On a Correspondence Between Two Kinds of Bilateral Proof-Systems

Abstract

The paper shows the deductive equivalence between bilateral (and multilateral) proof systems based on signed formulas and proof systems based on multiple derivability relations.

1. Introduction

Bilateralism is an approach to logic, and in particular to the theory of meaning of logical constants, that treats assertion and denial on a par: those are two primitive speech acts, not reducible to each other but coordinated.

Accordingly, in the theory of meaning known as proof-theoretic semantics (PTS) (see [1,2]), meaning-conferring proof-systems, mostly natural-deduction systems, are built over both speech acts.

Digression: The nomenclature of “speech act” is somewhat of a misnomer when applied to logic. After all, logic is not spoken… Thus, the speech act markers introduced below are formal speech markers, only drawing on the presence of “real” speech acts present in natural language.

As this nomenclature is widely used in the literature, I will keep its use.

(end of digression)

In [3] and in [4] the theory is extended to multilateralism, based on an arbitrary number of formal speech acts.

There are two, allegedly different, main approaches (There is a third approach, providing a bilateral reading to existing sequence calculi. See, for example, [5] This approach is irrelevant to my current purpose). How to devise a bilateral, or more generally, a multilateral, proof system.

• Signed formulas: Following Rumfitt [6], a non-itterable and non-embeddable sign is introduced for each speech act, and the rules of the proof-system operate on signed formulas. In the bilateral case, ‘+’ signifies formal assertion and ‘−’ signifies formal denial.
• Multiple derivability relations: Following [4,7], a derivability relation is introduced for each of a number of semantic notions, in correspondence to the same number of formal speech act, and rules of the system can arbitrarily mix those derivability relations. In the bilateral case, one speaks of the semantic notions of verification and falsification (or proof and co-proof), that are in correspondence with formal assertion and formal denial.

In [4], there is an argument in favor of the latter approach.

In this short paper, I show that, contra the argument in [4], the two approaches are merely deductively equivalent notational variants of each other.

In Section 2, I briefly delineate the technical details of multilateralism as multiple derivability relations, and in Section 3—the technical details of multilateralism as signed formulas derivability relation. In Section 4, I present the two general mappings establishing, for any bilateral logic, the deductive equivalence, and thereby my claim that the two approaches are merely notational variants. Iin the context of bilateral natural deduction, the existence of such a correspondence was noted by Drobyshevich [8], but confined to the case of two derivation relations only, and applied only to a specific logic, Wansing’s 2-int logic e.g., in [7].

2. Multilateralism as Multiple Derivability Relations

In this section, I briefly delineate the multilateralism as multiple derivability relations view of multilateral logic, following [4] with a slightly modified notation. Let $\phi$, $\psi$ range over object language formulas.

Definition 1 

(turnstile signed sequent [4]). A multiple derivability ($n+1)$-lateral turnstile signed sequent s has the form ${\Gamma }_1|{\Gamma }_2|\cdots |{\Gamma }_n|{\Gamma }_{n+1}{:}^i\phi$ for $1\le i\le n+1$, $n\ge 1$, where the ${\Gamma }_i$ are finite, possibly empty, multi-sets of formulas. We say that formulas in ${\Gamma }_i$ occur at position i.

I omit ‘$n+1$’ when clear from context.

Definition 2 

(multilateral multiple signed turnstile sequent calculus). An $n+1$-multilateral multiple derivations turnstile signed sequent calculus $S{C}^m$ is a non-empty set containing some axiomatic, initial turnstile signed sequents, and rules, ranged over by ρ, of the form $\frac{s_1\dots s_{\mathrm{m}}}{s}\left(\rho \right)$, where s and all $s_i$ $1\le i\le m$ are turnstile signed sequents.

There is the usual partition of rules to operational. featuring some specific logical constants, and structural, featuring no logical constant.

Definition 3 

(derivation). (1) Every instance of an initial sequent is a derivation. (2) Applications of sequent rules to instances of their schematic premise sequents as conclusions of derivations result in a derivation. If there is a derivation of a sequent s in a sequent calculus $S{C}^m$ we say that s is provable in $S{C}^m$ and denote this as ${⊢}_{S{C}^m}s$ (or just as $⊢s$ if the sequent calculus in question is clear).

3. Multilateralism as Signed Formulas Derivability Relation

In this section I briefly delineate the multilateralism as signed formulas derivability relation view of multilateral logic.

Consider a set $S=\{{\ast }_1,\cdots ,{\ast }_{n+1}\}$, $n\ge 1$, of signs, representing $n+1$ formal force markers.

Definition 4 

(signed formulas). A signed formula has the form ${\ast }_i\phi$, where ${\ast }_i\in S$ and φ is an object language formula.

For Γ a multi-set of object-language formulas and ${\ast }_i\in S$, let ${\Gamma }^{{\ast }_i}{=}^{df.}\left\{{\ast }_i\phi \right|\phi \in \Gamma \}$.

Let $\phi$, $\psi$ range over (unsigned) object language formulas, and $\alpha ,\beta$ range over signed formulas.

Definition 5 

(formula signed sequent). A formula signed sequent has the form $\Gamma :\alpha$, where $\Gamma$ is a multi-set of signed formulas and α is a signed formula.

Definition 6 

(formula signed sequent calculus)). A (multilateral) tformula signed sequent calculus $S{C}^s$ is a non-empty set containing some axiomatic, initial tformula signed sequents, and rules of the form $\frac{s_1\dots s_{\mathrm{m}}}{s}$ where s and all $s_i,1\le i\le m$ are formula signed sequents.

Derivations are defined as in Definition 3.

4. The Correspondences

In this section, I present two mappings, one from $S{C}^m$ to $S{C}^s$ and one from $S{C}^s$ to $S{C}^m$, establishing the deductive equivalence of the two systems. Thereby, I establish my claim that the alleged two kinds of multilateralism are just mutual notational variants.

4.1. From $S{C}^m$ to $S{C}^s$

Consider some given signed turnstile sequent calculus $S{C}_0^m$. I now show how to transform it to a deductively equivalent multilateral signed formula sequent calculus $S{C}_0^s$.

I start by presenting a mapping $⟦\cdots {⟧}^{m\Rightarrow s}$ from turnstile signed l sequents (Definition 1) to formula signed sequents (Definition 5).

$$⟦{\Gamma }_1|{\Gamma }_2|\cdots |{\Gamma }_n|{\Gamma }_{n+1}{:}^i\phi {⟧}^{m\Rightarrow s}{=}^{df.}{\cup }_{1\le j\le n+1}{\Gamma }^{{\ast }_j}:{\ast }_i\phi$$

(1)

where ${\Gamma }^{{\ast }_j}{=}^{df.}\left\{{\ast }_j\psi \right|\psi \in {\Gamma }_j\}$.

Remark 1. 

In the resulting signed formula sequent:

• Each formula in the l.h.s. context gets the sign corresponding to the position it occurs in.
• The formula in the r.h.s. context gets the sign of the sequent separator, ‘${:}^i$’.

Next, the mapping is pointwise extended to rules.

$$⟦\frac{s_1\dots s_{\mathrm{m}}}{s}{⟧}^{m\Rightarrow s}{=}^{df.}\frac{⟦s_1{⟧}^{m\Rightarrow s}\cdots ⟦s_m{⟧}^{m\Rightarrow s}}{⟦s{⟧}^{m\Rightarrow s}}$$

(2)

Example 1. 

In [4], a tetra-lateral multiple derivation sequent calculus $SN4mn$ is introduced for Nelson’s logic N4 [9]. Instead of numbering the sequent separators (arrows there), they are marked with $S=\{-,+,m,n\}$. Here ‘+’ means verification, ‘−’ means falsification, ‘m’ means ‘meaningful’ and ‘n’ - ‘nonsensical’.

The conditional falsification right rule is (in my notation):

$$\frac{{\Gamma }_1|{\Gamma }_2|{\Gamma }_3|{\Gamma }_4{:}^{+}\phi {\Gamma }_1|{\Gamma }_2\left|{\Gamma }_3\right|{\Gamma }_4{:}^{-}\psi }{{\Gamma }_1|{\Gamma }_2\left|{\Gamma }_3\right|{\Gamma }_4{:}^{-}\phi \to \psi }(\to r-)$$

This rule translates to

$$\frac{{\Gamma }_1^{-}\cup {\Gamma }_2^{+}\cup {\Gamma }_3^m\cup {\Gamma }_4^n:+\phi {\Gamma }_1^{-}\cup {\Gamma }_2^{+}\cup {\Gamma }_3^m\cup {\Gamma }_4^n:-\psi }{{\Gamma }_1^{-}\cup {\Gamma }_2^{+}\cup {\Gamma }_3^m\cup {\Gamma }_4^n:-(\phi \to \psi )}(\to r-)$$

And the corresponding left rule for falsifying the conditional is

$$\frac{\psi ,{\Gamma }_1|\phi ,{\Gamma }_2\left|{\Gamma }_3\right|{\Gamma }_4{:}^{\ast }\chi }{(\phi \to \psi ),{\Gamma }_1|{\Gamma }_2\left|{\Gamma }_3\right|{\Gamma }_4{:}^{\ast }\chi }(\to I-)$$

where ‘*’ ranges over S, abbreviating several rules to one. This rule translates to

$$\frac{(-\psi ,{\Gamma }_1^{-})\cup (+\phi ,{\Gamma }_2^{+})\cup {\Gamma }_3^m\cup {\Gamma }_4^n:\ast \chi }{({\Gamma }_1^{-},-(\phi \to \psi ))\cup {\Gamma }_2^{+}\cup {\Gamma }_3^m\cup {\Gamma }_4^n:\ast \chi }(\to l-)$$

The initial m-sequent is

$$\varnothing |\varnothing |p|\varnothing {:}^{m}p$$

Which translates to

$$m\left(p\right):m\left(p\right)$$

Now define the multilateral signed formula sequent calculus, denoted by $S{C}_0^s{=}^{df.}⟦S{C}^m{⟧}^{m\Rightarrow s}$, by taking as its rules the outcome of the mapping applied to the rules of $S{C}_0^m$. Note that the initial sequents of $S{C}_0^s$ are the outcome of the mapping applied to the initial sequents of $S{C}_0^m$.

The following theorem is proved by induction on the respective derivations. I omit the tedious details of the proof.

Theorem 1 

(deductive equivalence—I). For every φ, ${\Gamma }_1,...,{\Gamma }_{n+1}$ and every $\ast \in S$:

$$\begin{array}{c}{⊢}_{S{C}_0^m}{\Gamma }_1|\cdots |{\Gamma }_{n+1}{:}^{\ast }\phi \\ \mathrm{iff}\\ {⊢}_{S{C}_0^s}{\cup }_{1\le j\le n+1}\left[\right[{\Gamma }_j{\left]\right]}^{m\Rightarrow s}:\ast \phi \end{array}$$

(3)

4.2. From $S{C}^s$ to $S{C}^m$

Consider some given multilateral signed formula sequent calculus $S{C}_1^s$. I now show how to transform it to a deductively equivalent multilateral signed turnstile sequent calculus $S{C}_1^m$.

I start again by presenting a mapping $⟦\cdots {⟧}^{s\Rightarrow m}$ from signed formula sequents (Definition 5) to signed turnstile sequents (Definition 1).

$$⟦\Gamma :{\ast }_i\phi {⟧}^{s\Rightarrow m}{=}^{df.}{\Gamma }_1|{\Gamma }_2|\cdots |{\Gamma }_n|{\Gamma }_{n+1}{:}^i\phi$$

where ${\Gamma }_i{=}^{df.}\left\{\psi \right|{\ast }_i\psi \in \Gamma \}$.

Remark 2. 

In the resulting signed turnstile sequent:

• Each position consists of the formulas in Γ signed with the respective sign.
• The sequent separator inherits the sign of the formula in the r.h.s. context.

Next, the mapping is pointwise extended to rules.

$$⟦\frac{s_1\cdots s_m}{s}{⟧}^{s\Rightarrow m}{=}^{df.}\frac{⟦s_1{⟧}^{s\Rightarrow m}\cdots ⟧s_m{⟧}^{s\Rightarrow m}}{⟦s{⟧}^{s\Rightarrow m}}$$

(4)

Now define the multilateral signed turnstile sequent calculus, denoted by $S{C}_1^m{=}^{df.}⟦S{C}^s{⟧}^{s\Rightarrow m}$, by taking as its rules the outcome of the mapping applied to the rules of $S{C}_1^s$. Note that the initial sequents of $S{C}_0^s$ are the outcome of the mapping applied to the initial sequents of $S{C}_0^m$.

The following theorem is again proved by induction on the respective derivations. I again omit the tedious details of the proof.

Theorem 2 

(deductive equivalence—II). For every φ, Γ and every ${\ast }_i\in S$:

$$\begin{array}{c}{⊢}_{S{C}_1^s}\Gamma :{\ast }_i\phi \\ \mathrm{iff}\\ {⊢}_{S{C}_1^m}\left[\right[\Gamma :{\ast }_i\phi {\left]\right]}^{s\Rightarrow m}{:}^{{\ast }_i}\phi \end{array}$$

(5)

5. Conclusions

While allegedly the two approaches to bilateralism (and to multilateralism) in proof-theory, namely signed formulas and signed turnstile relations, are essentially different, I showed that this difference is only a matter of appearance. There are natural mappings that show that both kinds of systems are not more than deductively equivalent notational variants. So no approach has any essential proof-theoretic advantage over the other.

Still, they may be conceptual reasons to prefer one approach over the other. For example (I owe these observations to an anonymous referee of this journal), the signed formula calculus seems to be more handy for the formulation of coordination rules, the structural rules of bilateral/multilateral calculi. Also, Simonelli ([10] deploys a variant of this notation that involves schematizing over signs which enables one to do the meta-theory for bilateral systems at higher a level of generality, simplifying proofs in a natural and intuitive way.

Other philosophical considerations for a preference may still emerge.