The main goal of this entry is not only to present Peirce's accomplishments in extending both logic and logical systems, but also to explore the relation between these two novel developments.
For general discussion and study of temporalizing logical systems and properties of temporised logics, see Finger and Gabbay (1992 19966), Finger et al. (2002), Gabbay et al. (2003).
The strict implicational fragments of the logical systems S4 and S5 of modal logic are known as C4 and C5, respectively, and their Hilbert-style axiomatizations presuppose condensed detachment as their sole rule of inference.
A second type of application to logic is the merging of different types of logical systems, e.g. the formulation of systems with graded modalities.
From a technical viewpoint, there are several ways of combining models and logical systems and, in particular, of temporalising a logic: products, fusions, etc. (see the SEP entry on combining logics).
The modern approach, originally developed in the context of constructive mathematics and logic, has proved to be fruitful for the study, comparison and combination of various logical systems, such as paraconsistent, free, modal, and substructural logics.
If I am doing mathematical logic, I take great pains to work within one of those logical systems which are believed to be foolproof.
Definitions, which seem so peripheral to most logical systems, are in fact key to the potential power of Leśniewski's logic.
One of his logical systems became the basis for Ernst Schroeder's great three-volume treatise on logic, the Vorlesungen ueber die Algebra der Logik.
There's a tragedy here in applying logical systems to this very human thing.
They take a lot of time; they often require skill and perseverance, and although they are usually built around logical systems, sometimes they are not altogether fair.
