The present volume of the Handbook of the History of Logic brings
together two of the most important developments in 20th century
non-classical logic. These are many-valuedness and
non-monotonicity. On the one approach, in deference to vagueness,
temporal or quantum indeterminacy or reference-failure, sentences
that are classically non-bivalent are allowed as inputs and outputs
to consequence relations. Many-valued, dialetheic, fuzzy and
quantum logics are, among other things, principled attempts to
regulate the flow-through of sentences that are neither true nor
false. On the second, or non-monotonic, approach, constraints are
placed on inputs (and sometimes on outputs) of a classical
consequence relation, with a view to producing a notion of
consequence that serves in a more realistic way the requirements of
real-life inference.
Many-valued logics produce an interesting problem. Non-bivalent
inputs produce classically valid consequence statements, for any
choice of outputs. A major task of many-valued logics of all
stripes is to fashion an appropriately non-classical relation of
consequence.
The chief preoccupation of non-monotonic (and default) logicians is
how to constrain inputs and outputs of the consequence relation. In
what is called "left non-monotonicity," it is forbidden to add new
sentences to the inputs of true consequence-statements. The
restriction takes notice of the fact that new information will
sometimes override an antecedently (and reasonably) derived
consequence. In what is called "right non-monotonicity,"
limitations are imposed on outputs of the consequence relation.
Most notably, perhaps, is the requirement that the rule of
or-introduction not be givenfree sway on outputs. Also prominent is
the effort of paraconsistent logicians, both preservationist and
dialetheic, to limit the outputs of inconsistent inputs, which in
classical contexts are wholly unconstrained.
In some instances, our two themes coincide. Dialetheic logics are a
case in point. Dialetheic logics allow certain selected sentences
to have, as a third truth value, the classical values of truth and
falsity together. So such logics also admit classically
inconsistent inputs. A central task is to construct a right
non-monotonic consequence relation that allows for these
many-valued, and inconsistent, inputs.
The Many Valued and Non-Monotonic Turn in Logic is an indispensable
research tool for anyone interested in the development of logic,
including researchers, graduate and senior undergraduate students
in logic, history of logic, mathematics, history of mathematics,
computer science, AI, linguistics, cognitive science, argumentation
theory, and the history of ideas.
- Detailed and comprehensive chapters covering the entire range of
modal logic
- Contains the latest scholarly discoveries and interprative
insights that answers many questions in the field of logic
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