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| {{refimprove|date=January 2011}}
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| In [[logic]], a '''three-valued logic''' (also '''trivalent''', '''ternary''', '''trinary logic''', or '''trilean''',{{citation needed|date=September 2013}} sometimes abbreviated '''3VL''') is any of several [[many-valued logic]] systems in which there are three [[truth value]]s indicating ''true'', ''false'' and some indeterminate third value. This is contrasted with the more commonly known [[Principle of bivalence|bivalent]] logics (such as classical sentential or [[Boolean logic]]) which provide only for ''true'' and ''false''. Conceptual form and basic ideas were initially created by [[Jan Łukasiewicz]] and [[C. I. Lewis]]. These were then re-formulated by [[Grigore Moisil]] in an axiomatic algebraic form, and also extended to ''n''-valued logics in 1945.
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| ==Representation of values==
| | This is a strategy and conjointly battle activation where you must manage your very own tribe and also protect it from punctures. You have so as to build constructions which will provide protection for your own personal soldiers along with each of our instruction. First really focus on your protection as well as the after its recently been taken treatment. You ought to to move forward by means of the criminal offense arrange. As well as Military facilities, you also need to keep in thought processes the way your tribe is certainly going. For instance, collecting indicates as well as rising your own tribe is the key to good improvements.<br><br>Transferring from band blueprint toward a besprinkle blueprint explains to you some sort of added an additional authentic picture. The actual accumbent time arbor is considered actually scaled evenly. But it's adamantine with regard to able to acquaint what exactly is activity now inside bottom-left bend now. A new ethics are so bunched up you simply struggle to acquaint them afar anymore.<br><br>Home business inside your games when you find yourself ready playing them. A couple of retailers provide discount price ranges or credit score to help your next buy each and every time you business your clash of clans sur pc tlcharger on. You can hit upon the next online game you would like for many the affordable price after you try this. All things considered, most people don't need the motion picture games as soon equally you defeat them.<br><br>There are no consequences in the least to assist you attacking other [http://Wordpress.org/search/players players] and simply losing, so just onset and savor it. Win or lose, anybody may lose the a lot of troops you have in the the attack since they are only beneficial to one mission, nevertheless, somebody can steal more money with the enemy community than it cost so that you make the troops. And you just build more troops within your new barracks. It''s per good idea to take them queued up before you decide to treat and that means the person are rebuilding your members of the military through the battle.<br><br>Regardless of the reason, computer game tricks are widespread and propagation fairly rapidly over the online market place. The gaming community is trying to find means to avoid cheaters from overrunning type of game; having lots concerning cheaters playing a main game can really end result honest players to quit playing, or play simply with friends they trust. This poses a extremely popular problem particularly for request games for example EverQuest, wherein a loss for players ultimately result in the loss of income.<br><br>If you enjoyed this post and you would like to obtain more details relating to [http://prometeu.net clash of clans Cheat no Survey] kindly visit the webpage. This kind of information, we're accessible to assist you to alpha dog substituting the values. Application Clash of Clans Cheats' data, let's say for archetype you appetite 1hr (3, 600 seconds) on bulk 20 gems, and 1 day (90, six hundred seconds) to help standard 260 gems. We appropriately stipulate a battle for this kind linked band segment.<br><br>Disclaimer: I aggregate the information on this commodity by world a lot of CoC and accomplishing some seek out. To the best involving my knowledge, is it authentic combined with I accept amateur arrested all abstracts and computations. Nevertheless, it is consistently accessible which accept fabricated a aberration about or which the most important bold has afflicted rear end [http://pinterest.com/search/pins/?q=publication publication]. Use as part of your very own risk, Dislike accommodate virtually any guarantees. Please get in blow if the public acquisition annihilation amiss. |
| As with bivalent logic, truth values in ternary logic may be represented numerically using various representations of the [[ternary numeral system]]. A few of the more common examples are:
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| * in [[Balanced ternary]], each digit has one of 3 values: −1, 0, or +1; these values may also be simplified to −, 0, +, respectively.<ref>{{cite book | last = Knuth | first = Donald E. | authorlink = Donald Knuth | title = The Art of Computer Programming Vol. 2 | publisher = Addison-Wesley Publishing Company | year = 1981 | location = Reading, Mass. | pages = 190 }}</ref>
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| * in [[Redundant binary representation]], each digit can have a value of -1, 0, 0, or 1 (the value 0 has two different representations)
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| * in [[Ternary numeral system]], each [[numerical digit|digit]] is a '''trit''' ('''tr'''inary dig'''it''') having a value of: 0, 1, or 2
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| * in [[Skew binary number system]], only most-significant non-zero digit has a value 2, and the remaining digits have a value of 0 or 1
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| * 1 for ''true'', 2 for ''false'', and 0 for ''unknown'', ''unknowable''/''undecidable'', ''irrelevant'', or ''both''.<ref>{{cite journal | last = Hayes | first = Brian | title = Third Base | journal = American Scientist | volume = 89 | issue = 6 | pages = 490–494 | publisher = Sigma Xi, the Scientific Research Society | date = November–December 2001 | doi = 10.1511/2001.6.490 }}</ref>{{failed verification|date=October 2012}}
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| * 0 for ''false'', 1 for ''true'', and a third non-integer symbol such as # or ½ for the final value, also known as "maybe".<ref>{{cite book|title = The Penguin Dictionary of Mathematics. 2nd Edition| pages = 417| publisher = Penguin Books | year = 1998 |location = London, England}}</ref>
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| Inside a [[ternary computer]], ternary values are represented by [[ternary signal]]s.
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| This article mainly illustrates a system of ternary [[propositional logic]] using the truth values {''false, unknown,'' and ''true''}, and extends conventional Boolean [[connectives]] to a trivalent context. Ternary [[predicate logic]]s exist as well;{{Citation needed|date=February 2007}} these may have readings of the [[quantifier]] different from classical (binary) predicate logic, and may include alternative quantifiers as well.
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| == Logics ==
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| === Kleene logic ===
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| Below is a set of [[truth table]]s showing the logic operations for [[Stephen Cole Kleene|Kleene]]'s logic.
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| {| class="wikitable"
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| |-
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| ! rowspan="2" colspan="2" | ''A'' '''AND''' ''B'' !! colspan="3" | ''B''
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| |-
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| ! True !! Unknown !! False
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| |-
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| ! scope="row" rowspan="3" style="padding:0px 10px"| ''A''
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| ! scope="row" | True
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| | True || Unknown || False
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| |-
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| ! scope="row" | Unknown
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| | Unknown || Unknown || False
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| |-
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| ! scope="row" | False
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| | False || False || False
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| |}
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| | |
| {| class="wikitable"
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| |-
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| ! rowspan="2" colspan="2" | ''A'' '''OR''' ''B'' !! colspan="3" | ''B''
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| |-
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| ! True !! Unknown !! False
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| |-
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| ! scope="row" rowspan="3" style="padding:0px 10px"| ''A''
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| ! scope="row" | True
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| | True || True || True
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| |-
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| ! scope="row" | Unknown
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| | True || Unknown || Unknown
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| |-
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| ! scope="row" | False
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| | True || Unknown || False
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| |}
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| | |
| {| class="wikitable"
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| |-
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| ! A !! '''NOT''' A
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| |-
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| ! scope="row" | True
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| | False
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| |-
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| ! scope="row" | Unknown
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| | Unknown
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| |-
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| ! scope="row" | False
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| | True
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| |}
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| In these truth tables, the UNKNOWN state can be metaphorically thought of as a sealed box containing either an unambiguously TRUE or unambiguously FALSE value. The knowledge of whether any particular UNKNOWN state secretly represents TRUE or FALSE at any moment in time is not available. However, certain logical operations can yield an unambiguous result, even if they involve at least one UNKNOWN operand. For example, since TRUE OR TRUE equals TRUE, and TRUE OR FALSE also equals TRUE, one can infer that TRUE OR UNKNOWN equals TRUE, as well. In this example, since either bivalent state could be underlying the UNKNOWN state, but either state also yields the same result, a definitive TRUE results in all three cases.
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| If numeric values are assigned to FALSE, UNKNOWN and TRUE such that FALSE is less than UNKNOWN and UNKNOWN is less than TRUE, then A AND B AND C... = MIN(A, B, C ...) and A OR B OR C ... = MAX(A, B, C...).
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| Material implication for Kleene logic can be defined as:
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| <math> A \rightarrow B \ \overset{\underset{\mathrm{def}}{}}{=} \ \mbox{NOT}(A) \ \mbox{OR} \ B </math>, and its truth table is | |
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| {| class="wikitable"
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| |-
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| ! rowspan="2" colspan="2" | ''A'' '''→''' ''B'' !! colspan="3" | ''B''
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| |-
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| ! True !! Unknown !! False
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| |-
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| ! scope="row" rowspan="3" style="padding:0px 10px"| ''A''
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| ! scope="row" | True
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| | True || Unknown || False
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| |-
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| ! scope="row" | Unknown
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| | True || Unknown || Unknown
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| |-
| |
| ! scope="row" | False
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| | True || True || True
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| |}
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| which differs from that for Łukasiewicz logic (described below).
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| Kleene logic has no tautologies (valid formulas) because whenever all of the atomic components of a well-formed formula are assigned the value Unknown, the formula itself must also have the value Unknown. (And the only ''designated'' truth value for Kleene logic is True.) However, the lack of valid formulas does not mean that it lacks valid arguments and/or inference rules. An argument is semantically valid in Kleene logic if, whenever (for any interpretation/model) all of its premises are True, the conclusion must also be True. (Note that the [[paraconsistent logic|Logic of Paradox]] (LP) has the same truth tables as Kleene logic, but it has two ''designated'' truth values instead of one; these are: True and Both (the analogue of Unknown), so that LP does have tautologies but it has fewer valid inference rules.)<ref>http://www.uky.edu/~look/Phi520-Lecture7.pdf</ref>
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| === Łukasiewicz logic ===
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| {{further|Łukasiewicz logic}}
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| {{expand section|date=January 2011}}
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| 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. This section follows the presentation from Malinowski's chapter of the ''Handbook of the History of Logic'', vol 8.<ref>Grzegorz Malinowski, "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>
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| {| class="wikitable"
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| |-
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| ! rowspan="2" colspan="2" | ''A'' '''→''' ''B'' !! colspan="3" | ''B''
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| |-
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| ! True !! Unknown !! False
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| |-
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| ! scope="row" rowspan="3" style="padding:0px 10px"| ''A''
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| ! scope="row" | True
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| | True || Unknown || False
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| |-
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| ! scope="row" | Unknown
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| | True || True || Unknown
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| |-
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| ! scope="row" | False
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| | True || True || True
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| |}
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| In fact, using Łukasiewicz's implication and negation, the other usual connectives may be derived as:
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| * ''A'' ∨ ''B'' = (''A'' → ''B'') → ''B''
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| * ''A'' ∧ ''B'' = ¬(¬''A'' ∨ ¬ ''B'')
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| * ''A'' ↔ ''B'' = (''A'' → ''B'') ∧ (''B'' → ''A'')
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| It's also possible to derive a few other useful unary operators (first derived by Tarski in 1921):
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| * '''M'''''A'' = ¬''A'' → ''A''
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| * '''L'''''A'' = ¬'''M'''¬''A''
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| * '''I'''''A'' = '''M'''''A'' ∧ ¬'''L'''''A''
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| They have the following truth tables:
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| {| border="0"
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| |-
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| | valign="top" |
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| {| class="wikitable"
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| ! ''A'' !! M''A''
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| |-
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| ! scope="row" | False
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| | False
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| |-
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| ! scope="row" | Unknown
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| | True
| |
| |-
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| ! scope="row" | True
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| | True
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| |}
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| ||
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| {| class="wikitable"
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| ! ''A'' !! L''A''
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| |-
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| ! scope="row" | False
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| | False
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| |-
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| ! scope="row" | Unknown
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| | False
| |
| |-
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| ! scope="row" | True
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| | True
| |
| |}
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| ||
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| {| class="wikitable"
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| ! ''A'' !! I''A''
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| |-
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| ! scope="row" | False
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| | False
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| |-
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| ! scope="row" | Unknown
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| | True
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| |-
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| ! scope="row" | True
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| | False
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| |}
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| |}
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| 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..."
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| 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 ''A'' → ''A'' and ''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]], ''A'' ∨ ¬''A'', and the [[law of non-contradiction]], ¬(''A'' ∧ ¬''A'') are not tautologies in Ł3. However, using the operator '''I''' defined above, it is possible to state tautologies that are their analogues:
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| * ''A'' ∨ '''I'''''A'' ∨ ¬''A'' [law of excluded fourth]
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| * ¬(''A'' ∧ ¬'''I'''''A'' ∧ ¬''A'') [extended contradiction principle].
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| == Application in SQL ==
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| {{main|Null (SQL)}}
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| The database structural query language [[SQL]] implements ternary logic as a means of handling comparisons with [[Null (SQL)|NULL]] field content. The original intent of NULL in SQL was to represent missing data in a database, i.e. the assumption that an actual value exists, but that the value is not currently recorded in the database. SQL uses a common fragment of the Kleene K3 logic, restricted to AND, OR, and NOT tables. Notice that basing SQL on Łukasiewicz L3 logic would conduct to different results as {A AND B} and {A OR B} differ when A and B are both UNKNOWN: Łukasiewicz finds FALSE and TRUE respectively while Kleene finds UNKNOWN.<ref name="Klein">{{cite doi|10.1007/3-540-36596-6_7}}</ref>
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| In SQL, the intermediate value is intended to be interpreted as UNKNOWN. Explicit comparisons with NULL, including that of another NULL yields UNKNOWN. However this choice of semantics is abandoned for some set operations, e.g. UNION or INTERSECT, where NULLs are treated as equal with each other. Critics assert that this inconsistency deprives SQL of intuitive semantics in its treatment of NULLs.<ref name="Meyden">Ron van der Meyden, "[http://books.google.com/books?id=gF0b85IuqQwC&pg=PA344 Logical approaches to incomplete information: a survey]" in Chomicki, Jan; Saake, Gunter (Eds.) ''Logics for Databases and Information Systems'', Kluwer Academic Publishers ISBN 978-0-7923-8129-7, p. 344; [http://www.cse.unsw.edu.au/~meyden/research/indef-review.ps PS preprint] (note: page numbering differs in preprint from the published version)</ref> The SQL standard defines an optional feature called F571, which adds some unary operators, among which IS UNKNOWN corresponding to the Łukasiewicz '''I''' in this article. The addition of IS UNKNOWN to the other operators of SQL's three-valued logic makes the SQL three-valued logic [[functionally complete]],<ref>C. J. Date, ''Relational database writings, 1991-1994'', Addison-Wesley, 1995, p. 371</ref> meaning its logical operators can express (in combination) any conceivable three-valued logical function.
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| ==See also==
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| * [[Aymara language]] – a Bolivian language famous for using ternary rather than binary logic<ref>{{cite web|url=http://www.aymara.org/arpasi-idioma-aymara.html|title=El idioma de los aymaras|publisher=Aymara Uta|language=Spanish|accessdate=2013-08-20}}</ref>
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| * [[Binary logic (disambiguation)]]
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| * [[Boolean algebra (structure)]]
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| * [[Boolean function]]
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| * [[Digital circuit]]
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| * [[Four-valued logic]]
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| * [[Setun]] - an experimental Russian computer which was based on ternary logic
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| * [[Ternary numeral system]] (and [[Balanced ternary]])
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| * [[Three-state logic]]
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| ==References==
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| {{reflist}}
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| == Further reading ==
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| * {{cite book |last=Bergmann |first=Merrie |title=An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems |year=2008 |publisher=Cambridge University Press |isbn=978-0-521-88128-9 |url=http://www.cambridge.org/us/academic/subjects/philosophy/logic/introduction-many-valued-and-fuzzy-logic-semantics-algebras-and-derivation-systems?format=PB |accessdate=24 August 2013}}, chapters 5-9
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| == External links ==
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| * [http://www.wv.inf.tu-dresden.de/Teaching/SS-2011/mvl/mval.HANDOUT2.pdf Introduction to Many-Valued Logics] by Bertram Fronhöfer. Handout from a Technische Universität Dresden 2011 summer class. (Despite the title, this is almost entirely about three-valued logics.)
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| {{DEFAULTSORT:Ternary Logic}}
| |
| [[Category:Many-valued logic]]
| |
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