Modal operator

<h2 id="syntax-for-modal-operators">Syntax for modal operators</h2>

<p>The syntax rules for modal operators 
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
 and 
  
    
      
        ◊
      
    
    {\displaystyle \Diamond }
  
 are very similar to those for universal and existential <a href="/facts/Quantifier_(logic)/bgmvwxoo">quantifiers</a>; In fact, any formula with modal operators 
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
 and 
  
    
      
        ◊
      
    
    {\displaystyle \Diamond }
  
, and the usual <a href="/facts/Logical_connective/VkPhUs6M">logical connectives</a> in <a href="/facts/Propositional_calculus/MNt5v31V">propositional calculus</a> (
  
    
      
        ∧
        ,
        ∨
        ,
        ¬
        ,
        →
        ,
        ↔
      
    
    {\displaystyle \land ,\lor ,\neg ,\rightarrow ,\leftrightarrow }
  
) can be <a href="/facts/Rewriting/9PHlnnnJ">rewritten</a> to a <a href="/facts/De_dicto_and_de_re/QBbnjHc8"><i>de dicto</i></a> normal form, similar to <a href="/facts/Prenex_normal_form/D0UCl45n">prenex normal form</a>. One major caveat: Whereas the universal and existential quantifiers only binds to the <a href="/facts/Propositional_variable/NtAz1rfI">propositional variables</a> or the <a href="/facts/Predicate_variable/vdoPP5Yu">predicate variables</a> following the quantifiers, since the modal operators 
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
 and 
  
    
      
        ◊
      
    
    {\displaystyle \Diamond }
  
 quantifies over <a href="/facts/Accessibility_relation/p2Q34ASq">accessible</a> <a href="/facts/Possible_world/6jr7P8jV">possible worlds</a>, they will bind to any formula in their <a href="/facts/Scope_(logic)/MwDzrpCL">scope</a>. For example, 
  
    
      
        (
        ∃
        x
        (
        
          x
          
            2
          
        
        =
        1
        )
        )
        ∧
        (
        0
        =
        y
        )
      
    
    {\displaystyle (\exists x(x^{2}=1))\land (0=y)}
  
 is logically equivalent to 
  
    
      
        ∃
        x
        (
        
          x
          
            2
          
        
        =
        1
        ∧
        0
        =
        y
        )
      
    
    {\displaystyle \exists x(x^{2}=1\land 0=y)}
  
, but 
  
    
      
        (
        ◊
        (
        
          x
          
            2
          
        
        =
        1
        )
        )
        ∧
        (
        0
        =
        y
        )
      
    
    {\displaystyle (\Diamond (x^{2}=1))\land (0=y)}
  
 is not logically equivalent to 
  
    
      
        ◊
        (
        
          x
          
            2
          
        
        =
        1
        ∧
        0
        =
        y
        )
      
    
    {\displaystyle \Diamond (x^{2}=1\land 0=y)}
  
; Instead, 
  
    
      
        ◊
        (
        
          x
          
            2
          
        
        =
        1
        ∧
        0
        =
        y
        )
      
    
    {\displaystyle \Diamond (x^{2}=1\land 0=y)}
  
 is logically equivalent to 
  
    
      
        (
        ◊
        (
        
          x
          
            2
          
        
        =
        1
        )
        )
        ∧
        ◊
        (
        0
        =
        y
        )
      
    
    {\displaystyle (\Diamond (x^{2}=1))\land \Diamond (0=y)}
  
.
</p><p>When there are both modal operators and quantifiers in a formula, different order of an adjacent pair of modal operator and quantifier can lead to <a href="/facts/De_dicto_and_de_re/QBbnjHc8">different semantic meanings</a>; Also, when <a href="/facts/Multimodal_logic/hHwfwsXH">multimodal logic</a> is involved, different order of an adjacent pair of modal operators can also lead to different semantic meanings.
</p>
<h2 id="modality-interpreted">Modality interpreted</h2>

<p>There are several ways to <a href="/facts/Interpretation_(logic)/goPmjjQG">interpret</a> modal operators in modal logic, including at least:
<a href="/facts/Alethic_modality/BSsxl0N9">alethic</a>, <a href="/facts/Deontic_logic/3qHUf4cs">deontic</a>, <a href="/facts/Axiology/l6rLbmsK">axiological</a>, <a href="/facts/Epistemic_modal_logic/kRIBmdp5">epistemic</a>, and <a href="/facts/Doxastic_logic/fAQM342I">doxastic</a>.
</p>
<h3>Alethic</h3>
<p><a href="/facts/Alethic_modality/BSsxl0N9">Alethic</a> modal operators (M-operators) determine the fundamental conditions of <a href="/facts/Possible_worlds/6jr7P8jV">possible worlds</a>, especially <a href="/facts/Causality/Swh496X7">causality</a>, time-space parameters, and the action capacity of persons. They indicate the <a href="/facts/Logical_possibility/jGJ6gH7d">possibility</a>, <a href="/facts/Subjunctive_possibility/kqMBKFUj">impossibility</a> and <a href="/facts/Logical_truth/N0bUYhUW">necessity</a> of actions, states of affairs, events, people, and qualities in the possible worlds.
</p>
<h3>Deontic</h3>
<p><a href="/facts/Deontic_logic/3qHUf4cs">Deontic</a> modal operators (P-operators) influence the construction of possible worlds as proscriptive or prescriptive norms, i.e. they indicate what is prohibited, obligatory, or permitted.
</p>
<h3>Axiological</h3>
<p><a href="/facts/Axiology/l6rLbmsK">Axiological</a> modal operators (G-operators) transform the world's entities into values and disvalues as seen by a social group, a culture, or a historical period. Axiological modalities are highly subjective categories: what is good for one person may be considered as bad by another one.
</p>
<h3>Epistemic</h3>
<p><a href="/facts/Epistemic_logic/kRIBmdp5">Epistemic</a> modal operators (K-operators) reflect the level of knowledge, ignorance and belief in the possible world.
</p>
<h3>Doxastic</h3>
<p><a href="/facts/Doxastic_logic/fAQM342I">Doxastic</a> modal operators express belief in statements. 
</p>
<h3>Boulomaic</h3>
<p>Boulomaic modal operators express desire. 
</p>

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

<ol>
<li id="fn:1"><p>Garson, James (2021). "Modal Logic". The Stanford Encyclopedia of Philosophy (Summer 2021 ed.). Metaphysics Research Lab, Stanford University. Retrieved 5 February 2024. <a href="https://plato.stanford.edu/archives/sum2021/entries/logic-modal/" target="_blank">https://plato.stanford.edu/archives/sum2021/entries/logic-modal/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Modal operator open-in-new

Modal operator