Axiomatizing a Minimal Discussive Logic

In the paper we analyse the problem of axiomatizing the minimal variant of discussive logic denoted as D0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {D}}_{\textsf {0}}$$\end{document}. Our aim is to give its axiomatization that would correspond to a known axiomatization of the original discussive logic D2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {D}}_{\textsf {2}}$$\end{document}. The considered system is minimal in a class of discussive logics. It is defined similarly, as Jaśkowski’s logic D2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {D}}_{\textsf {2}}$$\end{document} but with the help of the deontic normal logic D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{D}$$\end{document}. Although we focus on the smallest discussive logic and its correspondence to D2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {D}}_{\textsf {2}}$$\end{document}, we analyse to some extent also its formal aspects, in particular its behaviour with respect to rules that hold for classical logic. In the paper we propose a deductive system for the above recalled discussive logic. While formulating this system, we apply a method of Newton da Costa and Lech Dubikajtis—a modified version of Jerzy Kotas’s method used to axiomatize D2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {D}}_{\textsf {2}}$$\end{document}. Basically the difference manifests in the result—in the case of da Costa and Dubikajtis, the resulting axiomatization is pure modus ponens-style. In the case of D0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {D}}_{\textsf {0}}$$\end{document}, we have to use some rules, but they are mostly needed to express some aspects of positive logic. D0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {D}}_{\textsf {0}}$$\end{document} understood as a set of theses is contained in D2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {D}}_{\textsf {2}}$$\end{document}. Additionally, any non-trivial discussive logic expressed by means of Jaśkowski’s model of discussion, applied to any regular modal logic of discussion, contains D0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {D}}_{\textsf {0}}$$\end{document}.


Introduction
Stanis law Jaśkowski proposed a model presenting an analysis of inconsistent theories within some consistent framework.A certain model of discussion was used since it permits intuitively present theories containing inconsistent statements.Moreover, acting on such an inconsistent basis no one would conclude from the information collected during the conversation, that everything follows; in other words, debaters can formulate pairwise contradictory theses, but none of the observers would agree that everything follows from such a discussion. 1 The role of an external observer in Jaśkowski's intuitive model consists in rendering a discussion assertion, which relies on preceding each thesis of the system with a proviso: "according to the view of one of the participants in the discussion" or "with a certain acceptable meaning of the words used".In Jaśkowski's motivation, such a role could be played by an impartial mediator who might understand this way the theses of particular participants in the discussion.So, from an intuitive point of view, the external observer should rather not be a discussant simultaneously.The socalled discussive connectives of conjunction and implication represent some aspects of communication acts holding between participants.
We will assume the reader's familiarity with the basic notion connected with normal modal logics.

Variants of Discussive Logics
Jaśkowski considered a discussion in which every one of two debaters can respond to each other.It can be treated as a simulation of the full accessibility relation determining the modal logic S5. 2 While defining the discussive logic D 2 you do not need all theses of S5 (see [5,[18][19][20]24]). On the other hand it is not the case that using any normal modal logic one obtains the very same D 2 .
From the intuitive point of view Jaśkowski's logic could be seen as 'democratic' in the sense that everyone is allowed to formulate a statement and everyone can respond to any statement made by anyone else.However, as we know from everyday experience, the situation may be different.One can observe that in some cases not everyone is in position to react to each statement.Such situations can be connected, for example, with some charisma of particular debaters.Hence, using the language of Kripke-style semantics, one can say that participants of a discussion are usually connected by some accessibility relation that determines which persons are in a position to react to the statements of a given debater.In this context, one can ask whether each accessibility relation is suitable to represent this aspect of discussion within this more general variant of Jaśkowski's model.The answer is obvious-not every accessibility relation.As an extremum one can indicate the empty relation that would not lead to any reactions.In the context of Jaśkowski's way of describing the current stage of a discussion by means of the point of view of the external observer who treats voices of a discussion as possible, it is also obvious that members of a discussive group who have the empty set of alternatives, i.e. as one could say, people whose opinions are not taken into account by others while formulating their own statements, are not included in the outcome of the discussion.Moreover, since we are interested in stating at the end what follows logically, we should be able to vary the considered point of view, so everyone who is meant to be a debater in the given discussion should be connected to some other participant.Semantically this means that we have to consider discussive groups in which everyone is connected with at least one member (the case of self-connection is not excluded).So seriality is the most general and weakest stipulation for groups of debaters meant to intuitively represent models used to semantically express the considered smallest variant of discussive logic.
In what follows we keep Jaśkowski's original understanding of discussive connectives of implication and conjunction.Jaśkowski's discussive implication → d is meant intuitively as saying: "if anyone states that p, then q" (see [7, p. 150, 1969], [8]), in the modal language: ♦p → q.Nowadays, discussive conjunction added to discussive language later in 1949 ( [9]) is usually treated as saying "p and someone said q" and in the modal language is translated as p ∧ ♦q.In the case of D 2 , the possibility operator meant to give an interpretation of discussive connectives in the modal language, intuitively is used to express statements presented by some participant during the discussion.In our interpretation we would like to rely on the relation that 'connects' debaters.Possibility is used by Jaśkowski also to simulate an evaluation made by an 'impartial arbiter'.Hence, according to Jaśkowski, theses of the discussive system have to be preceded by the stipulation: "if a thesis is recorded in a discursive system, its intuitive sense ought to be interpreted so as if it were preceded by the symbol Pos" ([7, p. 149, 1969], [8]), where 'Pos' was used to denote the possibility operator.
Preserving the described way of formulating discussive logic, one can define discussive logic by taking any normal modal logic or even any modal logic.Formulas of the discussive language For d become theses of some specific discussive system, if their translation into modal logic is a possible truth of the underlying modal system.In some cases, one can obtain even the empty set of theses, if a given modal logic does not have respective theses that could act as results of the translations used to define discussive logic.As an obvious example one can consider the logic K and try to define discussive logic on its basis.Since K has no thesis of the form ♦A, the resulting discussive logic would be the empty set.So, the necessary condition for a normal modal logic to be used to define a nontrivial discussive logic is to have at least one thesis of the form ♦A. But by monotonicity, such a logic contains the axiom (D): ♦(p → p).As one can easily see the same holds for regular modal logics.In particular, taking into account the standard semantics for regular logics, where frames with non-normal worlds are considered, one can observe that for any thesis ♦A of the normal deontic logic D, ♦A also belongs to D2 3 -the smallest regular logic containing the axiom (D).So, the intended discussive logic D 0 defined on the basis of the logic D (equivalently on the basis of D2) can be seen as the intersection of the family of all non-trivial discussive logics defined by means of normal or regular modal logics using Jaśkowski's translations.More precisely, D 0 is minimal in the following sense: consider any normal or regular modal logic L and define Jaśkowski's discussive logic obtained on the basis of L exactly in the same way as D 2 is defined on the basis of S5 (of course, with some fixed translations for discussive connectives, in our case with right discussive conjunction and discussive implication).If one sticks only to non-empty resulting discussive logics (as we mentioned, starting with K, the resulting discussive logic would be just the trivial logic-the empty set), D 0 would be contained in every such discussive logic or in other words D 0 would be the intersection of all these discussive logics.In the language of D 0 we take ¬, ∨, ∧ d , → d , ↔ d , however, if we would restrict the language, for example, to ¬, ∨, → d , we would obtain a logic-let us call it-D − 0 .Thus D − 0 would be even a weaker than D 0 logic, but due to the reduction of the language.
From this follows that D 0 is really the smallest logic in the class of all nonempty discussive logics defined on the basis of regular logics (notice that the family of normal modal logics is contained in the family of all regular logics).But taking into account the fact that weaker modal logics can have the same modal theses as a given normal logic, one could suspect that the minimality of D 0 can be saved also for some bigger classes of modal logics.However, this question could be the matter of some further research.Summarising the introduction, we would like to stress that the postulated minimality refers to the very specific class of logics, obtainable by a natural generalisation of the model of discussion developed by Jaśkowski.By taking other possible explications of 'discussiveness' expressed possibly by a modified model of discussion, one could obtain other 'discussive-like' minimal logics.
The object language of discussive logic is built out of propositional letters, truth-value operators '¬' and '∨', discussive implication (→ d ) and discussive conjunction (∧ d ) For d denotes the set of all discussive formulas, while letters A, B, C, etc. range over For d .
Let us recall that in Jaśkowski's intuitive model of D 2 every two debaters are in connection, while the antecedent of discussive implication is interpreted as saying: 'if anyone states that p'.The similar modal operation is applied to the whole formula, i.e. the possibility functor before the whole formula is meant as a kind of tool used by an external observer who acts as a judge.This external observer adjudicates the validity of a formula by referring to a given discussive group.Of course, since we are interested in setting the logical truth, we have to consider any discussive group treated as a model.Thus, formally, Jaśkowski's discussive logic D 2 is definable by means of S5 as follows: where i 1 is a translation of the discussive language to the modal one, in particular i 1 is a function from For d to For m , where we stipulate: We continue an investigation on the system D 0 given in [13] by the definition: The set D 0 is a logic: The set D 0 is closed under substitution and modus ponens with respect to → d .

Axiomatization of the Smallest Discussive Logic
As it was observed in [13], various classical theses fail to belong to D 0 .Also standard-for classical logic-inferences are not saved in the case of D 0 : To see this it is enough to take A := p, B := (p ∨ ¬p) ∧ d p and . These circumstances can be given as an explanation for the specific form of the given axiom system.Let us recall a result of the adaption of Kotas' method [11] (used by him to determine an axiomatization of D 2 ) that was used to axiomatize D 0 (see [13]).
Let Φ be a set of modal formulas.The result of 'removing' ♦ from elements of Φ will be denoted by ♦-Φ, while let Φ4 denote the set of all formulas resulting from adding before every element of Φ.Thus, we use the following notation: By definitions for any normal logic S ⊇ D: It is known that (see [24, p. 68]): For the sake of legibility let us denote for any A, B ∈ For d (resp.ϕ, ψ ∈ For m ), the formula ¬A ∨ B as A → c B (resp.¬ϕ ∨ ψ as ϕ → c ψ ).
Consider the following Frege-Lukasiewicz-Hilbert axiomatization of classical propositional logic: As in the case of sets of formulas, we use a similar custom for the case of names of formulas resulting from preceding a given formula with .In this way, for example, for formulas: we have respectively: We put Ω := {( Ai) : As it is known, every normal logic has (K ♦ ) as theses: Let us recall the logic D ([13]) being a set of theses with respect to the consequence relation determined by the set Ω as axioms, the substitution rule and the following ones: We know that: We recall an axiomatization of D 2 given in [21].It is indirectly an adaptation of an axiomatization given in [4] (which is further modified in [1]) and directly a correction of an axiomatization proposed in [3].The axiomatization given in [4] refers to a variant of discussive logic with left discussive conjunction.Non-adequacy of this axiomatization with respect to the original D 2 was observed by Ciuciura [3]. 5e recall the final axiomatization of D 2 given in [21].
where the only rule of inference is modus ponens (MP Standardly, as is the case of the usual associativity, using (D 13 ), (D  Next, similarly as in the case of the proof of (D 11 ) we get the announced result.
The proofs given above are minorly adapted versions of the proofs given in [1].We will use the idea of the first proof also in the case of D 0 for (D 0 3) on page 11.
For any formula A we use the following shortcuts: In particular, p and ⊥ p denote p → c p and ¬(p → c p), respectively.
Definition 5. Let D 0 be the set of theses with respect to an inference system determined by the set of axiom schemes: with the following rules: The proofs presented below are partially adapted proofs given in [4, Part II], but some other had to be done independently, given the very weak tools available in the considered system.
First, let us observe that the following rules are provable: One can see that (∧ d To obtain the second inference, it is enough to apply (Tr ax 1 ), (MP → d ) and For the case of (Syl ∨ ) we give the following inference: 1 and ( The below formula (D 0 1) will be used in the very same Lemma 6, but also in lemmas 7, 8 and 10.The formula (D 0 2) will be used to obtain (D 0 3) but also applied in Lemma 9.The formulas (D 0 3) and (D 0 4) will be needed in the proof of Lemma 8. Lemma 6.The following discussive formulas are theses of D 0 .
Ad (D 0 2) 1, 4 and (Tr − ) The next lemma provides some inferable rules.In particular, (⊥ − p ) will be used in the proof of Lemma 8, while (Weak ⊥ ) and (Weak → d ) in the proof of Lemma 9.

Lemma 7. The following rules are inferable on the basis of :
A Proof.Consider the following inferences.
Lemma 8.The following discussive formulas are theses of D 0 : and

¬(¬¬(A ∨ ¬B) ∨ B)
The below lemma will indicate provability of rules that correspond to rules of ♦-D in the axiomatization provided for Lemma 3. In particular, inferability of rules ( nec tr ), ( mp tr ), ( mp tr − ) and (pos tr ⇐ ) will be used in the proof of the final Theorem 13.

Lemma 9. The following rules are inferable on the basis of :
Proof.Ad ( nec tr ).
The translation i 2 : For m −→ For d given below is used in [4], another translation also denoted by i 2 was considered in [11].We refer to the axiomatization that arose from [4], thus, we also follow the respective translation.
1 i 2 (a) = a, for any propositional letter a, 2 for any ϕ, ψ ∈ For m : We will need the following lemma that will be used in the proof of Lemma 11: then
For the case of ( 14) consider the following sequence.
And finally for the case of ( 16) we have: The next Lemma will be used in the proof of Theorem 13.
Lemma 11.For every A ∈ For d the following formulas are theses of D 0 : Proof.The proof goes by induction on the complexity of a formula.The case of variables: by the definitions of i 1 and i 2 , i 2 (i 1 (a)) = a, hence due to Lemma 8 stating that (D 0 5): The case of negation: by the definitions of i 1 and i 2 , By the inductive hypothesis, we have both ¬(¬i 6), the inductive hypothesis and (Tr − ).Similarly using the other inductive hypothesis we infer The case of conjunction.By the definitions of i 1 and i 2 , By the inductive hypothesis we also have ¬(¬i Hence the required conditions hold by ( 12) and ( 13) given in Lemma 10.
The case of implication.By the definitions of i 2 and i 1 we obtain: Hence the required conditions hold by ( 14) and ( 15).The case of disjunction.By the definitions of i 2 and i 1 we have: Thus, the required fact follows by Lemma 10( 16).
Proof.First we will prove that each of the axioms belongs to D 0 .So, for a given axiom A we show that A ∈ D 0 .By the condition (4) on page 6 it is enough to show that i 1 (A) ∈ D.
To make notations shorter, i.e. to avoid the usage of values of the function i 1 , we will consider specific formulas, but not formula schemas.
Second, now we observe that each of the primitive rules leads from theses to theses of D 0 .
Taking into account that D 0 can be defined semantically, we could transfer the following theorem into a completeness or adequacy theorem for D 0 .To be more strict, applying the condition defining D 0 on page 5, using standard Kripke-style semantics for the normal modal logic D and straightforward semantical reading of conditions defining the translation i 1 , one could treat definition (def D 0 ) in semantic manners.
Theorem 13.For every thesis A ∈ D 0 there is a proof on the basis of , i.e.D 0 ⊆ D 0 .
Proof.Let us consider a formula A ∈ D 0 .By definition (def D 0 ), ♦i 1 (A) ∈ D and by Lemma 2, it is equivalent to the fact that i 1 (A) ∈ D. By Lemma 3 there is a proof of the formula i 1 (A) ∈ D on the basis of the system D .Consider a respective proof ϕ 1 , . . ., ϕ k = i 1 (A) ∈ D. Now, let us consider the sequence of values of the function i 2 of elements of the initial sequence: Observe that i 2 ( Ai) = (Ai tr ) for 1 i 12, but by Lemma 8, for every 1 i 12, (Ai tr ) is a thesis of D 0 .Next one can see that i 2 ( df ♦) = (D 0 15), i 2 ( D) = (D 0 14), i 2 ( K) = (D 0 16).Moreover, the translation of every rule among ( nec), ( mp), ( mp − ) and (pos ⇐ ) gives respectively ( nec tr ), ( mp tr ), ( mp tr − ) and (pos tr ⇐ ), but by Lemma 9, these rules are inferable for the considered system D 0 .So, by induction on the length of the proof we see that each element in the sequence (i 2 (ϕ i )) 1 i k is a thesis of D 0 .In particular, for i = k, we have (i 2 (i 1 (A))) ∈ D 0 , but by Lemma 11, we have ¬(¬i 2 (i 1 (A)) ∨ A) → d ⊥ p ∈ D 0 , so by ( mp tr − ), we conclude that A ∈ D 0 .

Towards the Embedding Procedure
The reason to base the system on the rules directly relying on discussive connectives is to be close to the formulation of D 2 with modus ponens for discussive as the only rule of inference.Below we will indicate other syntactic analogies, in particular between axioms from the axiomatizations of D 0 and D 2 .
Although only one axiom schema (D 13 ) from D 2 -axiomatization, given in [21] is D 0 -valid (D 0 13): ¬(A ∨ ¬A) → d B, other analogies are evident there.To explicate these analogies, let us denote a discussive formula of the form p ∧ d A as (A) ♦ d Using this shortcut we can rewrite the following axioms of the considered axiomatization of D 0 : The axiom: serves as an additional variant of the axiom (D 0 6) and also corresponds to (D 13 ) assuring it in the needed contexts.
In the context of (D 10 ), (D 11 ) and (D 15 ), a similar role is played by: which together with (D 0 11) and (D 0 14) allow to handling double negations.The axiom: naturally uses the idea of (D 18 ), however, for D 2 the order of conjuncts can be changed.Besides, corresponds in some way to (D 16 ) in the contexts of ¬ and ∨ with some additional needed transformations.In particular, due to (D 13 ) and positive logic, one can use as the consequent of (D 16) also ((¬B On the other hand, the negated formula in the antecedent of (D 0 18) corresponds to the formula A ∨ (¬B → d C)) (= F ), while one can see that on the basis of the axiomatic system of D 2 , formulas E and F are equivalent in the sense that (¬E ∨ F ) → d ⊥ p is inferable on the basis of D 2 . 6 The case of ), however due to limitations of D 0 , this proof cannot be conducted. 7 There are axioms whose content in D 2 is covered by positive logic, in particular the form of (D 0 10) and is connected with 'the cost' of modalities involved in discussive functors, so in its antecedent the classical formulation of conjunction has been used.
The axiom: Of course, what we are presenting here is not a formal proof of the formula (D018) on the basis of D2 but only some intuitions that show a kind of correspondence between the considered axioms.The full version of the proof of ¬(((¬⊥p on the basis of D2 requires quite few applications of axiom (D16).
7 Notice that due to Theorem 12 and definition of D0, as well as the fact that none of values of the function i1 at (D20), (D18), (D19) is a thesis of D, none of these formulas is a thesis of D0.And there are also other schemas, which correspond to positive logic: As regards the rules, their role is either to directly obtain positive inferences or-as in the case of (Tr ax 4 )-to simulate the use of positive logic.Interestingly, the rule (Add ∧ c ) is not valid for D 2 in general, but in the used context, needed cases are also legitimate for D 2 .
Hence, despite the weakness of discussive implication in D 0 observed on page 6,8 there are → d -theorems of D 2 that are provable on the basis of the given axiomatization of D 0 , or saying more, since D 0 is contained in D 2 , and due to the above-mentioned analogies, at least some proofs conducted on the basis of this axiomatization can be transferred into an inference on the basis of the axiomatization of D 0 -as an example, one can mention the proof of (D 0 3).
On the basis of D 2 the formula: , (D 11 ), positive logic and the thesis A ∨ ¬A.Again the proof cannot be repeated due to the weak part of positive logic that is valid for D 0 .
As we mentioned, one of the reasons and the aim was also to identify the smallest part which is in the same language as D 2 formalized in the language with right discussive conjunction, since such a language is nowadays treated as the intended one by Jaśkowski after an amendment presented by him in 1949.The aim of the current paper seemed to us to give an axiomatization that would correspond (in some way) to the axiomatization of D 2 that is given by the correction in [21] of the proposal in [3].Although, this 'correspondence' has not been defined, the presented in this section syntactic similarities between the axiomatisation of D 0 and axiomatisation of D 2 is proposed by us as a small justification for postulating a kind of a correlation between both systems, including the role of rules of D 0 needed to express in a way the behaviour of the positive part of classical logic.Notice, for example, that although p → d p is not a thesis of D 0 , p → d (p ∨ ¬p) ∧ d p = p → d (p) ♦ d is.
Since D 0 is a proper subsystem of D 2 , a natural question concerning the relationships between these two systems arises.Specifically, can we manage to embed D 2 into D 0 or is this impossible in principle?Having at hand axiomatizations of the above-mentioned systems we can try to address the problem.
First of all, in spite of the mentioned similarities between the two axiomatizations appearing in logical forms of axiom schemas, we cannot directly reuse any of D 2 -schemas in axiomatizations of D 0 .To see why this happens we shall extensively use modal counterparts of the systems.For instance, A → d (B → d A) is not a D 0 -thesis, because its translation into the modal language, ♦i 1 (A) → (♦i 1 (B) → i 1 (A)), is not a thesis of the modal system ♦-D, since the scheme ♦A → (♦B → A) is not a valid schema on the basis of D. 9 Fortunately, ♦-D = D, so it is convenient to use the existing proof-theoretical tools for D to check D 0 related facts.
It appears that if we add the constant p to the consequent of the above schema, which results in A → d p ∧ d (B → d A), we obtain an expression which is still not too far away from the original form but fits better for the purposes of the axiomatization of D 0 , since its translation, ♦A → p ∧ ♦(♦B → A), is a thesis of D. We can rewrite it in an equivalent form ♦A → ♦(♦B → A) to see that the point here is the "compensation" of the presence of a diamond in the antecedent in front of A.
But the situation can be slightly more complicated.However, we can observe that using the discussive translation of the modal scheme A ∧ d B → d B into the modal language, together with the application of simplifications valid for S5, we obtain (♦A ∧ ♦B) → ♦B.And this formula is also a thesis of D. can see that the usage of the mentioned simplifications of modal formulas can be repeated in the general way which leads to a transformation of a given discussive formula into a modal version without iterated modalities.Moreover, having modal formulas without iterated modalities we could use the results on relations between sets of theses without iterated modalities of subsystems of S5 [22].That's a guiding idea.
Following e.g., [22] we say that a modal formula involves iterated modalities iff some instance of ' ' or '♦' occurs within the scope of some other instance of ' ' or '♦'.We say A ∈ For m is at most of the first-degree10 iff it either does not contain any modal operator or contains a modal operator, but does not involve iterated modalities.Let 1 For m be the set of all at most the first-degree formulas.
A formula in For m is said to be in Modal Conjunctive Normal Form iff it is a conjunction (possibly degenerated), each conjunct of which is a disjunction (possibly degenerated) of classical formulas or formulas of the form α i , for some natural number i or a formula ♦α, where α i and α are classical formulas (see, e.g., [6]).Let MCNF be the set of all such formulas.
It is a well known fact that for any ϕ ∈ For m there is ϕ ∈ 1 For m such that ϕ ↔ ϕ ∈ S5 (see [6, p. 98]).One can easily see the same result holds for KD45.Although the above-mentioned ϕ is not determined uniquely, taking into account that all these formulas are equivalent on the basis of KD45, we can assume that under some order on the set For m , we can take the earliest respective formula under the given order.So, for any ϕ, let the above-described formula in 1 For m be denoted as m(ϕ).
As it is known, to define D 2 , one can use any modal logic which has the same theses beginning with '♦' as S5.Let S5 be the set of all modal logics such that L ∈ S5 iff ∀ A∈ ( ♦A ∈ L ⇐⇒ ♦A ∈ S5).It is known [2,15,16,24] that the logic S5 M -the smallest normal logic defining D 2 and simultaneously the smallest normal logic in S5 , is the smallest normal logic containing Moreover, since for any modal logic L: if S5 M ⊆ L ⊆ S5, then L ∈ S5 , so ♦S5 = ♦KD45 (see for example [16]).On the other hand, 1 KD45 = 1 D (see [22,23]) 11
in the context of ∨ and ¬.The proof of the sole formula is quite long, it requires the thesis (¬(¬A ∨ B) → d ⊥ p ) → d (A → d B) (on the basis of the axiomatization of D 2 one can prove it using (D 16 ), (D 12 ), (D 13 ), (D 10 ) and positive logic), while the postulated schema can be inferred using the formula ¬(¬(A → d B ) ∨ ¬(¬B ∧ d A)) → d ⊥ p (its proof can be obtained with the help of (D 13 ), (D 22 ), (D 15 ), (D 21 ) and positive logic).Similarly, on the basis of D 2 , the counterpart of our axiom (D 0 27) ¬(A ∨ (B ∧ d C)) → d (¬(A ∨ B) ∨ ¬(A ∨ ¬(C → d ⊥ p ))) ♦ d could be proved by using, among others, (D 20 ), (D 18 ), (D 19 corresponds to the axiom (D 9 ).speaking, it corresponds to¬(¬(A → d C ) ∨ ¬(B → d C )) → d (A ∨ B → d C )).One can easily see that by positive logic valid in D 2 it is enough on the side of D 2 to refer next to (D 9 ), additionally to the scheme ¬(¬A ∨ ¬B) → d (A ∧ d B), which can be easily obtained on the basis of the axiomatization of D 2 by (D 14 ), (D 4 )-(D 16 ), (D 11 ) and positive logic.
Consider the schemaA ∧ d B → d B from the list of axioms of D 2 .Its D 0 -analogue is A ∧ d B → d p ∧ d ( p ∧ d B).Why do we have a duplication of p now?Because i 1 (A ∧ d B → d B) = ♦(A ∧ ♦B) → B.As one can see, now the subformula B has a deeper "nested diamonds depth" in the antecedent, so we need ♦(A ∧ ♦B) → ♦♦B to convert the translation into a thesis of D.