$

#### 4.4. Hybrid Derivatives of “Ex Falso” and “Reductio ad Absurdum”

**Remark**

**7.**

**ND**obtained here. However, there is an essential distinction between the meanings of the two types of rules, e.g.,: whereas rejection of a sentence implies acceptance of its negation, and deductive refutation of the validity of a sentence does not imply deduction of the validity of its negation. That distinction is manifested e.g., by the rules $+\neg I$ and $-\neg E$ in [17] as compared to the hybrid derivative rules for ¬ obtained and employed here.

#### 4.5. Atomic Refutations and Monotonicity Rules

**refutation axiom scheme**${\mathsf{RefAx}}^{\mathsf{PL}}$:

- The rule ${\mathsf{Mon}}^{\u22a2}$:
**Monotonicity of**⊢$$\frac{\Gamma \u22a2\varphi ,\phantom{\rule{4pt}{0ex}}\Gamma \subseteq {\Gamma}^{\prime}}{{\Gamma}^{\prime}\u22a2\varphi},$$(Usually this rule is implicitly assumed in any traditional system of natural deduction.) - The rule ${\mathsf{Mon}}^{\u22a3}$:
**Anti-monotonicity of**⊣$$\frac{\Gamma \u22a3\varphi ,\phantom{\rule{4pt}{0ex}}{\Gamma}^{\prime}\subseteq \Gamma}{{\Gamma}^{\prime}\u22a3\varphi}.$$

**standard hybrid extension of ${\mathbf{ND}}^{\mathsf{PL}}$**.

## 5. Some Results about the Standard Hybrid Extension of ${\mathbf{ND}}^{\mathsf{PL}}$

#### 5.1. Soundness and Some Properties of ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$

**Proposition**

**3.**

- 1.
- Every rule of ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$ is sound.
- 2.
- ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$ is Ł-sound for $\mathsf{PL}$ and hence Ł-consistent.
- 3.
- If Γ is a satisfiable set of formulae, then $\Gamma \u22a2\perp $ is not derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$.

**Proof.**

**ND**for $\mathsf{PL}$ and Proposition 2. Proving the soundness of ${\mathsf{RefAx}}^{\mathsf{PL}}$, ${\mathsf{Mon}}^{\u22a2}$, and ${\mathsf{Mon}}^{\u22a3}$ for $\mathsf{PL}$ is quite routine, and I leave out the details.

**Lemma**

**1.**

- 1.
- If $\Gamma \u22a3\varphi $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$, then $\Gamma ,\neg \varphi \u22a3\varphi $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$.
- 2.
- If $\Gamma ,\varphi \u22a3\psi $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$, then $\Gamma \u22a3\varphi \to \psi $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$.
- 3.
- If $\Gamma \u22a2\varphi \to \psi $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$ and $\Gamma ,\varphi \u22a3\theta $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$, then $\Gamma ,\psi \u22a3\theta $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$.Consequently, if $\Gamma \u22a2\varphi \leftrightarrow \psi $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$, then $\Gamma ,\varphi \u22a3\theta $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$ iff $\Gamma ,\psi \u22a3\theta $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$.
- 4.
- If $\Gamma \u22a2\varphi \leftrightarrow \psi $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$, then $\Gamma ,\theta \u22a3\varphi $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$ iff $\Gamma ,\theta \u22a3\psi $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$.
- 5.
- $\Gamma ,{\psi}_{1},...,{\psi}_{k}\u22a3\theta $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$ iff $\Gamma ,{\psi}_{1}\wedge ...\wedge {\psi}_{k}\u22a3\theta $ is derivable in ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$.

**Proof.**

- Let $\Gamma \u22a3\varphi $ be derived in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$.Then, $\Gamma ,\neg \varphi \u22a3\perp $ is derived in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$, by ($\mathrm{HRAA}$).Hence, $\Gamma ,\neg \varphi \u22a3\varphi $ is derived in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$, by ($\neg {\mathrm{HE}}^{2}$).
- Suppose $\Gamma ,\varphi \u22a3\psi $ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$. Since $\Gamma ,\varphi \u22a2\varphi $ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$, we derive $\Gamma ,\varphi \u22a3\varphi \to \psi $ by ($\to \mathrm{HI}$). Then, by the Anti-Monotonicity rule ${\mathsf{Mon}}^{\u22a3}$, $\Gamma \u22a3\varphi \to \psi $ is derived in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$.
- Let $\Gamma \u22a2\varphi \to \psi $ be derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$.Since $(\varphi \to \psi )\to ((\psi \to \theta )\to (\varphi \to \theta ))$ is a classical tautology,$\Gamma \u22a2(\varphi \to \psi )\to ((\psi \to \theta )\to (\varphi \to \theta ))$ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$.Hence, by Modus Ponens, $\Gamma \u22a2(\psi \to \theta )\to (\varphi \to \theta )$ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$. (*)Now, suppose that $\Gamma ,\varphi \u22a3\theta $ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$.Then, by item 2, $\Gamma \u22a3\varphi \to \theta $ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$.Therefore, $\Gamma \u22a3\psi \to \theta $ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$ by ($\to {\mathrm{HE}}^{2}$) applied to the latter and (*). Then, finally, $\Gamma ,\psi \u22a3\theta $ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$, by ($\to {\mathrm{HE}}^{1}$).
- Let $\Gamma \u22a2\varphi \leftrightarrow \psi $ be derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$.Suppose that $\Gamma ,\theta \u22a3\varphi $ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$.Then, $\Gamma ,\u22a3\theta \to \varphi $ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$, by claim 2. (**)Since $(\varphi \leftrightarrow \psi )\to ((\theta \to \psi )\to (\theta \to \varphi ))$ is a classical tautology,$\Gamma \u22a2(\varphi \leftrightarrow \psi )\to ((\theta \to \psi )\to (\theta \to \varphi ))$ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$.Therefore, $\Gamma \u22a2(\theta \to \psi )\to (\theta \to \varphi )$ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$.Hence, $\Gamma ,\u22a3\theta \to \psi $ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$, by ($\to {\mathrm{HE}}^{2}$) applied to the latter and (**).Then, finally, $\Gamma ,\theta \u22a3\psi $ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$, by ($\to {\mathrm{HE}}^{1}$).
- It suffices to prove the claim when $k=2$ and then apply a straightforward induction.Suppose $\Gamma ,{\psi}_{1},{\psi}_{2}\u22a3\theta $ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$.Then, $\Gamma \u22a3({\psi}_{1}\to ({\psi}_{2}\to \theta ))$ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$, by applying claim 2 twice.Since $({\psi}_{1}\to ({\psi}_{2}\to \theta ))\leftrightarrow (({\psi}_{1}\wedge {\psi}_{2})\to \theta )$ is a classical tautology, $\Gamma \u22a3({\psi}_{1}\wedge {\psi}_{2})\to \theta $ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$, by claim 4.Then, finally, $\Gamma ,{\psi}_{1}\wedge {\psi}_{2}\u22a3\theta $ is derivable in ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$, by ($\to {\mathrm{HE}}^{1}$).The converse direction is similar.

**Lemma**

**2.**

**Proof.**

**Corollary**

**2.**

#### 5.2. Ł-Completeness and Ł-Adequacy of ${\mathcal{H}}^{s}({\mathit{ND}}^{\mathsf{PL}})$

**Theorem**

**1.**

**Proof.**

**Corollary**

**3.**

**Remark**

**8.**

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

**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:$$\frac{\u22a2\varphi ,\phantom{\rule{4pt}{0ex}}\u22a3\varphi}{\mathbf{F}},$$- ⊳
- “Ex (meta-)falso quodlibet”,
**EFQ**:$$\frac{\mathbf{F}}{\u22a2\varphi},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{\mathbf{F}}{\u22a3\varphi},$$ - ⊳
**Ł-Comp**: “Ł-completeness”:$$\frac{\begin{array}{c}\u22a2\varphi ]\\ \vdots \\ \mathbf{F}\end{array}}{\begin{array}{c}\u22a3\varphi \end{array}},$$- ⊳
**Ł-RAA**: “Ł-Reductio ad absurdum”$$\frac{\begin{array}{c}\u22a3\varphi ]\\ \vdots \\ \mathbf{F}\end{array}}{\begin{array}{c}\u22a2\varphi \end{array}}.$$

- Can any of these meta-rules strengthen the deductive power of a given (not complete) hybrid derivation system?
- In particular, can any of these bring about deductive completeness or Ł-completeness, when it does not hold without them?

**Remark**

**9.**

## 7. Conclusions

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

- Hybrid derivation systems put proofs and refutations on equal footing and thus enable their comparative study and of the development of meta-proof theory, where the interaction of the concepts of deduction and syntactic refutation for a given logic is the object of study.
- Hybrid derivation systems can yield purely deductive decision procedures, as indicated in Proposition 1 and illustrated for $\mathsf{PL}$ in Section 5.
- Hybrid derivation systems can capture important classes of non-valid formulae in recursively axiomatizable but undecidable logic, such as $\mathrm{FOL}$. They can also provide complete refutation systems for logical theories with co-r.e. validity. Typically, this is logic defined over a class of finite models, such as $\mathrm{FOL}$ in the finite or Medvedev’s logic of finite problems (see respectively [23,25] for R-complete refutation systems for these).
- Hybrid derivation systems can possibly provide more succinct proof systems. This hypothesis is yet to be tried and tested.

#### 7.2. Current and Future Work

- Develop and understand the general meta-proof theory of hybrid derivation systems.
- Design Ł-complete hybrid derivation systems for the intuitionistic propositional logic and for some important modal logic (extending such results from [3]) and for other non-classical logic.
- Extend/modify ${\mathcal{H}}^{s}({\mathbf{ND}}^{\mathsf{PL}})$ to hybrid derivation systems for classical and intuitionistic $\mathrm{FOL}$ that are R-complete for the non-validities in the finite. Characterise the set of refutable non-validities in these systems.
- Relate more explicitly hybrid derivation systems with tableaux systems. As the latter are designed to check satisfiability, i.e., non-validity of the negated input, they are naturally related to refutations and, hence, to hybrid derivation systems.
- Another potentially interesting direction (suggested by an anonymous referee) for related further research is to explore the relation between hybrid derivation systems and methods for proof certification [27].
- Last but not least: a challenge worth pursuing in this area would be to obtain new decidability results by designing Ł-adequate hybrid deductive systems for logic that is not yet known to be decidable, such as Medvedev’s logic.

## Funding

## Acknowledgments

## Conflicts of Interest

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Goranko, V.
Hybrid Deduction–Refutation Systems. *Axioms* **2019**, *8*, 118.
https://doi.org/10.3390/axioms8040118

**AMA Style**

Goranko V.
Hybrid Deduction–Refutation Systems. *Axioms*. 2019; 8(4):118.
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2019. "Hybrid Deduction–Refutation Systems" *Axioms* 8, no. 4: 118.
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