Logical connective

<h2 id="overview">Overview</h2>
In <a href="/facts/Formal_language/crDTyP8q">formal languages</a>, truth functions are represented by unambiguous symbols. This allows logical statements to not be understood in an ambiguous way. These symbols are called logical connectives, logical operators, propositional operators, or, in <a href="/facts/Classical_logic/ls62MAms">classical logic</a>, <a href="/facts/Truth_function/aH5kBhQ7">truth-functional</a> connectives. For the rules which allow new well-formed formulas to be constructed by joining other well-formed formulas using truth-functional connectives, see <a href="/facts/Well-formed_formula/YpTVlzuj">well-formed formula</a>.
Logical connectives can be used to link zero or more statements, so one can speak about <a href="/facts/Arity/zSVuwHvP">n-ary</a> logical connectives. The <a href="/facts/Boolean_algebra/VkCToAmg">boolean</a> constants True and False can be thought of as zero-ary operators. Negation is a unary connective, and so on.

<table><tbody><tr><th colspan="2">Symbol, name</th><th colspan="4">Truthtable</th><th>Venndiagram</th></tr><tr><th colspan="7">Zeroary connectives (constants)</th></tr><tr><td> ⊤ {\displaystyle \top } </td><td><a href="/facts/Truth/NyQmBtUC">Truth</a>/<a href="/facts/Tautology_(logic)/p82wWMaN">tautology</a></td><td colspan="4">1</td><td></td></tr><tr><td> ⊥ {\displaystyle \bot } </td><td><a href="/facts/False_(logic)/H6v5aztR">Falsity</a>/<a href="/facts/Contradiction/PQHH5a2J">contradiction</a></td><td colspan="4">0</td><td></td></tr><tr><th colspan="7">Unary connectives</th></tr><tr><td colspan="2"> p {\displaystyle p}  =</td><td colspan="2">0</td><td colspan="2">1</td></tr><tr><td></td><td>Proposition p {\displaystyle p} </td><td colspan="2">0</td><td colspan="2">1</td><td></td></tr><tr><td> ¬ {\displaystyle \neg } </td><td><a href="/facts/Negation/q5AQBvBj">Negation</a></td><td colspan="2">1</td><td colspan="2">0</td><td></td></tr><tr><th colspan="9">Binary connectives</th></tr><tr><td colspan="2"> p {\displaystyle p}  =</td><td>0</td><td>0</td><td>1</td><td>1</td></tr><tr><td colspan="2"> q {\displaystyle q}  =</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td> ∧ {\displaystyle \land } </td><td><a href="/facts/Logical_conjunction/62uyyn1d">Conjunction</a></td><td>0</td><td>0</td><td>0</td><td>1</td><td></td></tr><tr><td> ↑ {\displaystyle \uparrow } </td><td><a href="/facts/Sheffer_stroke/rpXH07u4">Alternative denial</a></td><td>1</td><td>1</td><td>1</td><td>0</td><td></td></tr><tr><td> ∨ {\displaystyle \vee } </td><td><a href="/facts/Logical_disjunction/jbI2EGlM">Disjunction</a></td><td>0</td><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td> ↓ {\displaystyle \downarrow } </td><td><a href="/facts/Logical_NOR/HKhNW8Qc">Joint denial</a></td><td>1</td><td>0</td><td>0</td><td>0</td><td></td></tr><tr><td> ↮ {\displaystyle \nleftrightarrow } </td><td><a href="/facts/Exclusive_or/FHyr7hGQ">Exclusive or</a></td><td>0</td><td>1</td><td>1</td><td>0</td><td></td></tr><tr><td> ↔ {\displaystyle \leftrightarrow } </td><td><a href="/facts/Logical_biconditional/AghYtFSh">Biconditional</a></td><td>1</td><td>0</td><td>0</td><td>1</td><td></td></tr><tr><td> → {\displaystyle \rightarrow } </td><td><a href="/facts/Material_conditional/hYf4b7AE">Material conditional</a></td><td>1</td><td>1</td><td>0</td><td>1</td><td></td></tr><tr><td> ↛ {\displaystyle \nrightarrow } </td><td><a href="/facts/Material_nonimplication/a3MtvWGl">Material nonimplication</a></td><td>0</td><td>0</td><td>1</td><td>0</td><td></td></tr><tr><td> ← {\displaystyle \leftarrow } </td><td><a href="/facts/Converse_implication/VhlUrxxx">Converse implication</a></td><td>1</td><td>0</td><td>1</td><td>1</td><td></td></tr><tr><td> ↚ {\displaystyle \nleftarrow } </td><td><a href="/facts/Converse_nonimplication/SGA5an6u">Converse nonimplication</a></td><td>0</td><td>1</td><td>0</td><td>0</td><td></td></tr><tr><td colspan='7"'><a href="/facts/Truth_function/aH5kBhQ7">More information</a></td></tr></tbody></table>
<h3>List of common logical connectives</h3>
Commonly used logical connectives include the following ones.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> 

<ul><li><a href="/facts/Negation/q5AQBvBj">Negation (not)</a>: 
 
 
 
 ¬
 
 
 {\displaystyle \neg }
 
, 
 
 
 
 ∼
 
 
 {\displaystyle \sim }
 
, 
 
 
 
 N
 
 
 {\displaystyle N}
 
 (prefix) in which 
 
 
 
 ¬
 
 
 {\displaystyle \neg }
 
 is the most modern and widely used, and 
 
 
 
 ∼
 
 
 {\displaystyle \sim }
 
 is also common;</li>
<li><a href="/facts/Logical_conjunction/62uyyn1d">Conjunction (and)</a>: 
 
 
 
 ∧
 
 
 {\displaystyle \wedge }
 
, 
 
 
 
 &
 
 
 {\displaystyle \&}
 
, 
 
 
 
 K
 
 
 {\displaystyle K}
 
 (prefix) in which 
 
 
 
 ∧
 
 
 {\displaystyle \wedge }
 
 is the most modern and widely used;</li>
<li><a href="/facts/Logical_disjunction/jbI2EGlM">Disjunction (or)</a>: 
 
 
 
 ∨
 
 
 {\displaystyle \vee }
 
, 
 
 
 
 A
 
 
 {\displaystyle A}
 
 (prefix) in which 
 
 
 
 ∨
 
 
 {\displaystyle \vee }
 
 is the most modern and widely used;</li>
<li><a href="/facts/Material_conditional/hYf4b7AE">Implication (if...then)</a>: 
 
 
 
 →
 
 
 {\displaystyle \to }
 
, 
 
 
 
 ⊃
 
 
 {\displaystyle \supset }
 
, 
 
 
 
 ⇒
 
 
 {\displaystyle \Rightarrow }
 
, 
 
 
 
 C
 
 
 {\displaystyle C}
 
 (prefix) in which 
 
 
 
 →
 
 
 {\displaystyle \to }
 
 is the most modern and widely used, and 
 
 
 
 ⊃
 
 
 {\displaystyle \supset }
 
 is also common;</li>
<li><a href="/facts/Logical_biconditional/AghYtFSh">Equivalence (if and only if)</a>: 
 
 
 
 ↔
 
 
 {\displaystyle \leftrightarrow }
 
, 
 
 
 
 ⊂
 
 
 
 ⊃
 
 
 {\displaystyle \subset \!\!\!\supset }
 
, 
 
 
 
 ⇔
 
 
 {\displaystyle \Leftrightarrow }
 
, 
 
 
 
 ≡
 
 
 {\displaystyle \equiv }
 
, 
 
 
 
 E
 
 
 {\displaystyle E}
 
 (prefix) in which 
 
 
 
 ↔
 
 
 {\displaystyle \leftrightarrow }
 
 is the most modern and widely used, and 
 
 
 
 ⊂
 
 
 
 ⊃
 
 
 {\displaystyle \subset \!\!\!\supset }
 
 is commonly used where 
 
 
 
 ⊃
 
 
 {\displaystyle \supset }
 
 is also used.</li></ul>
For example, the meaning of the statements it is raining (denoted by 
 
 
 
 p
 
 
 {\displaystyle p}
 
) and I am indoors (denoted by 
 
 
 
 q
 
 
 {\displaystyle q}
 
) is transformed, when the two are combined with logical connectives:

<ul><li>It is not raining (
 
 
 
 ¬
 p
 
 
 {\displaystyle \neg p}
 
);</li>
<li>It is raining and I am indoors (
 
 
 
 p
 ∧
 q
 
 
 {\displaystyle p\wedge q}
 
);</li>
<li>It is raining or I am indoors (
 
 
 
 p
 ∨
 q
 
 
 {\displaystyle p\lor q}
 
);</li>
<li>If it is raining, then I am indoors (
 
 
 
 p
 →
 q
 
 
 {\displaystyle p\rightarrow q}
 
);</li>
<li>If I am indoors, then it is raining (
 
 
 
 q
 →
 p
 
 
 {\displaystyle q\rightarrow p}
 
);</li>
<li>I am indoors if and only if it is raining (
 
 
 
 p
 ↔
 q
 
 
 {\displaystyle p\leftrightarrow q}
 
).</li></ul>
It is also common to consider the always true formula and the always false formula to be connective (in which case they are <a href="/facts/Nullary/zSVuwHvP">nullary</a>).

<ul><li><a href="/facts/Truth/NyQmBtUC">True</a> formula: 
 
 
 
 ⊤
 
 
 {\displaystyle \top }
 
, 
 
 
 
 1
 
 
 {\displaystyle 1}
 
, 
 
 
 
 V
 
 
 {\displaystyle V}
 
 (prefix), or 
 
 
 
 
 T
 
 
 
 {\displaystyle \mathrm {T} }
 
;</li>
<li><a href="/facts/False_(logic)/H6v5aztR">False</a> formula: 
 
 
 
 ⊥
 
 
 {\displaystyle \bot }
 
, 
 
 
 
 0
 
 
 {\displaystyle 0}
 
, 
 
 
 
 O
 
 
 {\displaystyle O}
 
 (prefix), or 
 
 
 
 
 F
 
 
 
 {\displaystyle \mathrm {F} }
 
.</li></ul>
This table summarizes the terminology:

<table><tbody><tr><th>Connective</th><th>In English</th><th>Noun for parts</th><th>Verb phrase</th></tr><tr><th>Conjunction</th><td>Both A and B</td><td>conjunct</td><td>A and B are conjoined</td></tr><tr><th>Disjunction</th><td>Either A or B, or both</td><td>disjunct</td><td>A and B are disjoined</td></tr><tr><th>Negation</th><td>It is not the case that A</td><td>negatum/negand</td><td>A is negated</td></tr><tr><th>Conditional</th><td>If A, then B</td><td>antecedent, consequent</td><td>B is implied by A</td></tr><tr><th>Biconditional</th><td>A if, and only if, B</td><td>equivalents</td><td>A and B are equivalent</td></tr></tbody></table>
<h3>History of notations</h3>
<ul><li>Negation: the symbol 
 
 
 
 ¬
 
 
 {\displaystyle \neg }
 
 appeared in <a href="/facts/Arend_Heyting/9sIh63gt">Heyting</a> in 1930<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> (compare to <a href="/facts/Gottlob_Frege/FXyStVYj">Frege</a>'s symbol ⫟ in his <a href="/facts/Begriffsschrift/9JWDX0EE">Begriffsschrift</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>); the symbol 
 
 
 
 ∼
 
 
 {\displaystyle \sim }
 
 appeared in <a href="/facts/Bertrand_Russell/SRpwNX2U">Russell</a> in 1908;<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> an alternative notation is to add a horizontal line on top of the formula, as in 
 
 
 
 
 
 p
 ¯
 
 
 
 
 {\displaystyle {\overline {p}}}
 
; another alternative notation is to use a <a href="/facts/Prime_(symbol)/3p9EvU5X">prime symbol</a> as in 
 
 
 
 
 p
 ′
 
 
 
 {\displaystyle p'}
 
.</li>
<li>Conjunction: the symbol 
 
 
 
 ∧
 
 
 {\displaystyle \wedge }
 
 appeared in Heyting in 1930<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> (compare to <a href="/facts/Giuseppe_Peano/VT2Wsmu7">Peano</a>'s use of the set-theoretic notation of <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a> 
 
 
 
 ∩
 
 
 {\displaystyle \cap }
 
<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>); the symbol 
 
 
 
 &
 
 
 {\displaystyle \&}
 
 appeared at least in <a href="/facts/Moses_Sch%25C3%25B6nfinkel/7p4TkTtA">Schönfinkel</a> in 1924;<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> the symbol 
 
 
 
 ⋅
 
 
 {\displaystyle \cdot }
 
 comes from <a href="/facts/George_Boole/fWJT2rek">Boole</a>'s interpretation of logic as an <a href="/facts/Elementary_algebra/973nufT7">elementary algebra</a>.</li>
<li>Disjunction: the symbol 
 
 
 
 ∨
 
 
 {\displaystyle \vee }
 
 appeared in <a href="/facts/Bertrand_Russell/SRpwNX2U">Russell</a> in 1908<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> (compare to <a href="/facts/Giuseppe_Peano/VT2Wsmu7">Peano</a>'s use of the set-theoretic notation of <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> 
 
 
 
 ∪
 
 
 {\displaystyle \cup }
 
); the symbol 
 
 
 
 +
 
 
 {\displaystyle +}
 
 is also used, in spite of the ambiguity coming from the fact that the 
 
 
 
 +
 
 
 {\displaystyle +}
 
 of ordinary <a href="/facts/Elementary_algebra/973nufT7">elementary algebra</a> is an <a href="/facts/Exclusive_or/FHyr7hGQ">exclusive or</a> when interpreted logically in a two-element <a href="/facts/Boolean_ring/lgko4uqq">ring</a>; punctually in the history a 
 
 
 
 +
 
 
 {\displaystyle +}
 
 together with a dot in the lower right corner has been used by <a href="/facts/Charles_Sanders_Peirce/z76dNqhM">Peirce</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li>
<li>Implication: the symbol 
 
 
 
 →
 
 
 {\displaystyle \to }
 
 appeared in <a href="/facts/David_Hilbert/1vhlHn4b">Hilbert</a> in 1918;<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>: 76  
 
 
 
 ⊃
 
 
 {\displaystyle \supset }
 
 was used by Russell in 1908<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> (compare to Peano's Ɔ the inverted C); 
 
 
 
 ⇒
 
 
 {\displaystyle \Rightarrow }
 
 appeared in <a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki</a> in 1954.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></li>
<li>Equivalence: the symbol 
 
 
 
 ≡
 
 
 {\displaystyle \equiv }
 
 in <a href="/facts/Gottlob_Frege/FXyStVYj">Frege</a> in 1879;<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> 
 
 
 
 ↔
 
 
 {\displaystyle \leftrightarrow }
 
 in Becker in 1933 (not the first time and for this see the following);<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> 
 
 
 
 ⇔
 
 
 {\displaystyle \Leftrightarrow }
 
 appeared in <a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki</a> in 1954;<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> other symbols appeared punctually in the history, such as 
 
 
 
 ⊃⊂
 
 
 {\displaystyle \supset \subset }
 
 in <a href="/facts/Gerhard_Gentzen/9ujN0ssI">Gentzen</a>,<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> 
 
 
 
 ∼
 
 
 {\displaystyle \sim }
 
 in Schönfinkel<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> or 
 
 
 
 ⊂⊃
 
 
 {\displaystyle \subset \supset }
 
 in Chazal, <a class="footnote-ref" id="fnref:19" href="#fn:19">19</a></li>
<li>True: the symbol 
 
 
 
 1
 
 
 {\displaystyle 1}
 
 comes from <a href="/facts/George_Boole/fWJT2rek">Boole</a>'s interpretation of logic as an <a href="/facts/Elementary_algebra/973nufT7">elementary algebra</a> over the <a href="/facts/Two-element_Boolean_algebra/8nIWUpwd">two-element Boolean algebra</a>; other notations include 
 
 
 
 
 V
 
 
 
 {\displaystyle \mathrm {V} }
 
 (abbreviation for the Latin word "verum") to be found in Peano in 1889.</li>
<li>False: the symbol 
 
 
 
 0
 
 
 {\displaystyle 0}
 
 comes also from Boole's interpretation of logic as a ring; other notations include 
 
 
 
 Λ
 
 
 {\displaystyle \Lambda }
 
 (rotated 
 
 
 
 
 V
 
 
 
 {\displaystyle \mathrm {V} }
 
) to be found in Peano in 1889.</li></ul>
Some authors used letters for connectives: 
 
 
 
 
 u
 .
 
 
 
 {\displaystyle \operatorname {u.} }
 
 for conjunction (German's "und" for "and") and 
 
 
 
 
 o
 .
 
 
 
 {\displaystyle \operatorname {o.} }
 
 for disjunction (German's "oder" for "or") in early works by Hilbert (1904);<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a> 
 
 
 
 N
 p
 
 
 {\displaystyle Np}
 
 for negation, 
 
 
 
 K
 p
 q
 
 
 {\displaystyle Kpq}
 
 for conjunction, 
 
 
 
 D
 p
 q
 
 
 {\displaystyle Dpq}
 
 for alternative denial, 
 
 
 
 A
 p
 q
 
 
 {\displaystyle Apq}
 
 for disjunction, 
 
 
 
 C
 p
 q
 
 
 {\displaystyle Cpq}
 
 for implication, 
 
 
 
 E
 p
 q
 
 
 {\displaystyle Epq}
 
 for biconditional in <a href="/facts/Jan_%25C5%2581ukasiewicz/1rlu9ej8">Łukasiewicz</a> in 1929.

<h3>Redundancy</h3>
Such a logical connective as <a href="/facts/Converse_implication/VhlUrxxx">converse implication</a> "
 
 
 
 ←
 
 
 {\displaystyle \leftarrow }
 
" is actually the same as <a href="/facts/Material_conditional/hYf4b7AE">material conditional</a> with swapped arguments; thus, the symbol for converse implication is redundant. In some logical calculi (notably, in <a href="/facts/Classical_logic/ls62MAms">classical logic</a>), certain essentially different compound statements are <a href="/facts/Logical_equivalence/ysCWmsxA">logically equivalent</a>. A less <a href="/facts/Triviality_(mathematics)/AqzUxokN">trivial</a> example of a redundancy is the classical equivalence between 
 
 
 
 ¬
 p
 ∨
 q
 
 
 {\displaystyle \neg p\vee q}
 
 and 
 
 
 
 p
 →
 q
 
 
 {\displaystyle p\to q}
 
. Therefore, a classical-based logical system does not need the conditional operator "
 
 
 
 →
 
 
 {\displaystyle \to }
 
" if "
 
 
 
 ¬
 
 
 {\displaystyle \neg }
 
" (not) and "
 
 
 
 ∨
 
 
 {\displaystyle \vee }
 
" (or) are already in use, or may use the "
 
 
 
 →
 
 
 {\displaystyle \to }
 
" only as a <a href="/facts/Syntactic_sugar/FYkNSdTI">syntactic sugar</a> for a compound having one negation and one disjunction.
There are sixteen <a href="/facts/Boolean_function/uT7E8pqC">Boolean functions</a> associating the input <a href="/facts/Truth_value/SY4e2w8l">truth values</a> 
 
 
 
 p
 
 
 {\displaystyle p}
 
 and 
 
 
 
 q
 
 
 {\displaystyle q}
 
 with four-digit <a href="/facts/Binary_numeral_system/a6JDSRPs">binary</a> outputs.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> These correspond to possible choices of binary logical connectives for <a href="/facts/Classical_logic/ls62MAms">classical logic</a>. Different implementations of classical logic can choose different <a href="/facts/Functional_completeness/ncyHAsHH">functionally complete</a> subsets of connectives.
One approach is to choose a minimal set, and define other connectives by some logical form, as in the example with the material conditional above.
The following are the <a href="/facts/Functional_completeness/ncyHAsHH">minimal functionally complete sets of operators</a> in classical logic whose arities do not exceed 2:

One element

{
        ↑
        }
      
    
    {\displaystyle \{\uparrow \}}
  
, 
  
    
      
        {
        ↓
        }
      
    
    {\displaystyle \{\downarrow \}}
  
.
Two elements

{
        ∨
        ,
        ¬
        }
      
    
    {\displaystyle \{\vee ,\neg \}}
  
, 
  
    
      
        {
        ∧
        ,
        ¬
        }
      
    
    {\displaystyle \{\wedge ,\neg \}}
  
, 
  
    
      
        {
        →
        ,
        ¬
        }
      
    
    {\displaystyle \{\to ,\neg \}}
  
, 
  
    
      
        {
        ←
        ,
        ¬
        }
      
    
    {\displaystyle \{\gets ,\neg \}}
  
, 
  
    
      
        {
        →
        ,
        ⊥
        }
      
    
    {\displaystyle \{\to ,\bot \}}
  
, 
  
    
      
        {
        ←
        ,
        ⊥
        }
      
    
    {\displaystyle \{\gets ,\bot \}}
  
, 
  
    
      
        {
        →
        ,
        ↮
        }
      
    
    {\displaystyle \{\to ,\nleftrightarrow \}}
  
, 
  
    
      
        {
        ←
        ,
        ↮
        }
      
    
    {\displaystyle \{\gets ,\nleftrightarrow \}}
  
, 
  
    
      
        {
        →
        ,
        ↛
        }
      
    
    {\displaystyle \{\to ,\nrightarrow \}}
  
, 
  
    
      
        {
        →
        ,
        ↚
        }
      
    
    {\displaystyle \{\to ,\nleftarrow \}}
  
, 
  
    
      
        {
        ←
        ,
        ↛
        }
      
    
    {\displaystyle \{\gets ,\nrightarrow \}}
  
, 
  
    
      
        {
        ←
        ,
        ↚
        }
      
    
    {\displaystyle \{\gets ,\nleftarrow \}}
  
, 
  
    
      
        {
        ↛
        ,
        ¬
        }
      
    
    {\displaystyle \{\nrightarrow ,\neg \}}
  
, 
  
    
      
        {
        ↚
        ,
        ¬
        }
      
    
    {\displaystyle \{\nleftarrow ,\neg \}}
  
, 
  
    
      
        {
        ↛
        ,
        ⊤
        }
      
    
    {\displaystyle \{\nrightarrow ,\top \}}
  
, 
  
    
      
        {
        ↚
        ,
        ⊤
        }
      
    
    {\displaystyle \{\nleftarrow ,\top \}}
  
, 
  
    
      
        {
        ↛
        ,
        ↔
        }
      
    
    {\displaystyle \{\nrightarrow ,\leftrightarrow \}}
  
, 
  
    
      
        {
        ↚
        ,
        ↔
        }
      
    
    {\displaystyle \{\nleftarrow ,\leftrightarrow \}}
  
.
Three elements

{
 ∨
 ,
 ↔
 ,
 ⊥
 }
 
 
 {\displaystyle \{\lor ,\leftrightarrow ,\bot \}}
 
, 
 
 
 
 {
 ∨
 ,
 ↔
 ,
 ↮
 }
 
 
 {\displaystyle \{\lor ,\leftrightarrow ,\nleftrightarrow \}}
 
, 
 
 
 
 {
 ∨
 ,
 ↮
 ,
 ⊤
 }
 
 
 {\displaystyle \{\lor ,\nleftrightarrow ,\top \}}
 
, 
 
 
 
 {
 ∧
 ,
 ↔
 ,
 ⊥
 }
 
 
 {\displaystyle \{\land ,\leftrightarrow ,\bot \}}
 
, 
 
 
 
 {
 ∧
 ,
 ↔
 ,
 ↮
 }
 
 
 {\displaystyle \{\land ,\leftrightarrow ,\nleftrightarrow \}}
 
, 
 
 
 
 {
 ∧
 ,
 ↮
 ,
 ⊤
 }
 
 
 {\displaystyle \{\land ,\nleftrightarrow ,\top \}}
 
.
Another approach is to use with equal rights connectives of a certain convenient and functionally complete, but not minimal set. This approach requires more propositional <a href="/facts/Axiom/mLjP14Mc">axioms</a>, and each equivalence between logical forms must be either an <a href="/facts/Axiom/mLjP14Mc">axiom</a> or provable as a theorem.
The situation, however, is more complicated in <a href="/facts/Intuitionistic_logic/zI7ulCnM">intuitionistic logic</a>. Of its five connectives, {∧, ∨, →, ¬, ⊥}, only negation "¬" can be reduced to other connectives (see <a href="/facts/False_(logic)/H6v5aztR">False (logic) § False, negation and contradiction</a> for more). Neither conjunction, disjunction, nor material conditional has an equivalent form constructed from the other four logical connectives.

<h2 id="natural-language">Natural language</h2>
The standard logical connectives of classical logic have rough equivalents in the grammars of natural languages. In <a href="/facts/English_language/tpkxGt7b">English</a>, as in many languages, such expressions are typically <a href="/facts/Grammatical_conjunction/bKVVIpob">grammatical conjunctions</a>. However, they can also take the form of <a href="/facts/Complementizer/IXl8kzvp">complementizers</a>, <a href="/facts/Verb/zp42buvn">verb</a> <a href="/facts/Suffix/Nlnft3NK">suffixes</a>, and <a href="/facts/Grammatical_particle/Da0LIXEc">particles</a>. The <a href="/facts/Denotation/oABEtLtA">denotations</a> of natural language connectives is a major topic of research in <a href="/facts/Formal_semantics_(natural_language)/FlWb7Hxp">formal semantics</a>, a field that studies the logical structure of natural languages.
The meanings of natural language connectives are not precisely identical to their nearest equivalents in classical logic. In particular, disjunction can receive an <a href="/facts/Exclusive_disjunction/FHyr7hGQ">exclusive interpretation</a> in many languages. Some researchers have taken this fact as evidence that natural language <a href="/facts/Semantics_(natural_language)/1L4WwLAP">semantics</a> is <a href="/facts/Nonclassical_logic/caR36T5n">nonclassical</a>. However, others maintain classical semantics by positing <a href="/facts/Pragmatics/KkK08yyI">pragmatic</a> accounts of exclusivity which create the illusion of nonclassicality. In such accounts, exclusivity is typically treated as a <a href="/facts/Scalar_implicature/FfMyjr5T">scalar implicature</a>. Related puzzles involving disjunction include <a href="/facts/Free_choice_inference/nH0TI7Lm">free choice inferences</a>, <a href="/facts/Hurford_disjunction/F5yhoK0o">Hurford's Constraint</a>, and the contribution of disjunction in <a href="/facts/Alternative_question/v6UgKvHN">alternative questions</a>.
Other apparent discrepancies between natural language and classical logic include the <a href="/facts/Paradoxes_of_material_implication/kIBSEESa">paradoxes of material implication</a>, <a href="/facts/Donkey_anaphora/oeGDFsL8">donkey anaphora</a> and the problem of <a href="/facts/Counterfactual_conditionals/GdfYSv1k">counterfactual conditionals</a>. These phenomena have been taken as motivation for identifying the denotations of natural language conditionals with logical operators including the <a href="/facts/Strict_conditional/HEgeQBHq">strict conditional</a>, the <a href="/facts/Variably_strict_conditional/GdfYSv1k">variably strict conditional</a>, as well as various <a href="/facts/Dynamic_semantics/QQvanPx6">dynamic</a> operators.
The following table shows the standard classically definable approximations for the English connectives.

<table><tbody><tr><th>English word</th><th>Connective</th><th>Symbol</th><th>Logical gate</th></tr><tr><td>not</td><td><a href="/facts/Negation/q5AQBvBj">negation</a></td><td> ¬ {\displaystyle \neg } </td><td><a href="/facts/Inverter_(logic_gate)/pGk3qGTI">NOT</a></td></tr><tr><td>and</td><td><a href="/facts/Logical_conjunction/62uyyn1d">conjunction</a></td><td> ∧ {\displaystyle \land } </td><td><a href="/facts/AND_gate/tVKQD0w1">AND</a></td></tr><tr><td>or</td><td><a href="/facts/Logical_disjunction/jbI2EGlM">disjunction</a></td><td> ∨ {\displaystyle \vee } </td><td><a href="/facts/OR_gate/ox2lHgIM">OR</a></td></tr><tr><td>if...then</td><td><a href="/facts/Material_conditional/hYf4b7AE">material implication</a></td><td> → {\displaystyle \rightarrow } </td><td><a href="/facts/IMPLY_gate/kpHxqy4J">IMPLY</a></td></tr><tr><td>...if</td><td><a href="/facts/Converse_implication/VhlUrxxx">converse implication</a></td><td> ← {\displaystyle \leftarrow } </td><td></td></tr><tr><td>either...or</td><td><a href="/facts/Exclusive_or/FHyr7hGQ">exclusive disjunction</a></td><td> ↮ {\displaystyle \nleftrightarrow } </td><td><a href="/facts/XOR_gate/WSfdqTcA">XOR</a></td></tr><tr><td>if and only if</td><td><a href="/facts/Logical_biconditional/AghYtFSh">biconditional</a></td><td> ↔ {\displaystyle \leftrightarrow } </td><td><a href="/facts/XNOR_gate/LW793PwY">XNOR</a></td></tr><tr><td>not both</td><td><a href="/facts/Sheffer_stroke/rpXH07u4">alternative denial</a></td><td> ↑ {\displaystyle \uparrow } </td><td><a href="/facts/NAND_gate/g6FUuhJT">NAND</a></td></tr><tr><td>neither...nor</td><td><a href="/facts/Logical_NOR/HKhNW8Qc">joint denial</a></td><td> ↓ {\displaystyle \downarrow } </td><td><a href="/facts/NOR_gate/3l3MqaMI">NOR</a></td></tr><tr><td>but not</td><td><a href="/facts/Material_nonimplication/a3MtvWGl">material nonimplication</a></td><td> ↛ {\displaystyle \nrightarrow } </td><td><a href="/facts/NIMPLY_gate/wUPhuHVV">NIMPLY</a></td></tr><tr><td>not...but</td><td><a href="/facts/Converse_nonimplication/SGA5an6u">converse nonimplication</a></td><td> ↚ {\displaystyle \nleftarrow } </td></tr></tbody></table>
<h2 id="properties">Properties</h2>
Some logical connectives possess properties that may be expressed in the theorems containing the connective. Some of those properties that a logical connective may have are:

<a href="/facts/Associativity/EQX8lV6r">Associativity</a>
Within an expression containing two or more of the same associative connectives in a row, the order of the operations does not matter as long as the sequence of the operands is not changed.
<a href="/facts/Commutativity/WaYxJLxd">Commutativity</a>
The operands of the connective may be swapped, preserving logical equivalence to the original expression.
<a href="/facts/Distributivity/0LNBrVJl">Distributivity</a>
A connective denoted by · distributes over another connective denoted by +, if a · (b + c) = (a · b) + (a · c) for all operands a, b, c.
<a href="/facts/Idempotence/khbSpexv">Idempotence</a>
Whenever the operands of the operation are the same, the compound is logically equivalent to the operand.
<a href="/facts/Absorption_Law/ffTPsQj7">Absorption</a>
A pair of connectives ∧, ∨ satisfies the absorption law if 
 
 
 
 a
 ∧
 (
 a
 ∨
 b
 )
 =
 a
 
 
 {\displaystyle a\land (a\lor b)=a}
 
 for all operands a, b.
<a href="/facts/Monotonicity/MjQK0YL2">Monotonicity</a>
If f(a1, ..., an) ≤ f(b1, ..., bn) for all a1, ..., an, b1, ..., bn ∈ {0,1} such that a1 ≤ b1, a2 ≤ b2, ..., an ≤ bn. E.g., ∨, ∧, ⊤, ⊥.
<a href="/facts/Affine_transformation/BD7yDr10">Affinity</a>
Each variable always makes a difference in the truth-value of the operation or it never makes a difference. E.g., ¬, ↔, 
 
 
 
 ↮
 
 
 {\displaystyle \nleftrightarrow }
 
, ⊤, ⊥.
<a href="/facts/Duality_(mathematics)/8jGuyQIU">Duality</a>
To read the truth-value assignments for the operation from top to bottom on its <a href="/facts/Truth_table/4KsmArDW">truth table</a> is the same as taking the complement of reading the table of the same or another connective from bottom to top. Without resorting to truth tables it may be formulated as g̃(¬a1, ..., ¬an) = ¬g(a1, ..., an). E.g., ¬.
Truth-preserving
The compound all those arguments are tautologies is a tautology itself. E.g., ∨, ∧, ⊤, →, ↔, ⊂ (see <a href="/facts/Validity_(logic)/oMGkywGC">validity</a>).
Falsehood-preserving
The compound all those argument are <a href="/facts/Contradiction/PQHH5a2J">contradictions</a> is a contradiction itself. E.g., ∨, ∧, 
 
 
 
 ↮
 
 
 {\displaystyle \nleftrightarrow }
 
, ⊥, ⊄, ⊅ (see <a href="/facts/Validity_(logic)/oMGkywGC">validity</a>).
<a href="/facts/Involution_(mathematics)/kKxYrUVa">Involutivity</a> (for unary connectives)
f(f(a)) = a. E.g. negation in classical logic.
For classical and intuitionistic logic, the "=" symbol means that corresponding implications "...→..." and "...←..." for logical compounds can be both proved as theorems, and the "≤" symbol means that "...→..." for logical compounds is a consequence of corresponding "...→..." connectives for propositional variables. Some <a href="/facts/Many-valued_logic/qNI26RdF">many-valued logics</a> may have incompatible definitions of equivalence and order (entailment).
Both conjunction and disjunction are associative, commutative and idempotent in classical logic, most varieties of many-valued logic and intuitionistic logic. The same is true about distributivity of conjunction over disjunction and disjunction over conjunction, as well as for the absorption law.
In classical logic and some varieties of many-valued logic, conjunction and disjunction are dual, and negation is self-dual, the latter is also self-dual in intuitionistic logic.

<h2 id="order-of-precedence">Order of precedence</h2>
As a way of reducing the number of necessary parentheses, one may introduce <a href="/facts/Precedence_rule/sWsoTzbH">precedence rules</a>: ¬ has higher precedence than ∧, ∧ higher than ∨, and ∨ higher than →. So for example, 
 
 
 
 P
 ∨
 Q
 ∧
 
 ¬
 R
 
 →
 S
 
 
 {\displaystyle P\vee Q\land {\neg R}\rightarrow S}
 
 is short for 
 
 
 
 (
 P
 ∨
 (
 Q
 ∧
 (
 ¬
 R
 )
 )
 )
 →
 S
 
 
 {\displaystyle (P\vee (Q\land (\neg R)))\rightarrow S}
 
.
Here is a table that shows a commonly used precedence of logical operators.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a><a class="footnote-ref" id="fnref:23" href="#fn:23">23</a>

<table><tbody><tr><th>Operator</th><th>Precedence</th></tr><tr><td> ¬ {\displaystyle \neg } </td><td>1</td></tr><tr><td> ∧ {\displaystyle \land } </td><td>2</td></tr><tr><td> ∨ {\displaystyle \vee } </td><td>3</td></tr><tr><td> → {\displaystyle \rightarrow } </td><td>4</td></tr><tr><td> ↔ {\displaystyle \leftrightarrow } </td><td>5</td></tr></tbody></table>
However, not all compilers use the same order; for instance, an ordering in which disjunction is lower precedence than implication or bi-implication has also been used.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a> Sometimes precedence between conjunction and disjunction is unspecified requiring to provide it explicitly in given formula with parentheses. The order of precedence determines which connective is the "main connective" when interpreting a non-atomic formula.

<h2 id="table-and-hasse-diagram">Table and Hasse diagram</h2>
The 16 logical connectives can be <a href="/facts/Partial_order/qb4a3FAX">partially ordered</a> to produce the following <a href="/facts/Hasse_diagram/EqeJbUH0">Hasse diagram</a>. 
The partial order is defined by declaring that 
 
 
 
 x
 ≤
 y
 
 
 {\displaystyle x\leq y}
 
 if and only if whenever 
 
 
 
 x
 
 
 {\displaystyle x}
 
 holds then so does 
 
 
 
 y
 .
 
 
 {\displaystyle y.}
 
 

<table><tbody><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td>    </td></tr></tbody></table>

<h2 id="applications">Applications</h2>
Logical connectives are used in <a href="/facts/Computer_science/lN3pEyPz">computer science</a> and in <a href="/facts/Set_theory/1ptfsvUP">set theory</a>.

<h3>Computer science</h3>
Main article: <a href="/facts/Logic_gate/RHLulvKI">Logic gate</a>
A truth-functional approach to logical operators is implemented as <a href="/facts/Logic_gate/RHLulvKI">logic gates</a> in <a href="/facts/Digital_circuit/3kJcKbkw">digital circuits</a>. Practically all digital circuits (the major exception is <a href="/facts/DRAM/koI9FGo5">DRAM</a>) are built up from <a href="/facts/Logical_NAND/rpXH07u4">NAND</a>, <a href="/facts/Logical_NOR/HKhNW8Qc">NOR</a>, <a href="/facts/Negation/q5AQBvBj">NOT</a>, and <a href="/facts/Logic_gate/RHLulvKI">transmission gates</a>; see more details in <a href="/facts/Truth_function/aH5kBhQ7">Truth function in computer science</a>. Logical operators over <a href="/facts/Bit_array/PAfzTEHB">bit vectors</a> (corresponding to finite <a href="/facts/Boolean_algebra_(structure)/cHbo65jm">Boolean algebras</a>) are <a href="/facts/Bitwise_operation/0gkqnUda">bitwise operations</a>.
But not every usage of a logical connective in <a href="/facts/Computer_programming/QmSsFnu0">computer programming</a> has a Boolean semantic. For example, <a href="/facts/Lazy_evaluation/mD3DxDkc">lazy evaluation</a> is sometimes implemented for P ∧ Q and P ∨ Q, so these connectives are not commutative if either or both of the expressions P, Q have <a href="/facts/Side_effect_(computer_science)/AU3GlLzG">side effects</a>. Also, a <a href="/facts/Conditional_(programming)/8gDQTqUD">conditional</a>, which in some sense corresponds to the <a href="/facts/Material_conditional/hYf4b7AE">material conditional</a> connective, is essentially non-Boolean because for if (P) then Q;, the consequent Q is not executed if the <a href="/facts/Antecedent_(logic)/8VVVqa06">antecedent</a> P is false (although a compound as a whole is successful ≈ "true" in such case). This is closer to intuitionist and <a href="/facts/Constructive_mathematics/1VtNuLwT">constructivist</a> views on the material conditional— rather than to classical logic's views.

<h3>Set theory</h3>
Main articles: <a href="/facts/Set_theory/1ptfsvUP">Set theory</a> and <a href="/facts/Axiomatic_set_theory/1ptfsvUP">Axiomatic set theory</a>
Logical connectives are used to define the fundamental operations of <a href="/facts/Set_theory/1ptfsvUP">set theory</a>,<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> as follows:

Set theory operations and connectives<table><tbody><tr><th>Set operation</th><th>Connective</th><th>Definition</th></tr><tr><td><a href="/facts/Intersection_(set_theory)/RPfUNJh9">Intersection</a></td><td><a href="/facts/Logical_conjunction/62uyyn1d">Conjunction</a></td><td> A ∩ B = { x : x ∈ A ∧ x ∈ B } {\displaystyle A\cap B=\{x:x\in A\land x\in B\}} <a class="footnote-ref" id="fnref:26" href="#fn:26">26</a><a class="footnote-ref" id="fnref:27" href="#fn:27">27</a><a class="footnote-ref" id="fnref:28" href="#fn:28">28</a></td></tr><tr><td><a href="/facts/Union_(set_theory)/KVVXKfWl">Union</a></td><td><a href="/facts/Logical_disjunction/jbI2EGlM">Disjunction</a></td><td> A ∪ B = { x : x ∈ A ∨ x ∈ B } {\displaystyle A\cup B=\{x:x\in A\lor x\in B\}} <a class="footnote-ref" id="fnref:29" href="#fn:29">29</a><a class="footnote-ref" id="fnref:30" href="#fn:30">30</a><a class="footnote-ref" id="fnref:31" href="#fn:31">31</a></td></tr><tr><td><a href="/facts/Complement_(set_theory)/yWUePIYY">Complement</a></td><td><a href="/facts/Negation/q5AQBvBj">Negation</a></td><td> A ¯ = { x : x ∉ A } {\displaystyle {\overline {A}}=\{x:x\notin A\}} <a class="footnote-ref" id="fnref:32" href="#fn:32">32</a><a class="footnote-ref" id="fnref:33" href="#fn:33">33</a><a class="footnote-ref" id="fnref:34" href="#fn:34">34</a></td></tr><tr><td><a href="/facts/Subset/SJGERmbA">Subset</a></td><td><a href="/facts/Material_conditional/hYf4b7AE">Implication</a></td><td> A ⊆ B ↔ ( x ∈ A → x ∈ B ) {\displaystyle A\subseteq B\leftrightarrow (x\in A\rightarrow x\in B)} <a class="footnote-ref" id="fnref:35" href="#fn:35">35</a><a class="footnote-ref" id="fnref:36" href="#fn:36">36</a><a class="footnote-ref" id="fnref:37" href="#fn:37">37</a></td></tr><tr><td><a href="/facts/Equality_(mathematics)/AQCuumLg">Equality</a></td><td><a href="/facts/Logical_biconditional/AghYtFSh">Biconditional</a></td><td> A = B ↔ ( ∀ X ) [ A ∈ X ↔ B ∈ X ] {\displaystyle A=B\leftrightarrow (\forall X)[A\in X\leftrightarrow B\in X]} <a class="footnote-ref" id="fnref:38" href="#fn:38">38</a><a class="footnote-ref" id="fnref:39" href="#fn:39">39</a><a class="footnote-ref" id="fnref:40" href="#fn:40">40</a></td></tr></tbody></table>
This definition of set equality is equivalent to the <a href="/facts/Axiom_of_extensionality/sz0G1dWV">axiom of extensionality</a>.

<h2 id="see-also">See also</h2>
<ul>
<li>Philosophy portal</li><li>Psychology portal</li></ul>

<ul><li><a href="/facts/Boolean_domain/9cYq9TC1">Boolean domain</a></li>
<li><a href="/facts/Boolean_function/uT7E8pqC">Boolean function</a></li>
<li><a href="/facts/Boolean_logic/VkCToAmg">Boolean logic</a></li>
<li><a href="/facts/Boolean-valued_function/vlJBGgUl">Boolean-valued function</a></li>
<li><a href="/facts/Catu%25E1%25B9%25A3ko%25E1%25B9%25ADi/tBu6q2yw">Catuṣkoṭi</a></li>
<li><a href="/facts/Dialetheism/oLRsG78y">Dialetheism</a></li>
<li><a href="/facts/Four-valued_logic/faGUrw8E">Four-valued logic</a></li>
<li><a href="/facts/List_of_Boolean_algebra_topics/zFWKrPUu">List of Boolean algebra topics</a></li>
<li><a href="/facts/Logical_conjunction/62uyyn1d">Logical conjunction</a></li>
<li><a href="/facts/Logical_constant/bvTHIVCM">Logical constant</a></li>
<li><a href="/facts/Modal_operator/UHtpPLuH">Modal operator</a></li>
<li><a href="/facts/Propositional_calculus/MNt5v31V">Propositional calculus</a></li>
<li><a href="/facts/Term_logic/Ke2jxPC0">Term logic</a></li>
<li><a href="/facts/Tetralemma/BXLtfxny">Tetralemma</a></li>
<li><a href="/facts/Truth_function/aH5kBhQ7">Truth function</a></li>
<li><a href="/facts/Truth_table/4KsmArDW">Truth table</a></li>
<li><a href="/facts/Truth_value/SY4e2w8l">Truth values</a></li></ul>

<h2 id="sources">Sources</h2>
<ul><li><a href="/facts/J%25C3%25B3zef_Maria_Boche%25C5%2584ski/nWsm4IKA">Bocheński, Józef Maria</a> (1959), A Précis of Mathematical Logic, translated from the French and German editions by Otto Bird, D. Reidel, Dordrecht, South Holland.</li>
<li>Chao, C. (2023). 数理逻辑：形式化方法的应用 [Mathematical Logic: Applications of the Formalization Method] (in Chinese). Beijing: Preprint. pp. 15–28.</li>
<li><a href="/facts/Herbert_Enderton/LVXAVgos">Enderton, Herbert</a> (2001). A Mathematical Introduction to Logic (2nd ed.). Boston, MA: Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-238452-3.</li>
<li><a href="/facts/L._T._F._Gamut/L1tYfkVr">Gamut, L.T.F</a> (1991). "Chapter 2". Logic, Language and Meaning. Vol. 1. University of Chicago Press. pp. 54–64. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/21372380">21372380</a>.</li>
<li><a href="/facts/Wolfgang_Rautenberg/bUWpeRmB">Rautenberg, W.</a> (2010). A Concise Introduction to Mathematical Logic (3rd ed.). <a href="/facts/New_York_City/YlyvFmVC">New York</a>: <a href="/facts/Springer_Science%252BBusiness_Media/nAesf6nT">Springer Science+Business Media</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4419-1221-3">10.1007/978-1-4419-1221-3</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4419-1220-6..</li>
<li>Humberstone, Lloyd (2011). The Connectives. MIT Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-262-01654-4.</li></ul>
<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Logical connectives.

<ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Propositional_connective">"Propositional connective"</a>. <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>. <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>. 2001 [1994].</li>
<li>Lloyd Humberstone (2010), "<a href="https://plato.stanford.edu/entries/connectives-logic/">Sentence Connectives in Formal Logic</a>", <a href="/facts/Stanford_Encyclopedia_of_Philosophy/LDuL5HW4">Stanford Encyclopedia of Philosophy</a> (An <a href="/facts/Abstract_algebraic_logic/QPCvnNSb">abstract algebraic logic</a> approach to connectives.)</li>
<li>John MacFarlane (2005), "<a href="https://plato.stanford.edu/entries/logical-constants/">Logical constants</a>", <a href="/facts/Stanford_Encyclopedia_of_Philosophy/LDuL5HW4">Stanford Encyclopedia of Philosophy</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Chao, C. (2023). 数理逻辑：形式化方法的应用 [Mathematical Logic: Applications of the Formalization Method] (in Chinese). Beijing: Preprint. pp. 15–28. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Heyting, A. (1930). "Die formalen Regeln der intuitionistischen Logik". Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse (in German): 42–56. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Denis Roegel (2002), 
A brief survey of 20th century logical notations (see chart on page 2). <a href="https://members.loria.fr/Roegel/loc/symboles-logiques-eng.pdf" target="_blank">https://members.loria.fr/Roegel/loc/symboles-logiques-eng.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Frege, G. (1879). Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a/S.: Verlag von Louis Nebert. p. 10. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Russell (1908) Mathematical logic as based on the theory of types (American Journal of Mathematics 30, p222–262, also in From Frege to Gödel edited by van Heijenoort). <a href="/wiki/Bertrand_Russell" target="_blank">/wiki/Bertrand_Russell</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Heyting, A. (1930). "Die formalen Regeln der intuitionistischen Logik". Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse (in German): 42–56. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Peano (1889) Arithmetices principia, nova methodo exposita. <a href="/wiki/Giuseppe_Peano" target="_blank">/wiki/Giuseppe_Peano</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Schönfinkel (1924) Über die Bausteine der mathematischen Logik, translated as On the building blocks of mathematical logic in From Frege to Gödel edited by van Heijenoort. <a href="/wiki/Moses_Sch%C3%B6nfinkel" target="_blank">/wiki/Moses_Sch%C3%B6nfinkel</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Russell (1908) Mathematical logic as based on the theory of types (American Journal of Mathematics 30, p222–262, also in From Frege to Gödel edited by van Heijenoort). <a href="/wiki/Bertrand_Russell" target="_blank">/wiki/Bertrand_Russell</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Peirce (1867) On an improvement in Boole's calculus of logic. <a href="/wiki/Charles_Sanders_Peirce" target="_blank">/wiki/Charles_Sanders_Peirce</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Hilbert, D. (1918). Bernays, P. (ed.). Prinzipien der Mathematik. Lecture notes at Universität Göttingen, Winter Semester, 1917-1918; Reprinted as Hilbert, D. (2013). "Prinzipien der Mathematik". In Ewald, W.; Sieg, W. (eds.). David Hilbert's Lectures on the Foundations of Arithmetic and Logic 1917–1933. Heidelberg, New York, Dordrecht and London: Springer. pp. 59–221. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Russell (1908) Mathematical logic as based on the theory of types (American Journal of Mathematics 30, p222–262, also in From Frege to Gödel edited by van Heijenoort). <a href="/wiki/Bertrand_Russell" target="_blank">/wiki/Bertrand_Russell</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Bourbaki, N. (1954). Théorie des ensembles. Paris: Hermann & Cie, Éditeurs. p. 14. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Frege, G. (1879). Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (in German). Halle a/S.: Verlag von Louis Nebert. p. 15. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Becker, A. (1933). Die Aristotelische Theorie der Möglichkeitsschlösse: Eine logisch-philologische Untersuchung der Kapitel 13-22 von Aristoteles' Analytica priora I (in German). Berlin: Junker und Dünnhaupt Verlag. p. 4. <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Bourbaki, N. (1954). Théorie des ensembles (in French). Paris: Hermann & Cie, Éditeurs. p. 32. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Gentzen (1934) Untersuchungen über das logische Schließen. <a href="/wiki/Gerhard_Gentzen" target="_blank">/wiki/Gerhard_Gentzen</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Schönfinkel (1924) Über die Bausteine der mathematischen Logik, translated as On the building blocks of mathematical logic in From Frege to Gödel edited by van Heijenoort. <a href="/wiki/Moses_Sch%C3%B6nfinkel" target="_blank">/wiki/Moses_Sch%C3%B6nfinkel</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Chazal (1996) : Éléments de logique formelle. <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Hilbert, D. (1905) [1904]. "Über die Grundlagen der Logik und der Arithmetik". In Krazer, K. (ed.). Verhandlungen des Dritten Internationalen Mathematiker Kongresses in Heidelberg vom 8. bis 13. August 1904. pp. 174–185. <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Bocheński (1959), A Précis of Mathematical Logic, passim. <a href="/wiki/J%C3%B3zef_Maria_Boche%C5%84ski" target="_blank">/wiki/J%C3%B3zef_Maria_Boche%C5%84ski</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">O'Donnell, John; Hall, Cordelia; Page, Rex (2007). Discrete Mathematics Using a Computer. Springer. p. 120. ISBN 9781846285981.. <a href="9781846285981" target="_blank">9781846285981</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. <a href="978-0-262-54364-4" target="_blank">978-0-262-54364-4</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Jackson, Daniel (2012). Software Abstractions: Logic, Language, and Analysis. MIT Press. p. 263. ISBN 9780262017152.. <a href="9780262017152" target="_blank">9780262017152</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Pinter, Charles C. (2014). A book of set theory. Mineola, New York: Dover Publications, Inc. pp. 26–29. ISBN 978-0-486-49708-2. <a href="978-0-486-49708-2" target="_blank">978-0-486-49708-2</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">"Set operations". www.siue.edu. Retrieved 2024-06-11. <a href="https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-operations.html" target="_blank">https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-operations.html</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">"1.5 Logic and Sets". www.whitman.edu. Retrieved 2024-06-11. <a href="https://www.whitman.edu/mathematics/higher_math_online/section01.05.html" target="_blank">https://www.whitman.edu/mathematics/higher_math_online/section01.05.html</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">"Theory Set". mirror.clarkson.edu. Retrieved 2024-06-11. <a href="https://mirror.clarkson.edu/isabelle/dist/library/HOL/HOL/Set.html" target="_blank">https://mirror.clarkson.edu/isabelle/dist/library/HOL/HOL/Set.html</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">"Set Inclusion and Relations". autry.sites.grinnell.edu. Retrieved 2024-06-11. <a href="https://autry.sites.grinnell.edu/csc208/readings/set-inclusion.html" target="_blank">https://autry.sites.grinnell.edu/csc208/readings/set-inclusion.html</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">"Set operations". www.siue.edu. Retrieved 2024-06-11. <a href="https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-operations.html" target="_blank">https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-operations.html</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">"1.5 Logic and Sets". www.whitman.edu. Retrieved 2024-06-11. <a href="https://www.whitman.edu/mathematics/higher_math_online/section01.05.html" target="_blank">https://www.whitman.edu/mathematics/higher_math_online/section01.05.html</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">"Complement and Set Difference". web.mnstate.edu. Retrieved 2024-06-11. <a href="https://web.mnstate.edu/peil/MDEV102/U1/S6/Complement3.htm" target="_blank">https://web.mnstate.edu/peil/MDEV102/U1/S6/Complement3.htm</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">"1.5 Logic and Sets". www.whitman.edu. Retrieved 2024-06-11. <a href="https://www.whitman.edu/mathematics/higher_math_online/section01.05.html" target="_blank">https://www.whitman.edu/mathematics/higher_math_online/section01.05.html</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Cooper, A. "Set Operations and Subsets – Foundations of Mathematics". Retrieved 2024-06-11. <a href="https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/" target="_blank">https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">"Basic concepts". www.siue.edu. Retrieved 2024-06-11. <a href="https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-basics.html" target="_blank">https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-basics.html</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">"1.5 Logic and Sets". www.whitman.edu. Retrieved 2024-06-11. <a href="https://www.whitman.edu/mathematics/higher_math_online/section01.05.html" target="_blank">https://www.whitman.edu/mathematics/higher_math_online/section01.05.html</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">Cooper, A. "Set Operations and Subsets – Foundations of Mathematics". Retrieved 2024-06-11. <a href="https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/" target="_blank">https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">"Basic concepts". www.siue.edu. Retrieved 2024-06-11. <a href="https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-basics.html" target="_blank">https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-basics.html</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">"1.5 Logic and Sets". www.whitman.edu. Retrieved 2024-06-11. <a href="https://www.whitman.edu/mathematics/higher_math_online/section01.05.html" target="_blank">https://www.whitman.edu/mathematics/higher_math_online/section01.05.html</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">Cooper, A. "Set Operations and Subsets – Foundations of Mathematics". Retrieved 2024-06-11. <a href="https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/" target="_blank">https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
</ol>

Logical connective open-in-new

Logical connective