In this paper, we explore modal logics over FDE from the point of view of both the Australian plan models with the Routley star operator and the American plan models with four semantic values. We show that, in general, the Australian plan modal logics are weaker, although the addition of a frame condition allows us to prove an equivalence between the two approaches. We then consider different S5-type logics, using accessibility relations that are equivalence relations or universal relations, on the two different plans. We show that there are three different logics that result. Next, we provide sequent systems for all of the logics and show that two of the systems enjoy cut elimination. Then, we show soundness and completeness with respect to appropriate classes of models for all the logics. Finally, we discuss some features of the logics and close by highlighting two open problems.
Bjørn Kjos-Hanssen observed that some of the semantic conditions which we consider in our framework for representing and dealing with contrary-to-duties still present problems, providing counterintuitive results in some scenarios. We consider two options for dealing with these problems, presenting the main advantages of each. In essence, both options allow us to deal with contrary-to-duty scenarios.
We isolate a canonical additive quotient associated with classical propositional logic. The construction starts from canonical proof decomposition in fractional semantics: to each formula A we associate the pair (A)=((A),(A)) , where (A) and (A) count, respectively, identity and complementary top-sequents in its canonical decomposition. Equality of these pairs, A≡ _B iff (A)=(B) , induces an equivalence relation on formulas, and conjunction descends to the resulting quotient, yielding a commutative semigroup. The semigroup turns out to be free commutative on two generators. After adjoining an identity element, it becomes isomorphic to (ℕ^2,+) ; in particular, every nonzero pair of natural numbers occurs as ((A),(A)) for some classical formula. This gives a canonical normal form for each equivalence class. Along the way, the defect coordinate yields a new single-clause route to Post-completeness. At the metatheoretic level, the first-order theory of the resulting monoid reduces effectively to Presburger arithmetic, and universal affine inequalities in the two coordinates admit an explicit validity criterion. The usual fractional value is recovered by normalizing these two counts. Although the quotient is intrinsically conjunctive, its disjunctive boundary can be characterized exactly by the reduced clause spectrum. Unlike the Lindenbaum–Tarski quotient, the evidence quotient retains multiplicity information while forgetting some literal information. The defect coordinate induces a discrete Lawvere cost semantics and the associated Lawvere quasi-metric geometry.
It is shown that Omori and Kapsner’s logic AM3 presented in Studia Logica (cf. [9]) is connexively unacceptable from a certain point of view. Then, we explore all alternative 3-valued acceptable connexive logics determined by each one of three important notions of negation.
