Intuitionistic Implication and Logics of Formal Inconsistency

Abstract

Logics of Formal Inconsistency ($LFI$ for short) are a class of paraconsistent logics that validate the principle of gentle explosion, meaning that any formula can be derived from the set of formulas: $\circ \alpha$, $\alpha$ and $\sim \alpha$. A unique feature of $LFI$ is the use of the symbol ‘∘’ to represent notions of consistency at the object-language level. These logics are simple in essence, built upon all the axiom schemas of positive classical logic, axioms for negation and the so-called ‘consistency operator’ ∘, with the only inference rule being detachment. In this paper, we propose an alternative foundation for $LFI$, which is the positive fragment of intuitionistic propositional logic. We present bi-valuational ‘Loparić-like’ semantics for the resulting logics and discuss their potential extensions.

1. Introduction

Let $Var$ denote the denumerable set of all propositional variables: p, q, $p_1$, $p_2$, …. The set $\mathcal{F}$ of formulas is standardly defined using propositional variables from $Var$ and the symbols ∼, ∨, ∧ and → for negation, disjunction, conjunction and implication, respectively. The connective of equivalence, $\alpha \leftrightarrow \beta$, is definable through implication and conjunction in the standard way. We expand the language with a unary connective ‘∘’. By placing the symbol ‘∘’ in front of a formula $\alpha$, we indicate that ‘$\alpha$ is consistent’. The fundamental idea behind this is not significantly different from that given to the law of non-contradiction in da Costa’s C-systems (see [1,2,3,4], for details). However, in contrast to da Costa’s systems, consistency is represented using a single connective (see [5,6] for a philosophical discussion on the topic).

We say that a calculus $\mathcal{C}$ (identified with the triple $\langle \mathcal{F},A_{\mathcal{C}},{⊢}_{\mathcal{C}}\rangle$ and determined by its set of axioms $A_{\mathcal{C}}$ which is included in $\mathcal{F}$) is paraconsistent if (1) the principle of explosion, $\{\alpha ,\sim \alpha \}{⊢}_{\mathcal{C}}\beta$, does not hold in $\mathcal{C}$, and (2) $\mathcal{C}$ is not trivial. We say that $\mathcal{C}$ is trivial iff for any formula $\alpha$, $\alpha$ is a thesis of $\mathcal{C}$. Logics of Formal Inconsistency satisfy these conditions. They also validate the principle of gentle explosion: $\{\circ \alpha ,\alpha ,\sim \alpha \}⊢\beta$. ‘A fundamental example of $LFI$ (…) where consistency is rendered expressible by means of a specific new primitive connective’ ([6], p. 3) is $mbC$. The calculus $mbC$ contains all the axiom schemas of the positive fragment of classical propositional calculus (abbreviated as $CP{C}^{+}$), specifically all instances of the following schemas:

($A1$) $\alpha \to (\beta \to \alpha )$

($A2$) $(\alpha \to (\beta \to \gamma \left)\right)\to \left(\right(\alpha \to \beta )\to (\alpha \to \gamma \left)\right)$

($A3$) $(\alpha \wedge \beta )\to \alpha$

($A4$) $(\alpha \wedge \beta )\to \beta$

($A5$) $\alpha \to (\beta \to (\alpha \wedge \beta \left)\right)$

($A6$) $\alpha \to (\alpha \vee \beta )$

($A7$) $\beta \to (\alpha \vee \beta )$

($A8$) $(\alpha \to \gamma )\to \left(\right(\beta \to \gamma )\to (\alpha \vee \beta \to \gamma \left)\right)$

($PL$) $\left(\right(\alpha \to \beta )\to \alpha )\to \alpha$,

and, additionally:

($ExM$) $\alpha \vee \sim \alpha$;

$\left(bcl\right)$ $\circ \alpha \to (\alpha \to (\sim \alpha \to \beta \left)\right)$.

The only inference rule is the detachment rule (MP) $\alpha \to \beta ,\alpha /\beta$.

Remark 1.

($PL$) is independent in $mbC$.

Proof. 

It is easy to check that the matrix ${\mathcal{M}}_{PL}=〈\{1,2,0\},\left\{1\right\},\sim ,\circ \to ,\vee ,\wedge 〉$ with 1 as the sole designated value, and the connectives given by the truth-tables:

	∼	∘		→	1	2	0		∨	1	2	0		∧	1	2	0
1	0	0		1	1	2	0		1	1	1	1		1	1	2	0
2	1	0		2	1	1	0		2	1	2	2		2	2	2	0
0	1	0		0	1	1	1		0	1	2	0		0	0	0	0

verifies the axioms ($A1$)–($A8$), ($ExM$), $\left(bcl\right)$ and (MP), but not ($PL$) when $\alpha$ is assigned the value 2 and $\beta$ takes the value 0. □

The intuitionistic variant of $mbC$, which is obtained by removing the axiom ($PL$) from $mbC$, was examined in Section 5.1.4 of [7]. However, it would be more interesting to extend $mbC$ with one of the following formulas: $\left(cl\right)$ $\sim (\alpha \wedge \sim \alpha )\to \circ \alpha$ or $\left(ci\right)$ $\sim \circ \alpha \to (\alpha \wedge \sim \alpha )$ (cf. Remarks 3 and 7 below) and subsequently discuss the intuitionistic variants of some well-known $LFI$ (see Section 3 and Section 4 below).

2. Calculi $\mathit{Ci}$ and $\mathit{Cil}$

The calculus $Ci$ as introduced in [6] (see Definition 102, p. 62) is defined by the postulates of $mbC$, $\left(ci\right)$, ($NN1$) $\sim \sim \alpha \to \alpha$ and (MP). Since $CP{C}^{+}$, ($ExM$), ($NN1$) and (MP) define the logic $C_{min}$ (see [8] for details), $Ci$ can be viewed as an extension of $C_{min}$ by two axioms involving the consistency connective: $\left(bcl\right)$ and $\left(ci\right)$.

Definition 1.

For any $\alpha \in \mathcal{F}$ and any $\mathsf{\Gamma }\subseteq \mathcal{F}$, we say that α is provable from $\mathsf{\Gamma }$ within $Ci$ (in symbols: $\mathsf{\Gamma }{⊢}_{Ci}\alpha$) iff there is a finite sequence of formulas, ${\beta }_1$, ${\beta }_2$, ⋯, ${\beta }_n$ such that ${\beta }_n=\alpha$ and for each $i⩽n$, either ${\beta }_i\in \mathsf{\Gamma }$, or ${\beta }_i$ ia an axiom of $Ci$, or for some $j,k⩽i$, we have ${\beta }_k={\beta }_j\to {\beta }_i$. A formula α is a thesis of $Ci$ iff α is provable from ∅ within $Ci$.

Before proceeding, let us review some results regarding $Ci$. Some of these results are quite evident, while others are well-established; therefore, they will be presented without proofs.

Remark 2.

The following formulas are provable in $Ci$:

$\left(IL\right)$$\alpha \to \alpha$

$\left(LoC\right)$$(\alpha \to (\beta \to \gamma \left)\right)\to (\beta \to (\alpha \to \gamma \left)\right)$

$\left(HS\right)$$(\alpha \to \beta )\to \left(\right(\beta \to \gamma )\to (\alpha \to \gamma \left)\right)$

$\left(C\right)$$(\alpha \to (\alpha \to \beta \left)\right)\to (\alpha \to \beta )$

$\left(LoE\right)$$(\alpha \wedge \beta )\to \gamma )\leftrightarrow (\alpha \to (\beta \to \gamma ))$

$\left(LE\right)$$(\alpha \to (\beta \wedge \gamma \left)\right)\to (\alpha \to (\gamma \to \beta \left)\right)$

$\left(ID\right)$$(\alpha \to \beta )\to (\sim \alpha \vee \beta )$

$\left(CM1\right)$$(\sim \alpha \to \alpha )\to \alpha$

$\left(CM2\right)$$(\alpha \to \sim \alpha )\to \sim \alpha$

$\left(C{M}^{★}\right)$ $(\sim (\alpha \wedge \sim \alpha )\to \alpha )\to \alpha$.

Lemma 1.

For every $\Gamma , \Delta \subseteq \mathcal{F}$ and $\alpha ,\beta \in \mathcal{F}$, we have:

(1) $\Gamma {⊢}_{Ci}\alpha$ iff for some finite $\Delta \subseteq \Gamma$, $\Delta {⊢}_{Ci}\alpha$

(2) If $\alpha \in \Gamma$, then $\Gamma {⊢}_{Ci}\alpha$

(3) If $\Gamma \subseteq \Delta$ and $\Gamma {⊢}_{Ci}\alpha$, then $\Delta {⊢}_{Ci}\alpha$

(4) If $\Delta {⊢}_{Ci}\alpha$ and, for every $\beta \in \Delta$ such that $\Gamma {⊢}_{Ci}\beta$, then $\Gamma {⊢}_{Ci}\alpha$

(5) If $\Gamma \cup \left\{\alpha \right\}{⊢}_{Ci}\beta$ and $\Delta {⊢}_{Ci}\alpha$, then $\Gamma \cup \Delta {⊢}_{Ci}\beta$,

in particular, if $\Gamma \cup \left\{\alpha \right\}{⊢}_{Ci}\beta$ and α is a thesis of $Ci$, then $\Gamma {⊢}_{Ci}\beta$.

Lemma 2.

For any $\Gamma \subseteq \mathcal{F}$ and $\alpha ,\beta \in \mathcal{F}$, we have:

($DIS$) if $\Gamma \cup \left\{\alpha \right\}{⊢}_{Ci}\gamma$ and $\Gamma \cup \left\{\beta \right\}{⊢}_{Ci}\gamma$, then $\Gamma \cup \{\alpha \vee \beta \}{⊢}_{Ci}\gamma$

($DT$) $\Gamma \cup \left\{\alpha \right\}{⊢}_{Ci}\beta$ iff $\Gamma {⊢}_{Ci}\alpha \to \beta$.

($RA1$) if $\Gamma \cup \left\{\alpha \right\}{⊢}_{Ci}\{\circ \beta ,\beta ,\sim \beta \}$, then $\Gamma {⊢}_{Ci}\sim \alpha$

($RA2$) if $\Gamma \cup \{\sim \alpha \}{⊢}_{Ci}\{\circ \beta ,\beta ,\sim \beta \}$, then $\Gamma {⊢}_{Ci}\alpha$,

where $\mathsf{\Gamma }{⊢}_{Ci}\{\circ \beta ,\beta ,\sim \beta \}$ stands for ‘$\mathsf{\Gamma }{⊢}_{Ci}\circ \beta$, $\mathsf{\Gamma }{⊢}_{Ci}\beta$ and $\mathsf{\Gamma }{⊢}_{Ci}\sim \beta$’.

The results, with the exception of ($RA1$) and ($RA2$) from Lemma 2, apply to every calculus discussed in this paper

Remark 3.

The axiom ($PL$) is dependent in $Ci$.

Proof. 

Note first that $(\circ \alpha \to \alpha )\to \left(\right((\sim \circ \alpha \to \alpha )\to \alpha )\to \alpha )$ is a thesis of $Ci$ (this can shown by applying ($DT$), $\left(ci\right)$, $\left(A3\right)$, $\left(HS\right)$ and (MP)). Then, we have: $(\alpha \to \beta )\to \alpha$ by ($DT$); $\circ \alpha \wedge \sim \alpha \to (\alpha \to \beta )$ by $\left(bcl\right)$, $\left(LoC\right)$, $\left(LoE\right)$, (MP); $\circ \alpha \to (\sim \alpha \to \alpha )$ by $\left(HS\right)$, $\left(LoE\right)$, (MP); $\circ \alpha \to \alpha$ by $\left(CM1\right)$, $\left(HS\right)$, (MP); $\sim \circ \alpha \vee \alpha$ by $\left(ID\right)$, (MP); $(\sim \circ \alpha \to \alpha )\to \alpha$ by ($A8$), $\left(LoC\right)$ $\left(IL\right)$, (MP); $\alpha$ by the thesis of $Ci$, (MP); and finally, $\left(\right(\alpha \to \beta )\to \alpha )\to \alpha$ by ($DT$). Notice, in passing, that the same proof can be provided to show that ($PL$) is derivable in $mCi$ (see Definition 75, [6], p. 52.) □

This suggests that $Ci$ may be regarded as a calculus ‘built upon’ $C_{\omega }$ of da Costa, defined by $\left(A1\right)$–$\left(A8\right)$, $\left(ExM\right)$, $\left(NN1\right)$ and (MP) (see [2,9,10], for details), rather than $C_{min}$.

Definition 2

([6], p. 63). A $Ci$-valuation is a function $v:\mathcal{F}⟶\{1,0\}$ that satisfies, for any $\alpha ,\beta \in \mathcal{F}$, all the conditions of $C_{min}$-valuation, i.e.,:

($v1$) $v(\alpha \wedge \beta )$ = 1 iff $v\left(\alpha \right)$ = 1 and $v\left(\beta \right)$ = 1

($v2$) $v(\alpha \vee \beta )$ = 1 iff $v\left(\alpha \right)$ = 1 or $v\left(\beta \right)$ = 1

($v3$) $v(\alpha \to \beta )$ = 1 iff $v\left(\alpha \right)$ = 0 or $v\left(\beta \right)$ = 1

($v4$) if $v(\sim \alpha )$ = 0, then $v\left(\alpha \right)$ = 1

($v5$) if $v\left(\sim \sim \alpha \right)$ = 1, then $v\left(\alpha \right)$ = 1,

and the following clauses:

($v6$) if $v(\circ \alpha )$ = 1, then $v\left(\alpha \right)$ = 0 or $v(\sim \alpha )$ = 0

($v7$) if $v\left(\sim \circ \alpha \right)$ = 1, then $v\left(\alpha \right)$ = $v(\sim \alpha )$ = 1.

Definition 3.

A formula α is a $Ci$-tautology iff for every $Ci$-valuation v, $v\left(\alpha \right)$ = 1. For any $\alpha \in \mathcal{F}$ and $\mathsf{\Gamma }\subseteq \mathcal{F}$, α is a semantic consequence of Γ (in symbols: $\mathsf{\Gamma }{\vDash }_{Ci}\alpha$) iff for any $Ci$-valuation v: if $v\left(\beta \right)$ = 1 for any $\beta \in \mathsf{\Gamma }$, then $v\left(\alpha \right)$ = 1.

Theorem 1.

For all $\mathsf{\Gamma }\subseteq \mathcal{F}$ and $\alpha \in \mathcal{F}$: $\mathsf{\Gamma }{⊢}_{Ci}\alpha$ iff $\mathsf{\Gamma }{\vDash }_{Ci}\alpha$.

By adding $\left(cl\right)$ $\sim (\alpha \wedge \sim \alpha )\to \circ \alpha$ to the postulates of $Ci$, we obtain the calculus $Cil$ (see [6] Definition 105, p. 64). $Cil$ has some interesting properties:

Remark 4.

The axiom $\left(ci\right)$ can be replaced by $\left(D{S}^{\circ }\right)$ $\circ \alpha \to (\sim \circ \alpha \to \beta )$.

Proof. 

$\left(ci\right)\Rightarrow \left(D{S}^{\circ }\right)$: $\circ \alpha$, $\sim \circ \alpha$ by ($DT$); $\alpha \wedge \sim \alpha$ by $\left(ci\right)$, (MP); $\alpha$, $\sim \alpha$ by ($A3$), ($A4$), (MP); $\beta$ by $\left(bcl\right)$, (MP); $\circ \alpha \to (\sim \circ \alpha \to \beta )$ by ($DT$).

$\left(D{S}^{\circ }\right)\Rightarrow \left(ci\right)$: $\sim \circ \alpha$ by ($DT$); $\circ \alpha \to (\alpha \wedge \sim \alpha )$ by $\left(D{S}^{\circ }\right)$, $\left(LoC\right)$, (MP); $\sim (\alpha \wedge \sim \alpha )\to (\alpha \wedge \sim \alpha )$ by $\left(cl\right)$, $\left(HS\right)$, (MP); $\alpha \wedge \sim \alpha$ by $\left(CM1\right)$, (MP); $\sim \circ \alpha \to (\alpha \wedge \sim \alpha )$ by ($DT$). □

Remark 5.

The axiom $\left(cl\right)$ can be replaced by ${\left(bcl\right)}^{★}$ $\sim (\alpha \wedge \sim \alpha )\to (\alpha \to (\sim \alpha \to \beta \left)\right)$.

Proof. 

$\left(cl\right)\Rightarrow {\left(bcl\right)}^{★}$ follows from applying ($DT$), $\left(cl\right)$, $\left(bcl\right)$ and (MP).

${\left(bcl\right)}^{★}\Rightarrow \left(cl\right)$: $\sim (\alpha \wedge \sim \alpha )$ by ($DT$); $(\alpha \wedge \sim \alpha )\to \circ \alpha$ by ${\left(bcl\right)}^{★}$, $\left(LoC\right)$, $\left(LoE\right)$, (MP); $\sim \circ \alpha \to \circ \alpha$ by $\left(ci\right)$, $\left(HS\right)$ and (MP); $\circ \alpha$ by $\left(CM1\right)$, (MP); and finally $\sim (\alpha \wedge \sim \alpha )\to \circ \alpha$ by ($DT$). □

As noted in the remarks above, $Cil$ can be defined by the axioms of $C_{\omega }$, $\left(bcl\right)$, (MP) and one of the following pairs of formulas: $\left(D{S}^{\circ }\right)$ and $\left(cl\right)$ or ${\left(bcl\right)}^{★}$ and $\left(ci\right)$.

Definition 4.

A $Cil$-valuation is a function $v:\mathcal{F}⟶\{1,0\}$ that satisfies, for any $\alpha ,\beta \in \mathcal{F}$, all the conditions of $C_{min}$-valuation, and additionally:

($v6$) if $v(\circ \alpha )$ = 1, then $v\left(\alpha \right)$ = 0 or $v(\sim \alpha )$ = 0

($v7$) if $v(\sim (\alpha \wedge \sim \alpha \left)\right)$ = 1, then $v(\circ \alpha )$ = 1

($v8$) if $v\left(\sim \circ \alpha \right)$ = 1, then $v(\circ \alpha )$ = 0.

Definition 5.

A formula α is a $Cil$-tautology iff for every $Cil$-valuation v, $v\left(\alpha \right)$ = 1. For any $\alpha \in \mathcal{F}$ and $\mathsf{\Gamma }\subseteq \mathcal{F}$, α is a semantic consequence of Γ (in symbols: $\mathsf{\Gamma }{\vDash }_{Cil}\alpha$) iff for any $Cil$-valuation v: if $v\left(\beta \right)$ = 1 for any $\beta \in \mathsf{\Gamma }$, then $v\left(\alpha \right)$ = 1.

Remark 6.

Given that v is a $Cil$-valuation. Then, v has the following properties: ($v4$)’ if $v\left(\alpha \right)$ = 0, then $v(\sim \alpha )$ = 1; ($v6$)’ $v(\circ \alpha )$ = 1 iff $v\left(\alpha \right)$ = 0 or $v(\sim \alpha )$ = 0; ($v8$)’ $v\left(\sim \circ \alpha \right)$ = 1 iff $v(\circ \alpha )$ = 0.

A slightly different bivaluation semantics for $Cil$ was proposed in [6], p. 67, where a $Cil$-valuation is defined as a function $v^{★}:\mathcal{F}⟶\{1,0\}$ that meets, for any $\alpha ,\beta \in \mathcal{F}$, all the conditions of $C_{min}$-valuation, along with the following additional clauses:

($v{6}^{★}$) if $v^{★}(\circ \alpha )$ = 1, then $v^{★}\left(\alpha \right)$ = 0 or $v^{★}(\sim \alpha )$ = 0

($v{8}^{★}$): if $v^{★}\left(\sim \circ \alpha \right)$ = 1, then $v^{★}\left(\alpha \right)$ = 1 and $v^{★}(\sim \alpha )$ = 1.

It is easy to see that the formula $\sim (p\wedge \sim p)\to \circ p$ being an instance of the axiom $\left(cl\right)$ is not valid for every $Cil$-valuation $v^{★}$. This indicates that the semantics proposed in [6] is an adequate semantics for $Ci$, but not for $Cil$.

Theorem 2.

For all $\Gamma \subseteq \mathcal{F}$ and $\alpha \in \mathcal{F}$: if $\Gamma {⊢}_{Cil}\alpha$, then $\mathsf{\Gamma }{\vDash }_{Cil}\alpha$.

The proof of soundness proceeds in the standard way, by induction on the length of a derivation in $Cil$. For the proof of completeness, we will employ a method based on the concept of maximal non-trivial sets of formulas (see, e.g., Section 2.2 of [7]). We begin by recalling the basic definitions and lemmas. The latter are generally well-known; therefore, they will be presented without proofs (see, e.g., [11,12], for further details).

Definition 6.

Let $\mathcal{C}$ = $\langle \mathcal{F},A_{\mathcal{C}},{⊢}_{\mathcal{C}}\rangle$ be a calculus satisfying Tarskian properties and $\Delta \subseteq \mathcal{F}$. We say that Δ is a closed theory of $\mathcal{C}$ if, and only if for any $\beta \in \mathcal{F}$: $\Delta {⊢}_{\mathcal{C}}\beta$ iff $\beta \in \Delta$. We say that Δ is maximal non-trivial with respect to $\alpha \in \mathcal{F}$ in $\mathcal{C}$, if, and only if (1) $\Delta /{⊢}_{\mathcal{C}}\alpha$, and (2) for every $\beta \in \mathcal{F}$, if $\beta \notin \Delta$ then $\Delta \cup \left\{\beta \right\}{⊢}_{\mathcal{C}}\alpha$.

Lemma 3

(Lemma 2.2.5 of [7]). Every maximal non-trivial set with respect to some formula is a closed theory.

Notice that the lemma holds for $Cil$. We also have:

Lemma 4.

For any maximal non-trivial set Δ with respect to α in $Cil$ the mapping $v:\mathcal{F}⟶\{1,0\}$ defined, for any $\delta \in \mathcal{F}$, as ($★$): $v\left(\delta \right)$ = 1 iff $\delta \in \Delta$, is a $Cil$-valuation.

Proof. 

The proof is by cases. The cases ($v1$) to ($v5$) are those of $C_{min}$ and are thus omitted.

($v6$): Assume, for a contradiction, that $v(\circ \beta )$ = 1, $v\left(\beta \right)$ = 1 and $v(\sim \beta )$ = 1. Then, by ($★$), we have $\circ \beta \in \Delta$, $\sim \beta \in \Delta$ and $\beta \in \Delta$, which entails, by Lemma 1(2), that $\Delta {⊢}_{Cil}\circ \beta$, $\Delta {⊢}_{Cil}\sim \beta$ and $\Delta {⊢}_{Cil}\beta$, that is, $\Delta {⊢}_{Cil}\{\circ \beta ,\sim \beta ,\beta \}$. Since ${⊢}_{Cil}$ is transitive, $\left(bcl\right)$ is an axiom of $Cil$ and ($DT$) of Lemma 2 holds true, then $\Delta {⊢}_{Cil}\alpha$. Recall that $\Delta$ is a closed theory, so $\alpha \in \Delta$. But, from the main assumption, we have $\alpha \notin \Delta$. This leads to a contradiction.

($v7$): Assume that $v(\sim (\beta \wedge \sim \beta \left)\right)$ = 1. Then, by ($★$), we have $\sim (\beta \wedge \sim \beta )\in \Delta$, which entails, by Lemma 1(2), that $\Delta {⊢}_{Cil}\sim (\beta \wedge \sim \beta )$. Since ${⊢}_{Cil}$ is transitive, $\left(cl\right)$ is a thesis of $Cil$ and ($DT$) holds in $Cil$, then $\Delta {⊢}_{Cil}\circ \beta$. Given that $\Delta$ is a closed theory, we have $\circ \beta \in \Delta$, which implies, by ($★$), that $v(\circ \beta )$ = 1.

($v8$): Suppose, for a contradiction, that $v\left(\sim \circ \beta \right)$ = 1 and $v(\circ \beta )$ = 1. Then, by ($★$), we have $\sim \circ \beta \in \Delta$ and $\circ \beta \in \Delta$, which entails $\Delta {⊢}_{Cil}\sim \circ \beta$ and $\Delta {⊢}_{Cil}\circ \beta$, that is, $\Delta {⊢}_{Cil}\{\sim \circ \beta ,\circ \beta \}$. Since ${⊢}_{Cil}$ is transitive, $\left(D{S}^{\circ }\right)$ is a thesis of $Cil$ and ($DT$) holds true, then $\Delta {⊢}_{Cil}\alpha$. Given that $\Delta$ is a closed theory, we have $\alpha \in \Delta$. But, from our main assumption, we know that $\alpha \notin \Delta$. This leads to a contradiction. □

Lemma 5

(Theorem 3.31 of [11]). For any $\mathsf{\Gamma }\subseteq \mathcal{F}$ and $\alpha \in \mathcal{F}$ such that $\mathsf{\Gamma }{⊬}_{\mathcal{C}}\alpha$, there is a maximal non-trivial set Δ with respect to α in $\mathcal{C}$ such that $\mathsf{\Gamma }\subseteq \Delta$.

Notice that Lindenbaum–Łoś theorem holds for any finitary calculus $\mathcal{C}$. Hence, the completeness for $Cil$ follows:

Theorem 3.

For all $\mathsf{\Gamma }\subseteq \mathcal{F}$ and $\alpha \in \mathcal{F}$: if $\mathsf{\Gamma }{\vDash }_{Cil}\alpha$, then $\mathsf{\Gamma }{⊢}_{Cil}\alpha$.

Proof. 

The proof is by contraposition. Suppose that $\mathsf{\Gamma }{⊬}_{Cil}\alpha$ and $\Delta$ is a maximal non-trivial set with respect to $\alpha$ in $Cil$ such that $\mathsf{\Gamma }\subseteq \Delta$. Then, we obtain that $\alpha \notin \Delta$. By Lemma 4, there is a $Cil$-valuation v such that $v\left(\alpha \right)$ = 0 and $v\left(\beta \right)$ = 1, for any $\beta \in \Delta$, which implies that $\Delta |{\vDash }_{Cil}\alpha$ and, in particular, $\mathsf{\Gamma } |{\vDash }_{Cil}\alpha$. This completes the proof. □

As a final remark in this section, let us observe that ($PL$) is redundant not only in $Ci$ (and in $Cil$), but also in an alternative weakening of $Cil$.

Remark 7.

($PL$) is dependent in the calculus obtained by dropping $\left(ci\right)$ from the set of axioms of $Cil$.

Proof. 

The proof follows a similar structure to that of Remark 3. The only significant difference is that after deriving $\circ \alpha \to \alpha$ by $\left(CM1\right)$, $\left(HS\right)$ and (MP), we obtain $\sim (\alpha \wedge \sim \alpha )\to \alpha$ by $\left(ci\right)$, $\left(HS\right)$, (MP); $\alpha$ by $\left(C{M}^{★}\right)$, (MP); and finally, we derive $\left(\right(\alpha \to \beta )\to \alpha )\to \alpha$ using ($DT$). □

3. Intuitionistic Implication and $\mathit{C}\mathit{i}\mathit{l}$

Although $Cil$ is axiomatized over the positive fragment of intuitionistic propositional logic, the negation-free fragment of $Cil$ corresponds to positive classical logic. Here, we suggest, by the example of $Cil$, how to ‘fix the asymmetry’. First, we will establish some auxiliary results.

Remark 8.

The axiom $\left(bcl\right)$ is equivalent to the triple of postulates: $\left(D{S}^{\circ }\right)$ $\circ \alpha \to (\sim \circ \alpha \to \beta )$ and $\left(i{c}^{\sim }\right)$ $\sim \alpha \wedge \sim \sim \alpha \to \sim \circ \alpha$ and $\left(c{c}^{\circ }\right)$ $\circ \alpha \wedge \alpha \to \sim \sim \alpha$.

Proof. 

$\left(bcl\right)\Rightarrow \left(D{S}^{\circ }\right)$: See $\left(ci\right)\Rightarrow \left(D{S}^{\circ }\right)$ of Remark 4; $\left(bcl\right)\Rightarrow \left(i{c}^{\sim }\right)$ and $\left(bcl\right)\Rightarrow \left(c{c}^{\circ }\right)$ follow from ($RA2$), ($A3$), ($A4$), $\left(NN1\right)$, $\left(bcl\right)$ and (MP).

$\left(D{S}^{\circ }\right),\left(i{c}^{\sim }\right),\left(c{c}^{\circ }\right)\Rightarrow \left(bcl\right)$: $\circ \alpha$, $\alpha$, $\sim \alpha$ by ($DT$); $\circ \alpha \wedge \alpha$ by ($A5$), (MP); $\sim \sim \alpha$ by $\left(c{c}^{\circ }\right)$, (MP); $\sim \alpha \wedge \sim \sim \alpha$ by ($A5$), (MP); $\sim \circ \alpha$ by $\left(i{c}^{\sim }\right)$, (MP); $\beta$ by $\left(D{S}^{\circ }\right)$, (MP); and finally $\circ \alpha \to (\alpha \to (\sim \alpha \to \beta \left)\right)$ by ($DT$). □

Lemma 6.

The axioms of $C_{\omega }$, $\left(cl\right)$, $\left(D{S}^{\circ }\right)$, $\left(i{c}^{\sim }\right)$, $\left(c{c}^{\circ }\right)$ and (MP) axiomatize $Cil$.

Remark 9.

The postulate $\left(c{c}^{\circ }\right)$ is independent of the other axioms.

Proof. 

The matrix $\mathcal{M}=〈\{1,2,0\},\{1,2\},\sim ,\circ \to ,\vee ,\wedge 〉$ with 1 and 2 as the designated values, and the connectives defined by the truth-tables:

	∼	∘		→	1	2	0		∨	1	2	0		∧	1	2	0
1	0	1		1	1	2	0		1	1	1	1		1	1	2	0
2	1	1		2	1	2	0		2	1	2	2		2	2	2	0
0	1	1		0	1	1	1		0	1	2	0		0	0	0	0

verifies the axioms of $C_{\omega }$, $\left(cl\right)$, $\left(D{S}^{\circ }\right)$, $\left(i{c}^{\sim }\right)$ and (MP), but not $\left(c{c}^{\circ }\right)$ when $\alpha$ is assigned the value 2. □

Let $iCil$ denote the calculus obtained by dropping $\left(c{c}^{\circ }\right)$ from the set of axiom schemas of $Cil$ (see Lemma 6). Observe that a weaker version of $\left(c{c}^{\circ }\right)$, i.e., $\circ \alpha \wedge \sim \alpha \to \sim \sim \sim \alpha$, is still provable in $iCil$.

Remark 10.

Neither $\left(PL\right)$, nor $\left(DL\right)$ $\alpha \vee (\alpha \to \beta )$, nor $\left(bcl\right)$ is provable in $iCil$.

Proof. 

The matrix ${\mathcal{M}}^{★}=〈\{1,2,0\},\left\{1\right\},\sim ,\circ \to ,\vee ,\wedge 〉$ with 1 as the sole designated value; negation, consistency connective, conjunction and disjunction interpreted as in $\mathcal{M}$, and implication defined by the truth-table:

→	1	2	0
1	1	2	0
2	1	1	0
0	1	1	1

validates the axioms of $iCil$ and (MP), but it falsifies $\left(PL\right)$, $\left(DL\right)$ and $\left(bcl\right)$ when $\alpha$ and $\beta$ are assigned the values 2 and 0, respectively. □

Remark 11.

$\left(ci\right)$ and $\left(bc{l}^{\sim }\right)$ $\circ \alpha \to (\sim \alpha \to (\sim \sim \alpha \to \beta \left)\right)$ are provable in $iCil$.

Proof. 

$\left(ci\right)$: $\sim \circ \alpha$ by ($DT$); $\circ \alpha \to (\alpha \wedge \sim \alpha )$ by $\left(D{S}^{\circ }\right)$, $\left(LoC\right)$, (MP); $\sim (\alpha \wedge \sim \alpha )\to (\alpha \wedge \sim \alpha )$ by $\left(cl\right)$, $\left(HS\right)$, (MP); $\alpha \wedge \sim \alpha$ by $\left(CM1\right)$, (MP); $\sim \circ \alpha \to (\alpha \wedge \sim \alpha )$ by ($DT$).

$\left(bc{l}^{\sim }\right)$: $\circ \alpha$, $\sim \alpha$, $\sim \sim \alpha$ by ($DT$); $\sim \alpha \wedge \sim \sim \alpha$ by ($A5$), (MP); $\sim \circ \alpha$ by $\left(i{c}^{\sim }\right)$, (MP); $\beta$ by $\left(D{S}^{\circ }\right)$, (MP); and finally, $\circ \alpha \to (\sim \alpha \to (\sim \sim \alpha \to \beta \left)\right)$ by ($DT$). □

Remark 12.

The axiom $\left(i{c}^{\sim }\right)$ can be replaced by $\left(bc{l}^{\sim }\right)$.

Proof. 

$\left(i{c}^{\sim }\right)\Rightarrow \left(bc{l}^{\sim }\right)$: See the proof of Remark 11; $\left(bc{l}^{\sim }\right)\Rightarrow \left(i{c}^{\sim }\right)$ follows from applying ($DT$), $\left(bc{l}^{\sim }\right)$, $\left(LoC\right)$, $\left(CM2\right)$, $\left(D{S}^{\circ }\right)$ and (MP). □

Clearly, $iCil$ can be defined by the axioms of $C_{\omega }$, $\left(cl\right)$, $\left(D{S}^{\circ }\right)$, $\left(bc{l}^{\sim }\right)$ and (MP). Moreover, $iCil$ enjoys intuitionistic (not classical) implication. A semantics for $iCil$ is established by reproducing the axiom schemas of $iCil$ in ‘Loparić-like’ semantics. The idea behind it traces back to the 1986 paper by Loparić, in which the semantics for $C_{\omega }$ was first introduced.

Definition 7

(cf. [9], p. 74). An $iCil$-semi-valuation is a function $s:\mathcal{F}⟶\{1,0\}$ that satisfies, for any $\alpha ,\beta \in \mathcal{F}$, all the conditions of $C_{\omega }$-semi-valuation, that is,

($s1$) $s(\alpha \wedge \beta )$ = 1 iff $s\left(\alpha \right)$ = 1 and $s\left(\beta \right)$ = 1

($s2$) $s(\alpha \vee \beta )$ = 1 iff $s\left(\alpha \right)$ = 1 or $s\left(\beta \right)$ = 1

($s3$) if $s(\alpha \to \beta )$ = 1, then $s\left(\alpha \right)$ = 0 or $s\left(\beta \right)$ = 1

($s4$) if $s(\alpha \to \beta )$ = 0, then $s\left(\beta \right)$ = 0

($s5$) if $s(\sim \alpha )$ = 0, then $s\left(\alpha \right)$ = 1

($s6$) if $s\left(\sim \sim \alpha \right)$ = 1, then $s\left(\alpha \right)$ = 1,

and the following clauses:

($s7$) if $s(\circ \alpha )$ = 1, then $s(\sim \alpha )$ = 0 or $s\left(\sim \sim \alpha \right)$ = 0

($s8$) if $s\left(\sim \circ \alpha \right)$ = 1, then $s(\circ \alpha )$ = 0

($s9$) if $s(\sim (\alpha \wedge \sim \alpha \left)\right)$ = 1, then $s(\circ \alpha )$ = 1.

Definition 8

(cf. [9], p. 74). An $iCil$-valuation is an $iCil$-semi-valuation for which the following holds: if $v({\alpha }_1\to ({\alpha }_2\to \cdots \to ({\alpha }_{n-1}\to {\alpha }_n)\cdots ))$ = 0, where $n\in \{1,2,3,\cdots \}$ and ${\alpha }_n\ne \beta \to \gamma$, then there is an $iCil$-semi-valuation s such that, for $1\le i<n$, $s\left({\alpha }_i\right)$ = 1 and $s\left({\alpha }_n\right)$ = 0.

Definition 9.

A formula α is an $iCil$-tautology iff for every $iCil$-valuation v, $v\left(\alpha \right)$ = 1. For any $\alpha \in \mathcal{F}$ and $\mathsf{\Gamma }\subseteq \mathcal{F}$, α is a semantic consequence of Γ (in symbols: $\mathsf{\Gamma }{\vDash }_{iCil}\alpha$) iff for any $iCil$-valuation v: if $v\left(\beta \right)$ = 1 for any $\beta \in \mathsf{\Gamma }$, then $v\left(\alpha \right)$ = 1.

There are two key distinctions between a $Cil$-valuation and an $iCil$-valuation (excluding the clause ($v6$) (resp. ($s7$)) where ‘$\circ \alpha$’ is assigned the value 1). Firstly, when the implication connective is assigned the value 0, Definition 4 states that the antecedent is assigned the value 1, while the consequent is assigned the value 0. In contrast, for an $iCil$-semi-valuation, if the implication is assigned the value 0, the consequent takes the value 0, but the antecedent is not assigned any logical value. Secondly, an $iCil$-valuation is established through a two-step process. This approach is significant because it allows for saving many useful tautologies that would otherwise be ‘lost’ without Definition 8. This two-step method is not required for a $Cil$-valuation.

Theorem 4.

For all $\mathsf{\Gamma }\subseteq \mathcal{F}$ and $\alpha \in \mathcal{F}$: if $\mathsf{\Gamma }{⊢}_{iCil}\alpha$, then $\mathsf{\Gamma }{\vDash }_{iCil}\alpha$.

Proof. 

We proceed in a manner similar to the proof of Theorem 4.4 in [9]. □

For completeness, we will apply the method based on the concept of saturated sets. To begin, we will review some auxiliary results.

Definition 10.

Let $\mathcal{C}$ = $\langle \mathcal{F},A_{\mathcal{C}},{⊢}_{\mathcal{C}}\rangle$ be a calculus and $\Delta \subseteq \mathcal{F}$. We say that Δ is saturated with respect to $\alpha \in \mathcal{F}$ in $\mathcal{C}$, or simply Δ is an α-saturated set, if (1) $\Delta /{⊢}_{\mathcal{C}}\alpha$, and (2) for every $\beta \in \mathcal{F}$, if $\beta \notin \Delta$ then $\Delta \cup \left\{\beta \right\}{⊢}_{\mathcal{C}}\alpha$.

Lemma 7

([9], Lemma 3.2). If $\mathsf{\Gamma } /{⊢}_{\mathcal{C}}\alpha$, then there is an α-saturated set Δ such that $\mathsf{\Gamma }\subset \Delta$.

Lemma 8

([9], Lemma 3.3). If Δ is an α-saturated set, then $\beta \in \Delta$ iff $\Delta {⊢}_{\mathcal{C}}\beta$, for every $\beta \in \mathcal{F}$.

It follows from the lemmas that:

Lemma 9

([9], Lemma 3.5). If Δ an α-saturated set and ${\alpha }_1\to ({\alpha }_2\to \cdots \to ({\alpha }_{n-1}\to {\alpha }_n)\cdots )\notin \Delta$, where $n\in \{1,2,\cdots \}$, then there is an ${\alpha }_n$-saturated set ${\Delta }^{*}$ such that $\Delta \cup \{{\alpha }_1,{\alpha }_2,\cdots {\alpha }_{n-1}\}\subset {\Delta }^{*}$.

Now, we need to prove an auxiliary lemma:

Lemma 10.

If Δ is α-saturated, then for any $\beta ,\gamma \in \mathcal{F}$, we have:

($\wedge 1$) if $\beta \wedge \gamma \in \Delta$, then $\beta \in \Delta$ and $\gamma \in \Delta$

($\wedge 0$) if $\beta \wedge \gamma \notin \Delta$, then $\beta \notin \Delta$ or $\gamma \notin \Delta$

($\vee 1$) if $\beta \vee \gamma \in \Delta$, then $\gamma \in \Delta$ or $\gamma \in \Delta$

($\vee 0$) if $\beta \vee \gamma \notin \Delta$, then $\gamma \notin \Delta$ and $\gamma \notin \Delta$

($\to 1$) if $\beta \to \gamma \in \Delta$, then $\beta \notin \Delta$ or $\gamma \in \Delta$

($\to 0$) if $\beta \to \gamma \notin \Delta$, then $\gamma \notin \Delta$

(∼) if $\sim \beta \notin \Delta$, then $\beta \in \Delta$

($\sim \sim$) if $\sim \sim \beta \in \Delta$, then $\beta \in \Delta$

($\circ 1$) if $\circ \beta \in \Delta$, then $\sim \beta \notin \Delta$ or $\sim \sim \beta \notin \Delta$

($\circ 0$) if $\sim (\beta \wedge \sim \beta )\in \Delta$, then $\circ \beta \in \Delta$

($\sim \circ 1$) if $\sim \circ \beta \in \Delta$, then $\circ \beta \notin \Delta$.

Proof. 

Cases ($\wedge 1$)–($\sim \sim$) are those of $C_{\omega }$ and hence omitted.

Case ($\circ 1$): Assume, for a contradiction, that $\circ \beta \in \Delta$, $\sim \beta \in \Delta$ and $\sim \sim \beta \in \Delta$. This implies, by Lemma 8, that $\Delta {⊢}_{iCil}\circ \beta$, $\Delta {⊢}_{iCil}\sim \beta$ and $\Delta {⊢}_{iCil}\sim \sim \beta$. By ($DT$), $\left(bc{l}^{\sim }\right)$ and the transitivity of ${⊢}_{iCil}$, we obtain $\Delta {⊢}_{iCil}\alpha$, which results in $\alpha \in \Delta$. But $\Delta$ is an $\alpha$-saturated set, so $\Delta /{⊢}_{iCil}\alpha$ and consequently, $\alpha \notin \Delta$. This leads to a contradiction.

Case ($\circ 0$): Suppose that $\sim (\beta \wedge \sim \beta )\in \Delta$. Then, by Lemma 8, we have $\Delta {⊢}_{iCil}\sim (\beta \wedge \sim \beta )$. By applying ($DT$), $\left(cl\right)$ and by the transitivity of ${⊢}_{iCil}$, we obtain that $\Delta {⊢}_{iCil}\circ \beta$. Since $\Delta$ is an $\alpha$-saturated set, then $\circ \beta \in \Delta$.

Case ($\sim \circ 1$): The proof is analogous to the proof of ‘Case ($\circ 1$)’. It requires $\left(D{S}^{\circ }\right)$ to be used. □

By applying Lemma 10, we obtain the following:

Lemma 11.

The characteristic function of an α-saturated set is an $iCil$-semi-valuation.

From Lemmas 9 and 11, along with the definition of $iCil$-valuation, we obtain the following result:

Lemma 12.

The characteristic function of an α-saturated set is an $iCil$-valuation.

Theorem 5.

For all $\mathsf{\Gamma }\subseteq \mathcal{F}$ and $\alpha \in \mathcal{F}$: if $\mathsf{\Gamma }{\vDash }_{iCil}\alpha$, then $\mathsf{\Gamma }{⊢}_{iCil}\alpha$.

Proof. 

Assume that $\mathsf{\Gamma } /{⊢}_{iCil}\alpha$. By Lemmas 7 and 8, there is an $\alpha$-saturated set $\Delta$ such that $\mathsf{\Gamma }\subset \Delta$ and $\alpha \notin \Delta$. Given that Lemma 12 holds for $iCil$, there is an $iCil$–valuation v such that $v\left(\alpha \right)$ = 0 and $v\left(\beta \right)$ = 1, for any $\beta \in \Delta$, which means that $\Delta |{\vDash }_{iCil}\alpha$ and in particular $\mathsf{\Gamma } |{\vDash }_{iCil}\alpha$. This completes the proof. □

As shown, removing $\left(c{c}^{\circ }\right)$ from the set of axiom schemas of $Cil$ yields a calculus with the positive intuitionistic basis. Now, consider an extension of $iCil$ by $\left(PL\right)$, $Ci{l}^{★}$ for brevity.

Remark 13.

$\left(c{c}^{\circ }\right)$ and $\left(bcl\right)$ are not provable in $Ci{l}^{★}$.

Proof. 

$\left(c{c}^{\circ }\right)$: See the proof Remark 9; $\left(bcl\right)$: Likewise, the matrix $\mathcal{M}$ invalidates $\left(bcl\right)$ if $\alpha$ and $\beta$ are assigned the values 2 and 0, respectively. □

Definition 11.

A $Ci{l}^{★}$-valuation is a function $v:\mathcal{F}⟶\{1,0\}$ that satisfies, for any $\alpha ,\beta \in \mathcal{F}$, all the conditions of $C_{min}$-valuation, and additionally:

($v6$) if $v(\circ \alpha )$ = 1, then $v(\sim \alpha )$ = 0 or $v\left(\sim \sim \alpha \right)$ = 0

($v7$) if $v(\sim (\alpha \wedge \sim \alpha \left)\right)$ = 1, then $v(\circ \alpha )$ = 1

($v8$) if $v\left(\sim \circ \alpha \right)$ = 1, then $v(\circ \alpha )$ = 0.

Definition 12.

A formula α is a $Ci{l}^{★}$-tautology iff for every $Ci{l}^{★}$-valuation v, $v\left(\alpha \right)$ = 1. For any $\alpha \in \mathcal{F}$ and $\mathsf{\Gamma }\subseteq \mathcal{F}$, α is a semantic consequence of Γ (in symbols: $\mathsf{\Gamma }{\vDash }_{Ci{l}^{★}}\alpha$) iff for any $Ci{l}^{★}$-valuation v: if $v\left(\beta \right)$ = 1 for any $\beta \in \mathsf{\Gamma }$, then $v\left(\alpha \right)$ = 1.

Theorem 6.

For all $\mathsf{\Gamma }\subseteq \mathcal{F}$ and $\alpha \in \mathcal{F}$: $\mathsf{\Gamma }{⊢}_{Ci{l}^{★}}\alpha$ iff $\mathsf{\Gamma }{\vDash }_{Ci{l}^{★}}\alpha$.

Proof. 

The proof is similar to the proofs of Theorem 1 and 2. □

4. $\mathit{iCila}$, $\mathit{iCilo}$ and More

Here, we consider some extensions of $iCil$ that differ from those presented in Section 3. The first extension, which refer to as $iCila$ from this point forward, is obtained by adding $\left(ca\right)$ $\circ \alpha \wedge \circ \beta \to \left(\sim \circ \right(\alpha ‡\beta )\to \gamma )$, where $‡\in \{\wedge ,\vee ,\to \}$, to the postulates of $iCil$. It is straightforward to demonstrate that the following two observations are true:

Remark 14

(cf. [6], p. 68). $\left(ca\right)$ is equivalent to the triple of axiom schemas: $\left(c{a}_1\right)$ $\circ \alpha \wedge \circ \beta \to \circ (\alpha \wedge \beta )$, $\left(c{a}_2\right)$ $\circ \alpha \wedge \circ \beta \to \circ (\alpha \vee \beta )$ and $\left(c{a}_3\right)$ $\circ \alpha \wedge \circ \beta \to \circ (\alpha \to \beta )$.

Remark 15.

Neither ($PL$) nor $\left(c{c}^{\circ }\right)$ is provable in $iCila$.

The negation-free part of $iCila$ is intuitionistic propositional logic. Therefore, a semantics for $iCila$ can be established quite easily by adding an additional clause to Definition 7.

Definition 13.

An $iCila$-semi-valuation is a function $s:\mathcal{F}⟶\{1,0\}$ that satisfies, for any $\alpha ,\beta \in \mathcal{F}$, all the conditions of $iCil$-semi-valuation, and the following clause: ($s10$) if $s(\circ \alpha )$ = $s(\circ \beta )$ = 1, then $s(\circ (\alpha ‡\beta \left)\right)$ = 1, where $‡\in \{\wedge ,\vee ,\to \}$.

An $iCila$-valuation, $iCila$-tautology and a semantic consequence in $iCila$ are defined similarly as in Definitions 8 and 9.

Theorem 7.

For all $\mathsf{\Gamma }\subseteq \mathcal{F}$ and $\alpha \in \mathcal{F}$: $\mathsf{\Gamma }{⊢}_{iCila}\alpha$ iff $\mathsf{\Gamma }{\vDash }_{iCila}\alpha$.

Proof. 

The proof is carried out similarly as for $iCil$ (see the proofs of Theorems 4 and 5). The key point is to show that the following lemma holds:

Lemma 13.

If Δ is α-saturated, then we have, for any $\beta ,\gamma \in \mathcal{F}$, that all the clauses of $iCil$ hold (see Lemma 10 above), and additionally: ($\circ ‡$) if $\circ \beta \in \Delta$ and $\circ \gamma \in \Delta$, then $\circ (\beta ‡\gamma )\in \Delta$, where $‡\in \{\wedge ,\vee ,\to \}$.

This can be performed by means of the axioms of $iCila$. □

The second extension, $iCilo$ for short, is obtained by adding $\left(co\right)$ $\circ \alpha \vee \circ \beta \to \left(\sim \circ \right(\alpha ‡\beta )\to \gamma )$, where $‡\in \{\wedge ,\vee ,\to \}$, to the postulates of $iCil$.

Remark 16

(cf. [6], pp. 71–72; [13], Section 3). $\left(co\right)$ is equivalent to the triple of axiom schemas: $\left(c{o}_1\right)$ $\circ \alpha \vee \circ \beta \to \circ (\alpha \wedge \beta )$, $\left(c{o}_2\right)$ $\circ \alpha \vee \circ \beta \to \circ (\alpha \vee \beta )$ and $\left(c{o}_3\right)$ $\circ \alpha \vee \circ \beta \to \circ (\alpha \to \beta )$.

From Remark 16 and the fact that $(\alpha \vee \beta \to \gamma )\to (\alpha \to \gamma )$ and $(\alpha \vee \beta \to \gamma )\to (\beta \to \gamma )$ are provable in $iCilo$, we deduce $\circ \alpha \to \circ (\alpha ‡\beta )$ and $\circ \beta \to \circ (\alpha ‡\beta )$, where $‡\in \{\wedge ,\vee ,\to \}$. This leads to counter-intuitive consequences; for example, a single consistent conjunct makes consistent the whole conjunction. Specifically, we have $\circ p_1\to \circ (p_1\wedge \cdots \wedge p_i\wedge \cdots \wedge p_m)$, for every $p_i,p_m\in Var$ such that $i\le m$ and $i,m\in \{1,2,3,...\}$.

Definition 14.

An $iCilo$-semi-valuation is a function $s:\mathcal{F}⟶\{1,0\}$ that satisfies, for any $\alpha ,\beta \in \mathcal{F}$, all the conditions of $iCil$-semi-valuation, and additionally: ($s9$) if $s(\circ \alpha )$ = 1 or $s(\circ \beta )$ = 1, then $s(\circ (\alpha ‡\beta \left)\right)$ = 1, where $‡\in \{\wedge ,\vee ,\to \}$.

An $iCilo$-valuation, $iCilo$-tautology and a semantic consequence in $iCilo$ are defined similarly as in Definitions 8 and 9.

Theorem 8.

For all $\mathsf{\Gamma }\subseteq \mathcal{F}$ and $\alpha \in \mathcal{F}$: $\mathsf{\Gamma }{⊢}_{iCilo}\alpha$ iff $\mathsf{\Gamma }{\vDash }_{iCilo}\alpha$.

Remark 17.

$iCila$ is a proper subsystem of $iCilo$.

Proof. 

By $\left(co\right)$, $(\alpha \vee \beta \to \gamma )\to (\alpha \wedge \beta \to \gamma )$ and (MP), we obtain $\left(ca\right)$. Conversely, there are theses of $iCilo$ (see, e.g., Remark 16) that are unprovable in $iCila$. □

Remark 18.

Enriching the set of axiom schemas of $iCila$ (resp. $iCilo$) with $\left(c{c}^{\circ }\right)$ results in obtaining $Cila$ (resp. $Cilo$).

The calculi $Cila$ and $Cilo$ are known in the literature and have been studied by various authors (see, e.g., Section 5.2 of [6] and Chapter 8 of [14]). It is important to emphasize that extending $iCila$ (resp. $iCilo$) with $\left(PL\right)$ does not yield $Cila$ (resp. $Cilo$). To clarify, let $Cil{a}^{★}$ (resp. $Cil{o}^{★}$) be defined by the postulates of $Ci{l}^{★}$, (MP) and $\left(ca\right)$ (resp. $\left(co\right)$).

Remark 19.

$\left(c{c}^{\circ }\right)$ and $\left(bcl\right)$ are provable in neither $Cil{a}^{★}$, nor $Cil{o}^{★}$.

Proof. 

See the proof of Remark 13. □

A bivaluation semantics for $Cil{a}^{★}$ (and $Cil{o}^{★}$) can be obtained from that for $Ci{l}^{★}$.

Definition 15.

A $Cil{a}^{★}$-valuation (resp. $Cil{o}^{★}$-valuation) is a function $v:\mathcal{F}⟶\{1,0\}$ that satisfies, for any $\alpha ,\beta \in \mathcal{F}$, all the conditions of $Ci{l}^{★}$-valuation, and additionally:

($v9$) if $v(\circ \alpha )$ = $v(\circ \beta )$ = 1, then $v(\circ (\alpha ‡\beta \left)\right)$ = 1 (resp. ($v9$) if $v(\circ \alpha )$ = 1 or $v(\circ \beta )$ = 1, then $v(\circ (\alpha ‡\beta \left)\right)$ = 1), where $‡\in \{\wedge ,\vee ,\to \}$.

The notions of tautology and semantic consequence are defined in the standard manner.

Theorem 9.

For all $\mathsf{\Gamma }\subseteq \mathcal{F}$ and $\alpha \in \mathcal{F}$: $\mathsf{\Gamma }{⊢}_X\alpha$ iff $\mathsf{\Gamma }{\vDash }_X\alpha$, where X stands for $Cil{a}^{★}$ or $Cil{o}^{★}$.

Proof. 

The proof is similar to the proofs of Theorem 1 and 2. □

5. Conclusions

This paper presents theoretical results on $LFI$ (for a discussion on the practical implications of $LFI$, particularly its potential applications in evolutionary databases, see [15]; for a discussion on paraconsistent reasoning in science, see, for example, [16]). It has been demonstrated that certain $LFI$, typically regarded as axiomatic extensions of $CP{C}^{+}$, can also be considered extensions of the positive fragment of intuitionistic propositional logic (see Appendix A). Although this concept has already been addressed in the literature (see, e.g., [7], pp. 187–191), neither the intuitionistic variant of $Cil$, $Cila$, nor $Cilo$ has been discussed thus far. As shown, the intuitionistic version of $mbC$ is obtained simply by removing ($PL$) from the axioms of $mbC$. This approach is not applicable to $Ci$ or its extensions, as the postulate ($PL$) is dependent on the other axioms (see Remark 3 above). Nevertheless, an alternative approach is available. First, it is essential to recognize that $\left(bcl\right)$ can be replaced by the triple of formulas: $\left(D{S}^{\circ }\right)$, $\left(i{c}^{\sim }\right)$ and $\left(c{c}^{\circ }\right)$. This assertion holds true not only for $Cil$ but also for $Ci$ (as evidenced by the proof in Remark 8, which depends solely on the rules of $Ci$), indicating that the axioms of $C_{\omega }$, $\left(D{S}^{\circ }\right)$, $\left(i{c}^{\sim }\right)$, $\left(c{c}^{\circ }\right)$ and (MP) axiomatize $Ci$. The postulate $\left(c{c}^{\circ }\right)$ is particularly important in this context: it is independent of the other axioms, and its removal from the set of axioms results in the intuitionistic variant of $Ci$ and its extensions: $Cil$, $Cila$ and $Cilo$. Additionally, as demonstrated in [17], a similar approach can be applied to da Costa’s hierarchy of $C_n$-calculi ($1⩽n<\omega$).