# Bilateral Connexive Logic

## Abstract

**:**

## 1. Introduction

- The outer negation in the ${A}_{i}$ axioms and the inner negation in those axioms have the same meaning. Note the difference from [7], in the context of dialectic logics. There, he considers two different negations, in contrast to my distinction between two meanings, depending on where the negation occurs.
- The outer negation in the consequent of the ${B}_{1}$ axiom and the inner negation in the consequent of this axioms have the same meaning.
- The (outer) negation in the consequent of the ${B}_{2}$ axiom and the (inner) negation in the antecedent of this axiom have the same meaning.

- Interpret the external negation in the ${A}_{i}$ axioms as a denial (of $\phi \to \neg \phi $ and $\neg \phi \to \phi $, respectively), and interpret the inner negation as ‘plain’ negation.
- Interpret similarly the two occurrences of negation in the ${B}_{i}$ axioms.

Some hypothetical propositions, however, are affirmative and others are negative […] Affirmatives are when we say “If it is A, it is B”, or “If it is not A, it is B”; negative “If it is A, it is not B”, “If it is not A, it is not B”. For it depends on the consequent whether the proposition is judged to be affirmative or negative.

- The notion of a bilateral connexive logic.
- The distinction in meaning between inner and outer occurrences of negation in the connexive axioms.

## 2. A Bilateral Connexive Natural Deduction Proof System

#### 2.1. Bilateral Logics

#### 2.2. Bilateral Natural Deduction Proof Systems

**Remark**

**1.**

- The prohibition of embedding the denial formal force marker makes bilateralism a suitable framework for my wish to distinguish the meaning of outer and inner occurrences of negation: the former can indicate denial, while the latter serves as ‘plain’ negation.
- The prohibition of iteration and embedding of the formal force markers is the source of the weak connexivity of the logic defined below. If not this prohibition, the ‘natural‘ bilateral reading of ${B}_{1}$ would have been$$\u22a2+(\phi \to \psi )\phantom{\rule{4pt}{0ex}}\to \phantom{\rule{4pt}{0ex}}-(\phi \to \neg \psi )$$

#### 2.3. Bilateral Connexive Natural-Deduction Proof Systems

**Remark**

**2.**

- As usual, discharged assumptions are enclosed between square brackets, indexed by a discharge index, linking the assumption to an instance of a rule actually discharging it. Note that both $[+\phi ]$ and $[-\phi ]$ can serve as discharged assumptions.
- A similar way of formulating the introduction of an assumption is attributed by von Plato [17] to Gentzen. The rule was intended to make the derivation of $\phi \supset \phi $ less awkward and was later abandoned by Gentzen.I revive this formulation here in order to make it explicit that φ, assumed in the premise, is also an explicit conclusion, carrying the same formal force indicator! Thus, when this rule is applied, it uses the assumption in the premise, allowing later discharge of the latter. We thus obtain
- The positive $I/E$-rules for the conditional are the usual intuitionistic $I/E$-rules.
- The negative $I/E$-rules for the conditional are the source of connexivity, reflecting the modification of the falsity conditions of the conditional according to what has become known as the Bochum plan [18].
- The negation rules are those of Rumfitt [8].
- Note how the distinction in meaning of inner and outer negation is reflected in the form of the rules for the denied conditional: inner negation occurs only in the premise, while outer negation occurs in the conclusion.Note that according to a common approach in proof-theoretic semantics (Dummett, Schroeder-Heister, Prawitz, and others), the meanings are captured by canonical derivations. For a detailed account, see [16]. The indicated difference in rules induces a difference in canonical derivations.

**Proposition**

**1**

**Proof.**

#### Establishing b-Connexivity

**Aristotle’s**b**-theses:****Boethius’**b**-theses:**

#### 2.4. Bilateral (in)coherence

## 3. Conclusions

- The logic induces a difference in meaning between inner and outer occurrences of negation in the connexive axioms.
- The logic allows incoherence—assertion and denial of the same formula while still being non-trivial.

## Funding

## Conflicts of Interest

## Notes

1 | While Aristotle’s and Boethius axioms originate in antiquity, the negative requirement was added by McCall [3] (p.417). |

2 | An exception is Ferguson’s paper [6]. |

3 | This notion of weak connexivity is different from the one, with the same name, found in Pizzi [9], where the inner conditional is taken as material: $(\phi \to \psi )\supset \neg (\phi \to \neg \psi )$. It also differs from Kapsner’s weak connexivity in [10,11]. It might be related to the weak connexivity in [12], where Boethius’ axiom also holds only in rule form. |

4 | A proof of non-triviality requires a definition of model theory in order to show non-derivability. I do not present here such a model theory, similarly to the general situation in bilateral logics over signed formulas, where model theories are not provided. Therefore, the claim of non-triviality, while seemingly correct, remains unproved. |

5 | See Note 4 above. |

6 | The same formula was used by McCall [3] to establish negation inconsistency in his connexive logic. |

7 | I omit here the explicit use of the ($Ass$)-rules. |

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Francez, N.
Bilateral Connexive Logic. *Logics* **2023**, *1*, 157-162.
https://doi.org/10.3390/logics1030008

**AMA Style**

Francez N.
Bilateral Connexive Logic. *Logics*. 2023; 1(3):157-162.
https://doi.org/10.3390/logics1030008

**Chicago/Turabian Style**

Francez, Nissim.
2023. "Bilateral Connexive Logic" *Logics* 1, no. 3: 157-162.
https://doi.org/10.3390/logics1030008