Abstract
This paper combines two classes of generalized logics, one of which is the class of weakly implicative logics introduced by Cintula and the other of which is the class of gaggle logics introduced by Dunn. For this purpose we introduce implicational tonoid logics. More precisely, we first define implicational tonoid logics in general and examine their relation to weakly implicative logics. We then provide algebraic semantics for implicational tonoid logics. Finally, we consider relational semantics, called Routley–Meyer–style semantics, for finitary those logics.
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Notes
It should be emphasized that these are not partially-ordered algebras in the standard sense which requires that the operations are preserved under the order. These are generalizations of this idea since each operation either preserves or inverts the order, and that can vary from place to place in the same operation. Thus e.g., \(a\le b\) implies \(b\rightarrow c\le a\rightarrow c\) and \(c\rightarrow a\le c\rightarrow b\). This led to the idea of a tonoid and the tonic types of its elements – see Section 2.1. Cintula’s weakly implicative logics do not have this property built into them.
Literally Galois connections involve two merely unary functions (perhaps identical). Negations with various properties can be viewed as Galois connections (see [17]), but gaggles allow for binary and, even more generally, n-ary operations. In [1], the Galois connections and residuation are generalized into “abstract residuation” and further “colligation.”
But clearly when \(\circ \) is commutative (i.e., \(a\circ b=b\circ a\)), then \(a\rightarrow b=b\leftarrow a.\)
In [1], the backward arrow and the forward arrow were respectively considered as the right and left residuals. However, following standard notations, here the forward arrow and the backward arrow are respectively considered as the right and left residuals. The last is sometimes written as \(b \rightsquigarrow c\), or something similar to avoid reversing the order.
Of course if one had a left residual instead, or as well, one would have to postulate that e is also a “left identity,” \(e\circ a=a.\)
Note that in relevance logic literature a theory including all the theorems of a logic L is called a regular theory. Here we just call such a theory an L-theory.
Note that, according to Cintula and Noguera in [5], Lindenbaum matrices have sentences as their elements, whereas Lindenbaum–Tarski matrices have equivalence classes as their elements. In this section, we follow their distinction. Note that implicational tonoid logics are algebraically complete, since implicational tonoid logics are also weakly implicative logics (see Theorem 2.10 above) and these logics are algebraically complete. Thus, in some sense, we do not have to introduce algebraic semantics for implicational tonoid logics in detail. However, so that the readers can get “the big picture,” here we describe algebraic semantics for implicational tonoid logics.
We borrow the name of labeled language from Jansana and Moraschini in [14].
We call it this because of the so-called Routley-Meyer ternary relational semantics for relevance logic. Similar semantics were produced by various authors at about the same time in the early 1970s, but this is the name that stuck, for good reasons. See [2] for a brief history.
The notation \(\pitchfork \) was first introduced with the same purpose in [1].
For all worlds \(w \in W\):
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\(v_{w}(\Box \varphi ) = 1\) if for all \(w' \in W\), if \(wRw'\), then \(v_{w'}(\varphi ) = 1\) and
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\(v_{w}(\Diamond \varphi ) = 1\) if there exists \(w' \in W\) such that \(wRw'\) and \(v_{w'}(\varphi ) = 1\).
For the notations in these evaluations, see e.g. [15].
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For all states of information \(x \in K\):
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\(v_{x}(\varphi \rightarrow \psi ) = 1\) if for all \(y, z \in K\), if Rxyz and \(v_{y}(\varphi ) = 1\), then \(v_{z}(\psi ) = 1\) and
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\(v_{w}(\varphi \circ \psi ) = 1\) if there exist \(y, z \in K\) such that Rxyz, \(v_{y}(\varphi ) = 1\), and \(v_{z}(\psi ) = 1\).
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Note that in (\(\#^{n}_{\Diamond +}\)) the index ‘\((\pm )\)’ of \(\#\) ambiguously denotes one of isotonicity \(+\) and antitonicity − of all the argument places of \(\#\) excepting its i-th one and the notation ‘\(\Vvdash \)’ ambiguously denotes one of their corresponding forcing \(\Vdash \) and non-forcing \(\not \Vdash \). Similarly for (\(\#^{n}_{\Diamond -}\)), (\(\#^{n}_{\Box +}\)), and (\(\#^{n}_{\Box -}\)).
We assume that the canonical relations \(R^{can}\)’s also preserve labeling and tonicity maps, i.e., \(R^{can}_{\Rightarrow }\) and \(R^{can}_{\#}\)’s preserve the labels and tonicities of the connectives \(\Rightarrow \) and \(\#\)’s given by the labeling and tonicity maps of the logic.
Note Schmidt theorem ensures that a logic L is finitary iff it is inductive in the sense that it is closed under unions of upwards directed families, i.e., families \(\mathcal {D} \ne \emptyset \) such that for every \(A, B \in \mathcal {D}\), there is \(C \in \mathcal {D}\) such that \(A \cup B \subseteq C\) (see Theorem 2.3.3 and Corollary 2.3.4 in [5] and Theorem 3 in [8]).
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Acknowledgements
I dedicate this paper to the late J. Michael Dunn. When I went to Indiana university as my sabbatical visit in 2017, Mike and I had a weekly academic meeting. During meetings, we planed to work together with this subject and the draft was completed when I came back to Korea in 2018. With several refinements, we finally submitted the manuscript to Logica Universalis in August 2020. Unfortunately, he passed away this April before I have the good news of its acceptance from Prof. Jean-Yves Beziau, the Editor-in-Chief of Logica Universalis, this September. As far as I know, he is one of the most warm-hearted and tender people. I will always remember Mike and our happy days in Bloomington (USA) and in Seoul and Jeonju (Korea). I miss him so much.
We wish to acknowledge the helpful comments we received from Katalin Bimbó and Petr Cintula, which certainly improved our paper. The authors also thank Jeonbuk National University and Indiana University for supporting our joint work during the first author’s sabbatical visit. This work (Yang) was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2019S1A5A2A01034874).
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Yang, E., Dunn, J.M. Implicational Tonoid Logics: Algebraic and Relational Semantics. Log. Univers. 15, 435–456 (2021). https://doi.org/10.1007/s11787-021-00288-z
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DOI: https://doi.org/10.1007/s11787-021-00288-z
Keywords
- (Implicational) tonoid logic
- Weakly implicative logic
- Algebraic semantics
- Relational semantics
- Gaggle logic