Another Look at Modality and Connexivity

Abstract

In this paper, I introduce a modal logic CS5, an expansion of the non-connexive propositional classical logic with modalities with a tweaked falsification condition, thereby rendered a connexive modal logic. The logic is defined with a hyper-sequent calculus, properly modified from that of classical S5.

Apendix

The characteristic axioms⁶ of connexive logics (see [4] and [10] for a general survey) are:

Aristotle’s axioms:.:

For every $\varphi$:

$$\begin{aligned} & A_{1}:\ {\vdash }\ \lnot (\varphi {\rightarrow }\lnot \varphi ) \end{aligned}$$

(5.21)

$$\begin{aligned} & A_{2}:\ {\vdash }\ \lnot (\lnot \varphi {\rightarrow }\varphi ) \end{aligned}$$

(5.22)

Boethius’ axioms:.:

For every $\varphi$ and $\psi$:

$$\begin{aligned} & B_{1}:\ {\vdash }\ (\varphi {\rightarrow }\psi ) {\rightarrow }\lnot (\varphi {\rightarrow }\lnot \psi ) \end{aligned}$$

(5.23)

$$\begin{aligned} & B_{2}:\ {\vdash }\ (\varphi {\rightarrow }\lnot \psi ) {\rightarrow }\lnot (\varphi {\rightarrow }\ \psi ) \end{aligned}$$

(5.24)

Also a negative characteristic is added, a non-derivability, to the effect that

$$\begin{aligned} (asym)\ \ {\nvdash }\ (\varphi {\rightarrow }\psi ) {\rightarrow }(\psi {\rightarrow }\varphi ) \end{aligned}$$

(5.25)

The condition (asym) is known as the asymmetry of the conditional, and is imposed to prevent interpreting the conditional as a biconditional.

Those axioms are not classical logic validities, rendering connexive logics as contra-classical. They have a strong intuitive appeal, reflecting properties of a “natural” conditional, closer to some uses of the indicative conditional in natural language.