We consider two formalisations of the notion of a compositionalsemantics for a language, and find some equivalent statements in termsof substitutions. We prove a theorem stating necessary and sufficientconditions for the existence of a canonical compositional semanticsextending a given partial semantics, after discussing what features onewould want such an extension to have. The theorem involves someassumptions about semantical categories in the spirit of Husserl andTarski.

A μ-algebra is a model of a first-order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms where μx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications.Standard μ-algebras are complete meaning that their lattice reduct is a complete lattice. We prove that any nontrivial quasivariety of μ-algebras contains a μ-algebra that has no embedding into a complete μ-algebra.We then focus on modal μ-algebras, i.e. algebraic (...) models of the propositional modal μ-calculus. We prove that free modal μ-algebras satisfy a condition–reminiscent of Whitman’s condition for free lattices–which allows us to prove that modal operators are adjoints on free modal μ-algebras, least prefixed points of Σ1-operations satisfy the constructive relation μx.f=logical and operatorn≥0fn. These properties imply the following statement: the MacNeille–Dedekind completion of a free modal μ-algebra is a complete modal μ-algebra and moreover the canonical embedding preserves all the operations in the class image of the fixed point alternation hierarchy. (shrink)

This paper exhibits a general and uniform method to prove axiomatic completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ, where x occurs only positively in γ, we obtain the flat modal fixpoint language by adding to the language of polymodal logic a connective γ for each γΓ. The term γ is meant to be interpreted as the least fixed point of the functional interpretation of the term γ. We consider the following (...) problem: given Γ, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language on Kripke structures. We prove two results that solve this problem.First, let be the logic obtained from the basic polymodal by adding a Kozen–Park style fixpoint axiom and a least fixpoint rule, for each fixpoint connective γ. Provided that each indexing formula γ satisfies a certain syntactic criterion, we prove this axiom system to be complete.Second, addressing the general case, we prove the soundness and completeness of an extension of . This extension is obtained via an effective procedure that, given an indexing formula γ as input, returns a finite set of axioms and derivation rules for γ, of size bounded by the length of γ. Thus the axiom system is finite whenever Γ is finite. (shrink)