The value of the four values. (English) Zbl 0928.03025
Summary: In his well-known paper “How computers should think” [in: G. Ryle (ed.), Contemporary aspects of philosophy (Oriel Press), 30-56 (1977)], N. D. Belnap argues that four-valued semantics is a very suitable setting for computerized reasoning. In this paper we vindicate this thesis by showing that the logical role that the four-valued structure has among Ginsberg’s bilattices is similar to the role that the two-valued algebra has among Boolean algebras. Specifically, we provide several theorems that show that the most useful bilattice-valued logics can actually be characterized as four-valued inference relations. In addition, we compare the use of three-valued logics with the use of four-valued logics, and show that at least for the task of handling inconsistent or uncertain information, the comparison is in favor of the latter.
MSC:
| 03B53 | Paraconsistent logics |
| 68T37 | Reasoning under uncertainty in the context of artificial intelligence |
| 03B50 | Many-valued logic |
| 68T27 | Logic in artificial intelligence |
| 03G25 | Other algebras related to logic |
Keywords:
paraconsistency; multiple-valued systems; preferential logics; reasoning; Ginsberg’s bilattices; bilattice-valued logics; four-valued inference relations; three-valued logics; four-valued logics; inconsistency; uncertaintyReferences:
| [1] | Anderson, A. R.; Belnap, N. D., (Entailment, Vol. 1 (1975), Princeton University Press: Princeton University Press Princeton, NJ) · Zbl 0323.02030 |
| [2] | Arieli, O.; Avron, A., Logical bilattices and inconsistent data, (Proceedings 9th IEEE Annual Symposium on Logic in Computer Science (1994), IEEE Press), 468-476 |
| [3] | Arieli, O.; Avron, A., Reasoning with logical bilattices, J. Logic, Language and Information, 5, 1, 25-63 (1996) · Zbl 0851.03017 |
| [4] | Arieli, O.; Avron, A., Bilattices and paraconsistency, (Proceedings First World Congress on Paraconsistency (1997)) · Zbl 0994.03021 |
| [5] | O. Arieli, A. Avron, A model-theoretic approach to recover consistent data from inconsistent knowledgebases, J. Automat. Reasoning, to appear.; O. Arieli, A. Avron, A model-theoretic approach to recover consistent data from inconsistent knowledgebases, J. Automat. Reasoning, to appear. · Zbl 0922.68113 |
| [6] | Avron, A., Natural 3-valued logics: characterization and proof theory, J. Symbolic Logic, 56, 1, 276-294 (1991) · Zbl 0745.03017 |
| [7] | Avron, A., The structure of interlaced bilattices, J. Math. Structures in Computer Science, 6, 287-299 (1996) · Zbl 0856.06005 |
| [8] | Belnap, N. D., A useful four-valued logic, (Epstein, G.; Dunn, J. M., Modern Uses of Multiple-Valued Logic (1977), Reidel Publishing Company: Reidel Publishing Company Boston), 7-73 · Zbl 0424.03012 |
| [9] | Belnap, N. D., How computer should think, (Ryle, G., Contemporary Aspects of Philosophy (1977), Oriel Press), 30-56 |
| [10] | da Costa, N. C.A., On the theory of inconsistent formal systems, Notre Dame J. Formal Logic, 15, 497-510 (1974) · Zbl 0236.02022 |
| [11] | Damasio, C. V.; Pereira, L. M., A model theory for paraconsistent logic programming, (Proceedings 7th Portuguese Conference on Artificial Intelligence. Proceedings 7th Portuguese Conference on Artificial Intelligence, Lecture Notes in AI, Vol. 990 (1995), Springer: Springer Berlin), 409-418 |
| [12] | Davis, M., Note on the mathematics of nonmonotonic reasoning, Artificial Intelligence, 13, 73-80 (1980) · Zbl 0435.68075 |
| [13] | D’Ottaviano, I. L.M., The completeness and compactness of a three-valued first-order logic, Revista Colombiana de Matematicas, XIX, 1-2, 31-42 (1985) |
| [14] | Dunn, J. M., Relevant logic and entailment, (Gabbay, D.; Guenthner, F., Handbook of Philosophical Logic, III (1986)) · Zbl 0875.03051 |
| [15] | Epstein, R. L., The Semantic Foundation of Logic, (Propositional Logics, Vol. I (1990), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Netherlands) · Zbl 0699.03001 |
| [16] | Fitting, M., Negation as refutation, (Proceedings 4th Annual Symposium on Logic in Computer Science (1989), IEEE Press), 63-70 · Zbl 0716.68024 |
| [17] | Fitting, M., Bilattices in logic programming, (Epstein, G., 20th International Symposium on Multiple-Valued Logic (1990), IEEE Press), 238-246 |
| [18] | Fitting, M., Kleene’s logic, generalized, J. Logic Comput., 1, 797-810 (1990) · Zbl 0744.03025 |
| [19] | Fitting, M., Bilattices and the semantics of logic programming, J. Logic Programming, 11, 2, 91-116 (1991) · Zbl 0757.68028 |
| [20] | Fitting, M., The family of stable models, J. Logic Programming, 17, 197-225 (1993) · Zbl 0798.68096 |
| [21] | Fitting, M., Kleene’s three-valued logics and their children, Fundamenta Informaticae, 20, 113-131 (1994) · Zbl 0804.03016 |
| [22] | Font, J. M.; Moussavi, M., Note on a six-valued extension of three-valued logic, J. Appl. Non-Classical Logic, 3, 2, 173-187 (1993) · Zbl 0806.03019 |
| [23] | Freund, M.; Lehmann, D.; Morris, P., Rationality, transitivity and contraposition, Artificial Intelligence, 52, 191-203 (1991) · Zbl 0773.03019 |
| [24] | Gabbay, D. M., Theoretical foundation for nonmonotonic reasoning in expert systems, (Apt, K. P., Proceedings NATO Advanced Study Inst. on Logic and Models of Concurent Systems (1985), Springer: Springer Berlin), 439-457 · Zbl 0581.68068 |
| [25] | Gelfond, M.; Lifschitz, V., The stable model semantics for logic programming, (Kowalski, R. A.; Bowen, K. A., Proceedings 5th Logic Programming Symposium (1988), MIT Press: MIT Press Cambridge, MA), 1070-1080 |
| [26] | Ginsberg, M. L., Multi-valued logics, (Ginsberg, M. L., Readings in Nonmonotonic Reasoning (1987), Morgan Kaufmann: Morgan Kaufmann Los Altos, CA), 251-258 |
| [27] | Ginsberg, M. L., Multivalued logics: a uniform approach to reasoning in AI, Computer Intelligence, 4, 256-316 (1988) |
| [28] | Kifer, M.; Lozinskii, E. L., A logic for reasoning with inconsistency, J. Automat. Reasoning, 9, 2, 179-215 (1992) · Zbl 0807.03019 |
| [29] | Kraus, S.; Lehmann, D.; Magidor, M., Nonmonotonic reasoning, preferential models and cumulative logics, Artificial Intelligence, 44, 1-2, 167-207 (1990) · Zbl 0782.03012 |
| [30] | Lehmann, D., Plausibility logic, (Proceedings 5th Conference of Computer Science Logic (1992), Springer: Springer Berlin), 227-241 · Zbl 0819.68125 |
| [31] | Lehmann, D.; Magidor, M., What does a conditional knowledge base entail?, Artificial Intelligence, 55, 1-60 (1992) · Zbl 0762.68057 |
| [32] | Lifschitz, V., Benchmark problems for formal nonmonotonic reasoning, (Proceedings 2nd International Workshop on Nonmonotonic Reasoning (1988), Springer: Springer Berlin), 202-219 · Zbl 0675.68057 |
| [33] | Makinson, D., General theory of cumulative inference, nonmonotonic reasoning, (Reinfrank, M., Lecture Notes in AI, Vol. 346 (1989), Springer: Springer Berlin), 1-18 · Zbl 0675.03007 |
| [34] | Makinson, D., General patterns in nonmonotonic reasoning, (Gabbay, D.; Hogger, C.; Robinson, J., Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 3 (1994), Oxford Science Publishers), 35-110 · Zbl 0804.03017 |
| [35] | McCarthy, J. M., Circumscription—a form of nonmonotonic reasoning, Artificial Intelligence, 13, 27-39 (1980) · Zbl 0435.68073 |
| [36] | Pelletier, F. J., Seventy-five problems for testing automatic theorem provers, J. Automat. Reasoning, 2, 191-216 (1986) · Zbl 0642.68147 |
| [37] | Priest, G., Reasoning about truth, Artificial Intelligence, 39, 231-244 (1989) · Zbl 0694.03021 |
| [38] | Priest, G., Minimally Inconsistent LP, Studia Logica, 50, 321-331 (1991) · Zbl 0748.03017 |
| [39] | Rozoner, L. I., On interpretation of inconsistent theories, Information Sciences, 47, 243-266 (1989) · Zbl 0677.03019 |
| [40] | Schlechta, K., Nonmonotonic logics—basic concepts, results, and techniques, (Lecture Notes in AI, Vol. 1187 (1997), Springer: Springer Berlin) · Zbl 1293.03001 |
| [41] | Schoter, A., Evidential bilattice logic and lexical inferences, J. Logic, Language and Information, 5, 1, 65-105 (1996) · Zbl 0851.03008 |
| [42] | Shoham, Y., A semantical approach to nonmonotonic logics, (Ginsberg, M. L., Readings in Nonmonotonic Reasoning (1987), Morgan Kaufmann: Morgan Kaufmann Los Altos, CA), 227-249 |
| [43] | Shoham, Y., Reasoning about Change: Time and Causation from the Standpoint of Artificial Intelligence (1988), MIT Press: MIT Press Cambridge, MA |
| [44] | Subrahmanian, V. S., Mechanical proof procedures for many valued lattice-based logic programming, J. Nonclassical Logic, 7, 7-41 (1990) · Zbl 0724.03012 |
| [45] | Subrahmanian, V. S., Paraconsistent disjunctive deductive databases, (Proceedings 20th International Symposium on Multiple Logic (1990), IEEE Press), 339-345 · Zbl 0745.68042 |
| [46] | Wagner, G., Disjunctive, deductive and active knowledge bases, (Technical Report 22/94 (1994), Freien Universität Berlin), (ver 1.0) |
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