Abstract
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.
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- 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.
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.
In Appendix we recall some chosen basic facts and notions concerning modal logic.
- 4.
If the classical conjunction were considered, one would have to add the following condition: (A ∧ B) ∙ = ⌜ A ∙ ∧ B ∙ ⌝ .
- 5.
In da Costa and Doria (1995) a similar relation was used, yet not for Ford, 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.
So notice that for the logic D 2 we have an analogous fact to Fact 9.A.1.
- 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.
For an explanation of the Lemmon code KX 1…X n or CX 1…X n see page 19.
- 9.
It was proved in Błaszczuk and Dziobiak (1975) that if L ∈ NS5 ◇, then L ⊆ S5.
- 10.
The name ‘CD45(1)’ is used in the sense of Segerberg (1971), vol. II. Notice that CD45 = KD45.
- 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.
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.
Notice that (b) implies (a).
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Nasieniewski, M., Pietruszczak, A. (2013). On Modal Logics Defining Jaśkowski’s D2-Consequence. In: Tanaka, K., Berto, F., Mares, E., Paoli, F. (eds) Paraconsistency: Logic and Applications. Logic, Epistemology, and the Unity of Science, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4438-7_9
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