Editing Many-valued logic (section)

=== 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>