On interpreting truth tables and relevant truth table logic.
Richard Sylvan · Notre Dame Journal of Formal Logic · 1992
Contrary to common mythology, the two-valued truth tables do not yield classical logic.Many contestable assumptions are required to reach classical logic.Indeed some assumptions are required to get anywhere logically.In between, and in other directions, lie several other logics.For, even logically, there are many ways in which the truth tables can themselves be interpreted.In particular, they can be variously read inferentially, in one direction or two, or they may be variously read semantically.Along inferential lines, Tennant's one-way reading is reconsidered.It is argued that the tables do not lead to the logics Tennant claims to reach but can lead to various other decidedly weak logics.Along more orthodox semantical lines, it is shown how the truth tables themselves do not exclude nonclassical situations but can allow for incomplete and inconsistent set-ups.So considered, they provide the framework for a four-valued relevant logic.A four-valued implication is grafted onto this framework, simply by generalising upon twovalued material implication artifice, to deliver the familiar system FDE of tautological entailment.Finally, for comparison, a less contrived semantics than pure truth tabular, a semantics due to Dunn, which now admits of ready higher degree extension, is supplied for FDE.It is commonplace nowadays to begin logic with the truth tables, the twovalued truth tables.These have two values, symbolized (say) as 1 and 0 -value 1 for "true", "on", "holds", "yes", "yin", etc., and value 0 for "false", "off", "fails", "no", "yang", etc.It is commonplace also to assume that these tables lead immediately to classical logic.But they do not: not without the "right" interpretation; not without both a considerable background and requisite assumptions.Consider the usual tables for "connectives" &, v, and ~ in isolation (an early choice is as to basic connectives), written one of the usual ways (already condensed a little from column forms):