In the past few decades, European and American logicians have attempted to provide mathematical foundations for logic and dialectic through formalisation, although logic has been related to dialectic since ancient times. Dialectical logic is a special term treated in Hegelian and Marxist thought.
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History
There have been pre-formal treatises on argument and dialectic, from authors such as Stephen Toulmin (The Uses of Argument), Nicholas Rescher (Dialectics), and van Eemeren and Grootendorst (Pragma-dialectics). One can include the communities of informal logic and paraconsistent logic.
Defeasibility
However, building on theories of defeasible reasoning (see John L. Pollock), systems have been built that define well-formedness of arguments, rules governing the process of introducing arguments based on fixed assumptions, and rules for shifting burden. Many of these logics appear in the special area of artificial intelligence and law, though the computer scientists' interest in formalizing dialectic originates in a desire to build decision support and computer-supported collaborative work systems.
Dialog games
Dialectic itself can be formalised as moves in a game, where an advocate for the truth of a proposition and an opponent argue. Such games can provide a semantics of logic, one that is very general in applicability.