consequence operator

1 Definition

Let $L$ be a set. A consequence operator on $L$ is amapping $C:𝒫(L)\to 𝒫(L)$which satisfies the following three properties:

1. For all $X\subseteq L$, it happens that $X\subseteq C(X)$.

2. $C\circ C=C$

3. For all $X,Y\subseteq L$, if $X\subseteq Y$, then $C(X)\subseteq C(Y)$

If, in addition, the following condition is satisfied, then a consequence operator $C$ is known as finitary . (Synonyms are “finite consequence operator” and “algebraic consequence operator”.)

• For all $X\in L$, it happens that $C(X)=\bigcup _{Y\in ℱ(X)}C(Y)$.

It is worth noting that, if the above condition is satisfied, then the thirdcondition of last paragraph becomes superfluous — as shown inhttp://planetmath.org/node/ 8678an attachment, it automatically followsfrom conitions 1 and 2 of last paragraph and the condition stated above.

A consequence operator $C$ such that $C(\mathrm{\varnothing })=\mathrm{\varnothing }$ is called axiomless. A consequence operator $C$ such that $C(\mathrm{\varnothing })\ne \mathrm{\varnothing }$ is called axiomatic.

2 Motivation

Alfred Tarksi introduced consequence operators as a way of discussing thenotion of conclusions following from premises in a general fashion. Supposethat the set $L$ consists of statements in some language. Then, given aset of statements $X$, let $C(X)$ be the set of all statements which can beinferred form statements of $X$.

The defining properties of “consequence operator” given above then expresssome fundamental facts about the process of inferring conclusions frompremises: Any statement can be concluded from itself. If a statement $s$follows from a set of premises $X$ and $Y$ is a superset of $X$, then$s$ also follows from $Y$. If one augments a set of premises by conclusionsderived from those premises, then one can only draw conclusions from thelarger set which could have been drawn from the original set of premises.Note that these conditions hold for a large class of logics, not justclassical logic of Aristotle, Boole, and Frege. However, they do not holdfor all logics — in particular, there are the so-called nonmonotoniclogics in which it is not always the case that, if $X\subset Y$, then$C(X)\subseteq C(Y)$.

In terms of this usage in logic, it is easy to understand the origin of theterms ”axiomatic” and ”axiomless”. An axiom in a logical theory is astatement which is assumed true without having to prove it from any otherstatement. Hence, an axiom is a consequence of the empty set, so we callconsequence operators which allow one to deduce conclusions from an emptyset of premises axiomatic.

The distinction of finitary consequence operators has to do with whether oneis permitted to draw a conclusion from an infinite set of premises whichcould not be drawn from any finite subset thereof. As for why one mightwant to do this, consider the following example. Suppose $X$ consists ofthe following statements:

• $1=1^2$

• $2=1^2+1^2$

• $3=1^2+1^2+1^2$

• $4=2^2$

• $5=2^2+1^2$

• $6=2^2+1^2+1^2$

• $7=2^2+1^2+1^2+1^2$

• $8=2^2+2^2$

• $9=3^2$

• $10=3^2+1^2$

• $11=3^2+1^2+1^2$

• $12=2^2+2^2+2^2$

• $13=3^2+2^2$

• $14=3^2+2^2+1^2$

• $\mathrm{\cdots }$

From $X$ one would like to be able to draw the conclusion “Any positiveinteger can be expressed as the sum of at most four squares.”. Thisconclusion, however, cannot be inferred from any proper subset of $X$, inparticular, from any finite subset of $X$. To make this conclusion wouldrequire a consequence operator which is not finitary.

3 Examples

1. To begin, there are two trivial consequence operators defined on any set.One is the identity operator $I:𝒫(L)\to 𝒫(L)$ defined as $I(X)=X$. The other is the constant operator $U:𝒫(L)\to 𝒫(L)$ defined as $U(X)=L$. It is perfectlystraightforward to check that these two operators satisfy the definingproperties of consequence operator and, furthermore, that they are bothfinitary consequence operators and that $I$ is axiomless whilst $U$ isaxiomatic. Trivial though they may be, these operators play an importantrole as exrtremal elements in the lattice of all consequence operatorsover a given set.

2. Next, we consider some less trivial consequence operators which can bedefined over an arbitrary set. Let $X$ and $Y$ be any two subsets of $L$.Then we may define operators $C_{\cap }(X,Y):𝒫(L)\to 𝒫(L)$and $C_{\cup }(X,Y):𝒫(L)\to 𝒫(L)$ as follows:

   $C_{\cap }(X,Y)(Z)$$=$

   $C_{\cup }(X,Y)(Z)$$=$

   It is shown that these are indeed consequence operators in an attachment tothis entry.

3. A much larger class of consequence operators may be defined as follows.Let $K$ be a subset of $𝒫(L)$ which includes $L$. Then, asshown in an http://planetmath.org/node/8671attachment, themap $C:𝒫(L)\to 𝒫(L)$, defined as

   $$C(X)=\cap \{Y\in K\mid X\subseteq Y\},$$

   is a consequence operator. As we shall see, all consequence operatorscan be obtained by this construction. In particular, the examples discussedabove can be obtained as follows: To obtain $I$, set $K=𝐏(L)$; toobtain $U$, set $K=\{L\}$; to obtain $C_{\cap }(X,Y)$, set

   $$K=\{Z\subseteq L\mid Y\cap Z=\mathrm{\varnothing }\}\cup \{X\cup Z\mid Z\subseteq L\wedge Y\cap Z\ne \mathrm{\varnothing }\};$$

   to obtain $C_{\cup }(X,Y)$, set

   $$K=\{Z\subseteq L\mid Y\cup Z\ne \mathrm{\varnothing }\}\cup \{X\cup Z\mid Z\subseteq L\wedge Y\cup Z=\mathrm{\varnothing }\}.$$

4. Turning to more specific examples, we have the example which inspired thedefinintion in the first place. Let $L$ be a set of logical expressionsconstructed from some set of sentence letters and predicate letters andthe usual connectives and quantifiers. Given a subset $X\subseteq L$,let $C(X)$ be the set of all expressions $\psi$ for which there existsa finite set of expressions ${\varphi }_1,\mathrm{\dots },{\varphi }_n$ such that$\mathrm{⌜}{\varphi }_1\wedge \mathrm{\cdots }\wedge {\varphi }_n\Rightarrow \psi \mathrm{⌝}$is a tautology. Note that this is a finitary consequence operator — itdoes not enable one to make the sort of deductions from infinite setsof premises described above.

5. This notion of consequence operator also applies to areas of mathematicsother than logic. For instance, suppose that $L$ is a vector space. Thenthe operator which assigns to a subset of $L$ the vector subspace which itspans is a consequence operator. This particular consequence operator isfinitary because if a vector $v$ belongs to the span of a set $X$, then$v$ can be expressed as a linear combination of a finite number of elementsof $X$.

6. The closure operator in topology is aconsequence operator. It is worth pointing out that not every consequenceoperator can be expressed as the closure operator for some topology becausethe closure operator satisfies some extra conditions beyond those whichdefine consequence operators. Typically, the closure operator is notfinitary because infinite subsets of topological spaces may have limitpoints.

4 Alternative Definition and Generalization

A consequence operator can be characterized by its fixed points. Givena consequence operator $C:𝒫(L)\to 𝒫(L)$,set $K=\{X\subseteq L\mid C(X)=X\}$. By the second definingproperty of consequence operator, we have $K=\{C(X)\mid X\in L\}$.One can show that

$$C(X)=\cap \{Y\in K\mid X\subseteq Y\}.$$

Conversely, suppose that $K$ is a subset of $L$ with the following minimum property:

• For every $X\in L$, there exists a $Y\in K$ suchthat $X\subseteq Y$ and if, for any $Z\in K$, if $X\subseteq Z$, then$Y\subseteq Z$.

Then the operator $C$ defined as

$$C(X)=\cap \{Y\in K\mid X\subseteq Y\}$$

is a consequence operator with $K$ as its set of fixed points.

One may also define consequence operators in the more general context ofa partially ordered set which may not be the power set of any set. Supposethat $⟨S,\le ⟩$ is a partially ordered set. Then we may definea consequence operator on this ordered set to be a map $C:S\to S$which satisfies the following three properties:

1. For all $X\in S$, it happens that $X\le C(X)$.

2. $C\circ C=C$

3. For all $X,Y\in S$, if $X\le Y$, then $C(X)\le C(Y)$

Such more general consequence operators arise frequently when we restrictattention to distinguished subsets of a set. As an example, we may considerthe following situation. Let $S$ be the set of linear subspaces of a Banachspace, ordered by inclusion. Then the operator $C:S\to S$ whichassigns to each subspace its Cauchy completion is a consequence operator.

As an example which does not arise this way, let $S=ℝ$ with theusual order. Then the ceiling function $\lceil \cdot \rceil :ℝ\to ℝ$ is a consequence operator.

For another example, let $S$ be the set of all fieldswith a countable number of elements. This set may be ordered as follows:$E\le F$ if and only if there exists a non-trivial morphism of $E$ into $F$. Then the operator which sends each field to its algebraic closure is aconsequence operator.

Title	consequence operator
Canonical name	ConsequenceOperator
Date of creation	2013-03-22 16:28:48
Last modified on	2013-03-22 16:28:48
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	40
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 03G25
Classification	msc 03G10
Classification	msc 03B22
Synonym	closure operator
Defines	finitary consequence operator
Defines	finite consequence operator
Defines	algebraic consequence operator
Defines	axiomatic consequence operator
Defines	axiomless consequence operator