Monotonicity

Non-monotonic logic

A non-monotonic logic is a formal logic whose consequence relation is not monotonic. Most studied formal logics have a monotonic consequence relation, meaning that adding a formula to a theory never produces a reduction of its set of consequences. Intuitively, monotonicity indicates that learning a...

 of entailment

Entailment

In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...

is a property of many logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions. In sequent calculi

Sequent calculus

In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi, systems LK and LJ, were introduced by Gerhard Gentzen in 1934 as a tool for studying natural deduction in...

 this property can be captured by an inference rule called weakening, or sometimes thinning, and in such systems one may say that entailment is monotone if and only if the rule is admissible. Logical systems with this property are occasionally called monotonic logics in order to differentiate them from non-monotonic logic

Non-monotonic logic

A non-monotonic logic is a formal logic whose consequence relation is not monotonic. Most studied formal logics have a monotonic consequence relation, meaning that adding a formula to a theory never produces a reduction of its set of consequences. Intuitively, monotonicity indicates that learning a...

s.

Weakening rule

To illustrate, starting from the natural deduction

Natural deduction

In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning...

 sequent

Sequent

In proof theory, a sequent is a formalized statement of provability that is frequently used when specifying calculi for deduction. In the sequent calculus, the name sequent is used for the construct which can be regarded as a specific kind of judgment, characteristic to this deduction system.-...

:

Γ C

weakening allows one to conclude:

Γ, A C

Non-monotonic logics

In most logics, weakening is either an inference rule or a metatheorem

Metatheorem

In logic, a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory.- Discussion :A formal...

 if the logic doesn't have an explicit rule. Notable exceptions are:

• Strict logic
  Strict logic
  Strict logic is essentially synonymous with relevant logic, though it can be characterized proof-theoretically as* ordinary logic without weakening, or* linear logic with contraction....
  
   or relevant logic, where every hypothesis must be necessary for the conclusion.
• Linear logic
  Linear logic
  Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter...
  
   which disallows arbitrary contraction
  Contraction
  Contraction may refer to:In physiology:* Muscle contraction, one that occurs when a muscle fiber lengthens or shortens** Uterine contraction, contraction of the uterus, such as during childbirth* Contraction, a stage in wound healing...
  
   in addition to arbitrary weakening.
• Bunched implications
  Bunched logic
  Bunched logic is a variety of substructural logic that, like linear logic, has classes of multiplicative and additive operators, but differs from usual proof calculi in having a tree-like context of hypotheses instead of a flat list-like structure; it is thus a calculus of deep inference. Sub-trees...
  
   where weakening is restricted to additive composition.
• Various types of default reasoning.
• Abductive reasoning
  Abductive reasoning
  Abduction is a kind of logical inference described by Charles Sanders Peirce as "guessing". The term refers to the process of arriving at an explanatory hypothesis. Peirce said that to abduce a hypothetical explanation a from an observed surprising circumstance b is to surmise that a may be true...
  
  , the process of deriving the most likely explanations of the known facts.
• Reasoning about knowledge, where statements specifying that something is not known need to be retracted when that thing is learned.

See also

• Contraction
  Idempotency of entailment
  Idempotency of entailment is a property of logical systems that states that one may derive the same consequences from many instances of a hypothesis as from just one...
  
  
• Exchange rule
• Substructural logic
  Substructural logic
  In logic, a substructural logic is a logic lacking one of the usual structural rules , such as weakening, contraction or associativity...