LOGICS OF GRADED CONSEQUENCE

Pelta [6]: Until now the construction of superficial many-valued logics, that is, logics with an arbitrary number (bigger than two) of truth values but always incorporating a binary consequence relation, has prevailed in investigations of logical many valuedness.

            It seems human brain does not always derive conclusions, certain to some degree, from a set of information, which are also certain to some degree, with full certainty. The prevalent prescriptions of logics do not handle that uncertainty of ‘deriving’ properly. The same concern was echoed in the lines of Parikh [5], where he mentioned [. . .] we seem to have come no closer to observationality by moving from two valued logic to real valued, fuzzy logic. A possible solution [. . .] is to use continuous valued logic not only for the object language but also for the metalanguage.” And, perhaps, Zadeh’s extended fuzzy logic [8] also could be counted as an account of the same concern.

            I shall present the formal theory of graded consequence (GCT), which is in existence from 1987 [1–4], as a general framework for the metatheory of a logic where deriving partially true conclusion from a set of partially true premises is also a matter of grade. The idea behind the notion of GCT is, given a set X of premises, whose truth/credibility are of matter of grade, and a prospective conclusion a, which is also true/believable to some extent, the process of deriving a from X, denoted by X |~ a, could also be a matter of grade. A graded consequence relation [2] is thus, a fuzzy relation (|~) from a set of all sets of formulae (P(F)) to the set of all formulae (F), satisfying

            (GC1) if a Î X then gr(X |~ a) = 1 (reflexivity/overlap),

            (GC2) if X Í Y then gr(X |~ a) £ gr(Y |~ a) (monotonicity/dilution), and

            (GC3) inf_bÎY gr(X |~ b) *_m gr(X ÈY |~ a) £ gr(X  |~ a) (cut),

where gr(X |~ a), the degree to which  a follows from X, is an element of a complete residuated lattice (L, Ù, Ù, *_m, ®_m, 0, 1). The semantic counterpart of the notion starts with a collection of fuzzy sets of formulae, say {T_i}_iÎI , which may be regarded as the initial context formed by a set of experts assigning values to the object level formulae.

            The discussion above gives an idea about the metatheory of GCT. We shall concentrate on the logic building part based on the metatheory of GCT.

            In this presentation we first present the idea of generating different logics of graded consequence, and show that the many-valued logics can be rediscovered following this scheme. Then we would try to exploit this general framework of GCT, which allows to have the flexibility of choosing different logical bases for different layers of decision making, in order to show a good connection with the key ideas of a decision support system [7], typically an interactive system between two agents, an human user and a decision making machine.

References

1. M. K. Chakraborty. Use of fuzzy set theory in introducing graded consequence in multiple valued logic. In M.M. Gupta and T. Yamakawa, editors, Fuzzy Logic in Knowledge-Based Systems, Decision and Control, pages 247–257. Elsevier Science Publishers, B.V.(North Holland), 1988.

2. M. K. Chakraborty. Graded consequence: further studies. Journal of Applied Non-Classical Logics, 5(2):127–137, 1995.

3. M. K. Chakraborty and S. Dutta. Graded consequence revisited. Fuzzy Sets and Systems, 161(14):1885–1905, 2010.

4. S. Dutta, S. Basu, and M.K. Chakraborty. Many-valued logics, fuzzy logics and graded consequence: a comparative appraisal. In K. Lodaya, editor, Proc. ICLA2013, LNCS 7750, pages 197–209. Springer, 2013.

5. R. Parikh. The problem of vague predicates. In R.S. Cohen and M. Wartofsky, editors, Language, Logic, and Method, pages 241–261. D. Ridel Publishing Company, Dordrecht, 1983.

6. C. Pelta. Wide sets, deep many-valuedness and sorites arguments. Mathware and Soft computing, 11:5–11, 2004.

7. R. Spiegel and Y. Nenh. An expert system supporting diagnosis in clinical diagnosis. In K. Morgan, J. Sanchez, C. A. Brebbia, and A Voiskounsky, editors, Human perspective in the internet society: culture, psychology and gender, pages 145–154. WIT press, 2004.

8. L.A. Zadeh. Toward extended fuzzy logic - a first step. Fuzzy Sets and Systems, 160:3175–3181, 2009.