Using known facts we give a simple characterization of the distributivity of lattices of finite length.

Even if a lattice L is not distributive, it is still possible that for particular elements x, y, z ∈ L it holds ∧z = ∨. If this is the case, we say that the triple is distributive. In this note we provide some sufficient conditions for the distributivity of a given triple.

In the paper we investigate Birkhoff’s conditions and. We prove that a discrete lattice L satisfies the condition ) if and only if L is a 4-cell lattice not containing a cover-preserving sublattice isomorphic to the lattice S*7. As a corollary we obtain a well known result of J. Jakub´ık from [6]. Furthermore, lattices S7 and S*7 are considered as so-called partially cover-preserving sublattices of a given lattice L, S7 ≪ L and S7 ≪ L, in symbols. It is shown (...) that an upper continuous lattice L satisfies if and only if L is a 4-cell lattice such that S7 ≪/ L. The final corollary is a generalization of Jakubík’s theorem for upper continuous and strongly atomic lattices. Keywords: Birkhoff’s conditions, semimodularity conditions, modular lattice, discrete lattices, upper continuous lattice, strongly atomic lattice, cover-preserving sublattice, cell, 4-cell lattice. (shrink)

In the paper we introduce two conditions and ) which are strengthenings of Birkhoff’s conditions. We prove that an upper continuous and strongly atomic lattice is distributive if and only if it satisfies and ). This result extends a theorem of R.P. Dilworth characterizing distributivity in terms of local distributivity and a theorem of M. Ward characterizing distributivity by means of covering diamonds.

A few side notes on Logic and Argumentation by Andrzej KisielewiczIn the paper we discuss selected philosophical theses presented in the book Logic and Argumentation. Practical Course in Critical Thinking by Andrzej Kisielewicz. In particular, we reflect on formal logic and practical reasoning, their merits and limitations, and we ask about a sensible compromise between the generality of the former and the usefulness of the latter.