Difference between revisions of "Lindenbaum method"
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Lindenbaum method is named after the Polish logician Adolf Lindenbaum who prematurely and without a clear trace disappeared in the turmoil of the Second World War at the age of about 37. (Cf.[15].) The method is based on the symbolic nature of formalized languages of deductive systems and opens a gate for applications of algebra to logic and, thereby, to [[Algebraic logic#Abstract algebraic logic|''Abstract algebraic logic'']]. | Lindenbaum method is named after the Polish logician Adolf Lindenbaum who prematurely and without a clear trace disappeared in the turmoil of the Second World War at the age of about 37. (Cf.[15].) The method is based on the symbolic nature of formalized languages of deductive systems and opens a gate for applications of algebra to logic and, thereby, to [[Algebraic logic#Abstract algebraic logic|''Abstract algebraic logic'']]. | ||
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+ | |||
== Lindenbaum's theorem == | == Lindenbaum's theorem == | ||
− | A formal propositional language, say $\mathcal{L}$, is understood as a nonempty set $\mathcal{ | + | A formal propositional language, say $\mathcal{L}$, is understood as a nonempty set $Vr_\mathcal{L}$ of symbols $p_0, p_1,... p_{\gamma}...$ called propositional variables and a finite set $\Pi$ of symbols $F_0, F_1,..., F_n$ called logical connectives. By $\overline{\overline{Vr_\mathcal{L}}}$ we denote the cardinality of $Vr_\mathcal{L}$. For each connective $F_i$, there is a natural number $\#(F_i)$ called the arity of the connective $F_i$. The notion of a statement (or a formula) is defined as follows: |
{| | {| | ||
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$\quad \quad \quad \quad A \in T_\mathcal{S} \Longleftrightarrow v(A)\in Cn_\mathcal{S}(\emptyset)$ for any valuation $v$.'' | $\quad \quad \quad \quad A \in T_\mathcal{S} \Longleftrightarrow v(A)\in Cn_\mathcal{S}(\emptyset)$ for any valuation $v$.'' | ||
− | |||
A matrix $\left<\mathfrak{A},\mathcal{F}\right>$ is said to be ''weakly adequate'' for a deductive system $\mathcal{S}$ if for any formula $A$, | A matrix $\left<\mathfrak{A},\mathcal{F}\right>$ is said to be ''weakly adequate'' for a deductive system $\mathcal{S}$ if for any formula $A$, | ||
$\quad \quad \quad \quad A \in T_\mathcal{S} \Longleftrightarrow v(A)\in \mathcal{F}$ for any valuation $v$. | $\quad \quad \quad \quad A \in T_\mathcal{S} \Longleftrightarrow v(A)\in \mathcal{F}$ for any valuation $v$. | ||
− | |||
Thus, according to Theorem 2, every structural system $\mathcal{S}$ has a weakly adequate $\mathcal{S}$-matrix of cardinality less than or equal to $\overline{\overline{\mathcal{V}}}+\aleph_0$. | Thus, according to Theorem 2, every structural system $\mathcal{S}$ has a weakly adequate $\mathcal{S}$-matrix of cardinality less than or equal to $\overline{\overline{\mathcal{V}}}+\aleph_0$. | ||
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Theorem 2 was discovered by A. Lindenbaum. Although this theorem was not published by the author, it had been known in Warsaw-Lvov logic circles at the time. In a published form it appeared for the first time in [9] without proof. Its proof appeared later on in the two independent publications of [8] and [6]. | Theorem 2 was discovered by A. Lindenbaum. Although this theorem was not published by the author, it had been known in Warsaw-Lvov logic circles at the time. In a published form it appeared for the first time in [9] without proof. Its proof appeared later on in the two independent publications of [8] and [6]. | ||
− | === References | + | == Wójcicki's theorems == |
+ | |||
+ | We get more {\mathcal{S}}-matrices, noticing the following. Let $\Sigma_\mathcal{S}$ be an $\mathcal{S}$-theory. The pair $\left<Fr_\mathcal{L},\Sigma_\mathcal{S} \right>$ is called a ''Lindenbaum matrix'' relative to $\mathcal{S}$. We observe that for any substitution $\sigma$, | ||
+ | |||
+ | $\quad \quad \quad \quad $ if $X \ \vdash_\mathcal{S} \ A$ and $\sigma[X] \subseteq \Sigma_\mathcal{S}$, then $\sigma(A) \in \Sigma_\mathcal{S}$}. | ||
+ | |||
+ | That is to say, any Lindenbaum matrix relative to a system $\mathcal{S}$ is an $\mathcal{S}$-model. | ||
+ | |||
+ | <br> | ||
+ | |||
+ | A deductive system $\mathcal{S}$ is said to be ''uniform'' if, given a set $X \subseteq Fr_\mathcal{S}$ and a consistent set $ Y \subseteq Fr_\mathcal{S}$, $X \cup Y \ \vdash_\mathcal{S} \ A$ and $Vr(Y) \cap Vr(A) = \emptyset$ imply $X \ \vdash_\mathcal{S} \ A$. A system $\mathcal{S}$ is ''couniform'' if for any collection $\{X_{i}\}_{i\in I}$ of formulas with $Vr(X_i) \cap Vr(X_j) = \emptyset$, providing $i \neq j$, if the set $\cup\{X_{i}\}_{i\in I}$ is inconsistent, then at least one $X_{i}$ is inconsistent as well. | ||
+ | |||
+ | '''Theorem 3''' (Wójcicki) | ||
+ | ''A structural deductive system $\mathcal{S}$ has a strongly adequate matrix if and only if $\mathcal{S}$ is both uniform and couniform.'' | ||
+ | |||
+ | For the "if" implication of the statement, the matrix of Theorem 2 is not enough. However, it is possible to extend the original language $\mathcal{L}$ to $\mathcal{L}^{+}$ in such a way that the ''natural extension'' $Cn_{\mathcal{S}^{+}}$ of $Cn_{\mathcal{S}}$ onto $\mathcal{L}^{+}$ allows one to define a Lindenbaum matrix $\left<\mathfrak{F}_{\mathcal{L}^{+}},Cn_{\mathcal{S}^{+}}(X)\right>$, for some $X \subseteq Fr_{\mathcal{L}^{+}}$, which is strongly adequate for $\mathcal{S}$. (Cf.[20] for detail.) | ||
+ | |||
+ | <br> | ||
+ | |||
+ | A pair $\left<\mathfrak{A}, where \ \{\mathcal{F}_{i}\}_{i\in I}\right>$, $\mathfrak{A}$ is an $\mathcal{L}$-algebra and each $\mathcal{F}_{i}\subseteq|\mathfrak{A}|$, is called a ''generalized matrix'' (or a $g$-''matrix'' for short). A $g$-matrix is a $g$-$\mathcal{S}$-''model'' (or a $g$-$\mathcal{S}$-''matrix'') if each $\left<\mathfrak{A},\mathcal{F}_{i}\right>$ is an $\mathcal{S}$-model. (In [4] a $g$-matrix is called an ''atlas''.) | ||
+ | |||
+ | ''Theorem 4''(Wójcicki) | ||
+ | ''For every structural deductive system $\mathcal{S}$, there is a $g$-$\mathcal{S}$-matrix $\mathcal{M}$ of cardinality $\overline{\overline{\mathcal{V}}}+\aleph_{0}$, which is strongly adequate for'' $\mathcal{S}$. | ||
+ | |||
+ | Indeed, let $\{\Sigma_\mathcal{S}\}$ be the collection of all $\mathcal{S}$-theories. Then the $g$-matrix $\left<Fr_\mathcal{L},\{\Sigma_\mathcal{S}\}\right>$ is strongly adequate for $\mathcal{S}$. (Cf.[20],[4] for detail.) | ||
+ | |||
+ | We note that, alternatively, one could use the notion of a''bundle of matrices''; a bundle is a set $\{\left<\mathfrak{A},\mathcal{F}_{i}\right> | i\in I \}$, where $\mathfrak{A}$ is an $\mathcal{L}$-algebra and each $\mathcal{F}_{i}$ is a filter of $\mathfrak{A}$. | ||
+ | |||
+ | |||
+ | === Historical remarks === | ||
+ | Theorem 3 was the result of the correction by R. Wójcicki of an erroneous assertion in [7], where the important question on the strong adequacy of a system was raised. | ||
+ | |||
+ | T. Smiley [14] was perhaps the first to propose $g$-matrices (known as ''Smiley matrices'') defined as pairs $\left<\mathfrak{A},Cn \right>$, where $\mathfrak{A}$ is an $\mathcal{L}$-algebra and an operator $Cn: \mathcal{P}(|\mathfrak{A}|) \rightarrow \mathcal{P}(|\mathfrak{A}|)$ satisfies the conditions $(c_1)-(c_3)$ (with $Cn$ instead if $Cn_\mathcal{S}$). Then, Smiley defined $x_1,..., x_n \ \vdash \ y$ if and only of $y \in Cn(\{x_1,...,x_n\})$, where it is assumed that $|\mathfrak{A}| \subseteq U$, where $U$ is a universal set of sentences. | ||
+ | |||
+ | == References == | ||
[1] Paul Bernays, ''Untersuchung des Aussagenkalküls der “Principia Mathematica”'', Math. Z. 25 (1926), 305–320. | [1] Paul Bernays, ''Untersuchung des Aussagenkalküls der “Principia Mathematica”'', Math. Z. 25 (1926), 305–320. |
Revision as of 18:50, 18 April 2013
Contents
Lindenbaum method (propositional language)
Lindenbaum method is named after the Polish logician Adolf Lindenbaum who prematurely and without a clear trace disappeared in the turmoil of the Second World War at the age of about 37. (Cf.[15].) The method is based on the symbolic nature of formalized languages of deductive systems and opens a gate for applications of algebra to logic and, thereby, to Abstract algebraic logic.
Lindenbaum's theorem
A formal propositional language, say $\mathcal{L}$, is understood as a nonempty set $Vr_\mathcal{L}$ of symbols $p_0, p_1,... p_{\gamma}...$ called propositional variables and a finite set $\Pi$ of symbols $F_0, F_1,..., F_n$ called logical connectives. By $\overline{\overline{Vr_\mathcal{L}}}$ we denote the cardinality of $Vr_\mathcal{L}$. For each connective $F_i$, there is a natural number $\#(F_i)$ called the arity of the connective $F_i$. The notion of a statement (or a formula) is defined as follows:
$(f_1)$ | Each variable $p\in\mathcal{V}$ is a formula; |
$(f_2)$ | If $F_i$ is a connective of the arity 0, then $F_i$ is a formula; |
$(f_3)$ | If $A_1, A_2,..., A_n$, $n\geq 1$, are formulas, and $F_n$ is a connective of arity $n$, then the symbolic expression $F_{n}A_{1}A_{2}... A_n$ is a formula; |
$(f_4)$ | A formula can be constructed only according to the rules $(f_1)-(f_3)$. |
The set of formulas will be denoted by $Fr_\mathcal{L}$ and $P(Fr_\mathcal{L})$ denotes the power set of $Fr_\mathcal{L}$. Given a set $X \subseteq Fr_\mathcal{L}$, we denote by $Vr(X)$ the set of propositional variables that occur in the formulas of $X$. Two formulas are counted equal if they are represented by two copies of the same string of symbols. (This is the key observation on which Theorem 1 is grounded.) Another key observation (due to Lindenbaum) is that $Fr_\mathcal{L}$ along with the connectives $\Pi$ can be regarded as an algebra of the similarity type associated with $\mathcal{L}$, which exemplifies an $\mathcal{L}$-algebra. We denote this algebra by $\mathfrak{F}_\mathcal{L}$. The importance of $\mathfrak{F}_\mathcal{L}$ can already be seen from the following observation.
Theorem 1. Algebra $\mathfrak{F}_\mathcal{L}$ is a free algebra of rank $\overline{\overline{\mathcal{V}}}$ with free generators $\mathcal{V}$ in the class $($variety$)$ of all $\mathcal{L}$-algebras. In other words, $\mathfrak{F}_\mathcal{L}$ is an absolutely free algebra of this class.
A useful feature of the set $Fr_\mathcal{L}$ is that it is closed under (simultaneous) substitution. More than that, any substitution $\sigma$ is an endomorphism
$\sigma: \mathfrak{F}_\mathcal{L}\longrightarrow \mathfrak{F}_\mathcal{L}$.
A monotone deductive system (or a deductive system or simply a system) is a relation between subsets and elements of $Fr_\mathcal{L}$. Each such system $\vdash_S$ is subject to the following conditions: For all $X,Y \subseteq \mathfrak{Fr}_\mathcal{L}$,
$(s_1)$ | if $A \in X$, then $X \ \vdash_\mathcal{S} \ A$; |
$(s_2)$ | if $X \ \vdash_\mathcal{S} \ B$ for all $B \in Y$, and $Y \ \vdash_\mathcal{S} \ A$, then $X \ \vdash_\mathcal{S} \ A$; |
$(s_3)$ | if $X \ \vdash_\mathcal{S} \ A$, then for every substitution $\sigma$, $\sigma[X] \ \vdash_\mathcal{S} \ \sigma(A)$. |
If $A$ is a formula and $\sigma$ is a substitution, $\sigma(A)$ is called a substitution instance of $A$. Thus, by $\sigma[X]$ above, one means the instances of the formulas of $X$ with respect to $\sigma$.
Given two sets $Y$ and $X$, we write
$\quad \quad \quad Y \sqsubseteq X $
if $Y$ is a finite (may be empty) subset of $X$.
A deductive system is said to be finitary if, in addition, it satisfies the following:
$(s_4)$ | if $X \ \vdash_\mathcal{S} \ A$, then there is $Y \sqsubseteq X$ such that $Y \ \vdash_\mathcal{S} \ A$. |
We note that the monotonicity property
$\quad \quad \quad \quad$ if $X \subseteq Y$ and $X \ \vdash_\mathcal{S} \ A$, then $Y \ \vdash_\mathcal{S} \ A$
is not postulated, because it follows from $(s_1)$ and $(s_2)$.
Each deductive system $\vdash_\mathcal{S}$ induces the (monotone structural) consequence operator $Cn_{\mathcal{S}}$ defined on the power set of $Fr_\mathcal{L}$ as follows: For every $X \subseteq Fr_\mathcal{L}$,
$\quad \quad \quad \quad A \in Cn_\mathcal{S}(X) \Longleftrightarrow X \ \vdash_\mathcal{S} \ A, \quad \quad \quad \quad \quad \quad \quad \quad (1)$
so that the following conditions are fulfilled: For all $X,Y \subseteq Fr_\mathcal{L}$ and any substitution $\sigma$,
$(c_1)$ | $X \subseteq Cn_\mathcal{S}(X);$ | (Reflexivity) |
$(c_2)$ | $Cn_\mathcal{S}(Cn_\mathcal{S}(X)) = Cn_\mathcal{S}(X);$ | (Idenpotency) |
$(c_3)$ | if $X \subseteq Y$, then $Cn_\mathcal{S}(X) \subseteq Cn_\mathcal{S}(Y);$ | (Monotonicity) |
$(c_4)$ | $\sigma[Cn_\mathcal{S}(X)] \subseteq Cn_\mathcal{S}(\sigma[X]).$ | (Structurality) |
If $\vdash_\mathcal{S}$ is finitary, then
$(c_5)$ | $Cn_\mathcal{S}(X) = \bigcup\lbrace Cn_\mathcal{S}(Y) \ | \ Y \Subset X \rbrace$ |
in which case $Cn_{\mathcal{S}}$ is called finitary.
Conversely, if an operator $Cn:\cal{P}(Fr_\mathcal{L})\rightarrow \cal{P}(Fr_\mathcal{L})$ satisfies the conditions $(c_1)-(c_4)$ (with $Cn$ instead of $Cn_\mathcal{S}$), then the equivalence
$\quad \quad \quad \quad X \ \vdash_\mathcal{S} \ A \Longleftrightarrow A \in {Cn}(X)$defines a deductive system, $\mathcal{S}$. Thus (1) allows one to use the deductive system and consequence operator (in a fixed formal language) interchangeably or even in one and the same context. For instance, we call $T_\mathcal{S} = Cn_\mathcal{L}(\emptyset)$ the set of theorems of the system $\vdash_\mathcal{S}$ (i.e. $\mathcal{S}$-theorems), and given a subset $X \subseteq Fr_\mathcal{S}$, $Cn-|mathcal{S}{X}$ is called the $\mathcal{S}$-theory generated by $X$. A subset $X \subseteq Fr_\mathcal{S}$, as well as the theory $Cn_\mathcal{S}(X)$, is called inconsistent if $Cn_\mathcal{S}(X) = Fr_\mathcal{S}$; otherwise both are consistent. Thus, given a system $\vdash_\mathcal{S}$, $T_\mathcal{S}$ is one of the system's theories; that is to say, if $X \subseteq T_\mathcal{S}$ and $X \vdash_\mathcal{S} A$, then $A \in T_\mathcal{S}$. This simple observation sheds light on the central idea of Lindenbaum method, which will be explained soon. For now, let us fix the ordered pair $\left<\mathcal{F}_\mathcal{L},T\mathcal{L}\right>$ and call it a Lindenbaum matrix. (The full definition will be given later.) We note that an operator $Cn$ satisfying $(c_1)-(c_3)$ can be obtained from a "closure system" over $Fr_\mathcal{L}$; that is for any subset $\cal{A}\subseteq P(Fr_\mathcal{L})$, which is closed under arbitrary intersection, we define:
$\quad \quad \quad \quad Cn_\mathcal{A}(X)=\cap \lbrace Y \ | \ X \subseteq Y \mbox{ and } Y \in \cal{A} \rbrace.$Another way of defining deductive systems is through the use of logical matrices. Given a language $\mathcal{L}$, a logical $\mathcal{L}$-matrix (or simply a matrix) is a pair $\mathcal{M} = \left<\mathfrak{A},\mathcal{F}\right>$, where $\mathfrak{A}$ is an $\mathcal{L}$-algebra and $\mathcal{F}\subseteq|\mathfrak{A}|$, where the latter is the universe of $\mathfrak{A}$. The set $\mathcal{F}$ is called the filter of the matrix and its elements are called designated. Given a matrix $\mathcal{M} = \left<\mathfrak{A},\mathcal{F}\right>$, the cardinality of $|\mathfrak{A}|$ is also the cardinality of $\mathcal{M}$.
Given a matrix $\mathcal{M}=\left<\mathfrak{A},\mathcal{F}\right>$, any homomorphism of $\mathfrak{A}$ into $\mathfrak{A}$ is called a valuation (or an assignment). Each such homomorphism can be obtained simply by assigning elements of $|\mathfrak{A}|$ to the variables of $Vr_\mathcal{L}$, since, by virtue of Theorem 1, any $v: Vr_\mathcal{L} \longrightarrow |\mathfrak{A}|$ can be extended uniquely to a homomorphism $\hat{v}: \mathfrak{A} \longrightarrow \mathfrak{A}$. Usually, $v$ is meant under a valuation (or an assignment) of variables in a matrix.
Now let $\sigma$ be a substitution and $v$ be any assignment in an algebra {\mathfrak{A}}. Then, defining
$\quad \quad \quad \quad v_{\sigma}=v\circ\sigma, \quad \quad \quad \quad \quad \quad \quad \quad (2)$
we observe that $v_{\sigma}$ is also an assignment in $\mathfrak{A}$.
With each matrix $\mathcal{M}=\left<\mathfrak{A},\mathcal{F}\right>$, we associate a relation $\models_\mathcal{M}$ between subsets of $Fr_\mathcal{L}$ and formulas of $Fr_\mathcal{L}$. Namely we define
$ \quad \quad \quad \quad X \ \models_\mathcal{M} \ A \Longleftrightarrow \text{ for every assignment } v, \text{ if } v[X]\subseteq \mathcal{F}, \text{ then } v(A)\in \mathcal{F}$.
Then, we observe that the following properties hold:
$(m_1)$ | if $A \in X$, then $X \ \models_\mathcal{M} \ A$ |
$(m_2)$ | if $X\models_\mathcal{M} B$ for all $B\in Y$, and $Y \ \models_\mathcal{M} \ A$, then $X \ \models_\mathcal{M} \ A.$ |
Also, with help of the definition (2), we derive the following:
$(m_3)$ | if $X \ \models_\mathcal{M} \ A$, then for every substitution $\sigma$, $\sigma[X] \ \models_\mathcal{M} \ \sigma(A)$. |
Comparing the condition $(m_1)-(m_3)$ with $(s_1)-(s_3)$, we conclude that every matrix defines a structural deductive system and hence, in view of (1), a structural consequence operator.
Given a system $\mathcal{S}$, suppose a matrix $\mathcal{M}=\left<\mathfrak{A},\mathcal{F}\right>$ satisfies the condition
$\quad \quad \quad \quad $ if $X \ \vdash_\mathcal{S} A$ and $v[X] \subseteq \mathcal{F}$, then $v(A) \in \mathcal{F} \quad \quad \quad \quad (3)$
Then the filter $\mathcal{F}$ is called an $\mathcal{S}$-filter and the matrix $\mathcal{M}$ is called an $\mathcal{S}$-matrix (or an $\mathcal{S}$-model). In view of (3), $\mathcal{S}$-matrices are an important tool in showing that $X \ \vdash_\mathcal{S} \ A$ does not hold. This idea has been employed in proving that one axiom is independent from a group of others in the search for an independent axiomatic system, as well as for semantic completeness results.
As Lindenbaum's famous theorem below explains, every structural system $\mathcal{S}$ has an $\mathcal{S}$-model.
Theorem 2. For any structural deductive system $\mathcal{S}$, the matrix $\left<Fr_\mathcal{L},Cn_\mathcal{S}(\emptyset)\right>$ is an $\mathcal{S}$-model. Moreover, for any formula $A$,
$\quad \quad \quad \quad A \in T_\mathcal{S} \Longleftrightarrow v(A)\in Cn_\mathcal{S}(\emptyset)$ for any valuation $v$.
A matrix $\left<\mathfrak{A},\mathcal{F}\right>$ is said to be weakly adequate for a deductive system $\mathcal{S}$ if for any formula $A$,
$\quad \quad \quad \quad A \in T_\mathcal{S} \Longleftrightarrow v(A)\in \mathcal{F}$ for any valuation $v$.
Thus, according to Theorem 2, every structural system $\mathcal{S}$ has a weakly adequate $\mathcal{S}$-matrix of cardinality less than or equal to $\overline{\overline{\mathcal{V}}}+\aleph_0$.
An $\mathcal{S}$-matrix is called strongly adequate for $\mathcal{S}$ if for any set $X \subseteq Fr_\mathcal{L}$ and any formula $A$,
$ \quad \quad \quad \quad X \ \vdash_\mathcal{S} \ A \Longleftrightarrow X \ \models_\mathcal{M} \ A. \quad \quad \quad \quad (4)$
We note that, if $\overline{\overline{\mathcal{V}}} \leq \aleph_{0}$, Theorem 2 cannot be improved to include strong adequacy of an denumerable matrix, for if $\mathcal{S} = IPC$ (intuitionistic propositional calculus), there is no denumerable matrix $\mathcal{M}$ with (4). (Cf.[21].)
Historical remarks
A. Tarski seems to be the first who promoted "the view of matrix formation as a general method of constructing systems" [9]. However, matrices had been employed earlier, e.g., by P. Bernays [q] and others either in the search for an independent axiomatic system or for defining a system different from classical logic. Also, later on J.C.C. McKinsey [10] used matrices to prove independence of logical connectives in intuitionistic propositional logic.
Theorem 2 was discovered by A. Lindenbaum. Although this theorem was not published by the author, it had been known in Warsaw-Lvov logic circles at the time. In a published form it appeared for the first time in [9] without proof. Its proof appeared later on in the two independent publications of [8] and [6].
Wójcicki's theorems
We get more {\mathcal{S}}-matrices, noticing the following. Let $\Sigma_\mathcal{S}$ be an $\mathcal{S}$-theory. The pair $\left<Fr_\mathcal{L},\Sigma_\mathcal{S} \right>$ is called a Lindenbaum matrix relative to $\mathcal{S}$. We observe that for any substitution $\sigma$,
$\quad \quad \quad \quad $ if $X \ \vdash_\mathcal{S} \ A$ and $\sigma[X] \subseteq \Sigma_\mathcal{S}$, then $\sigma(A) \in \Sigma_\mathcal{S}$}.
That is to say, any Lindenbaum matrix relative to a system $\mathcal{S}$ is an $\mathcal{S}$-model.
A deductive system $\mathcal{S}$ is said to be uniform if, given a set $X \subseteq Fr_\mathcal{S}$ and a consistent set $ Y \subseteq Fr_\mathcal{S}$, $X \cup Y \ \vdash_\mathcal{S} \ A$ and $Vr(Y) \cap Vr(A) = \emptyset$ imply $X \ \vdash_\mathcal{S} \ A$. A system $\mathcal{S}$ is couniform if for any collection $\{X_{i}\}_{i\in I}$ of formulas with $Vr(X_i) \cap Vr(X_j) = \emptyset$, providing $i \neq j$, if the set $\cup\{X_{i}\}_{i\in I}$ is inconsistent, then at least one $X_{i}$ is inconsistent as well.
Theorem 3 (Wójcicki) A structural deductive system $\mathcal{S}$ has a strongly adequate matrix if and only if $\mathcal{S}$ is both uniform and couniform.
For the "if" implication of the statement, the matrix of Theorem 2 is not enough. However, it is possible to extend the original language $\mathcal{L}$ to $\mathcal{L}^{+}$ in such a way that the natural extension $Cn_{\mathcal{S}^{+}}$ of $Cn_{\mathcal{S}}$ onto $\mathcal{L}^{+}$ allows one to define a Lindenbaum matrix $\left<\mathfrak{F}_{\mathcal{L}^{+}},Cn_{\mathcal{S}^{+}}(X)\right>$, for some $X \subseteq Fr_{\mathcal{L}^{+}}$, which is strongly adequate for $\mathcal{S}$. (Cf.[20] for detail.)
A pair $\left<\mathfrak{A}, where \ \{\mathcal{F}_{i}\}_{i\in I}\right>$, $\mathfrak{A}$ is an $\mathcal{L}$-algebra and each $\mathcal{F}_{i}\subseteq|\mathfrak{A}|$, is called a generalized matrix (or a $g$-matrix for short). A $g$-matrix is a $g$-$\mathcal{S}$-model (or a $g$-$\mathcal{S}$-matrix) if each $\left<\mathfrak{A},\mathcal{F}_{i}\right>$ is an $\mathcal{S}$-model. (In [4] a $g$-matrix is called an atlas.)
Theorem 4(Wójcicki) For every structural deductive system $\mathcal{S}$, there is a $g$-$\mathcal{S}$-matrix $\mathcal{M}$ of cardinality $\overline{\overline{\mathcal{V}}}+\aleph_{0}$, which is strongly adequate for $\mathcal{S}$.
Indeed, let $\{\Sigma_\mathcal{S}\}$ be the collection of all $\mathcal{S}$-theories. Then the $g$-matrix $\left<Fr_\mathcal{L},\{\Sigma_\mathcal{S}\}\right>$ is strongly adequate for $\mathcal{S}$. (Cf.[20],[4] for detail.)
We note that, alternatively, one could use the notion of abundle of matrices; a bundle is a set $\{\left<\mathfrak{A},\mathcal{F}_{i}\right> | i\in I \}$, where $\mathfrak{A}$ is an $\mathcal{L}$-algebra and each $\mathcal{F}_{i}$ is a filter of $\mathfrak{A}$.
Historical remarks
Theorem 3 was the result of the correction by R. Wójcicki of an erroneous assertion in [7], where the important question on the strong adequacy of a system was raised.
T. Smiley [14] was perhaps the first to propose $g$-matrices (known as Smiley matrices) defined as pairs $\left<\mathfrak{A},Cn \right>$, where $\mathfrak{A}$ is an $\mathcal{L}$-algebra and an operator $Cn: \mathcal{P}(|\mathfrak{A}|) \rightarrow \mathcal{P}(|\mathfrak{A}|)$ satisfies the conditions $(c_1)-(c_3)$ (with $Cn$ instead if $Cn_\mathcal{S}$). Then, Smiley defined $x_1,..., x_n \ \vdash \ y$ if and only of $y \in Cn(\{x_1,...,x_n\})$, where it is assumed that $|\mathfrak{A}| \subseteq U$, where $U$ is a universal set of sentences.
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Lindenbaum method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lindenbaum_method&oldid=29658