Monotonicity of entailment is a property of many
logical system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A form ...
s that states that the hypotheses of any derived fact may be freely extended with additional assumptions. In
sequent calculi this property can be captured by an inference rule called weakening, or sometimes thinning, and in such systems one may say that
entailment
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one ...
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
A non-monotonic logic is a formal logic whose conclusion relation is not monotonic. In other words, non-monotonic logics are devised to capture and represent defeasible inferences (cf. defeasible reasoning), i.e., a kind of inference in which rea ...
s.
Weakening rule
To illustrate, consider the
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. This contrasts with Hilbert-style systems, which instead use axiom ...
sequent
In mathematical logic, a sequent is a very general kind of conditional assertion.
: A_1,\,\dots,A_m \,\vdash\, B_1,\,\dots,B_n.
A sequent may have any number ''m'' of condition formulas ''Ai'' (called " antecedents") and any number ''n'' of asse ...
:
Γ
C
That is, on the basis of a list of assumptions Γ, one can prove C. Weakening, by adding an assumption A, allows one to conclude:
Γ, A
C
For example, the syllogism "All men are mortal. Socrates is a man. Therefore Socrates is mortal." can be weakened by adding a premise: "All men are mortal. Socrates is a man. Cows produce milk. Therefore Socrates is mortal." The validity of the original conclusion is not changed by the addition of premises.
Non-monotonic logics
In most logics, weakening is either an inference rule or a
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 ...
if the logic doesn't have an explicit rule. Notable exceptions are:
*
Strict logic
In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusive ...
or
relevant logic Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics. It is generally, but ...
, where every hypothesis is necessary for the conclusion.
*
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. Although the logic has also be ...
which disallows arbitrary
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. This property can be captured by a structural rule called contraction, and in ...
.
See also
*
Contraction
Contraction may refer to:
Linguistics
* Contraction (grammar), a shortened word
* Poetic contraction, omission of letters for poetic reasons
* Elision, omission of sounds
** Syncope (phonology), omission of sounds in a word
* Synalepha, merged ...
*
Exchange rule
In proof theory, a structural rule is an inference rule that does not refer to any logical connective, but instead operates on the judgment or sequents directly. Structural rules often mimic intended meta-theoretic properties of the logic. Logic ...
*
Substructural logic
In logic, a substructural logic is a logic lacking one of the usual structural rules (e.g. of classical and intuitionistic logic), such as weakening, contraction, exchange or associativity. Two of the more significant substructural logics are ...
*
No-cloning theorem In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theore ...
Logical consequence
Theorems in propositional logic
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