what role they play in inferences. One rule of inference states that, if something is necessary, then it is
also possible. This means that  follows from . Another principle states that if a proposition is
necessary then its negation is impossible and vice versa. This means that  is equivalent to .[128]
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.[129] 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.[129] In epistemology, epistemic modal logic is used to
represent the ideas of knowing something in contrast to merely believing it to be the case.[130]
Higher-order logics extend classical logic not by using modal operators but by introducing new forms of
quantification.[131] Quantifiers correspond to terms like "all" or "some". In classical first-order logic,
quantifiers are only applied to individuals. The formula " " (some apples are
sweet) is an example of the existential quantifier " " applied to the individual variable " ". 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
" ". In this case, the existential quantifier is applied to the predicate variable
" ".[132] The added expressive power is especially useful for mathematics since it allows for more
succinct formulations of mathematical theories.[43] 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.[133]
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.[134]
Intuitionistic logic is a restricted version of classical logic.[135] 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  follows from . 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  is true.[135] 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.[136]
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.[137] 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.[138]
Higher order logic
Deviant