---
title: "Logic"
chunk: 8/11
source: "https://en.wikipedia.org/wiki/Logic"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:56:39.118518+00:00"
instance: "kb-cron"
---

Modal logic is an extension of classical logic. In its original form, sometimes called "alethic modal logic", it introduces two new symbols: 
  
    
      
        ◊
      
    
    {\displaystyle \Diamond }
  
 expresses that something is possible while 
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
 expresses that something is necessary. For example, if the formula 
  
    
      
        B
        (
        s
        )
      
    
    {\displaystyle B(s)}
  
 stands for the sentence "Socrates is a banker" then the formula 
  
    
      
        ◊
        B
        (
        s
        )
      
    
    {\displaystyle \Diamond B(s)}
  
 articulates the sentence "It is possible that Socrates is a banker". To include these symbols in the logical formalism, modal logic introduces new rules of inference that govern what role they play in inferences. One rule of inference states that, if something is necessary, then it is also possible. This means that 
  
    
      
        ◊
        A
      
    
    {\displaystyle \Diamond A}
  
 follows from 
  
    
      
        ◻
        A
      
    
    {\displaystyle \Box A}
  
. Another principle states that if a proposition is necessary then its negation is impossible and vice versa. This means that 
  
    
      
        ◻
        A
      
    
    {\displaystyle \Box A}
  
 is equivalent to 
  
    
      
        ¬
        ◊
        ¬
        A
      
    
    {\displaystyle \lnot \Diamond \lnot A}
  
.
Other forms of modal logic introduce similar symbols but associate different meanings with them to apply modal logic to other fields. For example, deontic logic concerns the field of ethics and introduces symbols to express the ideas of obligation and permission, i.e. to describe whether an agent has to perform a certain action or is allowed to perform it. The modal operators in temporal modal logic articulate temporal relations. They can be used to express, for example, that something happened at one time or that something is happening all the time. In epistemology, epistemic modal logic is used to represent the ideas of knowing something in contrast to merely believing it to be the case.

==== Higher order logic ====

Higher-order logics extend classical logic not by using modal operators but by introducing new forms of quantification. Quantifiers correspond to terms like "all" or "some". In classical first-order logic, quantifiers are only applied to individuals. The formula "
  
    
      
        ∃
        x
        (
        A
        p
        p
        l
        e
        (
        x
        )
        ∧
        S
        w
        e
        e
        t
        (
        x
        )
        )
      
    
    {\displaystyle \exists x(Apple(x)\land Sweet(x))}
  
" (some apples are sweet) is an example of the existential quantifier "
  
    
      
        ∃
      
    
    {\displaystyle \exists }
  
" applied to the individual variable "
  
    
      
        x
      
    
    {\displaystyle x}
  
". In higher-order logics, quantification is also allowed over predicates. This increases its expressive power. For example, to express the idea that Mary and John share some qualities, one could use the formula "
  
    
      
        ∃
        Q
        (
        Q
        (
        M
        a
        r
        y
        )
        ∧
        Q
        (
        J
        o
        h
        n
        )
        )
      
    
    {\displaystyle \exists Q(Q(Mary)\land Q(John))}
  
". In this case, the existential quantifier is applied to the predicate variable "
  
    
      
        Q
      
    
    {\displaystyle Q}
  
". The added expressive power is especially useful for mathematics since it allows for more succinct formulations of mathematical theories. But it has drawbacks in regard to its meta-logical properties and ontological implications, which is why first-order logic is still more commonly used.

=== Deviant ===

Deviant logics are logical systems that reject some of the basic intuitions of classical logic. Because of this, they are usually seen not as its supplements but as its rivals. Deviant logical systems differ from each other either because they reject different classical intuitions or because they propose different alternatives to the same issue.
Intuitionistic logic is a restricted version of classical logic. It uses the same symbols but excludes some rules of inference. For example, according to the law of double negation elimination, if a sentence is not not true, then it is true. This means that 
  
    
      
        A
      
    
    {\displaystyle A}
  
 follows from 
  
    
      
        ¬
        ¬
        A
      
    
    {\displaystyle \lnot \lnot A}
  
. This is a valid rule of inference in classical logic but it is invalid in intuitionistic logic. Another classical principle not part of intuitionistic logic is the law of excluded middle. It states that for every sentence, either it or its negation is true. This means that every proposition of the form 
  
    
      
        A
        ∨
        ¬
        A
      
    
    {\displaystyle A\lor \lnot A}
  
 is true. These deviations from classical logic are based on the idea that truth is established by verification using a proof. Intuitionistic logic is especially prominent in the field of constructive mathematics, which emphasizes the need to find or construct a specific example to prove its existence.
Multi-valued logics depart from classicality by rejecting the principle of bivalence, which requires all propositions to be either true or false. For instance, Jan Łukasiewicz and Stephen Cole Kleene both proposed ternary logics which have a third truth value representing that a statement's truth value is indeterminate. These logics have been applied in the field of linguistics. Fuzzy logics are multivalued logics that have an infinite number of "degrees of truth", represented by a real number between 0 and 1.
Paraconsistent logics are logical systems that can deal with contradictions. They are formulated to avoid the principle of explosion: for them, it is not the case that anything follows from a contradiction. They are often motivated by dialetheism, the view that contradictions are real or that reality itself is contradictory. Graham Priest is an influential contemporary proponent of this position and similar views have been ascribed to Georg Wilhelm Friedrich Hegel.

=== Informal ===