Mortensen Logics

In a paraconsistent context where formulas have three admissible assignments, and assuming the standard properties with respect to the “classical” assignments, that is

Among his many contributions in logic and philosophy, Chris Mortensen introduced a connexive logic commonly known as ‘M3V’. M3V is obtained by adding a special conditional to González-Asenjo/Priest’s LP. Such conditional is structurally the same as the one used by Anderson and Belnap in [2] to show the consistency of the logic E and, in particular, to show how to block the paradox of necessity, i.e. to avoid validating formulas of the form $X>(Y>Z)$, with $>$ an entailment connective, $X$ a contingent truth and $(Y>Z)$ a logical truth.¹¹1A logic containing M3V was developed around the same time by Peña to cope with comparatives, gradables and vagueness. See [35] for a summary of his results and [34] for a more friendly exposition of them. Among its most notable features, besides its being connexive, M3V is negation-inconsistent and it validates the negation of every conditional.

But Mortensen has also studied and applied extensively other non-connexive logics. On the one hand there is closed set logic, CSL, a paraconsistent logic motivated by dualizing open set logic, i.e. intuitionistic logic. CSL has notoriously been found defective in lacking a conditional connective because in it there is no connective $\copyright$ such that $A\copyright B$ is untrue if $A$ is true and $B$ untrue, as one would expect from a conditional. The two most obvious candidates, $\neg A\vee B$ and $\neg(A\wedge\neg B)$ are true when $A$ is true and $B$ is untrue, delivering thus countermodels to Detachment.²²2Mortensen has always argued that this is not a serious defect, especially when it comes to doing mathematics with CSL. We will not address this issue here. The fact is that there is no such connective in the logic; how bad is that is a different discussion. On the other hand, in [25] he proposed another logic, which later Marcos [18] modified to obtain a variant of Sette’s logic, identified and called P² by Marcos.

In this paper, we analyze and compare systematically the connexive variants of CSL and P², obtained by adding the M3V conditional to them. Our main observations are two. First, that the inconsistency of M3V is exacerbated in the connexive variant of closed set logic, while it is attenuated in the connexive variant of the Sette-like P². Second, that the M3V conditional is, unlike other conditionals, connexively stable, meaning that it validates the core connexive schemas when combined with the main paraconsistent negations.

The plan of the paper is as follows. In Section 2 we present some preliminary, general notions that will be useful for the remainder of the paper. In Section 3 we present M3V and mention some of its properties; some of them are already well-known, but others are noticed here for the first time. In Section 4 we introduce cCSL3, closed set logic restricted to three admissible interpretations, like M3V, enriched with the E-conditional. We give some of its most notable features, including likenesses and differences with M3V. There we show that, unlike other conditionals, the E-conditional is connexively stable with respect to both $\sim$ and $\neg$. Finally, in Section 5 we present cP². It shares the $\{\sim,\rightarrow_{\_}{\tiny{\textbf{E}}}\}$-fragment with M3V, but still they differ in ways that are significant for connexive logicians.

Let $A$ and $B$ arbitrary formulas of a given formal language, and $\Gamma$ a set of formulas of that language. In this paper, logical consequence is understood as truth-preservation from premises to conclusions in all interpretations, that is:

• $\Gamma\models_{\_}{\tiny{\textbf{L}}}A$ if and only if, for all $\sigma$, if $1\in\sigma(B)$ for all $B\in\Gamma$ then $1\in\sigma(A)$

The logic M3V was introduced, although not with that name, in [24] (the name was given in [21], presumably to mean “Mortensen’s 3-valued connexive logic”). The following truth tables, with $V_{\_}{\tiny{\textbf{M3V}}}=\{2,1,0\}$ and $D^{+}=\{2,1\}$, characterize M3V:

It must be noted that Mortensen’s satisfiability conditions for the conditional are structurally the same as the ones used by Anderson and Belnap in [2] to show the consistency of the logic E, hence the subscript. In particular, they showed how to block the paradox of necessity, i.e. to avoid validating formulas of the form $X>(Y>Z)$, where $X$ is a contingent truth and $(Y>Z)$ is a logical truth.³³3A logic containing M3V was developed around the same time by Peña to cope with comparatives, gradables and vagueness. See [35] for a summary of his results and [34] for a more friendly exposition of them.

The three-valued nature of Mortensen’s logic, along with the number of elements in $D^{+}$ and the evaluation conditions for negation motivate the representation of Mortensen’s 2, 1, 0 as three subsets of the set of classical values $\{1,0\}$, namely $\{1\}$, $\{1,0\}$ and $\{0\}$, respectively, leaving the remaining subset $\{\ \}$ aside as in the two-valued relational semantics for LP:

Applying the mechanical procedure described in [32] for turning truth tables employing three of the four truth values of FDE into Dunn conditions (i.e., pairs of positive and negative conditions in terms of containing or not containing the classical values 0 or 1), we define a relation $\sigma$, which takes formulas as its domain and the set of truth values $\{1,0\}$ as its codomain.

Then, the positive condition describes the cases in which $1\in\sigma(X)$, and the negative condition describes the cases in which $0\in\sigma(X)$. From the truth tables above we can infer that the conditions for the implication-free fragment of the language are standard, and that the clauses for $\rightarrow_{\_}{\tiny{\textbf{E}}}$ are as follows:

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  $1\in\sigma(A\rightarrow_{\_}{\tiny{\textbf{E}}}B)$ if a and only if $1\notin A$, or $0\notin B$, or both $0\in A$ and $1\in B$

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  $0\in\sigma(A\rightarrow_{\_}{\tiny{\textbf{E}}}B)$ if and only if $1\in\sigma(A)$ or $0\in\sigma(A)$ and either $1\in\sigma(B)$ or $0\in\sigma(B)$

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  Unlike LP, M3V validates unrestricted Detachment.

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  It is connexive.

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  It is contradictory. As witnesses, consider $(A\wedge\sim\!A)\rightarrow_{\_}{\tiny{\textbf{E}}}A$ and $\sim\!((A\wedge\sim\!A)\rightarrow_{\_}{\tiny{\textbf{E}}}A)$.

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  All conditionals are false in M3V. The falsity condition for the conditional is but a sophisticated way of expressing $0\in\sigma(A\rightarrow_{\_}{\tiny{\textbf{E}}}B)$, which implies that $\models_{\_}{\tiny{\textbf{M3V}}}\sim\!(A\rightarrow_{\_}{\tiny{\textbf{E}}}B)$, for any $A$ and $B$.

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  Though all conditionals are false in M3V, some of them are true as well. Simply consider a conditional where both antecedent and consequent are just true. The conditional is false, yet true as well.

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  $\models_{\_}{\tiny{\textbf{M3V}}}\sim\!(A\rightarrow_{\_}{\tiny{\textbf{E}}}B)$ implies $\models_{\_}{\tiny{\textbf{M3V}}}\sim\!(A\rightarrow_{\_}{\tiny{\textbf{E}}}\sim\!B)$, by a simple substitution in the consequent. Due to the validity of the latter, we say that M3V is ultra-Abelardian.⁴⁴4Claudio Pizzi has urged the connexive logic community not to multiply the principles with names of ancient philosophers. However, that plays a role in keeping a healthy logical memory. Peter Abelard held that conditionals express natures and that natures are characterized positively. For example, he believed that it would not be part of a human’s nature to not be a stone, although being an animal would be. (For details see [19].) Thus, for him, no conditional of the form $A\rightarrow\sim\!B$, where $A$ is necessarily positive —that is, its main connective is not a negation— and $\sim\!B$ is not a subformula of $A$, is true on pain of contradiction. Omitting the constraints on $A$ and $\sim\!B$ would lead to ultra-Abelardianism.

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  Almost obvious given the validity of $\sim\!(A\rightarrow_{\_}{\tiny{\textbf{E}}}B)$, but even more overlooked, is the fact that M3V validates some schemas from Abelian logic, namely the Centering principles⁵⁵5Nonetheless, it does not validate the Meyer-Slaney relativity axiom (schema), characteristic of purely implicative Abelian logics: $\not\models_{\_}{\textbf{\tiny{M3V}}}((A\rightarrow_{\_}{\tiny{\textbf{E}}}B)\rightarrow_{\_}{\tiny{\textbf{E}}}B)\rightarrow_{\_}{\tiny{\textbf{E}}}A$ (For a countermodel, let $\sigma(A)=\{0\}$ and $\sigma(B)=\{1\}$.) The validity of $\sim\!(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)$ demands moreover a comparison with Meyer and Martin’s SI$\sim$I —see [23]—, where such schema is valid too. In that logic, $(C\rightarrow D)\rightarrow((A\rightarrow C)\rightarrow(A\rightarrow D))$ and $(A\rightarrow C)\rightarrow((C\rightarrow D)\rightarrow(A\rightarrow D))$, both object-language expressions of transitivity, are valid, but their negations are not. Nevertheless, since all conditionals are false in M3V, the negation of these forms of transitivity is valid as well.:

  $\models_{\_}{\tiny{\textbf{M3V}}}\sim\!(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)$

  $\models_{\_}{\tiny{\textbf{M3V}}}\sim\!(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)\leftrightarrow_{\_}{\tiny{\textbf{E}}}(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)$

  (This provides other witnesses of negation-inconsistency, namely $A\rightarrow_{\_}{\tiny{\textbf{E}}}A$ and $\sim\!(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)$.

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  M3V is not hyper-connexive. Suppose it were, and that the Converse of Boethius hold. By ultra-Abelardianism and Detachment, $A\rightarrow B$ would be valid, but it is not. (A similar argument can be run using the Converse of the Variant of Boethius and the falsity of all conditionals.)

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  Francez’s logics (see [11]; see also [12] and [13]) have been the only recognized nexive logics so far. But M3V is nexive too, as a consequence of all negated conditionals being true. It is not hyper-nexive, though. (The proof is similar to the proof that it is not hyper-connexive.)

From the above, perhaps the most surprising feature is the fact that all conditionals are false in M3V. Indeed, one could argue that M3V is an interesting logic in so far as having arbitrary false conditionals, among many otherwise familiar properties, is an interesting feature for a logic to have. Nonetheless, this may require some philosophical elucidation.

The first thing to be said is that Cantwell’s logic for conditional negation CN and M3V are inter-definable. In particular, the E-conditional is the contraposable conditional defined with the conditional in CN; see [33]. Thus, one could attempt to build upon the intuitive features of CN to obtain some extra-logical support for M3V. True, the intuitiveness of the basic notions do not transfer immediately to the derived notions, but it could be a start.

We do not follow that route, though. In our view, it is not unreasonable to have a logic in which all conditionals are false. On the one hand, tradition has it that certain syllogisms that are deemed valid often lack some tacit premise. For example, from “Every human is mortal” infer “I am mortal”, where premise “I am a human” is tacit, i.e. it is a suppressed or unstated truth or piece of information not mentioned explicitly yet being part of the argument so that the conclusion indeed follows. This kind of argument is called enthymeme by Aristotle (Rethoric, 1357a16-21) and the implication relation between its premises and its conclusions is called enthymematic implication by Sylvan [38, p. 142]. Following this line of thought, M3V might be considered as a logic of enthymematic implication, i.e. as a logic about conditional arguments that strictly speaking are invalid, since they always lack some antecedent, premise or background information in order to hold (i.e. in order to entail the conclusion or consequent), but which may also be accepted as valid sotto voce, prima facie or ceteris paribus.

On the other hand, connexive logic has been intimately attached to counterfactual notions from its very (contemporary) beginnings. (See [3].). This is relevant because, for example, Alan Hájek has long argued, in still unpublished but much read work, for the idea that most counterfactuals are false. (See [15].) According to him, the indeterminism and indeterminacy associated with most counterfactuals entail their falsehood. Yet, counterfactual reasoning seems to play an important role in science, and ordinary speakers judge many counterfactuals that they utter to be true. Thus, M3V could be regarded as both a (zero-order) formalization of a radical version of Hájek’s ideas on the falsity of counterfactual conditionals, while also capturing the idea that some of them need to be true.⁶⁶6There are of course many ways to address Hájek’s challenge, and many of them that do not require a contradictory logic. Here we simply suggest that M3V can be taken as a formalization of a certain form of that debate. For another proposal in the connexive vicinity to address Hájek’s challenge, see [16].

Finally, Meyer and Martin wanted to provide a logic for Aristotle’s syllogistic, which was irreflexive. In their logic SI$\sim$I⁷⁷7They do not call it in that way, though. However, we simply indicate what further axiom schemas are added to the basis S, with ‘I’ standing for $A\rightarrow A$, and ‘$\sim$I’ for $\sim\!(A\rightarrow A)$., $A\rightarrow A$ was treated as a borderline case, both a fallacy with no valid instances (due to the irreflexivity of entailment) and a validity (because of the truth-preservation account of entailment), hence the validity of both $A\rightarrow A$ and $\sim\!(A\rightarrow A)$. One could explore the idea that implication or entailment are relations so demanding that no sentences can be ever in that relation, hence the validity of $\sim\!(A\rightarrow B)$. However, as in the case of SI$\sim$I, one could argue that, for theoretical simplicity, in this case, the functional approach, the truth of some instances of $A\rightarrow B$ are required as well.

We know that all what we have said is far from convincing. However, making a full case for the conceptual usefulness of M3V is beyond our aims. We merely expressed some ideas to take this logic as more than a mathematical curiosity.

The logic that we call ‘closed set logic’ was introduced algebraically in [26] and subsequently studied in [27], [28] and [29].⁸⁸8Although the ideas underlying it are older, going back at least to [22]. The first systematic treatment of that logic on its own was the proof-theoretical analysis in [14]. We focus here in the restriction to three interpretations, CSL3, defined by the following tables:

It is common wisdom that there is no conditional in CSL3. Consider a connective defined as follows:

But consider the expansion cCSL3, which adds the E-conditional, one of the 2⁴ mentioned above, to CSL3. Let us point out some of cCSL3’s main features, starting with those involving just its $\{\neg,\rightarrow_{\_}{\tiny{\textbf{E}}}\}$-fragment.

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  cCSL3 validates unrestricted Detachment.

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  All conditionals are false in cCSL3 and so $\models_{\_}{\tiny{\textbf{cCSL3}}}\neg(A\rightarrow_{\_}{\tiny{\textbf{E}}}B)$, for any $A$ and $B$.

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  It is connexive.

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  It follows that if cCSL3 is connexive and all conditionals are false in it, cCSL3 is contradictory, just as M3V. As witnesses, consider $A\rightarrow_{\_}{\tiny{\textbf{E}}}A$ and $\neg(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)$.

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  cCSL3 is ultra-Abelardian.

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  cCSL3 does not validate exactly the same centering principles as M3V. One has

  $\models_{\_}{\tiny{\textbf{cCSL3}}}\neg(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)$

  $\models_{\_}{\tiny{\textbf{cCSL3}}}(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)\rightarrow_{\_}{\tiny{\textbf{E}}}\neg(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)$

  but also

  $\not\models_{\_}{\tiny{\textbf{cCSL3}}}\neg(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)\rightarrow_{\_}{\tiny{\textbf{E}}}(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)$

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  The above implies that cCSL3 also lacks the Deduction Property. In fact, every logical truth in cCSL3 entails any other logical truth, in particular, $\neg(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)\models_{\_}{\tiny{\textbf{cCSL3}}}A\rightarrow_{\_}{\tiny{\textbf{E}}}A$, yet $\not\models_{\_}{\tiny{\textbf{cCSL3}}}\neg(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)\rightarrow_{\_}{\tiny{\textbf{E}}}(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)$.

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  The invalidity of $\neg(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)\rightarrow_{\_}{\tiny{\textbf{E}}}(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)$ generalizes. Since any conditional of the form $X\rightarrow_{\_}{\tiny{\textbf{E}}}Y$ is false and any conditional of the form $\neg(W\rightarrow_{\_}{\tiny{\textbf{E}}}Z)$ is just true, it follows that no conditional of the form $\neg(W\rightarrow_{\_}{\tiny{\textbf{E}}}Z)\rightarrow_{\_}{\tiny{\textbf{E}}}(X\rightarrow_{\_}{\tiny{\textbf{E}}}Y)$ is valid.¹⁰¹⁰10And the validity of $(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)\rightarrow_{\_}{\tiny{\textbf{E}}}\neg(A\rightarrow_{\_}{\tiny{\textbf{E}}}A)$ also generalizes: every conditional of the form $(X\rightarrow_{\_}{\tiny{\textbf{E}}}Y)\rightarrow_{\_}{\tiny{\textbf{E}}}\neg(W\rightarrow_{\_}{\tiny{\textbf{E}}}Z)$ is valid.

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  It is clear now that cCSL3 and M3V validate different arguments. As another witness, consider $A\models_{\_}{\textbf{M3V}}\sim\sim\!A$ but $A\not\models_{\_}{\textbf{cCSL3}}\neg\neg A$.

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  cCSL3 is not hyper-connexive. The argument is as for M3V.

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  cCSL3 is nexive, just as M3V. And like M3V, it is not hyper-nexive. Again, the proof is an adaptation of the proof that M3V is not hyper-connexive.

A natural question at this point is whether $\neg$ is definable in M3V. It is not. It could be defined as $\sim\!\circ(A\rightarrow_{\_}{\tiny{\textbf{E}}}\sim\!\circ\circ\!A)$, with $\circ$ a consistency connective:

The list of properties above does not highlight enough some features of cCSL3, especially around connexive principles:

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  $\neg(A\rightarrow_{\_}{\tiny{\textbf{E}}}B)$ is just true in all interpretations in cCSL3; $\sim\!(A\rightarrow_{\_}{\tiny{\textbf{E}}}B)$ is true in all interpretations in M3V, but it is also false under some of them. This has consequences for the connexive principles, as we will see.

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  Recall that, in M3V, Aristotle’s Theses are true under all interpretations, although there are some interpretations under which they are also false. That is not the case in cCSL3: Aristotle’s Theses are just true.

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  Boethius’ Theses are true under all interpretations in M3V, although they are also false under all interpretations. That is the case as well in cCSL3, with the difference that in this logic, the negations of Boethius’ Theses are just true.

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  Both $(A\wedge\sim\!A)\rightarrow_{\_}{\tiny{\textbf{E}}}A$ and $\sim\!((A\wedge\sim\!A)\rightarrow_{\_}{\tiny{\textbf{E}}}A)$ are valid in M3V, they are both true and false in all interpretations. But although both $(A\wedge\neg A)\rightarrow_{\_}{\tiny{\textbf{E}}}A$ and $\neg((A\wedge\neg A)\rightarrow_{\_}{\tiny{\textbf{E}}}A)$ are valid in cCSL3, the latter is just true in all interpretations.

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  More generally: If both $X$ and $\sim\!X$ are valid in M3V, then $\neg X$ is just true in cCSL3, unlike $\sim\!X$ in M3V, even if $X$ fails to be valid in cCSL3. (The proof is straightforward. For schemas exemplifying this, recall the ones for the failure of the Deduction Property.)

Finally, an attractive feature of the E-conditional should be mentioned: unlike some well-known connexive conditionals in the literature, it is stable under changes of negation. Let us make that more precise.

Let a standard negation be a unary connective $N$ satisfying that $\sigma(NA)=\{1\}$ if $\sigma(A)=\{0\}$, and $\sigma(NA)=\{0\}$ if $\sigma(A)=\{1\}$. If a standard negation $N$ is such that, in a logic L, $A,NA\not\models_{\_}{\tiny{\textbf{L}}}B$, we will call it a standard paraconsistent negation. Let us define the type of standard paraconsistent negations (TSPN) as the set of all such connectives definable according to a set of admissible evaluations. In the present context, TSPN only has two connectives: $\sim$ and $\neg$.

Now, let us say that a conditional $A>B$ is connexively stable with respect to TSPN iff

Note that Francez’s conditional, from [12], restricted to three admissible interpretations, i.e.

There is one more logic due partly to Mortensen, but also partly to Marcos. In [25], Mortensen introduced a logic called ‘C${}_{\_}{0.2}$’ characterized by the following tables:

Marcos [18] suggested to replace the interpretation $\{\ \}$ by the interpretation $\{1,0\}$ —or, in his original terms, to make the value 2 designated along with 1—, and put $\sim$ instead of $\neg$, to get the logic P², whose tables look like these:

Now, to get a connexive variant of this, cP², replace the P-conditional with the E-conditional, i.e. obtain a logic characterized by the following tables:

M3V and cP² coincide in the $\{\sim,\rightarrow_{\_}{\tiny{\textbf{E}}}\}$-fragment, but they differ in ways that are significant for connexive logicians. Consider the following (non-core) connexive principles:

In this paper, we took two interests of Mortensen, connexivity and certain brands of paraconsistency, and combined them into single logics. Although connexivity is at least a matter of two, negation and the conditional, the E-conditional of Mortensen’s M3V excels among other conditional in validating the connexive schemas even when combined with other (paraconsistent) negations. Also, some features of M3V are exacerbated when a different negation is used. For example, in M3V all negated conditionals are true, but also sometimes false, whereas changing the negation leads to the result that negated conditionals are just true, never false.

There are at least five ways in which this work can be continued. First, when presented in slightly different ways, a logic might exhibit more interesting features. As we mentioned, the logics CN and M3V are inter-definable; it would be worth take a look at the logics defined here with other conditionals definable on them. Second, one could enrich the languages here with consistency connectives to make a comparison with the LFIs. Third, one could try to get both negations in a single language and study the effect of that on connexive principles. Fourth, at least in the E-conditional , Mortensen suggested to couple closed set logic with different notions of logical consequence. This would allow, among other things, to discriminate between schemas that are true under all interpretations and those that are just true under all interpretations. This would give rise to “exactly true” or “non-falsity” versions of the logics above, which have been studied in the vicinity of FDE. (See for example [36] and [37], but also [9] for a discussion closer to the present context.) Speaking of that, and finally, one can move the entire discussion on top of FDE to work with more admissible interpretations. That would augment the number of logical and conceptual distinctions available to work with.