Vagueness being an important area of investigation in many fields (e.g. logic, philosophy, computer science) nowadays, various logics have emerged during the past few decades to address the topic e.g., fuzzy logics, rough logics and the theory of graded consequence. But in most of the existing literature, be it of logic or philosophy or computer applications vagueness is thought to occur at the level of object-language only. The metalogical notions are usually considered to be crisp. There was a claim made by Pelta in [11] that many-valuedness at the metalevel had been first explicitly considered by Marraud in [10]. But this claim is incorrect. There is a series of papers viz. [2,3,4,5] dealing with the theory of gradation of metalinguistic concepts like ‘consequence’, ‘consistency’, ‘tautologihood’, ‘completeness’ and the like. A Gentzen-type axiomatization for the consequence relation was introduced in [2] as follows:

  

A graded consequence relation ~ is a fuzzy relation from P(F) to F satisfying

i.        if α Î X then gr (X  ~ α) = 1 (overlap)

ii.      if X Í Y then gr (X  ~ α) ≤ gr  (Y  ~ α)  (dilution)

and     iii.  inf gr (X ~ β) Ù gr (X ∪ Z  ~ α) ≤  gr (X  ~ α) (cut)

                  βÎZ

where P(F) is the power set of F, the set of well formed formulas, X, Y Î P(F), α, β Î F and gr(X ~ α) is an element of a Heyting algebra giving the ‘grade’ to which α is a consequence of the premise X.

The notion was further generalized in [5] by the use of the product operator ‘*’ of a residuated lattice as the and-ing operator instead of the usual lattice meet (Ù). In the same paper, the notion of graded consistency was introduced. The idea of partial consistency or consistency to a degree is frequent in everyday encounters. The axioms of graded (in)consistency are taken as:

i.        if X Í Y then INCONS (X) ≤ INCONS (Y),

ii.      INCONS (X ∪ Y) * INCONS (X ∪ {~β) ≤ INCONS (X) for any βÎ Y

and     iii.   k ≤ INCONS ({α, ~ α}) for any wff α and some k, 0<k.

where INCONS (X) is an element of the residuated lattice representing the ‘degree’ of inconsistency of the set X.

The interface between the notions viz. graded consequence and graded consistency has been investigated.

            Although in Pavelka’s seminal work [1] there are traces of these ideas, the main theme of his work being different, the above issues did not surface. Pavelka axiomatized consequence operator C in the Tarskian tradition in the following way:

i.        X Í  C(X),

ii.      X Í Y implies C(X) Í C(Y)

iii.    CC(X) = C(X)

where X is a fuzzy subset of premises and C(X) is the fuzzy subset of consequences of X.

The question, however, is

“What does the value C(X)(α) indicate? Is it the degree to which α follows from X? That is, is it a value of the process of derivation or is it a value that is assigned to α through the derivation process? C is after all a crisp function on the domain of fuzzy subsets.”

The conceptual difference of the two makes significant difference in organizing a comprehensive theory of vagueness. Those who have followed and further developed the line of thought of Pavelka e.g. Hajek [9], Novak [6], Godo et al. [9] etc. have not taken up this issue either, except perhaps Gerla [7]  who devoted a section on graded consequence in his book and Gottwald [12] who moved closer to this idea.

            A comparison between Pavelka’s approach and the approach of graded consequence and a scrutiny of the two reveal that

1)      these are not identical methods,

2)      there are confusions and arbitrariness in the usage of algebraic operators for evaluating values of meta-level expressions,

and more importantly,

3)      three levels are involved in the discourse of such logics, viz.

a)      object level, comprising of well formed formulae with terms interpreted generally but not necessarily, as vague notions;

b)     meta level I, required to talk about object level items, the predicates etc. of this level being in general vague as well;

c)      meta level II, required to talk about and/or define the notions of meta level I.

These three levels are blurred in the discourses.

           

The present tutorial is an attempt to remove these confusions by

·        clarifying the three levels,

·        using appropriate symbolisms (viz. quotation mark ‘’ for naming at level I and quotation mark ⌐ ¬ for naming at level II) to distinguish the levels,

·        identifying the predicates, functions, connectives and quantifiers required at each level,

·        probing into the interrelation, both syntactic and semantic, between the levels.

The set of axioms for graded consequence under such symbolism takes the form

i.        gr ⌐‘α’ Î ‘X’¬   ≤   gr ⌐ ‘X’  ~ ‘α’ ¬

ii.      gr ⌐‘X’ Í ‘Y’ & ‘X’  ~ ‘α’¬   ≤   gr ⌐‘Y’  ~ ‘α’¬

iii.    gr ⌐"v ( v Î ‘Z’ → ‘X’ ~ v) & ‘X ∪ Z’ ~ ‘α’¬   ≤   gr ⌐‘X’ ~ ‘α’¬

for all ‘α’, ‘X’, ‘Y’, ‘Z’.

The expressions with the mark ‘’ are terms of meta level I while expressions with the mark ⌐ ¬ are names of wffs of level I in level II.

Axioms of graded inconsistency can also be presented similarly:

i.        gr  ⌐‘X’ Í ‘Y’ & INCONS (‘X’)¬ ≤ gr ⌐INCONS (‘Y’)¬

ii.      gr ⌐INCONS (‘X∪Y’) & INCONS (‘X∪{~β}’¬ ≤ gr ⌐INCONS (‘X’)¬

for any ‘β’ Î ‘Y’

iii.    there is some k, 0<k, s.t. k ≤ gr ⌐INCONS (‘{α, ~α}’¬ for any ‘α’.

Names of (truth) values such as ⌐k¬ or ⌐0¬ are also required at meta level II but dropped for simplicity in writing the above axioms. Similar is the case for the sign ≤. The complete list of symbols used at each level shall be presented in the full text of the paper.

            Besides dispelling of above mentioned confusions, this study helps in the assessment of the two approaches viz. [1] and [2] with respect to the incorporation of degrees at the meta levels. At the metalevel II, all the notions are crisp. The study also suggests a general methodology for construction of logics with several tiers in which sentences of one tier ‘speak’ about objects of the next lower tier.

   The tutorial is planned as follows :

     Session 1  :  Introducing the notion.

     Session 2  :  Camparison with Fuzzy Logics.

    Session 3  :  Illustratrative examples with indications towards applications.