The Core Logic Paradox

TL;DR

This paper proves that Tennant's logical system, known as Core logic, entails a paradox called the Core logic paradox, highlighting a fundamental inconsistency within this logical framework.

Contribution

It introduces and formally proves the existence of the Core logic paradox within Tennant's intuitionistic relevant logic system.

Findings

Identification of the Core logic paradox in Tennant's system

Formal proof demonstrating the paradox's entailment

Implications for the consistency of Core logic

Abstract

This paper provides a proof that Tennant's logical system entails a paradox that is called Core logic paradox, in reference to the new name given by Tennant to his intuitionistic relevant logic.

Tables

Table 1. Table 1: Rules of sequent calculus for propositional Core logic [ 5 ]

\tab@multicol \tab@bgroup            Ax.   $A⊢$$A$ \tab@egroup
$\Delta ⊢$$A$  $L\neg$   $\neg A,\Delta ⊢$	$A,\Delta ⊢$  $R\neg$       $\Delta ⊢$$\neg A$
$\Delta ⊢$$C$  $L\wedge$   $A\wedge B,\Delta \backslash \{A,B\}⊢$$C$                   where $\Delta \cap \{A,B\}\ne \varnothing$	$\Delta ⊢$$A$         $\Gamma ⊢$$B$  $R\wedge$        $\Delta ,\Gamma ⊢$$A\wedge B$
$A,\Delta ⊢$$C/\perp$         $B,\Gamma ⊢$$C/\perp$  $L\vee$           $A\vee B,\Delta ,\Gamma ⊢$$C/\perp$	$\Delta ⊢$$A$  $R\vee$   $\Delta ⊢$$A\vee B$   $\Delta ⊢$$B$  $R\vee$   $\Delta ⊢$$A\vee B$
$\Delta ⊢$$A$         $B,\Gamma ⊢$$C$  $L\to$        $A\to B,\Delta ,\Gamma ⊢$$C$	$A,\Delta ⊢$  $R{\to }_{\left(a\right)}$       $\Delta ⊢$$A\to B$          $\Delta ⊢$$B$  $R{\to }_{\left(b\right)}$   $\Delta \backslash \left\{A\right\}⊢$$A\to B$

Equations

$$\neg A,A⊬B$$

$$⊢\neg A\to (A\to B)$$

$$\Delta ⊣⊢\top \wedge \Delta$$

$$\neg A\to (A\to B),\neg A,A⊬B$$

$$\neg A\to (A\to B),\neg A,A⊢B$$

Full text

1 Two basic claims of Core logic

The idiosyncrasy of Tennant’s logical system lies mainly in this pair of claims:

1. (1)

The First Lewis Paradox that is the sequent $\lnot A,A\vdash B$ is not provable in Core logic, therefore the formula

[TABLE]

is claimed as being valid in Core logic.111Tennant [5, page 195] wrote $\lnot A,A\nvDash B$, because of the completeness claimed for Core logic, formula (1) is justified. 2. (2)

Every intuitionistic logical theorem is provable in Core logic, therefore

[TABLE]

is provable in Core logic.

The slogan “relevance at the level of the turnstile” [5, p. 15, p. 41, p. 121, p. 263] explains this surprising feature of Tennant’s logical system.

The proofs of lemmas and theorem given in the next section are based on the following rules of sequent calculus for propositional Core logic.