On Modal Logics Defining Jaśkowski’s D₂-Consequence

Jaśkowski’s logic D ₂ (as a set of formulae) was formulated with the help of the modal logic S5 (see Jaśkowski, Stud Soc Sci Torun I(5):57–77, 1948; Stud Soc Sci Torun I(8):171–172, 1949). In Furmanowski (Stud Log 34:39–43, 1975), Perzanowski (Rep Math Log 5:63–72, 1975), Nasieniewski and Pietruszczak (Bull Sect Logic 37(3–4):197–210, 2008) it was shown that to define D ₂ one can use normal and regular logics weaker than S5. In his paper Jaśkowski used a deducibility relation which we will denote by⊢ _{D 2} and which fulfilled the following condition: A ₁,…,A _n ⊢; _{D 2} B iff \(\ulcorner \lozenge {A}_{1}^{\bullet }\rightarrow (\ldots \rightarrow (\lozenge {A}_{n}^{\bullet }\rightarrow \lozenge {B}^{\bullet })\ldots \,)\urcorner \in \mathbf{S5}\), where (−)^∙ is a translation of discussive formulae into the modal language. We indicate the weakest normal and the weakest regular modal logic which define D ₂ -consequence.

1. 1.

   For n = 0 we inquire whether the sentence \(\mathfrak{Q}\) is valid in the discussive logic, i.e. whether the modal sentence \(\lozenge {\mathfrak{Q}}^{\bullet }\) is valid in S5.

2. 2.

   Notice that for n = 1 and any m > 0 a sentence \(\ulcorner ({\mathfrak{p}}_{1} \wedge \cdots \wedge {\mathfrak{p}}_{m}){\rightarrow }^{\mathrm{d}}\mathfrak{Q}\urcorner \) has a form (a)^d as well as a form (b)^d, for \({\mathfrak{P}}_{1} := \ulcorner {\mathfrak{p}}_{1} \wedge \cdots \wedge {\mathfrak{p}}_{m}\urcorner \). Thus, it can be treated as expressing the external point of view where only one participant is considered.

3. 3.

   In Appendix we recall some chosen basic facts and notions concerning modal logic.

4. 4.

   If the classical conjunction were considered, one would have to add the following condition: (A ∧ B)^∙ =  ⌜ A ^∙ ∧ B ^∙ ⌝ .

5. 5.

   In da Costa and Doria (1995) a similar relation was used, yet not for For^d, but for a modal language enriched with some discussive connectives. However, in this modal language the discussive conjunction was defined as follows: ⌜ (A ∧ ^​d B) ↔ ( ◊ A ∧ B) ⌝ . But, as it was proved in Ciuciura (2005), for a new transformation  − ^∗ such that (A ∧ ^​d B)^∗ =  ⌜ ◊ A ^∗ ∧ B ^∗ ⌝ , we obtain another discussive logic D ₂ ^∗ which differs from D ₂ .

6. 6.

   So notice that for the logic D ₂ we have an analogous fact to Fact 9.A.1.

7. 7.

   As it is well known, in all regular logics (and so in normal ones) the formula ⌜ ◊ ⊤ ⌝ is equivalent to the formula (D) (see Lemma 9.A.7). The smallest normal logic containing (D) (equivalently ⌜ ◊ ⊤ ⌝ ) is denoted by ‘KD’ or simply by ‘D’. We have, D ⊊ S5 ^M.

8. 8.

   For an explanation of the Lemmon code KX ₁…X _n or CX ₁…X _n see page 19.

9. 9.

   It was proved in Błaszczuk and Dziobiak (1975) that if L ∈ NS5 _◇, then L ⊆ S5.

10. 10.

    The name ‘CD45(1)’ is used in the sense of Segerberg (1971), vol. II. Notice that CD45 = KD45.

11. 11.

    We have also a proof of the following fact without the use of Lemma 9.A.9. Firstly, by Lemma 9.A.8(ii), (5c) ∈ CD4; so CN ¹ 5 _c 5(1) ⊆ CD45(1). Secondly, 5 ^◇ (1) belongs to C5 _c 5(1), so by US we have: ‘ □ ⊤ → ( ◊ □ p → □ ◊ □ p)’. Moreover, by { 5(1)}, RM, (K) and PL, we obtain: ‘ □ □ ⊤ → ( □◊□ p → □ □ p)’. So, by PL, we receive: ‘( □ □ ⊤ ∧ □ ⊤ ) → ( ◊ □ p → □ □ p)’. Hence, by (5c) and PL, we get ‘( □ □ ⊤ ∧ □ ⊤ ) → ( □ p → □ □ p)’. Hence, by (N ¹), PL and RM, we have that (4) ∈ CN ¹ 5 _c 5(1). Thus, CD45(1) ⊆ CN ¹ 5 _c 5(1), since by Lemma 9.A.8(i), (D) ∈ C5 _c .

12. 12.

    In Bull and Segerberg (1984) and Chellas and Segerberg (1996) the symbol ‘ ◊ ’ is only an abbreviation of ‘ ¬ □  ¬’. In the present paper ‘ ◊ ’ is a primary symbol, thus, we have to admit an axiom of the form (rep). Theses of this form are equivalent to the usage of ‘ ◊ ’ as the abbreviation of ‘ ¬ □  ¬’.

13. 13.

    Notice that (b) implies (a).

Authors and Affiliations

1. Department of Logic, Nicolaus Copernicus University, Toruń, Poland

   Marek Nasieniewski & Andrzej Pietruszczak

Corresponding author

Correspondence to Marek Nasieniewski .

Editors and Affiliations

1. , Department of Philosophy, University of Auckland, Arts 2, 18 Symonds Street, Auckland, 1142, New Zealand

   Koji Tanaka

2. , Department of Philosophy, University of Aberdeen, King's College, High Street, Aberdeen, AB24 3UB, United Kingdom

   Francesco Berto

3. , Department of Philosophy, Victoria University of Wellington, Murphy 518, Kelburn Parade, Wellington, New Zealand

   Edwin Mares

4. , Department of Education, University of Cagliari, Via Is Mirrionis 1, Cagliari, 09123, Italy

   Francesco Paoli

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