A General Schema for Bilateral Proof Rules

Abstract

Bilateral proof systems, which provide rules for both affirming and denying sentences, have been prominent in the development of proof-theoretic semantics for classical logic in recent years. However, such systems provide a substantial amount of freedom in the formulation of the rules, and, as a result, a number of different sets of rules have been put forward as definitive of the meanings of the classical connectives. In this paper, I argue that a single general schema for bilateral proof rules has a reasonable claim to inferentially articulating the core meaning of all of the classical connectives. I propose this schema in the context of a bilateral sequent calculus in which each connective is given exactly two rules: a rule for affirmation and a rule for denial. Positive and negative rules for all of the classical connectives are given by a single rule schema, harmony between these positive and negative rules is established at the schematic level by a pair of elimination theorems, and the truth-conditions for all of the classical connectives are read off at once from the schema itself.

Appendix

In this appendix, I lay out the formal system officially and provide proofs of the results stated in the body of the paper.

1.1 Basic Set-Up

I define two sequent calculi. The first is what I’ll call “BK$_+$,” which contains the unrestricted Containment axiom schema and the rule of Conclusion of Incoherence:

where

For the purposes of the present paper, I treat the left side of the sequents as sets of signed formulas. Left sides could alternatively be treated as multi-sets or sequences, and it’s worth noting that, if one does opt for such a treatment, Contraction is eliminable in this system, like Ketonen’s, but I do not deal with this complication for the purposes of the present paper.

Proposition 1.1

The structural rules of Cut and Weakening:

are admissible in BK$_+$.

Proof

One can derive Cut as follows:

and, given Cut, one can derive $\Gamma , B \vdash A$ from $\Gamma \vdash A$ along with an instance of Containment as follows:

$\square$

Proposition 1.2

The connective rules of BK$_+$ are invertible.

Proof

Given any formula of the form $\Gamma \vdash \varvec{c} \langle \varphi \circ \psi \rangle$ or $\Gamma \vdash \varvec{c}^* \langle \varphi \circ \psi \rangle$, we can derive its premises from instances of Containment as follows:

$\square$

The second sequent calculus, which I just call ${\textbf {BK}}$ is the same as BK$_+$, but without the CI rule and with CO restricted to the cases where $\Gamma$ contains only atomic signed formulas and A is a signed atomic formula. The first set of results show that CI and non-atomic CO are eliminable in BK$_+$, and thus, admissible in BK. I leave out the cases involving negation, as the negation rules are not novel to this system and it’s easy to show that the main results hold for them.

1.2 Elimination Proofs and Consequences

Proposition 2.1

Conclusion of Incoherence is eliminable in BK$_+$.

Proof

Proceeds analogously to standard Cut-Elimination, but at this higher level of generality.¹ I’ll refer to the schematic formula “A,” in the above CI schema, which gets eliminated through an application of CI, the “CI formula” (analogously to the “Cut formula”). We induct primarily on CI formula weight with a secondary induction on CI height, where:

Formula Weight: The weight of a sentence $\varphi$, $w(\varphi )$, is defined inductively where $w(p) = 1$ and $w(\varphi \circ \psi ) = w(\varphi ) + w(\psi ) + 1$. The weight of a signed formula is simply the weight of the sentence that is signed.

CI Height: The height of an application of CI is the sum of the heights of the proofs of the premises.

Importantly, uses of RV are not taken to contribute to proof height, since it does not modify the complexity of the formulas and applications of RV can always be immediately undone through another application of RV. In this way, RV is treated analogously to Exchange in standard treatments of sequent calculi in which sequents are treated as involving sequences of formulas rather than sets or multi-sets. It is possible to simplify the proofs so that one doesn’t need to deal with applications of RV by working the equivalent solely left-sided system laid out below, yet, since it is an important philosophical thesis of this paper that doing one’s metatheory at this higher level of generality actually makes good conceptual sense in the bilateralist terms I’ve laid out, I have done the CI-Elimination proof directly in the main system proposed here.

For the proof, we show six ways in which a proof involving CI can be transformed to one with either lesser CI height or CI formula weight. The outline of the proof in which these transformations figure is as follows:

Primary Induction: On CI formula weight:

• 1. Base Case: CI on atoms is eliminable, proven by:

  • 1. Secondary Induction: On CI height:

    • 1. Base Case: CI on atoms of height 0 is eliminable, proven by Case Zero.

    • 2. Inductive Step: If CI on atoms of height n is eliminable, CI on atoms of height $n+1$ is eliminable, proven by Cases One to Four.

• 3. Inductive Step: If CI on formulas of weight n is eliminable, then CI on formulas of weight $n+1$ is eliminable, proven by Cases One through Six.

Elaborating this a bit, Case Zero shows that, when both premises of CI are axioms, so too is the Conclusion. Cases One through Four show that CI height can be reduced in any case where the CI formula is not principal in both premises (a formula is said to be principal in a premise of a rule if the last rule applied was to derive that formula). Since the CI formula will never be principal in the case where it is atomic (since it won’t be derived at all), this suffices to establish the inductive step of the secondary induction. For the primary inductive step, if the CI formula is not principal in both premises, then some series of transformations of type One through Four, will transform it into a proof in which the CI formula is principal in both premises, and then a transformation of type Five or Six will reduce the weight of the CI formula.

Case Zero: CI of height 0, where both premises are axioms or follow from an axiom via a single application of RV (since applications of RV don’t affect proof height, such sequents are counted as axioms for our purposes here). If the left premise is an axiom, then either $A \in \Gamma$ or is some formula B such that $B \in \Gamma$ and $B^* \in \Gamma$. If the right premise is an axiom, then either $A^* \in \Delta$ or is some formula B such that $B \in \Delta$ and $B^* \in \Delta$. So, if both premises are axioms, then either $A \in \Gamma , \Delta$ and $A^* \in \Gamma , \Delta$ or there’s some formula B such that $B \in \Gamma , \Delta$ and $B^* \in \Gamma , \Delta$. Either way, $\Gamma , \Delta \vdash$ is an axiom.

Case One: CI formula is not principal in the left premise, where $\Gamma = \Gamma ', \varvec{c} \langle \varphi \circ \psi \rangle$:

We have a CI of height $n+1 +m$. We can push applications of CI up the proof tree as follows:

to get a CI of lesser height $n+m$.

Case Two: CI formula is not principal in the left premise, where $\Gamma = \Gamma ', \varvec{c}^* \langle \varphi \circ \psi \rangle$:

We have a CI of height $\textit{max}(n, m) + 1 + k$. We can push applications of CI up the proof tree as follows:

to get two CIs of lesser heights $n + k$ and $m +k$

Case Three: CI formula is not principal in the right premise, where $\Delta = \Delta ', \varvec{c} \langle \varphi \circ \psi \rangle$. Exactly analogous to Case One.

Case Four: CI formula is not principal in the right premise, where $\Delta = \Delta ', \varvec{c}^* \langle \varphi \circ \psi \rangle$. Exactly analogous to Case Two.

Case Five: CI formula is principal in both premises, where $A = \varvec{c} \langle \varphi \circ \psi \rangle$

We have a CI of height $\textit{max}(n, m) + 1 + k+1$. We transform the proof tree as follows:²

Here, we have CIs of heights $m +k$ and $n + \textit{max}(m, k)$, and the latter is not necessarily lesser than the original CI height, but the weight of the CI formula has decreased in both cases.

Case Six: CI formula is principal in both premises, where $A = \varvec{c}^* \langle \varphi \circ \psi \rangle$. Exactly analogous to Case Five. $\square$

Proposition 2.2

BK$_+$ is consistent in the sense that, if $\vdash A$ is derivable, then $\vdash A^*$ is not derivable.

Proof

Note first that we cannot derive the empty sequent. Since all of the axioms are of the form $\Gamma , A \vdash A$ (where A is importantly not null), the only way to derive the empty sequent would be through applying the only simplifying rule, CI. Since CI is eliminable, the empty sequent cannot be derived. Suppose now $\vdash A$ and $\vdash A^*$. Then, by CI, we could derive the empty sequent. Since the empty sequent is not derivable, it follows that if $\vdash A$ then $\vdash A^*$ is not derivable. $\square$

Proposition 2.3

Adding any of the connectives of BK$_+$ to a language $L_0$ to yield a language L constitutes a conservative extension of $L_0$ in the sense that, where $\Gamma \cup \{A\}$ contains only formulas of $L_0$, if $\Gamma \nvdash _{L_0} A$, then $\Gamma \nvdash _{L} A$.

Proof

Follows directly from the fact that the only simplifying rule is CI, and this rule is eliminable. $\square$

Proposition 2.4

The axiom schema of Containment can be limited to atoms.

Proof

The proof involves two inductions. For the first induction, we show that any sequent of the $\Gamma , A \vdash A$, where A is atomic, is derivable by induction on complexity of the most complex formulas in $\Gamma$. The base case is immediate as an instance of atomic Containment. For the inductive step, we suppose the most complex formulas in $\Gamma$ are of weight $n+1$ and show that a sequent of the form $\Gamma ', \varvec{c} \langle \varphi \circ \psi \rangle , A, \vdash A$ or $\Gamma ', \varvec{c}^* \langle \varphi \circ \psi \rangle , A, \vdash A$ can be derived from some number of sequents of the form $\Gamma '', A \vdash A$ in which the most complex formulas in $\Gamma ''$ are of weight n:

For the second induction, we show that any sequent of the form $\Gamma , A \vdash A$ is derivable by induction on the complexity of A. The base case is already established. For the inductive step, we suppose that A is complexity $n+1$ and show we can derive $\Gamma , A \vdash A$ from some number of sequents of the form $\Gamma ', B \vdash B$ where B is complexity n. Whether A is of the form $\varvec{c} \langle \varphi \circ \psi \rangle$ or $\varvec{c}^* \langle \varphi \circ \psi \rangle$, the following derivation establishes this:

$\square$

Proposition 2.5

CI, non-atomic CO, Cut, and Weakening are all admissible in BK.

Proof

Since BK is just BK$_+$ without CI and non-atomic CO, and we’ve just shown that these rules are eliminable in BK$_+$, they are admissible in BK. Since Cut and Weakening are admissible given these rules (Proposition 1.1), they too are admissible in BK. $\square$

Proposition 2.6

The connective rules of BK are invertible.

Proof

Proposition 1.2 establishes that the connective rules are invertible, given non-atomic CO and CI. Since these structural rules are admissible in BK, the connective rules of BK are invertible. $\square$

1.3 Equivalence with K, Soundness and Completeness

Consider the following solely left-sided version of BK, which I call BK$_\text {ls}$:

Proposition 3.1

BK$_\text {ls}$ is equivalent to BK in that any BK proof corresponds to a unique BK$_\text {ls}$ proof and any BK$_\text {ls}$ proof corresponds to an equivalence class of BK proofs under Reversal.

Proof

Straightforward by induction on proof height (where, once again, we do not take applications of Reversal to contribute to proof height). For the base case, any instance of the axiom schema of BK$_\text {ls}$ of the form $\Gamma , A, A^* \vdash$, is obtained by Reversal from a BK axiom of the form $\Gamma , A \vdash A$. For the inductive step, we suppose that we’ve shown the correspondence of proofs up to height n, and we show that proofs correspond at height $n+1$ by showing that, for any application of a rule of one system, a Reversed form of the conclusion sequent can be obtained, via a rule in other system, from a Reversed form of the premise sequent(s). $\square$

Proposition 3.2

The fragment of BK$_\text {ls}$ consisting in the rules for negation, conjunction, disjunction, and the conditional is a notational variant of Ketonen [29] multiple conclusion sequent calculus, K:

Proof

We can provide a one-to-one translation schema to show that the two systems are simply notational variants. To translate a K sequent of the form $\Gamma \vdash \Delta$ to a BK$_\text {ls}$ sequent of the form $\Gamma ' \vdash$ let $\Gamma ' = \{+ \langle \varphi \rangle \mid \varphi \in \Gamma \} \cup \{- \langle \varphi \rangle \mid \varphi \in \Delta \}$. Conversely, to translate a BK$_\text {ls}$ sequent of the form $\Gamma ' \vdash$ to a K sequent of the form $\Gamma \vdash \Delta$ let $\Gamma = \{ \varphi \mid + \langle \varphi \rangle \in \Gamma '\}$ and $\Delta = \{ \varphi \mid - \langle \varphi \rangle \in \Gamma '\}$ .

Remark

As an interesting side-note, this notation precisely capture’s Restall’s [43] bilateral reading of multiple conclusion sequents according to which a sequent of the form $\Gamma \vdash \Delta$ expresses that the position consisting in affirming everything in $\Gamma$ and denying everything in $\Delta$ is incoherent.

Let us define the basic semantic notions needed to state soundness and completeness:

Correctness Function: The correctness function [] is a function from $\{+, -\}$ to $\{1, 0\}$ mapping $+$ to 1 and − to 0.

Correctness: Taking some stance $\varvec{a}$ towards some sentence $\varphi$ is correct, relative to some valuation v, just in case $[\varvec{a}] = v(\varphi )$.

Classical Valuations: Let a classical valuation v be any function from $\mathcal {L} \rightarrow \{1, 0\}$ such that

• 1. $\forall p \in \mathcal {A}$, $v(p) = 1$ or $v(p)=0$

• 2. $v(\lnot \varphi ) = {\left\{ \begin{array}{ll} 1, &{} \text {if}\ v(\varphi ) = 0 \\ 0, &{} \text {if}\ v(\varphi ) = 1 \end{array}\right. }$

• 3. $v(\varphi \circ \psi ) = {\left\{ \begin{array}{ll} [\varvec{c}], &{} \text {if}\ v(\varphi ) = [\varvec{a}] \text { and } v(\psi ) = [\varvec{b}] \\ {[}\varvec{c}^*{]}, &{} \text {if}\ v(\varphi ) = {[}\varvec{a}^*{]} \text { or } v(\psi ) ={[}\varvec{b}^*{]} \end{array}\right. }$

Classical Unsatisfiability: A set of signed formulas $\Gamma$ is classically unsatisfiable, $\Gamma \vDash$, just in case there is no classical valuation v such that all of the stances in $\Gamma$ are correct.

Classical Validity: An inference $\Gamma : A$ is classically valid, $\Gamma \vDash A$, just in case there is no classical valuation v such that all of the stances in $\Gamma$ are correct and A is incorrect.

Proposition 3.3

BK$_\text {ls}$ proves $\Gamma \vdash$ just in case $\Gamma \vDash$

Proof

Equivalence with K, which is known to be sound and complete, suffices to establish this for the standard connectives. For all the non-standard connectives, a direct proof of soudness and completeness is easily obtained by schematizing a proof of soundness and completeness for K. $\square$

Proposition 3.4

BK proves $\Gamma \vdash A$ just in case $\Gamma \vDash A$

Proof

Given the equivalence of BK and BK$_\text {ls}$ under Reversal, it suffices just to point out that $\Gamma : \varphi$ is classically valid just in case $\Gamma , \varphi ^*$ is classically unsatisfiable. $\square$