Abstract
This paper highlights the importance of a strategy for semantic analysis initiated by J. Michael Dunn, known in the literature as the “American Plan.” The key insight of the plan relies on allowing under-determined and over-determined logical valuations, which prove to be essential for a logical analysis of information structures. The main directions in the development of this fundamental idea are explained, and an implementation of the possible generalization thereof is briefly reviewed, culminating in the notion of a multi-consequence logic.
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Notes
- 1.
It should be pointed out that classical principles of bi-valence and unique-valence were occasionally criticized long before Dunn’s work, see e.g., (Łukasiewicz 1920, 1993). However, it was in fact Dunn, who not only challenged particular principles, but also initiated a ground-breaking research program (paradigm) on semantic analysis, in which abandoning certain classical principles turned out to be not a starting point, but rather the effect of more general philosophical considerations.
- 2.
Meyer and Routley (Sylvan) are inseparably associated with Australia owing to their long and fruitful service at the Australian National University, even though Meyer originally came from the United States and Routley from New Zealand.
- 3.
This definition shows that \(X_1\) represents the “falsity domain” of the corresponding propositional surrogate, while \(X_2\) is its “truth domain.” If one were to interpret the pair \((X_1,X_2)\) as a “possible world semantics,” then \(X_1\) and \(X_2\) would be the sets of “worlds” in which the corresponding proposition are false and true respectively.
- 4.
On truth-values and their importance for logic and philosophy see, e.g., (Shramko and Wansing 2014).
- 5.
By Definitions 2.1 and 2.2, I try in a way to systematize and regularize the bilattice-terminology which is sometimes incoordinate [or “not uniform,” see (Mobasher et al. 2000, p. 111)] in works by different authors. Moreover, these definitions are formulated in such a way as to enable further generalizations when it comes to trilattices and multilattices, see subsequent sections.
- 6.
Note again that in the first degree entailment systems, a consequence is standardly considered to be a relation between (single) formulas, with a usual generalization in mind to a relation between sets of formulas (so that \(\{\,A_1,\ldots ,A_m\,\}\vdash \{\,B_1,\ldots ,B_n\,\}\) can be represented by \(A_1\wedge \cdots \wedge A_m\vdash B_1\vee \cdots \vee B_n\)).
- 7.
Following a suggestion by Odintsov in (2009) I reverse here the falsity ordering as compared to its definition in (Shramko and Wansing 2005). As Odintsov remarks, such a reversion allows us to define logical connectives and entailment relations for the “truth-language” and “falsity-language” in a homomorphic and uniform way.
- 8.
In (Shramko and Wansing 2006) this result was extended to the infinite case, showing that Belnap’s strategy of generalizing the set \(\mathbf {2}=\{\,T,F\,\}\) of classical truth-values not only is coherent but stabilizes. At any stage, no matter how far it goes, the logic of the truth (non-falsity) order is again First Degree Entailment.
- 9.
Odintsov and Wansing denote their system \(\mathbf {BiCalc}\), but I prefer to retain the original label as more instructive. I also slightly modify the formulation from (Odintsov and Wansing 2015) to minimize the set of axioms and rules, and to visualize its further generalization.
- 10.
Moreover, for certain orderings it could be useful to consider combined inversion operations, so that, e.g., 23-inversion would invert simultaneously both \(\sqsubseteq _2\) and \(\sqsubseteq _3\), leaving the other partial orders untouched, but I skip this subject here, cf. (Shramko and Wansing 2006, p. 411).
- 11.
Precisely in the sense in which a generalized truth value of a “higher degree” may contain several truth values of a “lower degree” or fail to contain some (maybe all) of them.
- 12.
Interestingly, Arieli and Avron also admit the possibility that “more than one consequence relation is relevant,” thereby enabling “the use of corresponding implication connectives,” that “allow us also to express higher-order connections among those relations” (Arieli and Avron 1996, p. 44).
- 13.
For a single premiss–single conclusion case these conditions can be reformulated as follows: \(B\wedge A\vdash A\); \(B\vdash A\Rightarrow B\wedge C\vdash A\); \(B\vdash A \text{ and } C\wedge A \vdash D\Rightarrow B\wedge C\vdash D\), which obviously hold for every \(\vdash _j\) of \(\mathbf {FDE}_n^n\).
- 14.
For a comparison of defining consequence (entailment) relations through designated truth values and trough logical orderings, see e.g., (Wansing and Shramko 2008a).
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Acknowledgments
My work on this paper is a part of Marie Curie project PIRSES-GA-2012-318986 Generalizing Truth Functionality within the Seventh Framework Programme for Research funded by EU. I am grateful to Heinrich Wansing, Sergei Odintsov and Norihiro Kamide for helpful comments.
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Shramko, Y. (2016). Truth, Falsehood, Information and Beyond: The American Plan Generalized. In: Bimbó, K. (eds) J. Michael Dunn on Information Based Logics. Outstanding Contributions to Logic, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-29300-4_11
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