Editing Many-valued logic (section)

== Examples ==
{{main|Three-valued logic|Four-valued logic|Nine-valued logic}}

=== Kleene (strong) {{math|''K''<sub>3</sub>}} and Priest logic {{math|''P''<sub>3</sub>}} ===

[[Stephen Cole Kleene|Kleene]]'s "(strong) logic of indeterminacy" {{math|''K''<sub>3</sub>}} (sometimes <math>K_3^S</math>) and [[Graham Priest|Priest]]'s "logic of paradox" add a third "undefined" or "indeterminate" truth value {{math|I}}. The truth functions for [[negation]] (¬), <!--(strong)--> [[logical conjunction|conjunction]] (∧), <!--(strong)--> [[disjunction]] (∨), <!--(strong)--> [[material conditional|implication]] ({{underset|''K''|→}}), and <!--(strong)--> [[biconditional]] ({{underset|''K''|↔}}) are given by:<ref>{{harv|Gottwald|2005|p=19}}</ref>
{| cellpadding="0"
|- valign="bottom"
|
{| class="wikitable" style="text-align:center;"
|-
! width="25" | ¬
! width="25" | &nbsp;
|-
! {{math|T}} 
| {{math|F}}
|-
! {{math|I}} 
| {{math|I}}
|-
! {{math|F}} 
| {{math|T}}
|}
||
||
||
{| class="wikitable" style="text-align: center;"
|-
! width="25" | ∧
! width="25" | {{math|T}} 
! width="25" | {{math|I}} 
! width="25" | {{math|F}}
|-
! {{math|T}} 
| {{math|T}} || {{math|I}} || {{math|F}}
|-
! {{math|I}} 
| {{math|I}} || {{math|I}} || {{math|F}}
|-
! {{math|F}} 
| {{math|F}} || {{math|F}} || {{math|F}}
|}
||
||
||
{| class="wikitable" style="text-align: center;"
|-
! width="25" | ∨
! width="25" | {{math|T}} 
! width="25" | {{math|I}} 
! width="25" | {{math|F}}
|-
! {{math|T}} 
| {{math|T}} || {{math|T}} || {{math|T}}
|-
! {{math|I}} 
| {{math|T}} || {{math|I}} || {{math|I}}
|-
! {{math|F}} 
| {{math|T}} || {{math|I}} || {{math|F}}
|}
||
||
||
{| class="wikitable" style="text-align: center;"
|-
! width="25" | {{underset|''K''|→}}
! width="25" | {{math|T}} 
! width="25" | {{math|I}} 
! width="25" | {{math|F}}
|-
! {{math|T}} 
| {{math|T}} || {{math|I}} || {{math|F}}
|-
! {{math|I}} 
| {{math|T}} || {{math|I}} || {{math|I}}
|-
! {{math|F}} 
| {{math|T}} || {{math|T}} || {{math|T}}
|}
||
||
||
{| class="wikitable" style="text-align: center;"
|-
! width="25" | {{underset|''K''|↔}}
! width="25" | {{math|T}} 
! width="25" | {{math|I}} 
! width="25" | {{math|F}}
|-
! {{math|T}} 
| {{math|T}} || {{math|I}} || {{math|F}}
|-
! {{math|I}} 
| {{math|I}} || {{math|I}} || {{math|I}}
|-
! {{math|F}} 
| {{math|F}} || {{math|I}} || {{math|T}}
|}
|}

The difference between the two logics lies in how [[tautology (logic)|tautologies]] are defined. In {{math|''K''<sub>3</sub>}} only {{math|T}} is a ''designated truth value'', while in {{math|''P''<sub>3</sub>}} both {{math|T}} and {{math|I}} are (a logical formula is considered a tautology if it evaluates to a designated truth value). In Kleene's logic {{math|I}} can be interpreted as being "underdetermined", being neither true nor false, while in Priest's logic {{math|I}} can be interpreted as being "overdetermined", being both true and false. {{math|''K''<sub>3</sub>}} does not have any tautologies, while {{math|''P''<sub>3</sub>}} has the same tautologies as classical two-valued logic.<ref>{{cite book
|last= Humberstone
|first= Lloyd
|date= 2011
|title= The Connectives
|url= https://archive.org/details/connectives00humb
|url-access= limited
|location= Cambridge, Massachusetts
|publisher= The MIT Press
|pages= [https://archive.org/details/connectives00humb/page/n219 201]
|isbn= 978-0-262-01654-4
}}</ref>

=== Bochvar's internal three-valued logic ===

Another logic is Dmitry Bochvar's "internal" three-valued logic <math>B_3^I</math>, also called Kleene's weak three-valued logic. Except for negation and biconditional, its truth tables are all different from the above.<ref name="Bergmann 2008 80">{{harv|Bergmann|2008|p=80}}</ref>

{|
|
{| class="wikitable" style="text-align: center;"
|-
! width="25" | {{underset|+|∧}}
! width="25" | {{math|T}} 
! width="25" | {{math|I}} 
! width="25" | {{math|F}}
|-
! {{math|T}} 
| {{math|T}} || {{math|I}} || {{math|F}}
|-
! {{math|I}} 
| {{math|I}} || {{math|I}} || {{math|I}}
|-
! {{math|F}} 
| {{math|F}} || {{math|I}} || {{math|F}}
|}
||
||
||
{| class="wikitable" style="text-align: center;"
! width="25" | {{underset|+|∨}}
! width="25" | {{math|T}}  
! width="25" | {{math|I}} 
! width="25" | {{math|F}}
|-
! {{math|T}}  
| {{math|T}} || {{math|I}} || {{math|T}}
|-
! {{math|I}} 
| {{math|I}} || {{math|I}} || {{math|I}}
|-
! {{math|F}} 
| {{math|T}} || {{math|I}} || {{math|F}}
|}
||
||
||
{| class="wikitable" style="text-align: center;"
|-
! width="25" | {{underset|+|→}}
! width="25" | {{math|T}}  
! width="25" | {{math|I}} 
! width="25" | {{math|F}}
|-
! {{math|T}} 
| {{math|T}} || {{math|I}} || {{math|F}}
|-
! {{math|I}} 
| {{math|I}} || {{math|I}} || {{math|I}}
|-
! {{math|F}} 
| {{math|T}} || {{math|I}} || {{math|T}}
|}
|}

The intermediate truth value in Bochvar's "internal" logic can be described as "contagious" because it propagates in a formula regardless of the value of any other variable.<ref name="Bergmann 2008 80"/>

=== Belnap logic ({{math|''B''<sub>4</sub>}}) ===
[[Nuel Belnap|Belnap]]'s logic {{math|''B''<sub>4</sub>}} combines {{math|''K''<sub>3</sub>}} and {{math|''P''<sub>3</sub>}}. The overdetermined truth value is here denoted as ''B'' and the underdetermined truth value as ''N''.

{|
|
{| class="wikitable" style="text-align:center;"
|-
! width="25" | {{math|''f''<sub>&not;</sub>}}
! width="25" | &nbsp;
|-
! {{math|T}} 
| {{math|F}} 
|-
! {{math|B}} 
| {{math|B}} 
|-
! {{math|N}} 
| {{math|N}} 
|-
! {{math|F}} 
| {{math|T}} 
|}
||
||
||
{| class="wikitable" style="text-align:center;"
|-
! width="25" | {{math|''f''<sub>∧</sub>}}
! width="25" | {{math|T}} 
! width="25" | {{math|B}} 
! width="25" | {{math|N}} 
! width="25" | {{math|F}} 
|-
! {{math|T}} 
| {{math|T}} || {{math|B}} || {{math|N}} || {{math|F}} 
|-
! {{math|B}} 
| {{math|B}} || {{math|B}} || {{math|F}} || {{math|F}} 
|-
! {{math|N}} 
| {{math|N}} || {{math|F}} || {{math|N}} || {{math|F}} 
|-
! {{math|F}} 
| {{math|F}} || {{math|F}} || {{math|F}} || {{math|F}} 
|}
||
||
||
{| class="wikitable" style="text-align:center;"
|-
! width="25" | {{math|''f''<sub>&or;</sub>}}
! width="25" | {{math|T}} 
! width="25" | {{math|B}} 
! width="25" | {{math|N}} 
! width="25" | {{math|F}} 
|-
! {{math|T}} 
| {{math|T}} || {{math|T}} || {{math|T}} || {{math|T}} 
|-
! {{math|B}} 
| {{math|T}} || {{math|B}} || {{math|T}} || {{math|B}}
|-
! {{math|N}} 
| {{math|T}} || {{math|T}} || {{math|N}} || {{math|N}}
|-
! {{math|F}} 
| {{math|T}} || {{math|B}} || {{math|N}} || {{math|F}} 
|}
|}

=== Gödel logics ''G<sub>k</sub>'' and ''G''<sub>∞</sub> ===

In 1932 [[Kurt Gödel|Gödel]] defined<ref>{{cite journal
 | last = Gödel | first = Kurt
 | title = Zum intuitionistischen Aussagenkalkül
 | journal = Anzeiger der Akademie der Wissenschaften in Wien
 | date = 1932 | issue = 69 | pages = 65f
}}</ref> a family <math>G_k</math> of many-valued logics, with finitely many truth values <math>0, \tfrac{1}{k - 1}, \tfrac{2}{k - 1}, \ldots, \tfrac{k - 2}{k - 1}, 1</math>, for example <math>G_3</math> has the truth values <math>0, \tfrac{1}{2}, 1</math> and <math>G_4</math> has <math>0, \tfrac{1}{3}, \tfrac{2}{3}, 1</math>. In a similar manner he defined a logic with infinitely many truth values, <math>G_\infty</math>, in which the truth values are all the [[real number]]s in the interval <math>[0, 1]</math>. The designated truth value in these logics is 1.

The conjunction <math>\wedge</math> and the disjunction <math>\vee</math> are defined respectively as the [[minimum]] and [[maximum]] of the operands:

: <math>\begin{align}
  u \wedge v &:= \min\{u, v\} \\
    u \vee v &:= \max\{u, v\}
\end{align}</math>

Negation <math>\neg_G</math> and implication <math>\xrightarrow[G]{}</math> are defined as follows:

: <math>\begin{align}
  \neg_G u &= \begin{cases}
                1, & \text{if }u = 0 \\
                0, & \text{if }u > 0
              \end{cases} \\[3pt]
  u \mathrel{\xrightarrow[G]{}} v &= \begin{cases}
                             1, & \text{if }u \leq v \\
                             v, & \text{if }u > v
                           \end{cases}
\end{align}</math>

Gödel logics are completely axiomatisable, that is to say it is possible to define a logical calculus in which all tautologies are provable. The implication above is the unique [[Heyting implication]] defined by the fact that the suprema and minima operations form a complete lattice with an infinite distributive law, which defines a unique [[complete Heyting algebra]] structure on the lattice.

=== Łukasiewicz logics {{mvar|L<sub>v</sub>}} and {{math|''L''<sub>∞</sub>}}===

Implication <math>\xrightarrow[L]{}</math> and negation <math>\underset{L}{\neg}</math> were defined by [[Jan Łukasiewicz]] through the following functions:

: <math>\begin{align}
             \underset{L}{\neg} u &:= 1 - u \\
  u \mathrel{\xrightarrow[L]{}} v &:= \min\{1, 1 - u + v\}
\end{align}</math>

At first Łukasiewicz used these definitions in 1920 for his three-valued logic <math>L_3</math>, with truth values <math>0, \frac{1}{2}, 1</math>. In 1922 he developed a logic with infinitely many values <math>L_\infty</math>, in which the truth values spanned the real numbers in the interval <math>[0, 1]</math>. In both cases the designated truth value was 1.<ref>{{cite book
|last1= Kreiser |first1= Lothar
|last2 = Gottwald |first2 = Siegfried
|last3 = Stelzner |first3 = Werner
|date= 1990
|title= Nichtklassische Logik. Eine Einführung
|location= Berlin
|publisher= Akademie-Verlag
|pages= 41ff – 45ff
|isbn= 978-3-05-000274-3
}}</ref>

By adopting truth values defined in the same way as for Gödel logics <math>0, \tfrac{1}{v-1}, \tfrac{2}{v-1}, \ldots, \tfrac {v-2} {v-1}, 1</math>, it is possible to create a finitely-valued family of logics <math>L_v</math>, the abovementioned <math>L_\infty</math> and the logic <math>L_{\aleph_0}</math>, in which the truth values are given by the [[rational number]]s in the interval <math>[0,1]</math>. The set of tautologies in <math>L_\infty</math> and <math>L_{\aleph_0}</math> is identical.

=== Product logic {{math|Π}} ===

In product logic we have truth values in the interval <math>[0,1]</math>, a conjunction <math>\odot</math> and an implication <math>\xrightarrow [\Pi]{}</math>, defined as follows<ref>Hajek, Petr: ''Fuzzy Logic''. In: Edward N. Zalta: ''The Stanford Encyclopedia of Philosophy'', Spring 2009. ([http://plato.stanford.edu/archives/spr2009/entries/logic-fuzzy/])</ref>

: <math>\begin{align}
                          u \odot v &:= uv \\
  u \mathrel{\xrightarrow[\Pi]{}} v &:=
    \begin{cases}
                1, & \text{if } u \leq v \\
      \frac{v}{u}, & \text{if } u > v
    \end{cases}
\end{align}</math>

Additionally there is a negative designated value <math>\overline{0}</math> that denotes the concept of ''false''. Through this value it is possible to define a negation <math>\underset{\Pi}{\neg}</math> and an additional conjunction <math>\underset{\Pi}{\wedge}</math> as follows:

: <math>\begin{align}
  \underset{\Pi}{\neg} u &:=
    u \mathrel{\xrightarrow[\Pi]{}} \overline{0} \\
  u \mathbin{\underset{\Pi}{\wedge}} v &:=
    u \odot \left(u \mathrel{\xrightarrow[\Pi]{}} v\right)
\end{align}</math>

and then <math>u \mathbin{\underset{\Pi}{\wedge}} v = \min\{u, v\}</math>.

=== Post logics ''P<sub>m</sub>'' ===

In 1921 [[Emil Leon Post|Post]] defined a family of logics <math>P_m</math> with (as in <math>L_v</math> and <math>G_k</math>) the truth values <math>0, \tfrac 1 {m-1}, \tfrac 2 {m-1}, \ldots, \tfrac {m-2} {m-1}, 1</math>. Negation <math>\underset{P}{\neg}</math> and conjunction <math>\underset{P}{\wedge}</math> and disjunction <math>\underset{P}{\vee}</math>  are defined as follows:

: <math>\begin{align}
               \underset{P}{\neg} u &:= \begin{cases}
                                                            1, & \text{if } u = 0 \\
                                          u - \frac{1}{m - 1}, & \text{if } u \not= 0
                                        \end{cases} \\[6pt]
  u \mathbin{\underset{P}{\wedge}} v &:= \min\{u,v\} \\[6pt]
    u \mathbin{\underset{P}{\vee}} v &:= \max\{u,v\}
\end{align}</math>

=== Rose logics ===

In 1951, Alan Rose defined another family of logics for systems whose truth-values form [[lattice (order theory)|lattice]]s.<ref>{{cite journal|title=Systems of logic whose truth-values form lattices|journal=Mathematische Annalen|volume=123|date=December 1951|pages=152–165|doi=10.1007/BF02054946|last1=Rose|first1=Alan|s2cid=119735870}}</ref>