# A Fundamental Non-Classical Logic

## Abstract

**:**

## 1. Introduction

Without going nearly so far as to claim that the ability to follow the introduction and elimination rules is all there is to grasping the meaning of ’and’, one can still appreciate that the validity of the introduction and elimination rules is a central semantic fact about ’and’.If we are asked what is the meaning of the word ’and’, at least in the purely conjunctive sense (as opposed to, e.g., its colloquial use to mean ’and then’), the answer is said to be completely given by saying that (i) from any pair of statements P and Q, we can infer the statement formed by joining P to Q with ’and’ (which statement we hereafter describe as ’the statement P-and-Q’), that (ii) for any conjunctive statement P-and-Q we can infer P, and (iii) from P-and-Q we can always infer Q. Anyone who has learnt to perform these inferences knows the meaning of ’and’, for there is simply nothing more to knowing the meaning of ’and’ than being able to perform these inferences. (p. 38)

- (1)
- It’s raining but it might not be raining ($p\wedge \diamond \neg p$)

- (2)
- either it’s raining or it’s not, and it might be raining and it might not be raining ($(p\vee \neg p)\wedge \diamond p\wedge \diamond \neg p$)

- (3)
- it’s raining and it might not be, or it’s not raining and it might be($(p\wedge \diamond \neg p)\vee (\neg p\wedge \diamond p)$),

- if $\phi \u22a2\chi $ and $\psi \u22a2\chi $, then $\phi \vee \psi \u22a2\chi $,

- if $\alpha \wedge \phi \u22a2\chi $ and $\alpha \wedge \psi \u22a2\chi $, then $\alpha \wedge (\phi \vee \psi )\u22a2\chi $, or
- if $\alpha ,\phi \u22a2\chi $ and $\alpha ,\psi \u22a2\chi $, then $\alpha ,(\phi \vee \psi )\u22a2\chi $.

**Remark 1.1.**

**Remark 1.2.**

## 2. Fitch-Style Natural Deduction

- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof and $\tau $ is a proof, then $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\tau \rangle $ is a proof.
- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof and ${\sigma}_{i},{\sigma}_{j}$ are formulas, then $\langle {\sigma}_{1},\dots ,{\sigma}_{n},{\sigma}_{i}\wedge {\sigma}_{j}\rangle $ is a proof (∧I).
- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof and ${\sigma}_{i}$ is a formula of the form $\phi \wedge \psi $, then $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\phi \rangle $ and $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\psi \rangle $ are proofs (∧E).
- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof and ${\sigma}_{i}$ is a formula, then for any formula $\phi $, both$\langle {\sigma}_{1},\dots ,{\sigma}_{n},{\sigma}_{i}\vee \phi \rangle $ and $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\phi \vee {\sigma}_{i}\rangle $ are proofs (∨I).
- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof, ${\sigma}_{i}$ is a formula of the form $\phi \vee \psi $, ${\sigma}_{n-1}$ is a sequence beginning with $\phi $ and ending with $\chi $, and ${\sigma}_{n}$ is a sequence beginning with $\psi $ and ending with $\chi $, then $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\chi \rangle $ is a proof (∨E).
- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof, ${\sigma}_{i}$ is a formula $\psi $, and ${\sigma}_{n}$ is a sequence beginning with $\phi $ and ending with $\neg \psi $, then $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\neg \phi \rangle $ is a proof (¬I).
- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof and ${\sigma}_{i}$ and ${\sigma}_{j}$ are formulas of the form $\phi $ and $\neg \phi $, respectively, then for any formula $\psi $, $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\psi \rangle $ is a proof (¬E).

This formulation of ¬ introduction is admissible in Fitch’s system, thanks to his Reiteration rule; but Fitch [14] states his ¬ introduction rule in a way that requires a pair of contradictory formulas to appear in the subproof that starts with $\phi $.4 To accomplish what we accomplish with ¬I, Fitch would reiterate $\psi $ into the subproof beginning with $\phi $ to obtain a contradiction between $\psi $ and $\neg \psi $ within the subproof. But we can disassociate Reiteration, which we do not allow (recall the cautionary Figure 1), from ¬ introduction. The idea of Reiteration is that if $\psi $ was derived just before a subproof beginning with $\phi $, then ψ still holds under the assumption of φ. By contrast, when applying our ¬I rule, we prove that the negation of $\psi $ holds under the assumption of φ, and then since we know that ψ holds prior to the assumption of φ, we deduce $\neg \phi $.5if from the assumption of φ, you derive the negation of another formula derived just before the assumption, then conclude $\neg \phi $.

**Definition 2.1.**

1. $\phi \u22a2\phi $ | 8. if $\phi \u22a2\psi $ and $\psi \u22a2\chi $, then $\phi \u22a2\chi $ |

2. $\phi \wedge \psi \u22a2\phi $ | |

3. $\phi \wedge \psi \u22a2\psi $ | 9. if $\phi \u22a2\psi $ and $\phi \u22a2\chi $, then $\phi \u22a2\psi \wedge \chi $ |

4. $\phi \u22a2\phi \vee \psi $ | |

5. $\phi \u22a2\psi \vee \phi $ | 10. if $\phi \u22a2\chi $ and $\psi \u22a2\chi $, then $\phi \vee \psi \u22a2\chi $ |

6. $\phi \u22a2\neg \neg \phi $ | |

7. $\phi \wedge \neg \phi \u22a2\psi $ | 11. if $\phi \u22a2\psi $, then $\neg \psi \u22a2\neg \phi $. |

**Proposition 2.2.**

**Proposition 2.3.**

**Proof.**

**Figure 4.**Given a proof from $\phi \wedge \psi $ to ⊥, which easily yields a proof from $\phi \wedge \psi $ to $\neg \psi $, Reiteration would permit the construction of a proof from $\psi $ to $\neg \phi $.

**Figure 5.**To modify Gentzen-style natural deduction rules to match our dropping of Reiteration from Fitch-style natural deduction, for ∨E the only open assumptions of ${\mathcal{D}}_{1}$ and ${\mathcal{D}}_{2}$ may be $\phi $ and $\psi $, respectively; for ¬I the only open assumption of ${\mathcal{D}}_{1}$ may be $\phi $.

**Theorem 2.4**

## 3. Algebras

**Definition 3.1.**

**Definition 3.2.**

pre | proto | ultraweak pseudo | weak pseudo | pseudo | ortho | |
---|---|---|---|---|---|---|

$a\le b\Rightarrow \neg b\le \neg a$ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

$\neg 1=0$ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

$\neg 0=1$ | ✓ | ✓ | ✓ | ✓ | ✓ | |

$a\wedge \neg a=0$ | ✓ | ✓ | ✓ | ✓ | ||

$a\le \neg \neg a$ | ✓ | ✓ | ✓ | ✓ | ||

$a\wedge b=0\Rightarrow b\le \neg a$ | ✓ | |||||

$\neg \neg a\le a$ | ✓ |

**Remark 3.3.**

**Lemma 3.4.**

- 1.
- If ¬ is a semicomplementation, then ¬ is anti-inflationary: $a\nleqq \neg a$ for all nonzero $a\in L$. If ¬ is antitone and anti-inflationary, then ¬ is a semicomplementation.
- 2.
- ¬ satisfies antitonicity and double negation introduction iff for all $a,b\in L$, $a\le \neg b$ implies $b\le \neg a$.
- 3.
- ¬ is an orthocomplementation iff ¬ is a weak pseudocomplementation satisfying double negation elimination: $\neg \neg a\le a$ for all $a\in L$.

**Proof.**

**Figure 7.**${\mathbf{N}}_{5}$ equipped with a pseudocomplementation (

**left**), a weak pseudocomplementation (

**middle**), and a protocomplementation (

**right**), indicated by dashed arrows. Arrows for $\neg 0=1$ and $\neg 1=0$ are omitted.

**Figure 8.**The Benzene ring ${\mathbf{O}}_{6}$ equipped with an orthocomplementation (

**left**) and pseudocomplementation (

**right**), indicated by dashed arrows. Arrows for $\neg 0=1$ and $\neg 1=0$ are omitted.

**Definition 3.5.**

**Proposition 3.6.**

**Proof.**

**Proposition 3.7.**

**Proposition 3.8.**

## 4. Relational Representation and Semantics

#### 4.1. From Relational Frames to Lattices with Negation

**Definition 4.1.**

**Remark 4.2.**

**Proposition 4.3.**

**Proposition 4.4.**

- 1.
- the operation ${c}_{\u22b2}:\wp \left(X\right)\u27f6\wp \left(X\right)$ defined by$${c}_{\u22b2}\left(A\right)=\{x\in X\mid \forall {x}^{\prime}\u22b2x\phantom{\rule{0.277778em}{0ex}}\exists {x}^{\u2033}\u22b3{x}^{\prime}:\phantom{\rule{0.166667em}{0ex}}{x}^{\u2033}\in A\}$$
- 2.
- the operation ${\neg}_{\u22b2}:\wp \left(X\right)\u27f6\wp \left(X\right)$ defined by$${\neg}_{\u22b2}A=\{x\in X\mid \forall y\u22b2x\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}y\notin A\}$$

**Proposition 4.5.**

- 1.
- the ${c}_{\u22b2}$-fixpoints ordered by ⊆ form a complete lattice $\mathfrak{L}(X,\u22b2)$ with meet and join calculated as in Proposition 4.3;
- 2.
- ${\neg}_{\u22b2}$ is a precomplementation on $\mathfrak{L}(X,\u22b2)$;
- 3.
- if ⊲ is reflexive, then ${\neg}_{\u22b2}$ is a protocomplementation on $\mathfrak{L}(X,\u22b2)$.

**Definition 4.6.**

**Lemma 4.7.**

- 1.
- the 0 of $\mathfrak{L}(X,\u22b2)$ is the set of absurd states, also equal to ${\neg}_{\u22b2}1$;
- 2.
- ${\neg}_{\u22b2}0=1$ iff there is no $y\in X$ and absurd $x\in X$ with $x\u22b2y$.

**Proof.**

**Remark 4.8.**

**Remark 4.9.**

**Example 4.10.**

- from any $x\in X\backslash A$, you can step forward along an arrow to a state ${x}^{\prime}$ that cannot step backward along an arrow into A.

**Figure 9.**Reflexive frame representations of the lattice expansions in Figure 7.

**Figure 10.**Reflexive frame representations of the lattice expansions in Figure 8.

**Definition 4.11.**

- 1.
- x pre-refines y if for all $z\in X$, $z\u22b2x$ implies $z\u22b2y$;
- 2.
- x post-refines y if for all $z\in X$, $x\u22b2z$ implies $y\u22b2z$;
- 3.
- x refines y if x pre-refines and post-refines y;
- 4.
- x is compossible with y if there is a non-absurd $w\in X$ that refines x and pre-refines y.

**Lemma 4.12.**

**Proof.**

**Theorem 4.13.**

- 1.
- $(L,\neg )$ is a complete Heyting algebra with pseudocomplementation ¬ iff $(L,\neg )$ is isomorphic to $(\mathfrak{L}(X,\u22b2),{\neg}_{\u22b2})$ for a relational frame $(X,\u22b2)$ in which ⊲ is reflexive and compossible.
- 2.
- $(L,\neg )$ is a complete ortholattice with orthocomplementation ¬ iff $(L,\neg )$ is isomorphic to $(\mathfrak{L}(X,\u22b2),{\neg}_{\u22b2})$ for a relational frame $(X,\u22b2)$ in which ⊲ is reflexive and symmetric.
- 3.
- $(L,\neg )$ is a complete Boolean algebra with Boolean negation ¬ iff $(L,\neg )$ is isomorphic to $(\mathfrak{L}(X,\u22b2),{\neg}_{\u22b2})$ for a relational frame $(X,\u22b2)$ in which ⊲ is reflexive, symmetric, and compossible.

**Proposition 4.14.**

- (a) for all ${c}_{\u22b2}$-fixpoints A, we have $A\cap {\neg}_{\u22b2}A=0$;(b) for all non-absurd $x\in X$, there is a $z\u22b2x$ that pre-refines x.
- (a) for all ${c}_{\u22b2}$-fixpoints A, we have $A\subseteq {\neg}_{\u22b2}{\neg}_{\u22b2}A$;(b) pseudosymmetry: for all $x\in X$ and $y\u22b2x$, there is a $z\u22b2y$ that pre-refines x.
- (a) for all ${c}_{\u22b2}$-fixpoints $A,B$, if $A\cap B=0$, then $A\subseteq {\neg}_{\u22b2}B$.(b) weak compossibility: for all $x\in X$ and $y\u22b2x$, there is a non-absurd z that pre-refines y and x.
- (a) for all ${c}_{\u22b2}$-fixpoints A, we have ${\neg}_{\u22b2}{\neg}_{\u22b2}A\subseteq A$;(b) for all $x\in X$ and $y\u22b2x$, there is a ${y}^{\prime}\u22b2x$ such that for all $z\in X$, if $z\u22b2{y}^{\prime}$ then $y\u22b2z$.

**Proof.**

**Remark 4.15.**

**Remark 4.16.**

**Definition 4.17.**

**Example 4.18.**

**Definition 4.19.**

- 1.
- $\mathcal{M},x\u22a9p$ iff $x\in V\left(p\right)$;
- 2.
- $\mathcal{M},x\u22a9\neg \phi $ iff for all ${x}^{\prime}\u22b2x$, $\mathcal{M},{x}^{\prime}\u22ae\phi $;
- 3.
- $\mathcal{M},x\u22a9\phi \wedge \psi $ iff $\mathcal{M},x\u22a9\phi $ and $\mathcal{M},x\u22a9\psi $;
- 4.
- $\mathcal{M},x\u22a9\phi \vee \psi $ iff $\forall {x}^{\prime}\u22b2x$$\exists {x}^{\u2033}\u22b3{x}^{\prime}$: $\mathcal{M},{x}^{\u2033}\u22a9\phi $ or $\mathcal{M},{x}^{\u2033}\u22a9\psi $.

**Lemma 4.20.**

**Example 4.21.**

#### 4.2. Discrete Representation of Lattices with Negation

_{⊲}-fixpoints the given lattice embeds. The following definition and result are from [21] with some details expanded.

**Definition 4.22.**

- 1.
- if $a\nleqq b$, then there is a $(c,d)\in P$ with $c\le a$ and $c\nleqq b$;
- 2.
- for all $(c,d)\in P$, if $c\nleqq b$, then there is a $({c}^{\prime},{d}^{\prime})\u22b2(c,d)$ such that for all$({c}^{\u2033},{d}^{\u2033})\u22b3({c}^{\prime},{d}^{\prime})$, we have ${c}^{\u2033}\nleqq b$.

**Proposition 4.23.**

- 1.
- f is a complete embedding of L into $\mathfrak{L}(P,\u22b2)$;
- 2.
- if L is complete, then f is an isomorphism from L to $\mathfrak{L}(P,\u22b2)$.

**Proof.**

**Theorem 4.24.**

- 1.
- If ¬ is a precomplementation on L, then where$$P=\left\{\right(a,\neg a)\mid a\in L\}\cup \left\{\right(1,b)\mid b\in \mathrm{\Lambda}\},$$
- 2.
- If ¬ is a protocomplementation on L, then where$$P=\left\{\right(a,\neg a)\mid a\in L,a\ne 0\}\cup \left\{\right(1,b)\mid b\in \mathrm{\Lambda},b\ne 1\},$$
- 3.
- If ¬ is an ultraweak pseudocomplementation on L, then where$$P=\left\{\right(a,\neg a)\mid a\in \mathrm{V}\}\cup \left\{\right(1,b)\mid b\in \mathrm{\Lambda}\},$$
- 4.
- If ¬ is a weak pseudocomplementation on L, then where$$P=\left\{\right(a,\neg a)\mid a\in \mathrm{V},a\ne 0\}\cup \left\{\right(1,b)\mid b\in \mathrm{\Lambda},b\ne 1\},$$

**Proof.**

**Example 4.25.**

**Figure 12.**A lattice with weak pseudocomplementation (

**left**) represented by a pseudosymmetric reflexive frame (

**right**) (with reflexive loops assumed but not shown) as in Theorem 4.24.4.

**Remark 4.26.**

**Theorem 4.27.**

- 1.
- $\phi {\u22a2}_{\mathsf{pre}}\psi $ if and only if $\phi {\vDash}_{\mathbb{K}}\psi $;
- 2.
- $\phi {\u22a2}_{\mathsf{pro}}\psi $ if and only if $\phi {\vDash}_{\mathbb{R}}\psi $;
- 3.
- $\phi {\u22a2}_{\mathsf{para}}\psi $ if and only if $\phi {\vDash}_{\mathbb{P}}\psi $ (resp. $\phi {\vDash}_{\mathbb{S}}\psi $);
- 4.
- $\phi {\u22a2}_{\mathsf{F}}\psi $ if and only if $\phi {\vDash}_{\mathbb{PR}}\psi $ (resp. $\phi {\vDash}_{\mathbb{SR}}\psi $);
- 5.
- $\phi {\u22a2}_{\mathsf{psu}}\psi $ if and only if $\phi {\vDash}_{\mathbb{WCR}}\psi $.

**Proof.**

**Proposition 4.28.**

**Proof.**

**Theorem 4.29.**

**Proof.**

#### 4.3. Topological Representation of Lattices with Negations

**Theorem 4.30.**

- 1.
- an embedding of $(L,\neg )$ into $(\mathfrak{L}\left(\mathsf{FI}(L,\neg )\right),{\neg}_{\u22b2})$ and
- 2.
- an isomorphism from L to the subalgebra of $(\mathfrak{L}\left(\mathsf{FI}(L,\neg )\right),{\neg}_{\u22b2})$ consisting of ${c}_{\u22b2}$-fixpoints that are compact open in the space $\mathsf{S}\left(L\right)$.

**Proof.**

**Remark 4.31.**

**Proposition 4.32.**

**Proof.**

#### 4.4. Modal Translations

**S4**[94,95], the modal logic of reflexive and transitive frames. In a similar spirit, Goldblatt [16] gave a full and faithful embedding of orthologic into the normal modal logic

**KTB**, the modal logic of reflexive and symmetric frames. Below we will give a full and faithful embedding of our logic ${\u22a2}_{\mathsf{F}}$ into the extension of the minimal temporal logic ${\mathbf{K}}_{t}$ [96] (Def. 4.33) with the reflexivity axiom $Hq\to q$ and the pseudosymmetry axiom $Hq\to HPHq$ (or $FHq\to PHq$), based on viewing ⊲ in our frames as the temporal relation. We call this logic ${\mathbf{K}}_{t}\mathbf{TP}$. The pseudosymmetry axiom $\mathbf{P}$ is Sahlqvist and hence canonical [96] (Thm. 4.42), so ${\mathbf{K}}_{t}\mathbf{TP}$ is complete with respect to the class of pseudosymmetric reflexive frames. In fact, the canonical frame for ${\mathbf{K}}_{t}\mathbf{TP}$ [96] (Def. 4.34) is strongly pseudosymmetric. For where $\mathrm{\Gamma}$ and $\mathrm{\Sigma}$ are maximally consistent sets and R is the canonical relation, we claim that if $\mathrm{\Gamma}R\mathrm{\Sigma}$, then

**Proposition 4.33.**

**Proposition 4.34.**

**Proof.**

## 5. Quantification

- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof, ${\sigma}_{i}$ is a formula $\phi $, and v does not occur free in ${\sigma}_{1}$, then $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\forall v\phi \rangle $ is a proof (∀I).
- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof, ${\sigma}_{i}$ is a formula of the form $\forall v\phi $, and u is substitutable for v in $\phi $, then $\langle {\sigma}_{1},\dots ,{\sigma}_{n},{\phi}_{u}^{v}\rangle $ is a proof (∀E).
- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof, ${\sigma}_{i}$ is a formula of the form ${\phi}_{u}^{v}$, and u is substitutable for v in $\phi $, then $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\exists v\phi \rangle $ is a proof (∃I).
- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof, ${\sigma}_{i}$ is a formula of the form $\exists v\phi $, ${\sigma}_{n}$ is a proof beginning with $\phi $ and ending with $\psi $, and v does not occur free in $\psi $, then $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\psi \rangle $ is a proof (∃E).

**Theorem 5.1**

**.**It is decidable in double exponential time whether $\phi {\u22a2}_{\mathsf{FQ}}\psi $.

- $\mathcal{M},x{\u22a9}_{g}P({v}_{1},\dots ,{v}_{n})$ iff $x\in V(P,g\left({v}_{1}\right),\dots ,g\left({v}_{n}\right))$;
- clauses for ¬, ∧, and ∨ as before;
- $\mathcal{M},x{\u22a9}_{g}\forall v\phi $ iff $\forall h{\sim}_{v}g$, $\mathcal{M},x{\u22a9}_{h}\phi $;
- $\mathcal{M},x{\u22a9}_{g}\exists v\phi $ iff $\forall {x}^{\prime}\u22b2x$$\exists {x}^{\u2033}\u22b3{x}^{\prime}$$\exists h{\sim}_{v}g$: $\mathcal{M},{x}^{\u2033}{\u22a9}_{h}\phi $.

**Lemma 5.2.**

**Proof.**

**Theorem 5.3.**

**Proof.**

## 6. Comments on Conditionals

- 1.
- $y\in B$;
- 2.
- $\exists z\u22b2y$: $z\in B$;
- 3.
- $\exists z\u22b3y$: $z\in B$;
- 4.
- $\exists z\u22b2y$: $z\in A\cap B$;
- 5.
- $\exists z\u22b3y$: $z\in A\cap B$.

**Definition 6.1.**

- 1.
- $1\to a\le a$;
- 2.
- $a\wedge b\le a\to b$;
- 3.
- $a\to b\le a\to (a\wedge b)$;
- 4.
- if $b\le a$, then $a\to (b\to c)\le b\to c$;
- 5.
- if $b\le c$, then $a\to b\le a\to c$.

**Proposition 6.2.**

**Proof.**

**Theorem 6.3.**

**Proof.**

**Theorem 6.4.**

**Proof.**

## 7. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof given R and $\tau $ is a proof given $R\phantom{\rule{0.166667em}{0ex}}\cup \phantom{\rule{0.166667em}{0ex}}\{{\sigma}_{i}\mid {\sigma}_{i}\text{a formula}\}$, then $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\tau \rangle $ is a proof given R.
- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof given R and $\phi \in R$, then $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\phi \rangle $ is a proof given R (Reiteration).
- closure conditions for ∧I, ∧E, ∨I, ∨E, ¬I, and ¬E as in Section 2 with ’proof’ replaced by ’proof given R’.

- If $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof given R, ${\sigma}_{i}$ is a formula of the form $\psi $, and ${\sigma}_{n}$ is a sequence beginning with $\neg \phi $ and ending with $\neg \psi $, then $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\phi \rangle $ is a proof given R (RAA).

## Appendix B

- $\neg 0=1$ turns into $b\to b=1$;
- $a\wedge \neg a\le 0$ turns into $a\wedge (a\to b)\le b$;
- $a\le \neg \neg a$ turns into $a\le (a\to b)\to b$;
- $a\wedge c\le 0\Rightarrow a\le \neg c$ turns into $a\wedge c\le b\Rightarrow a\le c\to b$.

**Lemma B.1.**

- (a) for all ${c}_{\u22b2}$-fixpoints B, we have $B{\to}_{\u22b2}B=1$;(b) for all $x\in X$ and $y\u22b2x$, there is a $z\u22b3y$ that pre-refines y.
- (a) for all ${c}_{\u22b2}$-fixpoints $A,B$, we have $A\cap \left(A{\to}_{\u22b2}B\right)\subseteq B$;(b) right pre-interpolation: for all $x\in X$ and $y\u22b2x$, there is a $z\u22b2x$ that post-refines y and pre-refines x.
- (a) for all ${c}_{\u22b2}$-fixpoints $A,B$, we have $A\subseteq \left(A{\to}_{\u22b2}B\right){\to}_{\u22b2}B$;(b) left pre-interpolation: for all $x\in X$ and $y\u22b2x$, there is a $z\u22b2y$ that post-refines y and pre-refines x.
- (a) for all ${c}_{\u22b2}$-fixpoints $A,B,C$, if $A\cap C\subseteq B$, then $A\subseteq C{\to}_{\u22b2}B$;(b) left post-extendability: for all $x\in X$ and $y\u22b2x$, there is a $z\u22b3y$ that pre-refines y and x.

**Proof.**

**Definition B.2.**

- 1.
- $a=1\to a$;
- 2.
- $a\to (a\to b)\le a\to b$;
- 3.
- if $a\le b$, then $b\to c\le a\to c$;
- 4.
- if $a\le b$, then $c\to a\le c\to b$.

**Lemma B.3.**

**Proof.**

**Definition B.4.**

**Lemma B.5.**

- 1.
- for all $a,b\in L$, $a\le (a\to b)\to b$;
- 2.
- for all $a,b,c\in L$, if $a\le c\to b$, then $c\le a\to b$.

**Proof.**

**Theorem B.6.**

- 1.
- If → is a preimplication on L, then where$$P=\left\{\right(a,a\to b)\mid a,b\in L\},$$
- 2.
- If → is a protoimplication on L, then where$$P=\left\{\right(a,a\to b)\mid a,b\in L,a\nleqq b\},$$
- 3.
- If → is an ultraweak pseudoimplication on L, then where$$P=\left\{\right(a,a\to b)\mid a\in \mathrm{V},b\in L\}\cup \left\{\right(1,1\to b)\mid b\in \mathrm{\Lambda}\},$$
- 4.
- If → is a weak pseudoimplication on L, then where$$P=\left\{\right(a,a\to b)\mid a\in \mathrm{V},b\in L,a\nleqq b\},$$
- 5.
- If → is a relative pseudocomplementation on L, then where$$P=\left\{\right(a,a\to b)\mid a\in \mathrm{V},b\in \mathrm{\Lambda},a\nleqq b\},$$

**Proof.**

**Theorem B.7.**

- 1.
- an embedding of $(L,\to )$ into $(\mathfrak{L}\left(\mathsf{FI}(L,\to )\right),{\to}_{\u22b2})$ and
- 2.
- an isomorphism from L to the subalgebra of $(\mathfrak{L}\left(\mathsf{FI}(L,\to )\right),{\to}_{\u22b2})$ consisting of ${c}_{\u22b2}$-fixpoints that are compact open in the space $\mathsf{S}\left(L\right)$.

**Proof.**

## Notes

1 | This is in contrast to ’It’s raining but I don’t know it’, which is infelicitous to assert but does not embed like a contradiction; e.g., it is fine in the antecedent of a conditional such as ’If it’s raining but I don’t know it, I’ll be surprised when I get wet’. For a review of evidence that the badness of (1) is not merely pragmatic, see [8] (Section 2.1). |

2 | On the importance of this distinction concerning side assumptions in relation to the idea that the introduction and elimination rules for ∨ should be in “harmony” with each other, see [12] (p. 229). |

3 | To avoid ambiguity, assume formulas are constructed in such a way that no formula is a sequence beginning with a formula. |

4 | Note that if one does derive a pair of contradictory formulas in a subproof that starts with $\phi $, then by ¬E one can derive the negation of a formula derived just before the assumption of the subproof, so our ¬I rule is applicable. |

5 | Note that our ¬I rule produces proofs of the form $\langle \dots ,\psi ,\dots ,\langle \phi ,\dots ,\neg \psi \rangle ,\neg \phi \rangle $ but not $\langle \dots ,\psi ,\dots ,\langle \chi ,\dots ,\langle \phi ,\dots ,\neg \psi \rangle ,\neg \phi \rangle \rangle $ (where $\psi $ is not an element of the subproof beginning with $\chi $). If we were to strengthen ¬I so as to allow the intervention of the additional assumption $\chi $ as in the preceding sequence, then we could commit the same mistakes to which Reiteration leads as in Section 1. Indeed, we could reiterate any negated formula into a subproof: given a formula $\neg \psi $ immediately preceding a subproof $\sigma $ beginning with $\chi $, to reiterate $\neg \psi $ into $\sigma $, create a subproof ${\sigma}^{\prime}$ inside $\sigma $ such that ${\sigma}^{\prime}$ begins with $\psi $, from which we can prove $\neg \neg \psi $, contradicting the $\neg \psi $ occurring before the assumptions of $\chi $ and $\psi $, in which case the strengthened rule would allow us to conclude $\neg \psi $ after ${\sigma}^{\prime}$. Then a restricted version of pseudocomplementation would hold: if $\phi \wedge \neg \psi \u22a2\perp $, then $\neg \psi \u22a2\neg \phi $. But then from the fact that $\diamond \neg p\wedge \neg \neg p$ (“It might be that it isn’t raining, but it’s not the case that it isn’t raining”) is contradictory, we could prove using the restricted version of pseudocomplementation and other properties of negation that $\diamond \neg p\u22a2\neg p$. |

6 | By contrast, a Glivenko-style theorem [31] stating that $\phi {\u22a2}_{\mathsf{O}}\psi $ iff $\phi {\u22a2}_{\mathsf{F}}\neg \neg \psi $ does not hold, because $\neg \neg p\wedge \neg \neg q{\u22a2}_{\mathsf{O}}p\wedge q$ but $\neg \neg p\wedge \neg \neg q{\u22ac}_{\mathsf{F}}\neg \neg (p\wedge q)$, as we show semantically in Section 3. |

7 | We do not have ⊥ as a primitive in our language, so we formulate ¬E as follows: proofs of $\phi $ and $\neg \phi $ may be joined with a new root labeled by any formula $\psi $, forming a proof that inherits all the open assumptions of the two proofs. |

8 | |

9 | Ultraweak pseudocomplementations are equivalent to what Dunn and Zhou [44] call quasi-minimal negations with the added assumption that $\neg 1=0$ (see Remark 3.3). |

10 | |

11 | Another definition of c, building in monotonicity, is $c\left(A\right)=\{x\in X\mid \exists B\in N(x):B\subseteq A\}$. |

12 | |

13 | It follows that accepting A entails rejecting $\neg A$. The ideas that accepting A is inconsistent with rejecting A and that accepting $\neg A$ entails rejecting A will follow from the key conditions on frames for fundamental logic. |

14 | If $x\u22b2y$ and y accepts A, so $y\in A$, then x does not reject A by definition. Conversely, if $x\overline{)\u22b2}y$, then using Proposition 4.4, y accepts the proposition ${c}_{\u22b2}\left(\left\{y\right\}\right)$ but x rejects it given $x\overline{)\u22b2}y$. |

15 | Given this definition of the closure operation, a candidate definition of morphism between $(X,\u22b2)$ and $({X}^{\prime},{\u22b2}^{\prime})$ is a map $f:X\u27f6{X}^{\prime}$ such that (i) $y\u22b2x$ implies $f\left(y\right){\u22b2}^{\prime}f\left(x\right)$, and (ii) if ${y}^{\prime}{\u22b2}^{\prime}f\left(x\right)$, then $\exists y\u22b2x$$\forall z\u22b3y$$f\left(z\right){\u22b3}^{\prime}{y}^{\prime}$. Condition (ii) guarantees that if ${A}^{\prime}$ is a fixpoint of ${c}_{{\u22b2}^{\prime}}$, then ${f}^{-1}\left[{A}^{\prime}\right]$ is a fixpoint of ${c}_{\u22b2}$. For suppose ${x}^{\prime}\notin {f}^{-1}\left[{A}^{\prime}\right]$, so $f\left({x}^{\prime}\right)\notin {A}^{\prime}$. Then since ${A}^{\prime}$ is a fixpoint of ${c}_{{\u22b2}^{\prime}}$, there is a ${y}^{\prime}{\u22b2}^{\prime}f\left({x}^{\prime}\right)$ such that for all ${z}^{\prime}{\u22b3}^{\prime}{y}^{\prime}$, we have ${z}^{\prime}\notin {A}^{\prime}$. By (ii), $\exists y\u22b2x$$\forall z\u22b3y$$f\left(z\right){\u22b3}^{\prime}{y}^{\prime}$, which by the previous sentence implies $\exists y\u22b2x$$\forall z\u22b3y$$f\left(z\right)\notin {A}^{\prime}$ and hence $z\notin {f}^{-1}\left[{A}^{\prime}\right]$. This shows that ${f}^{-1}\left[{A}^{\prime}\right]$ is a fixpoint of ${c}_{\u22b2}$. If we want morphisms that also preserve negation, then ${f}^{-1}\left[{\neg}_{{\u22b2}^{\prime}}{A}^{\prime}\right]\subseteq {\neg}_{\u22b2}{f}^{-1}\left[{A}^{\prime}\right]$ follows from (i), and ${\neg}_{\u22b2}{f}^{-1}\left[{A}^{\prime}\right]\subseteq {f}^{-1}\left[{\neg}_{{\u22b2}^{\prime}}{A}^{\prime}\right]$ follows from the additional condition (iii) that if ${y}^{\prime}{\u22b2}^{\prime}f\left(x\right)$, then $\exists y\u22b2x$$\forall {z}^{\prime}{\u22b2}^{\prime}f\left(y\right)$${z}^{\prime}{\u22b2}^{\prime}{y}^{\prime}$. For if $x\notin {f}^{-1}\left[{\neg}_{{\u22b2}^{\prime}}{A}^{\prime}\right]$, so $f\left(x\right)\notin {\neg}_{{\u22b2}^{\prime}}{A}^{\prime}$, then there is a ${y}^{\prime}{\u22b2}^{\prime}f\left(x\right)$ with ${y}^{\prime}\in {A}^{\prime}$. Then we claim for the $y\u22b2x$ given by (iii) that $f\left(y\right)\in {A}^{\prime}$; for by (iii), $f\left(y\right)\in {c}_{{\u22b2}^{\prime}}\left(\left\{{y}^{\prime}\right\}\right)$, and since ${y}^{\prime}\in {A}^{\prime}$, we have ${c}_{{\u22b2}^{\prime}}\left(\left\{{y}^{\prime}\right\}\right)\subseteq {c}_{{\u22b2}^{\prime}}\left({A}^{\prime}\right)={A}^{\prime}$. Hence $x\notin {\neg}_{\u22b2}{f}^{-1}\left[{A}^{\prime}\right]$. |

16 | |

17 | Note that in this setting, ’⊥’ and ’⊤’ are arguably no longer appropriate symbols to abbreviate $p\wedge \neg p$ and $\neg (p\wedge \neg p)$. |

18 | This shows that ${\u22a2}_{\mathsf{pre}}$ is complete with respect to bounded lattices with precomplementations satisfying $\neg 0=1$. This depends on the fact that we do not have primitive symbols ⊥ and ⊤ interpreted as 0 and 1 in our language. If we had such symbols in a language ${\mathcal{L}}_{\perp ,\top}$ with corresponding rules $\perp \u22a2\phi $ and $\phi \u22a2\top $ in the definition of ${\u22a2}_{{\mathsf{pre}}_{\perp ,\top}}$, then ${\u22a2}_{{\mathsf{pre}}_{\perp ,\top}}$ would not be complete with respect to lattices with precomplementations satisfying $\neg 0=1$, and the Lindenbaum-Tarski algebra of ${\u22a2}_{{\mathsf{pre}}_{\perp ,\top}}$ would be bounded in the first place. |

19 | When defining a proof given a set R of reiterables as in Appendix A, ∀I states that if $\langle {\sigma}_{1},\dots ,{\sigma}_{n}\rangle $ is a proof given R, ${\sigma}_{i}$ is a formula $\phi $, and v does not occur free in ${\sigma}_{1}$ or in any formula in R, then $\langle {\sigma}_{1},\dots ,{\sigma}_{n},\forall v\phi \rangle $ is a proof given R. |

20 | Recall Theorem 4.13.1. In compossible reflexive frames, a definition used in [21] (Thm. 2.21(i)) that is equivalent to options 3 and 5 is that $x\in A\to B$ iff for every y that pre-refines x, if $y\in A$, then $y\in B$. Toward proving the equivalence, first a lemma about Modus Ponens under option 3: if $x\in A$ and $x\in A\to B$, then $x\in B$. For if $y\u22b2x$, then by compossibility, there is a z that refines y and pre-refines x; since ⊲ is reflexive and z pre-refines x, we have $z\u22b2x$ and $z\in A$ by Lemma 4.12. Given $x\in A\to B$, $z\u22b2x$, and $z\in A$, there is a $w\in B$ with $z\u22b2w$. Then since z post-refines y, we have $y\u22b2w$. Thus, we have shown that $\forall y\u22b2x$$\exists w\u22b3y$: $w\in B$, so $x\in B$. Now for the equivalence, suppose $x\in A\to B$ according to option 3. Further suppose that y pre-refines x, and $y\in A$. Then $y\in A\to B$ by Lemma 4.12, so $y\in B$ by the Modus Ponens lemma, so $x\in A\to B$ according to the definition from [21]. Conversely, suppose $x\in A\to B$ according to that definition, which obviously validates Modus Ponens. Further suppose $y\u22b2x$ and $y\in A$. Then by compossibility, there is a z that refines y and pre-refines x, and by reflexivity, $z\u22b2z$. Hence $y\u22b2z$, $z\in A$, and $z\in A\to B$, so $z\in A\cap B$ by Modus Ponens, so $x\in A\to B$ according to options 3 and 5. |

21 | By contrast, we note that the implication in algebras for Visser’s [102] basic propositional logic is not necessarily a preconditional or preimplication, since it can violate $1\to a\le a$. |

22 | The equivalent condition is that if $y\u25c2x$, then $\exists {x}^{\prime}\u22b2x$$\forall {x}^{\u2033}\u22b3{x}^{\prime}$$\exists z\u25c2{x}^{\u2033}$: z pre-refines y. To see this is sufficient, suppose $x\notin {\neg}_{\u25c2}A$, so there is a $y\u25c2x$ with $y\in A$. Then by the condition, $\exists {x}^{\prime}\u22b2x$$\forall {x}^{\u2033}\u22b3{x}^{\prime}$$\exists z\u25c2{x}^{\u2033}$: z pre-refines y. Since z pre-refines y and A is a ${c}_{\u22b2}$-fixpoint, $z\in A$ by Lemma 4.12, so ${x}^{\u2033}\notin {\neg}_{\u25c2}A$. Thus, assuming $x\notin {\neg}_{\u25c2}A$, we have $\exists {x}^{\prime}\u22b2x$$\forall {x}^{\u2033}\u22b3{x}^{\prime}$, ${x}^{\u2033}\notin {\neg}_{\u25c2}A$, which shows that ${\neg}_{\u25c2}A$ is a ${c}_{\u22b2}$-fixpoint. For necessity, suppose the condition does not hold. Let $A={c}_{\u22b2}\left(\left\{y\right\}\right)$, which is the set of states that pre-refine y. Then $x\notin {\neg}_{\u25c2}A$ but $\forall {x}^{\prime}\u22b2x$$\exists {x}^{\u2033}\u22b3{x}^{\prime}$: ${x}^{\u2033}\in {\neg}_{\u25c2}A$, so ${\neg}_{\u25c2}A$ is not a ${c}_{\u22b2}$-fixpoint. |

23 | Applying the discrete representation of Section 4.2 to complete lattices with modalities raises additional issues, such as the requirement that □ (resp. $A\to (\xb7)$) be completely multiplicative (see [21] (Section 4)). |

24 | The introduction and elimination rules for the intuitionistic implication → can obviously be added in the same style. |

25 | Returning to the issue of morphisms broached in Note 15, a candidate notion of morphism between relational frames that also preserves ${\to}_{\u22b2}$ is a map f that satisfies (i) and (ii) from Note 15 plus two extra conditions for ${\to}_{\u22b2}$. First recall (iii) from Note 15, expressed in the language of Definition 4.11: if ${y}^{\prime}{\u22b2}^{\prime}f\left(x\right)$, then $\exists y\u22b2x$: $f\left(y\right)$ pre-refines ${y}^{\prime}$. This ensures ${\neg}_{\u22b2}{f}^{-1}\left[{A}^{\prime}\right]\subseteq {f}^{-1}\left[{\neg}_{{\u22b2}^{\prime}}{A}^{\prime}\right]$. To ensure ${f}^{-1}\left[{A}^{\prime}\right]{\to}_{\u22b2}{f}^{-1}\left[{B}^{\prime}\right]\subseteq {f}^{-1}\left[{A}^{\prime}{\to}_{{\u22b2}^{\prime}}{B}^{\prime}\right]$, we strengthen (iii) to (iii${}^{+}$): if ${y}^{\prime}{\u22b2}^{\prime}f\left(x\right)$, then $\exists y\u22b2x$: $f\left(y\right)$ refines ${y}^{\prime}$. For suppose $x\in {f}^{-1}\left[{A}^{\prime}\right]{\to}_{\u22b2}{f}^{-1}\left[{B}^{\prime}\right]$. To show $f\left(x\right)\in {A}^{\prime}{\to}_{{\u22b2}^{\prime}}{B}^{\prime}$, suppose ${y}^{\prime}{\u22b2}^{\prime}f\left(x\right)$ and ${y}^{\prime}\in {A}^{\prime}$. Then picking y as in (iii${}^{+}$), since $f\left(y\right)$ pre-refines ${y}^{\prime}$, we have $f\left(y\right)\in {A}^{\prime}$ by Lemma 4.12. Hence $y\in {f}^{-1}\left[{A}^{\prime}\right]$, which with $y\u22b2x$ and $x\in {f}^{-1}\left[{A}^{\prime}\right]{\to}_{\u22b2}{f}^{-1}\left[{B}^{\prime}\right]$ implies there is a $z\u22b3y$ with $z\in {f}^{-1}\left[{B}^{\prime}\right]$, so $f\left(z\right)\in {B}^{\prime}$. Then from $z\u22b3y$ we have $f\left(z\right){\u22b3}^{\prime}f\left(y\right)$ by (i), and then since $f\left(y\right)$ post-refines ${y}^{\prime}$, we have $f\left(z\right){\u22b3}^{\prime}{y}^{\prime}$. Thus, we have shown that for all ${y}^{\prime}{\u22b2}^{\prime}f\left(x\right)$ with ${y}^{\prime}\in {A}^{\prime}$, there is a ${z}^{\prime}{\u22b3}^{\prime}{y}^{\prime}$ with ${z}^{\prime}\in {B}^{\prime}$, so $f\left(x\right)\in {A}^{\prime}{\to}_{\u22b2}{B}^{\prime}$. Finally, to ensure ${f}^{-1}\left[{A}^{\prime}{\to}_{{\u22b2}^{\prime}}{B}^{\prime}\right]\subseteq {f}^{-1}\left[{A}^{\prime}\right]{\to}_{\u22b2}{f}^{-1}\left[{B}^{\prime}\right]$, consider (iv) (and compare it with (iii)): if ${y}^{\prime}{\u22b3}^{\prime}f\left(x\right)$, then $\exists y\u22b3x$: $f\left(y\right)$ pre-refines ${y}^{\prime}$. We will apply (iv) with a change of variables: if ${z}^{\prime}{\u22b3}^{\prime}f\left(y\right)$, then $\exists z\u22b3y$: $f\left(z\right)$ pre-refines ${z}^{\prime}$. Now suppose $f\left(x\right)\in {A}^{\prime}{\to}_{{\u22b2}^{\prime}}{B}^{\prime}$. To show $x\in {f}^{-1}\left[{A}^{\prime}\right]{\to}_{\u22b2}{f}^{-1}\left[{B}^{\prime}\right]$, suppose $y\u22b2x$ and $y\in {f}^{-1}\left[{A}^{\prime}\right]$, so $f\left(a\right)\in {A}^{\prime}$. By (i), we have $f\left(y\right){\u22b2}^{\prime}f\left(x\right)$. Then since $f\left(x\right)\in {A}^{\prime}{\to}_{{\u22b2}^{\prime}}{B}^{\prime}$, there is a ${z}^{\prime}{\u22b3}^{\prime}f\left(y\right)$ such that ${z}^{\prime}\in {B}^{\prime}$. Then taking z as in (iv), we have $f\left(z\right)\in {B}^{\prime}$ by Lemma 4.12, so $z\in {f}^{-1}\left[{B}^{\prime}\right]$. Thus, we have shown that for all $y\u22b2x$ with $y\in {f}^{-1}\left[{A}^{\prime}\right]$, there is a $z\u22b3y$ with $z\in {f}^{-1}\left[{B}^{\prime}\right]$, so $x\in {f}^{-1}\left[{A}^{\prime}\right]{\to}_{\u22b2}{f}^{-1}\left[{B}^{\prime}\right]$. |

26 | A referee informed me that this idea is what led Meyer and Slaney [109] to their Abelian Logic by generalizing the classical axiom $\neg \neg a\to a$ to $\left(\right(a\to b)\to b)\to a$. |

27 | For parts 2 and 4 when $\mathrm{V}=L$, we can take $z=x$, in which case $(z,z\to ({x}^{\prime}\to {y}^{\prime}))$ pre-refines $(x,x\to y)$ and vice versa, so a strong right pre-interpolation property holds. |

28 | For parts 3 and 4 when $\mathrm{V}=L$, we can take $z=x$, in which case $(z,z\to ({x}^{\prime}\to {y}^{\prime}))$ pre-refines $(x,x\to y)$ and vice versa, so a strong left pre-interpolation property holds. |

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**Figure 11.**A valuation on the reflexive frame on the right of Figure 9.

**Table 2.**$f\left(n\right)$ is the number of algebras of size n up to isomorphism in the given class.

$\mathit{f}\left(2\right)$ | $\mathit{f}\left(3\right)$ | $\mathit{f}\left(4\right)$ | $\mathit{f}\left(5\right)$ | $\mathit{f}\left(6\right)$ | $\mathit{f}\left(7\right)$ | $\mathit{f}\left(8\right)$ | $\mathit{f}\left(9\right)$ | $\mathit{f}\left(10\right)$ | |
---|---|---|---|---|---|---|---|---|---|

lattices with weak pseudocomplementation | 1 | 1 | 3 | 9 | 38 | 187 | 1130 | 7914 | 63,782 |

lattices | 1 | 1 | 2 | 5 | 15 | 53 | 222 | 1078 | 5994 |

pseudocomplemented lattices | 1 | 1 | 2 | 4 | 10 | 29 | 99 | 391 | 1357 |

distributive lattices | 1 | 1 | 2 | 3 | 5 | 8 | 15 | 26 | 47 |

ortholattices | 1 | 0 | 1 | 0 | 2 | 0 | 5 | 0 | 15 |

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Holliday, W.H.
A Fundamental Non-Classical Logic. *Logics* **2023**, *1*, 36-79.
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Holliday WH.
A Fundamental Non-Classical Logic. *Logics*. 2023; 1(1):36-79.
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2023. "A Fundamental Non-Classical Logic" *Logics* 1, no. 1: 36-79.
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