.. _L3:

********************************************
L{L3} - Łukasiewicz 3-valued Logic
********************************************

L{L3} is a three-valued logic with values V{T}, V{F}, and V{N}. It is
similar to {@K3}, but with a different Conditional.

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

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

.. module:: pytableaux.logics.l3

.. _l3-semantics:
.. _l3-model:

Semantics
=========

.. _l3-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{L3} is the singleton: { V{T} }

.. _l3-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{L3} does not have a separate `Assertion` operator, but we include a table
and rules for it, for cross-compatibility.

.. truth-tables::
  :operators: Assertion

.. _l3-predication:

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

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

.. _l3-quantification:

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

.. rubric:: Existential

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

.. rubric:: Universal

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

.. _l3-consequence:

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

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

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

.. _l3-system:

Tableaux
========

L{L3} 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:

.. _l3-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
=====

* :term:`Conditional Identity` holds: |=>| :s:`A $ A`.

References
==========

* Łukasiewicz, J. (1920). On 3-valued logic. Ruch Filozoficzny 5, 169-171.

* Łukasiewicz, J. (1957). Aristotle's Syllogistic from the Standpoint of Modern
    Formal Logic. Oxford: Clarendon Press

.. cssclass:: hidden

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