Editing Three-valued logic (section)

=== Łukasiewicz logic ===
{{further|Łukasiewicz logic}}
The Łukasiewicz Ł3 has the same tables for AND, OR, and NOT as the Kleene logic given above, but differs in its definition of implication in that "unknown implies unknown" is '''true'''. This section follows the presentation from Malinowski's chapter of the ''Handbook of the History of Logic'', vol 8.<ref>Grzegorz Malinowski, "[https://books.google.com/books?id=3TNj1ZkP3qEC&dq=%22Many-valued+Logic+and+its+Philosophy%22&pg=PA13 Many-valued Logic and its Philosophy]" in Dov M. Gabbay, John Woods (eds.) ''Handbook of the History of Logic Volume 8. The Many Valued and Nonmonotonic Turn in Logic'', Elsevier, 2009</ref>

Material implication for Łukasiewicz logic truth table is
{| style="border-spacing: 10px 0;" align="center"
|- valign="bottom"
|
{| class="wikitable" style="text-align:center;"
|+ IMP{{sub|Ł}}(A, B)
! rowspan="2" colspan="2" | A → B
! colspan="3" |B
|-
! width="25" {{no|F}}
! width="25" | U
! width="25" {{yes|T}}
|-
! rowspan="3" |A
! scope="row" {{no|F}}
| {{yes|T}}
| {{yes|T}}
| {{yes|T}}
|-
! scope="row" | U
| U
| {{yes|T}}
| {{yes|T}}
|-
! scope="row" width="25" {{yes|T}}
| {{no|F}}
| U
| {{yes|T}}
|}
|
{| class="wikitable" style="text-align:center;"
|+ IMP{{sub|Ł}}(A, B), MIN(1, 1−A+B)
! rowspan="2" colspan="2" | A → B
! colspan="3" |B
|-
! width="25" | −1
! width="25" | 0
! width="25" | +1
|-
! rowspan="3" |A
! scope="row" | −1
|  +1
|  +1
|  +1
|-
! scope="row" | 0
| 0
|  +1
| +1
|-
! scope="row" width="25" | +1
| −1
| 0
|  +1
|}
|}

In fact, using Łukasiewicz's implication and negation, the other usual connectives may be derived as:
* {{math|1=''A'' ∨ ''B'' = (''A'' → ''B'') → ''B''}}
* {{math|1=''A'' ∧ ''B'' = ¬(¬''A'' ∨ ¬ ''B'')}}
* {{math|1=''A'' ⇔ ''B'' = (''A'' → ''B'') ∧ (''B'' → ''A'')}}

It is also possible to derive a few other useful unary operators (first derived by Tarski in 1921):
{{citation needed|date=June 2021}}
* {{math|1='''M'''''A'' = ¬''A'' → ''A''}}
* {{math|1='''L'''''A'' = ¬'''M'''¬''A''}}
* {{math|1='''I'''''A'' = '''M'''''A'' ∧ ¬'''L'''''A''}}

They have the following truth tables:
{| style="border-spacing: 10px 0;" align="center"
|- valign="bottom"
|
{| class="wikitable" style="text-align:center;"
! width="25" | {{mvar|A}}
! width="25" | {{math|1=M''A''}}
|-
! scope="row" {{no|F}}
| {{no|F}}
|-
! scope="row" | U
| {{yes|T}}
|-
! scope="row" {{yes|T}}
| {{yes|T}}
|}
|
{| class="wikitable" style="text-align:center;"
! width="25" | {{mvar|A}}
! width="25" | {{math|1=L''A''}}
|-
! scope="row" {{no|F}}
| {{no|F}}
|-
! scope="row" | U
| {{no|F}}
|-
! scope="row" {{yes|T}}
| {{yes|T}}
|}
|
{| class="wikitable" style="text-align:center;"
! width="25" | {{mvar|A}}
! width="25" | {{math|1=I''A''}}
|-
! scope="row" {{no|F}}
| {{no|F}}
|-
! scope="row" | U
| {{yes|T}}
|-
! scope="row" {{yes|T}}
| {{no|F}}
|}
|}
M is read as "it is not false that..." or in the (unsuccessful) Tarski–Łukasiewicz attempt to axiomatize [[modal logic]] using a three-valued logic, "it is possible that..." L is read "it is true that..." or "it is necessary that..." Finally I is read "it is unknown that..." or "it is contingent that..."

In Łukasiewicz's Ł3 the [[designated value]] is True, meaning that only a proposition having this value everywhere is considered a [[tautology (logic)|tautology]]. For example, {{math|1=''A'' → ''A''}} and {{math|1=''A'' ↔ ''A''}} are tautologies in Ł3 and also in classical logic. Not all tautologies of classical logic lift to Ł3 "as is". For example, the [[law of excluded middle]], {{math|1=''A'' ∨ ¬''A''}}, and the [[law of non-contradiction]], {{math|1=¬(''A'' ∧ ¬''A'')}} are not tautologies in Ł3. However, using the operator {{math|1='''I'''}} defined above, it is possible to state tautologies that are their analogues:
* {{math|1=''A'' ∨ '''I'''''A'' ∨ ¬''A''}}  ([[law of excluded fourth]])
* {{math|1=¬(''A'' ∧ ¬'''I'''''A'' ∧ ¬''A'')}} ([[extended contradiction principle]]).