Two-layer modal logics: from fuzzy logics to a general framework

Mathematical Fuzzy Logic (MFL) started as the study of logics based on left-continuous t-norms,1 most prominently Łukasiewicz logic Ł, Gödel–Dummett logic G, Product logic Π, Hájek logic BL, and the system MTL introduced by Esteva and Godo. In the last years, the scope of MFL has been progressively expanded by considering weaker logical systems characterized by their completeness with respect to a semantics of linearly algebras (such as chains endowed with a uninorm, or a non-commutative tnorm to interpret &, or other kinds of linearly ordered residuated lattices). There have been some proposals for a general framework to deal in a uniform way with this growing family of logics. The first one appeared in [12] when Hájek and Cintula introduced that classes of core and 4-core fuzzy logics, understood as axiomatic expansions of MTL and MTL4, resp. A wider framework, encompassing weaker systems and stronger expansions, has been proposed in [3] based on the notion of weakly implicative semilinear logic. In both approaches MFL retains what we can arguably see as its defining feature, namely the study of systems of non-classical logics with a semantics based on (linearly ordered) scales of degrees of truth. This is what makes these logics specially suited for the study of gradual aspects of vagueness and imprecision, found in sentences like ‘it is heavily raining’ or ‘that man is tall’. A conceptually different issue, that of uncertainty, has also been addressed inside MFL. The main idea, introduced in [13] and later developed by Hájek in his monograph [11], is that one could use probability to determine the truth degree of statements such as ‘tomorrow it will probably rain’ or ‘the probability that tomorrow it will rain is high’. Indeed, one takes classical logic and its formulae φ to describe crisp events, introduces a new modal operator P which can be applied on them to create atomic modal formulae Pφ which may be read as ‘probably φ’ (or better ‘the probability of φ is high’), and finally these atomic modal formulae are combined by using the connectives of Łukasiewicz logic. What we obtain is a two-layer modal fuzzy logic built on atomic formulae Pφ whose truth values are given by a probability measure. Several works have followed this idea with variations. In [9] Godo, Esteva and Hájek replaced Łukasiewicz logic on the second layer by ŁΠ, but kept classical logic for non-modal formulae. The logic ŁΠ, with its expanded language, enabled them to deal with conditional probability. Flaminio and Montagna also considered conditional probability in [7], and Godo and Marchioni investigated coherent conditional probabilities in [10]. Marchioni also proposed a class of logics of uncertainty in [14] with different kinds of measures (besides probability) to quantify the uncertainty of events. In all of these works classical logic has been kept as the underlying logic for non-modal formulae. However, if one wants to deal with uncertainty and vagueness at once, i.e. with the probability of vague events, as in ‘tomorrow it will probably rain heavily’, the two-layer paradigm can still be useful provided that the underlying classical logic is substituted by a fuzzy logic. This idea has been also investigated in some works, as [5] where finite Łukasiewicz systems Łn are taken as the logics of vague events. Other recent works along these lines are surveyed in [6]. There has even been a first attempt at an abstract theory of two-layer modal fuzzy logics in [15]; but it is restricted to the family of core fuzzy logics which, as argued above, has become too narrow to contain the current scope of MFL.