For (finitary) deductive systems, we formulate a signature-independent abstraction of the emph{weak excluded middle law} (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a qua
sivariety $mathsf{K}$ algebraizes a deductive system $,vdash$. We prove that, in this case, if $,vdash$ has a WEML (in the general sense) then every relatively subdirectly irreducible member of $mathsf{K}$ has a greatest proper $mathsf{K}$-congruence; the converse holds if $,vdash$ has an inconsistency lemma. The result extends, in a suitable form, to all protoalgebraic logics. A super-intuitionistic logic possesses a WEML iff it extends $mathbf{KC}$. We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of $mathbf{S4}$ has a global consequence relation with a WEML iff it extends $mathbf{S4.2}$, while every axiomatic extension of $mathbf{R^t}$ with an IL has a WEML.
The logics RL, RP, and RG have been obtained by expanding Lukasiewicz logic L, product logic P, and Godel--Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining r
esults that stand in contrast to the known situation in L, P, and G. Namely, RL is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of RG we provide a base of admissible rules. We leave the problem open whether the variety of rational Godel algebras is Q-universal.
It is proved that epimorphisms are surjective in a range of varieties of residuated structures, including all varieties of Heyting or Brouwerian algebras of finite depth, and all varieties consisting of Goedel algebras, relative Stone algebras, Sugih
ara monoids or positive Sugihara monoids. This establishes the infinite deductive Beth definability property for a corresponding range of substructural logics. On the other hand, it is shown that epimorphisms need not be surjective in a locally finite variety of Heyting or Brouwerian algebras of width 2. It follows that the infinite Beth property is strictly stronger than the so-called finite Beth property, confirming a conjecture of Blok and Hoogland.
A notion of interpretation between arbitrary logics is introduced, and the poset Log of all logics ordered under interpretability is studied. It is shown that in Log infima of arbitrarily large sets exist, but binary suprema in general do not. On the
other hand, the existence of suprema of sets of equivalential logics is established. The relations between Log and the lattice of interpretability types of varieties are investigated.
Abstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic L is associated with a matrix semantics Mod*(L). This paper is a contribution to the sys
tematic study of the so-called truth sets of the matrices in Mod*(L). In particular, we show that the fact that the truth sets of Mod*(L) can be defined by means of equations with universally quantified parameters is captured by an order-theoretic property of the Leibniz operator restricted to deductive filters of L. This result was previously known for equational definability without parameters. Similarly, it was known that the truth sets of Mod*(L) are implicitly definable if and only if the Leibniz operator is injective on deductive filters of L over every algebra. However, it was an open problem whether the injectivity of the Leibniz operator transfers from the theories of L to its deductive filters over arbitrary algebras. We show that this is the case for logics expressed in a countable language, and that it need not be true in general. Finally we consider an intermediate condition on the truth sets in Mod*(L) that corresponds to the order-reflection of the Leibniz operator.
