Abstract
Minimal logic, i.e., intuitionistic logic without the ex falso principle, is investigated in its original form with a negation symbol instead of a symbol denoting the contradiction. A Kripke semantics is developed for minimal logic and its sublogics with a still weaker negation by introducing a function on the upward closed sets of the models. The basic logic is a logic in which the negation has no properties but the one of being a unary operator. A number of extensions is studied of which the most important ones are contraposition logic and negative ex falso, a weak form of the ex falso principle. Completeness is proved, and the created semantics is further studied. The negative translation of classical logic into intuitionistic logic is made part of a chain of translations by introducing translations from minimal logic into contraposition logic and intuitionistic logic into minimal logic, the latter having been discovered in the correspondence between Johansson and Heyting. Finally, as a bridge to the work of Franco Montagna a start is made of a study of linear models of these logics.
Keywords
Classical Logic Intuitionistic Logic Canonical Model Propositional Variable Kripke Model1 Introduction
In this paper, we study minimal logic in its two equivalent formulations: one with a basic symbol for the contradiction the other with a basic symbol for negation. Given a countable set of propositional variables, the formulation used nowadays is based on the propositional language of the positive fragment of intuitionistic logic, i.e., \(\mathcal {L}^{+}=\{\wedge ,\vee ,\rightarrow \}\), with an additional propositional constant f, representing falsum. In this setting, negation of \(\varphi \) is defined as \(\varphi \rightarrow f\) and denoted by \(\lnot \varphi \). The significant difference between minimal and intuitionistic logic is that the former does not consider the ex falso quodlibet axiom as a valid axiom. If IPC \(^{+}\) denotes the positive fragment of intuitionistic logic, minimal logic has the same axioms as IPC \(^{+}\), and hence, f does not have the same properties as the intuitionistic \(\bot \). We write MPC \(_f\) for this formalization of minimal logic.
The other formulation of minimal logic makes use of the language \(\mathcal {L}^{+}\cup \{\lnot \}\), where the unary symbol \(\lnot \) represents negation. Thus, we denote with MPC \(_{\lnot }\) the system axiomatized by the IPC \(^{+}\) axioms and the additional axiom \((p\rightarrow q)\wedge (p\rightarrow \lnot q)\rightarrow \lnot p\). This version of minimal logic is the one originally proposed by Johansson (1937), and even before, in a language with only \(\rightarrow \) and \(\lnot \), by Kolmogorov (1925). Completeness with respect to our Kripkestyle semantics is proved for both versions of minimal logic.
The main purpose of the paper is to study a weak form of negation, considering subsystems of minimal logic while keeping the IPC \(^{+}\) axioms fixed. We call such forms of negation subminimal negation. So, we use the term subminimal negation in a nontechnical sense. This term has been used before by Vakarelov (2005, 2006) with a more restricted meaning. We will return to this point later. We define a semantics of negation by means of an auxiliary persistent function N, a different approach than previous authors such as Došen (1999) and Vakarelov. This alternative Kripke semantics leads to a basic system N, in which negation has no properties but the one of being a ‘function.’ A canonical model is defined in order to prove completeness. Among the extensions of N studied here, the one axiomatized by \((p\rightarrow q)\rightarrow (\lnot q\rightarrow \lnot p)\) and denoted as CoPC is the most striking. We succeed in interpreting minimal logic in CoPC. To some extent, we connect with Franco Montagna’s work, by considering the extensions of these logics by way of LC: \((p\rightarrow q)\vee (q\rightarrow p)\). Such extensions represent weakening of the Gödel–Dummett logic. We conclude stating some remarks and ideas for further research.
The ex falso quodlibet or, as it is called in paraconsistent settings, the law of explosion (Carnielli et al. 2007), is the logical law expressing that any statement can be proved from a contradiction (or a falsehood). Classical logic, CPC, intuitionistic logic and many other systems consider ex falso to be valid. However, there has not always been widespread agreement about this. Some supporters of an intuitionistic standpoint, such as the early (Kolmogorov 1925), rejected ex falso. According to him, ex falso asserts something about a consequence of something ‘impossible,’ and hence, it is unacceptable. But, since Heyting’s formalization of intuitionistic logic (Heyting 1934), it has been assumed as an axiom for such a system. In paraconsistent logic, it is necessary to reject ex falso, in order to allow for inconsistent theories and ‘accept’ contradictions^{1}. We present in this paper minimal logic, CoPC and its subsystems as paraconsistent variations of intuitionistic logic.
2 Intuitionistic logic
The propositional language of \(\mathsf{IPC}\) consists of a set P of propositional variables \(\{p_0, p_1, p_2, \dots \}\), the propositional constants \(\bot , \top \) and the set of binary connectives \(\mathcal {L}^{+}(P)\). For any formula \(\varphi \), its negation \(\lnot \varphi \) is defined as \(\varphi \rightarrow \bot \) (see Troelstra and van Dalen 2014). In practice, it is often more convenient to conceive formulas as containing both \(\lnot \) and \(\bot \), and to add \(\top \). We take the axioms of \(\mathsf{IPC}\) as in Troelstra and van Dalen (2014).
2.1 Kripke semantics for Intuitionistic logic
Definition 1
A propositional Kripke frame of \(\mathsf{IPC}\) is a pair \(\mathfrak {F}=(W, R)\), where W is a nonempty set of possible worlds and R is a partial order.
For \(w\in W\), R(w) denotes the upward closed set generated by w. Note that for every \(v \in W\), wRv iff \(v\in R(w)\).

\(w \vDash p \Leftrightarrow w \in V(p)\)

\(w \nvDash \bot \)

\(w \vDash \varphi \wedge \psi \Leftrightarrow w\vDash \varphi \text { and } w\vDash \psi \)

\(w \vDash \varphi \vee \psi \Leftrightarrow w\vDash \varphi \text { or } w\vDash \psi \)

\(w \vDash \varphi \rightarrow \psi \Leftrightarrow \forall v ( ( wRv \text { and } v\vDash \varphi ) \Rightarrow v\vDash \psi )\)
Defining \(\lnot \varphi \) as \(\varphi \rightarrow \bot \), we get \(w \vDash \lnot \varphi \Leftrightarrow \forall v( wRv \Rightarrow v\not \vDash \varphi )\). We write \(V(\varphi )\) for \(\{w  w\vDash \varphi \}\). We may emphasize a valuation V by writing \(\vDash _V\) for \(\vDash \), and sometimes we may stress the particular model and write \(\vDash _{\mathfrak {M}}\). We say that \(\varphi \) is valid on \(\mathfrak {M}=(W,R,V)\) if \(w\vDash \varphi \) for every \(w\in W\), and that \(\varphi \) is valid in \(\mathfrak {F}\) if \(\varphi \) is valid on every \(\mathfrak {M}\) on \(\mathfrak {F}\). We say that the set of formulas \(\varGamma \) is valid on \(\mathfrak {M}\) if each \(\varphi \in \varGamma \) is valid on \(\mathfrak {M}\).
Lemma 1
 1.
(Persistency) If wRv and \(w\models \varphi \), then \(v\models \varphi \)
 2.
(Locality) If \(V {\upharpoonright } R(w)=V' {\upharpoonright } R(w)\), then \(w \vDash _V \varphi \text { iff } w \vDash _{V'} \varphi \)
Proof
Straightforward by induction on the structure of \(\varphi \). \(\square \)
Theorem 1
(Soundness and Completeness of IPC)
Given a set of IPC formulas \(\varGamma \), then \(\varGamma \vdash _\mathsf{IPC} \varphi \) if and only if \(\varphi \) is valid in all Kripke models of \(\varGamma \) for \(\mathsf{IPC}\).
The proof goes via a canonical model, defined as follows.
Definition 2

\(\mathcal {W}:=\{\varDelta  \varDelta \text { is a consistent theory with the \textit{disjunction}} \text { \textit{property}: }\) \(\forall \varphi , \psi \,(\varphi \vee \psi \in \varDelta \Rightarrow \varphi \in \varDelta \text { or } \psi \in \varDelta ) \},\)

\(\mathcal {R}:= \,\subseteq ,\)

Valuation \(\mathcal {V}\): \(\varDelta \in \mathcal {V}(p) \Leftrightarrow p \in \varDelta .\)
3 Minimal logic
3.1 Minimal logic as MPC \(_f\)
The propositional language \(\mathcal {L}_f(P)\) consists of the language of IPC \(^{+}\) to which a propositional constant f representing ‘falsum’ is added. Negation \(\lnot \varphi \) is defined as \(\varphi \rightarrow f\). The axioms for minimal logic with f are just the axioms of IPC \(^{+}\).
Definition 3
A propositional Kripke frame of \(\mathsf{MPC}_f\) is a triple \(\mathfrak {F}=(W, R, F)\), where W is a nonempty set of possible worlds, R is a partial order and \(F\subseteq W\) is an upward closed set, intended to be \(\{w\in W \,\, w\vDash f\}\).
The following proposition, known to Johansson, is easy to prove.
Proposition 1
It follows that the notion of contradiction expressed by f in MPC \(_f\) will be available in MPC \(_{\lnot }\) as \(\lnot p\wedge \lnot \lnot p\).
3.2 Minimal logic as MPC \(_{\lnot }\)
The considered axiom was explicitly used by Johansson (1937) in his original article. However, it was previously introduced by Kolmogorov in the article that has been included in the book ‘From Frege to Gödel: a source book in mathematical logic,’ a collection by Heijenoort (1967). Kolmogorov says: ‘The usual principle of contradiction: A judgment cannot be true and false, cannot be formulated in terms of an arbitrary judgment, implication, and negation. Our principle contains something else: namely, from it, together with the first axiom of implication, there follows the principle of reductio ad absurdum.’
From the axiom the principles of negative ex falso and absorption of negation, as we will call them, readily follow.
Lemma 2
 1.
\(\mathsf{MPC}_{\lnot }\vdash p\wedge \lnot p\rightarrow \lnot q,\)
 2.
\(\mathsf{MPC}_{\lnot }\vdash (p\rightarrow \lnot p)\rightarrow \lnot p.\)
Lemma 3
For every \(\varDelta \in \mathcal {W}\), \(\varDelta \in \mathcal {F}\) if and only if \(\lnot \psi \in \varDelta \) for all \(\,\psi \).
Proof
The righttoleft direction of the statement is trivial. We focus on the other direction. Assume \(\varDelta \) to be in \(\mathcal {F}\), and consider an arbitrary formula \(\psi \). The definition of \(\mathcal {F}\) gives us the existence of a contradiction in \(\varDelta \), i.e., there is a formula \(\varphi \) in \(\varDelta \), whose negation is also an element of \(\varDelta \). The formulas \(\varphi \) and \(\lnot \varphi \) both being logical consequences of \(\varDelta \), imply \(\varDelta \vdash \varphi \wedge \lnot \varphi \). Lemma 2(1) leads us to \(\varDelta \,{\vdash }\,\lnot \psi \), via an application of modus ponens. The set \(\varDelta \) is a theory, and hence, \(\lnot \psi \in \varDelta \). \(\square \)
Completeness is proved as for intuitionistic logic. It is sufficient to prove that for any theory in the canonical model, membership relation and truth relation coincide. We prove the induction step concerning the negation \(\lnot \varphi \).
Proof
The left to right goes by contraposition. Assume \(\lnot \varphi \not \in \varDelta \), for \(\varDelta \in \mathcal {W}\). This gives us \(\varDelta \not \vdash \lnot \varphi \). By Lemma 2(2), \(\varDelta \vdash (\varphi \rightarrow \lnot \varphi )\rightarrow \lnot \varphi \). Thus, \(\varDelta \not \vdash \varphi \rightarrow \lnot \varphi \). This is equivalent to saying that the formula \(\lnot \varphi \) is not a logical consequence of the set \(\varDelta \cup \{\varphi \}\). From the standard Lindenbaum type lemma, we get the existence of a theory \(\varGamma \in \mathcal {W}\), extending \(\varDelta \cup \{\varphi \}\) and not containing \(\lnot \varphi \). Apply now Lemma 3, to get that \(\varGamma \) is not an element of \(\mathcal {F}\). Moreover, \(\varGamma \vDash \varphi \) by induction hypothesis. The last two results are equivalent to \(\varGamma \not \vDash \lnot \varphi \). The canonical model \(\mathcal {M}_\lnot \) being persistent, we conclude \(\varDelta \not \vDash \lnot \varphi \).
For the righttoleft direction, we proceed directly. Suppose \(\lnot \varphi \in \varDelta \), and consider an arbitrary \(\subseteq \)successor \(\varGamma \) of \(\varDelta \). Assume \(\varGamma \vDash \varphi \). The induction hypothesis gives us \(\varphi \in \varGamma \). We assumed \(\lnot \varphi \) to be an element of \(\varDelta \), and hence, of \(\varGamma \). Both \(\varphi \) and \(\lnot \varphi \) being in \(\varGamma \), we conclude \(\varGamma \in \mathcal {F}\). Therefore, \(\varDelta \vDash \lnot \varphi \) as desired. \(\square \)
4 Basic subminimal logic: N
The propositional language coincides with the one for minimal logic with negation \(\lnot \). The semantics of negation is defined in terms of an auxiliary persistent function N. Different axioms attribute different properties to such a function. The aim of the Kripke semantics is that a negated formula \(\lnot \varphi \) is true in a world if and only if that world is in the image of \(V(\varphi )\) under N.
The basic logic N is axiomatized by \((p\leftrightarrow q)\rightarrow (\lnot p\leftrightarrow \lnot q)\) (N).
Definition 4
A propositional Kripke frame is a triple \(\mathfrak {F}=(W, R, N)\), where W is a nonempty set of possible worlds, R is a partial order on W and N is a function \(N: \mathcal {U}(W) \rightarrow \mathcal {U}(W)\), where \(\mathcal {U}(W)\) is the set of all upward closed subsets of W.

P1: \( w\in N(U) \Leftrightarrow w\in N(U\cap R(w))\), with R(w) the upward closed set generated by w.

P2: If \(w\in N(U)\), then,\(\text { for all } v \text { such that } wRv, \, v\in N(U)\).
Definition 5
Given a frame \(\mathfrak {F}=(W,R,F)\) and a world \(w\in W\), the subframe \(\mathfrak {F}_w\) generated by w is defined on the set of worlds R(w), with the function \(N_w(U)=N(U)\cap R(w)\), for every upward closed set U.
Similarly, \(\mathfrak {M}_w\) is defined on the basis of the model \(\mathfrak {M}\).
Lemma 4
Given \(v\in R(w)\), then: \(v\vDash _{\mathfrak {M}_w}\varphi \) if and only if \(v\vDash _{\mathfrak {M}}\varphi \).
Proof
We only unfold the induction step of the proof concerning the negation. Indeed, \(v\vDash _{\mathfrak {M}_w}\varphi \) is equivalent to \(v\in N_w(V_w(\varphi ))\), which means \(v\in N(V_w(\varphi ))\cap R(w)\), and it is equivalent to \(v\in N(V(\varphi )\cap R(w))\cap R(w)\) (by induction hypothesis). By P1, this is equivalent to \(v\in N(V(\varphi )\cap R(v))\cap R(w)\) which, again by P2, is just \(v\in N(V(\varphi ))\), as desired. \(\square \)
Lemma 5
\(\mathcal {N}\) satisfies P1 and P2.
Proof

P1: To show: \(\varDelta \in \mathcal {N}(U)\) if and only if \(\varDelta \in \mathcal {N}(U\cap \mathcal {R}(\varDelta ))\). Note that \(\varDelta \in \mathcal {N}(U)\) means \(\,U\,{\cap }\, \mathcal {R}(\varDelta )\,{=}\,\llbracket \varphi \rrbracket \,{\cap }\, \mathcal {R}(\varDelta )\) and \(\lnot \varphi \in \varDelta \), for some \(\varphi \). This is equivalent to: \((U\,{\cap }\, \mathcal {R}(\varDelta ))\,{\cap }\, \mathcal {R}(\varDelta )\,{=}\,\llbracket \varphi \rrbracket \,{\cap }\, \mathcal {R}(\varDelta )\) and \(\lnot \varphi \in \varDelta \) for the same \(\varphi \), by associativity of \(\cap \). The latter means exactly \(\varDelta \in \mathcal {N}(U\cap \mathcal {R}(\varDelta ))\), and hence, we proved the desired equivalence.

P2: To show: if \(\varDelta \in \mathcal {N}(U)\) and \(\varDelta \subseteq \varDelta '\) hold, then \(\varDelta '\in \mathcal {N}(U)\). Assume the antecedent and note that this means \(\,U\,{\cap }\, \mathcal {R}(\varDelta )\,{=}\,\llbracket \varphi \rrbracket \,{\cap }\, \mathcal {R}(\varDelta )\) and \(\lnot \varphi \in \varDelta \), for some \(\varphi \). By the inclusion \(\varDelta \subseteq \varDelta '\), we get \(\lnot \varphi \in \varDelta '\). Moreover, \(\varDelta \subseteq \varDelta '\) if and only if \(\mathcal {R}(\varDelta ')=\mathcal {R}(\varDelta )\cap \mathcal {R}(\varDelta ')\). This, by associativity of \(\cap \), implies \(\,U\,{\cap }\, \mathcal {R}(\varDelta ')\,{=}\,\llbracket \varphi \rrbracket \,{\cap }\, \mathcal {R}(\varDelta ')\). Therefore, \(\varDelta '\in \mathcal {N}(U)\).\(\square \)
Theorem 2
The basic logic of unary operator N is complete with respect to the class of Kripke models defined above.
Proof
By contraposition we prove: if \(\varGamma \,{\nvdash _\mathsf{N}}\, \varphi \), then \(\varDelta \nvDash \varphi \), for some \(\varDelta \) containing \(\varGamma \) in the canonical model. First we show by induction on \(\varphi \) that, for any \(\varDelta \) in the canonical model, \(\varDelta \vDash \varphi \Leftrightarrow \varphi \in \varDelta \). We only treat the negation case. We need to prove that \(\varDelta \vDash \lnot \varphi \Leftrightarrow \lnot \varphi \in \varDelta \).
\((\Rightarrow )\) Assume \(\varDelta \vDash \lnot \varphi \). So, \(\varDelta \in \mathcal {N}(\llbracket \varphi \rrbracket )\). By definition, there is a formula \(\psi \) such that \(\llbracket \psi \rrbracket \,{\cap }\,\mathcal {R}(\varDelta )=\llbracket \varphi \rrbracket \,{\cap }\,\mathcal {R}(\varDelta )\) and \(\lnot \psi \in \varDelta \). Then, for all extensions \(\varGamma \) of \(\varDelta \), \(\varphi \in \varGamma \) if and only if \(\psi \in \varGamma \). As in \(\mathsf{IPC}\), \(\varphi \leftrightarrow \psi \in \varDelta \). By the axiom N, \(\lnot \varphi \leftrightarrow \lnot \psi \in \varDelta \) as well. So, it follows that \(\lnot \varphi \in \varDelta \).
\((\Leftarrow )\) Assume \(\lnot \varphi \in \varDelta \). Then \(\exists \psi \,\,(\,\llbracket \varphi \rrbracket \,{\cap }\, \mathcal {R}(\varDelta )\,{=}\,\llbracket \psi \rrbracket \,{\cap }\, \mathcal {R}(\varDelta )\) and \(\lnot \psi \in \varDelta )\), namely \(\psi :=\varphi \). Hence, \(\varDelta \in \mathcal {N}(\llbracket \varphi \rrbracket )\) and, by induction hypothesis, \(\varDelta \in \mathcal {N}(\mathcal {V}(\varphi ))\), and hence, \(\varDelta \vDash \lnot \varphi \). \(\square \)
It is worth remarking here that the axiom N is exactly what is needed to prove the substitution theorem, \(\vdash _\mathsf{N}\!(\varphi _1\leftrightarrow \varphi _2)\rightarrow (\psi [\varphi _1/p]\leftrightarrow \psi [\varphi _2/p])\).
A few words in connection with Vakarelov (2005, 2006) are in order at this point. He considers a weak negation in combination with a strong negation in the sense of Nelson (1949), which makes it somewhat difficult to compare to our work. One of the systems he studied restricted to the weak negation can be seen as having the axiom \(\lnot \varphi \leftrightarrow ((\varphi \rightarrow f)\wedge t)\) with additionally f implying t. For negation only this logic will be an extension of N, and in fact of CoPC, and a subsystem of MPC. We did not study it carefully yet.
5 Extensions of N
We present some extensions of the basic logic N. Each of the additional axioms will enrich the semantic function N with a different property.
5.1 Axioms of negation
 1.
Absorption of negation: \((p\rightarrow \lnot p)\rightarrow \lnot p\)
 2.
Contraposition: \((p\rightarrow q)\rightarrow (\lnot q\rightarrow \lnot p)\)
 3.
Negative ex falso: \((p\wedge \lnot p)\rightarrow \lnot q\)
 4.
Double negation: \(p\rightarrow \lnot \lnot p\)
 5.
Distribution over conjunction: \(\lnot (p\wedge q)\rightarrow (\lnot p\vee \lnot q)\)
In Sect. 7.3, we will give a semantic proof of the fact that absorption of negation does not follow from contraposition. We already saw in Lemma 2 (1) that negative ex falso follows from contraposition.
Remark 1
Note that the contraposition instance that we are considering, denoted as CoPC, is the one valid in intuitionistic logic, while the instance \((\lnot q\rightarrow \lnot p)\rightarrow (p\rightarrow q)\) is not. Moreover, from the latter, the law of explosion follows. Thus, a logic in which \((\lnot q\rightarrow \lnot p)\rightarrow (p\rightarrow q)\) is accepted is no longer paraconsistent.
In what follows, we denote axiom 1 as An, and axiom 3 as NeF. We prove that minimal logic can also be axiomatized by CoPC\(\,+\,\)An. We study the logic CoPC, axiomatized by contraposition, and we will see later on that minimal logic and CoPC are closely related systems.
Proposition 2
Minimal logic MPC \(_{\lnot }\) can be equivalently axiomatized by CoPC \(+\) An. In other words, MPC \(=\) CoPC \(+\) An.
Proof
We first show that \((p\rightarrow q)\wedge (p\rightarrow \lnot q)\rightarrow \lnot p\) is a theorem of CoPC \(\,{+}\,\)An.
From CoPC, we have \((p \rightarrow \lnot q)\wedge (p\rightarrow q) \rightarrow (\lnot q \rightarrow \lnot p)\)
By transitivity, we obtain \((p \rightarrow \lnot q)\wedge (p\rightarrow q) \rightarrow (p \rightarrow \lnot p)\)
Because of An, we have \((p \rightarrow \lnot q)\wedge (p\rightarrow q) \rightarrow \lnot p\)
 In \(\mathsf{MPC}\), we prove CoPC.
\( \vdash _\mathsf{MPC}\)
\((p\rightarrow \lnot q)\wedge (p\rightarrow q) \rightarrow \lnot p\)
\(\vdash _\mathsf{MPC}\)
\( \lnot q\wedge (p\rightarrow q) \rightarrow \lnot p\)
By commutativity of \(\wedge \), we obtain
\( \vdash _\mathsf{MPC}\)
\( (p\rightarrow q)\wedge \lnot q \rightarrow \lnot p\)
Thus follows
\( \vdash _\mathsf{MPC}\)
\((p\rightarrow q) \rightarrow (\lnot q \rightarrow \lnot p)\)
 In \(\mathsf{MPC}\), we prove An.\(\square \)
\( \vdash _\mathsf{MPC}\)
\((p\rightarrow \lnot q)\wedge (p\rightarrow q) \rightarrow \lnot p\)
Changing q into p, we obtain
\(\vdash _\mathsf{MPC}\)
\((p\rightarrow \lnot p)\wedge (p\rightarrow p) \rightarrow \lnot p\)
\(\vdash _\mathsf{MPC}\)
\((p\rightarrow \lnot p) \rightarrow \lnot p\)
In a similar way, it can be shown that minimal logic is equivalent to N \(+\) NeF \(+\) An.
5.2 Contraposition logic: CoPC
The Kripkestyle semantics for this system is exactly the same as in N. An additional requirement for the function N needs to be specified. Indeed, the semantic function N needs to satisfy P1, P2 and antimonotonicity:
\(\mathbf P_\mathsf{CoPC}\): For all U, \(U'\in \mathcal {U}(W)\), if \(U\subseteq U'\), then \(N(U')\subseteq N(U).\)
So, assume the lemma holds for \(\varphi \) and all \(\varGamma \), and let \(\lnot \varphi \in \varDelta \). We have to show \(\varDelta \vDash \lnot \varphi \), i.e., \(\varDelta \in \mathcal {N}(\llbracket \varphi \rrbracket )\). Take any \(\psi \) such that \(\llbracket \psi \rrbracket \cap R(\varDelta ) \subseteq \llbracket \varphi \rrbracket \); we need to show that \(\lnot \psi \in \varDelta \). We can easily see that \(\psi \,\,{\rightarrow }\,\,\varphi \) has to be a member of \(\varDelta \) because otherwise by use of a Lindenbaum lemma \(\varDelta \) would have an extension containing \(\psi \) but not \(\varphi \). But the axiom \(((\psi \rightarrow \varphi )\rightarrow (\lnot \varphi \rightarrow \lnot \psi ))\) is in \(\varDelta \) and by assumption \(\lnot \varphi \) also. Therefore, indeed \(\lnot \psi \in \varDelta \).
5.3 Negative ex falso: NeF
The Kripke semantics is just the same as for the basic logic N, with the additional requirement for the function N
\(\mathbf P_\mathsf{NeF}\): For all U, \(U'\in \mathcal {U}(W)\), \(U\cap N(U)\subseteq N(U')\)
For both contraposition logic and negative ex falso logic, the finite model property holds. For the proof, theories within an adequate set have been used.
6 Relation between CoPC and minimal logic
We begin this section by giving an example of a derivation in CoPC.
Proposition 3
Proof
The following is a Hilbertstyle derivation in CoPC.
By NeF  \(\vdash \)  \((p\wedge \lnot p)\rightarrow \lnot \lnot p\) 
by \(\mathsf{IPC}^+\)  \(\vdash \)  \(p\rightarrow (\lnot p\rightarrow \lnot \lnot p)\) 
by \(\mathsf{CoPC}\)  \( \vdash \)  \(p\rightarrow (\lnot \lnot \lnot p\rightarrow \lnot \lnot p)\) 
by \(\mathsf{IPC}^+\)  \(\vdash \)  \(\lnot \lnot \lnot p\rightarrow (p\rightarrow \lnot \lnot p)\) 
by \(\mathsf{CoPC}\)  \(\vdash \)  \(\lnot \lnot \lnot p\rightarrow (\lnot \lnot \lnot p\rightarrow \lnot p)\) 
by \(\mathsf{IPC}^+\)  \(\vdash \)  \(\lnot \lnot \lnot p\rightarrow \lnot p\) 
From this, we get that we do not need more than 3 negations in CoPC.^{2}
Corollary 1
\(\mathsf{CoPC}\vdash \lnot \lnot \lnot \lnot p\leftrightarrow \lnot \lnot p.\)
Proof
The two directions of the proof go as follows.
(\(\Rightarrow \)) Substitute \(\lnot p\) for p in Proposition 3.
(\(\Leftarrow \)) Apply \(\mathsf CoPC\) to Proposition 3. \(\square \)
6.1 Translating MPC into CoPC
In the first part of this section, we present a translation of minimal logic into contraposition logic. Presenting later a translation of intuitionistic logic into minimal logic, we get a ‘chain’ of interpretations between contraposition logic and classical logic.
Recall that the ‘negative’ translation from classical logic into intuitionistic logic ensures that IPC has at least the same expressive power and consistency strength of classical logic (Troelstra and van Dalen 2014). A similar thing happens with Gödel’s translation of \(\mathsf{IPC}\) into the modal logic S4. Here, we establish a similar translation from minimal logic into CoPC.
Consider \(\sim \varphi :=\varphi \rightarrow \lnot \varphi \). We define a translation such that \((\lnot \varphi )^{\sim }:=\sim \varphi ^{\sim }\), while every other connective is left unchanged (i.e., \((\varphi \circ \psi )^{\sim }:=\varphi ^{\sim }\circ \psi ^{\sim }\), for \(\circ \in \{\wedge ,\vee ,\rightarrow \}\), and also every atom stays the same).
Theorem 3
Proof
It is worth noticing that the considered translation works also for the negative ex falso logic (instead of CoPC), and even for the basic logic N.
6.2 A translation of Intuitionistic logic into MPC
Gaspar (2013) uses a closely related Friedman–Dragalin type translation, translating propositional variables p into \(p\vee \bot \). It is better behaved and works also for the consequence relations, not only for theorems such as the Heyting–Johansson translation. The idea behind this translation is the same as in the Heyting–Johansson translation but extended from proofs of implications to all proofs.
7 Linear frames
In this section, we want to analyze the frames of our systems in which the LCaxiom, i.e., \((p\rightarrow q)\vee (q\rightarrow p)\), is valid. For each logic, the class of frames satisfying the considered formula corresponds to the class of upwards linear frames (Fig. 1).
7.1 Linear frames in minimal logic

If \(w\not \in F\), \(n(w)=F.\)

If \(w\in F\), n(w) is the whole set, i.e., \(n(w)=W.\)
7.2 Linear frames in subminimal systems

If \(w\not \in F\), then \(n(w)=F,\)

If \(w\in F\), then \(n(w)\supseteq R(w).\)

If \(w\in F\), then \(n(w)\supseteq {F}.\)
7.3 Counterexamples
In the last part of this section, we give two examples to show how the different axioms we are considering are logically related to each other.
Proposition 4
Absorption of negation An is not a theorem in CoPC.
Proof
(Fig. 3) The idea is that we consider a linear finite CoPC frame in which the set F is a proper subset of W and, for every upward closed set U, \(N(U)=F\). In this way, by assigning a valuation \(V(p)\subseteq F\) for some propositional variable p, we get that every world \(v\not \in {F}\) does not force \(\lnot p\), while it forces the implication \(p\rightarrow \lnot p\). Observe that a frame in which \(N(U)={F}\) for every U is indeed a CoPC frame. The only thing we need to check is the locality condition, given that the other two properties trivially hold. Also locality is quite trivial, given that \(w\in N(U)\) if and only if \(w\in {F}\), which again would be equivalent to \(w\!\in \! N(U\cap R(w))\). \(\square \)
Proposition 5
Contraposition CoPC is not a theorem of N.
Proof
(Fig. 4) For obtaining an N model in which CoPC does not hold, it is enough to consider an arbitrary finite linear frame such that \(n(w)=R(w)\) for every world. For the sake of simplicity, let \(\bar{w}\) be the greatest world in the frame, and assign a valuation such that \(V(p)=\{\bar{w}\}\) and \(V(q)=R(v)\), where \(v\ne \bar{w}\), for some propositional variables p, q. Indeed, the world v forces the implication \(p\rightarrow q\). On the other hand though, \(\lnot q\) is true in v, while \(\lnot p\) is not. Therefore, CoPC is not valid on the considered frame. Note again that the function N defined as we did is persistent. Moreover, whenever \(w\in N(R(v))\) for some v, this means that \(R(w)\subseteq R(v)\), and hence, \(w\in N(R(v)\cap R(w))\) amounts to \(w\in N(R(w))=n(w)\), which is true by definition. For the other direction, again, saying that \(w\in N(R(v)\cap R(w))\) for some v implies that \(R(w)\subseteq R(v)\cap R(w)\) which indeed means \(R(w)\subseteq R(v)\). The definition on N implies \(w\in n(v)=N(R(v))\), as desired. \(\square \)
8 Conclusions and further research
The main purpose of this paper was to explore and analyze minimal logic with negation as a primitive and its subminimal subsystems with a weaker negation. We concentrate on a basic logic \(\mathsf N\) where negation is just a unary operator without additional properties, and on two of its extensions: contraposition logic and negative ex falso. The semantics of negation is defined in terms of a persistent function N on the set of upward closed sets of a Kripke model. Completeness can be proved by means of canonical models. We show that CoPC interprets MPC by means of a sound translation, and complete the chain of translations from CoPC to CPC by a translation of IPC into MPC appearing in the correspondence between Johansson and Heyting in 1935.
For future work, the first step is allowing the negation function N to be partial (compare to neighborhood models of modal logic Došen 1989; Kracht and Wolter 1999). This produces more natural and general canonical models. The corresponding algebras for a study of duality are Heyting algebras (see, e.g., Bezhanishvili et al. 2016). There is a close relationship between our locality condition and the algebraic notion of compatible function of Caicedo and Cignoli (2001) (see also Ertola et al. 2007) that needs to be clarified.
The abovementioned translations are effective for firstorder logic as well, and in general there are many interesting questions about firstorder logic. It is also already clear that the systems are very suitable for introduction of cutfree sequent systems to prove properties such as interpolation.
The study of the models of weak Gödel–Dummett logic, which provides a bridge to the work of Franco Montagna, can be extended by looking at the behavior of the logics on the models (0,1] and [0,1]. Here also the algebras and the proof theory (Metcalfe and Montagna (2007)) seem well worth studying.
Finally, the structure of the lattice of all logics between \(\mathsf N\) and minimal logic is intriguing. Certainly it will contain infinitely many logics.
Footnotes
 1.
Kolmogorov and Johansson’s minimal intuitionistic logic is introduced as MIL in the ‘big manifesto’ paper on paraconsistency, in Carnielli et al. (2007).
 2.
We thank Lex Hendriks for these observations.
 3.
This correspondence has been studied with van der Molen (2016).
 4.
Similarly, there are no restrictions by locality on N(U) for \(U=\emptyset \).
Notes
Acknowledgments
We thank Benno van den Berg, Marta Bílková, Rodolfo Ertola, Lex Hendriks and Anne Troelstra for helpful discussions on the subject of this paper. We thank an unknown referee for valuable comments.
Compliance with ethical standards
Conflict of interest
All authors declare that they have no conflict of interest
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