.. _G3:

*************************************
L{G3} - Gödel 3-valued Logic
*************************************

L{G3} is a 3-valued logic, with values V{T}, V{F}, and V{N}. It features a
classical-like negation, and {@L3}-like conditional.

.. contents:: Contents
  :local:
  :depth: 2

------------------------

.. module:: pytableaux.logics.g3

.. _g3-semantics:
.. _g3-model:

Semantics
=========

.. _g3-truth-values:

Truth Values
------------

Common labels for the values include:

.. include:: include/k3/value-table.rst

.. rubric:: Designated Values

The set of *designated values* for L{G3} is the singleton: { V{T} }

.. _g3-truth-tables:

Truth Tables
------------

.. include:: include/truth_table_blurb.rst

.. truth-tables::
  :operators: Negation, Conjunction, Disjunction, Conditional

.. rubric:: Defined Operators

.. include:: include/bicond_define.rst

.. include:: include/bicond_table.rst

.. include:: include/material_defines.rst

.. include:: include/material_tables.rst

.. rubric:: Compatibility Tables

L{G3} does not have a separate `Assertion` operator, but we include a table
and rules for it, for cross-compatibility.

.. truth-tables::
  :operators: Assertion

.. _g3-predication:

Predication
-----------

.. include:: include/k3/m.predication.rst

.. _g3-quantification:

Quantification
--------------

.. rubric:: Existential

.. include:: include/fde/m.existential.rst

.. rubric:: Universal

.. include:: include/fde/m.universal.rst

.. _g3-consequence:

Consequence
-----------

**Logical Consequence** is defined in terms of the set of *designated* values
{ V{T} }:

.. include:: include/fde/m.consequence.rst

.. _g3-system:

Tableaux
========

L{G3} tableaux are built similary to L{FDE}.

Nodes
-----

.. include:: include/fde/nodes_blurb.rst

Trunk
-----

.. include:: include/fde/trunk_blurb.rst

.. tableau::
  :build-trunk:
  :prolog:

Closure
-------

.. tableau-rules::
  :group: closure
  :titles:
  :legend:
  :doc:

.. _g3-rules:

Rules
--------

.. include:: include/fde/rules_blurb.rst

.. tableau-rules::
  :docflags:
  :group: operator
  :exclude: non_native

.. tableau-rules::
  :docflags:
  :group: quantifier

.. tableau-rules::
  :docflags:
  :title: Compatibility Rules
  :group: operator
  :include: non_native

Notes
=====

References
==========

- Rescher, Nicholas. `Many-valued Logic`_. United Kingdom, McGraw-Hill, 1969.

.. rubric:: Futher Reading

- Heyting, Arend. `Intuitionism`_: An Introduction. Netherlands, North-Holland, 1966.

.. _Many-valued Logic: https://www.google.com/books/edition/_/ZyTXAAAAMAAJ?hl=en
.. _Intuitionism: https://www.google.com/books/edition/Intuitionism/qfp_-Fo9yWMC?hl=en&gbpv=0

.. cssclass:: hidden

.. autoclass:: Rules()
  :members:
  :undoc-members: