hcju/proofwikips huggingface dataset:
Definition:Distributed Term of Categorical Syllogism/Predicate
Let $\mathbf{\Phi} \left({S, P}\right)$ be a categorical statement, expressed in abbreviated form. Let $\mathbf{\Phi}$ be a negative categorical statement. Then $S$ is described as being '''distributed'''.
Definition:Undistributed Term of Categorical Syllogism
A term in a syllogism is '''undistributed''' if the categorical statement in which it occurs does not refer to the whole of the class designated by the term. === Undistributed Subject === {{:Definition:Undistributed Term of Categorical Syllogism/Subject}} === Undistributed Predicate === {{:Definition:Undistributed Term of Categorical Syllogism/Predicate}}
Definition:Undistributed Term of Categorical Syllogism/Predicate
Let $\mathbf{\Phi} \left({S, P}\right)$ be a categorical statement, expressed in abbreviated form. Let $\mathbf{\Phi}$ be an affirmative categorical statement. Then $P$ is described as being '''undistributed'''.
Definition:Undistributed Term of Categorical Syllogism/Subject
Let $\mathbf{\Phi} \left({S, P}\right)$ be a categorical statement, expressed in abbreviated form. Let $\mathbf{\Phi}$ be a particular categorical statement. Then $S$ is described as being '''undistributed'''.
Definition:Distributed Term of Categorical Syllogism/Examples
=== Universal Affirmative === {{:Definition:Distributed Term of Categorical Syllogism/Examples/Universal Affirmative}} === Universal Negative === {{:Definition:Distributed Term of Categorical Syllogism/Examples/Universal Negative}} === Particular Affirmative === {{:Definition:Distributed Term of Categorical Syllogism/Examples/Particular Affirmative}} === Particular Negative === {{:Definition:Distributed Term of Categorical Syllogism/Examples/Particular Negative}} This can be tabulated as follows: :$\begin{array}{rcl} \mathbf{A} & (S, & P) \\ & d & u \end{array} \qquad \begin{array}{rcl} \mathbf{E} & (S, & P) \\ & d & d \end{array}$ :$\begin{array}{rcl} \mathbf{I} & (S, & P) \\ & u & u \end{array} \qquad \begin{array}{rcl} \mathbf{O} & (S, & P) \\ & u & d \end{array}$ where $d$ denotes a distributed term and $u$ denotes an undistributed term.
Definition:Non-Trivial Subgroup
Let $\struct {G, \circ}$ be a group whose identity is $e$. A '''non-trivial subgroup''' of $\struct {G, \circ}$ is a subgroup $\struct {H, \circ}$ of $G$ such that $H \ne \set e$.
Definition:Non-Trivial Element
Let $\struct {G, \circ}$ be a group whose identity is $e$. A '''non-trivial element''' of $G$ is an element of $G$ distinct from $e$.
Definition:Commutative/Set
Let $\struct {S, \circ}$ be an algebraic structure. Let $X \subseteq S$ be a subset of $S$ such that: :$\forall a, b \in X: a \circ b = b \circ a$ That is, every element of $X$ commutes with every other element. Then $X$ is a '''commuting set of elements''' of $S$.
Definition:Proper Subgroup/Non-Trivial
Let $\struct {H, \circ}$ be a subgroup of $\struct {G, \circ}$ such that $\set e \subset H \subset G$, that is: :$H \ne \set e$ :$H \ne G$ Then $\struct {H, \circ}$ is a '''non-trivial proper subgroup of $\struct {G, \circ}$'''.
Definition:Group Type
An equivalence class under group isomorphism is a '''(group) type'''.
Definition:Real Function/Definition by Formula
A function $f: S \to T$ can be considered as a '''formula''' which tells us how to determine what the value of $y \in T$ is when we have selected a value for $x \in S$. === As an Equation === {{:Definition:Real Function/Definition by Equation}}
Definition:Real Function/Definition by Equation
It is often convenient to refer to an equation or formula as though it were a function. What is meant is that the equation ''defines'' the function; that is, it specifies the rule by which we obtain the value of $y$ from a given $x$. === Square Function === For example, let $x, y \in \R$. {{:Definition:Real Square Function|Square Function}} We may express this as $y = x^2$, and use this equation to ''define'' this function. This may be conceived as: :For each $x \in \R$, the number $y \in \R$ assigned to it is that which we get by squaring $x$. Another useful notation is: :$\forall x \in \R: x \mapsto x^2$
Definition:Nu Function
The '''$\nu$ function''' is the function $\nu: \Z_{>0} \to \Z_{>0}$ is defined as: :$\forall n \in \Z_{>0}: \map \nu n = $ the number of types of group of order $n$
Definition:Ring of Sets/Definition 1
A '''ring of sets''' $\RR$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \text {RS} 1_1         | lc= Non-Empty:         | m = \RR \ne \O }} {{axiom | n = \text {RS} 2_1         | lc= Closure under Intersection:         | q = \forall A, B \in \RR         | m = A \cap B \in \RR }} {{axiom | n = \text {RS} 3_1         | lc= Closure under Symmetric Difference:         | q = \forall A, B \in \RR         | m = A * B \in \RR }} {{end-axiom}}
Definition:Ring of Sets/Definition 2
A '''ring of sets''' $\RR$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \text {RS} 1_2         | lc= Empty Set:         | m = \O \in \RR }} {{axiom | n = \text {RS} 2_2         | lc= Closure under Set Difference:         | q = \forall A, B \in \RR         | m = A \setminus B \in \RR }} {{axiom | n = \text {RS} 3_2         | lc= Closure under Union:         | q = \forall A, B \in \RR         | m = A \cup B \in \RR }} {{end-axiom}}
Definition:Limit Inferior of Sequence of Sets/Definition 1
Let $\sequence {E_n : n \in \N}$ be a sequence of sets. Then the '''limit inferior''' of $\sequence {E_n : n \in \N}$, denoted $\ds \liminf_{n \mathop \to \infty} E_n$, is defined as: {{begin-eqn}} {{eqn | l = \liminf_{n \mathop \to \infty} E_n       | o = :=       | r = \bigcup_{n \mathop = 0}^\infty \bigcap_{i \mathop = n}^\infty E_n }} {{eqn | r = \paren {E_0 \cap E_1 \cap E_2 \cap \ldots} \cup \paren {E_1 \cap E_2 \cap E_3 \cap \ldots} \cup \cdots }} {{end-eqn}}
Definition:Limit Superior of Sequence of Sets/Definition 1
Let $\sequence {E_n : n \in \N}$ be a sequence of sets. Then the '''limit superior''' of $\sequence {E_n: n \in \N}$, denoted $\ds \limsup_{n \mathop \to \infty} E_n$, is defined as: {{begin-eqn}} {{eqn | l = \limsup_{n \mathop \to \infty} E_n       | o = :=       | r = \bigcap_{i \mathop = 0}^\infty \bigcup_{n \mathop = i}^\infty E_n }} {{eqn | r = \paren {E_0 \cup E_1 \cup E_2 \cup \ldots} \cap \paren {E_1 \cup E_2 \cup E_3 \cup \ldots} \cap \ldots }} {{end-eqn}}
Definition:Limit Superior of Sequence of Sets/Definition 2
Let $\sequence {E_n: n \in \N}$ be a sequence of sets. Then the '''limit superior''' of $\sequence {E_n: n \in \N}$, denoted $\ds \limsup_{n \mathop \to \infty} E_n$, is defined as: :$\ds \limsup_{n \mathop \to \infty} E_n = \set {x : x \in E_i \text { for infinitely many } i}$
Definition:Limit Inferior of Sequence of Sets/Definition 2
Let $\sequence {E_n : n \in \N}$ be a sequence of sets. Then the '''limit inferior''' of $\sequence {E_n : n \in \N}$, denoted $\ds \liminf_{n \mathop \to \infty} E_n$, is defined as: :$\ds \liminf_{n \mathop \to \infty} E_n := \set {x: x \in E_i \text { for all but finitely many } i}$
Definition:Ring of Sets/Definition 3
A '''ring of sets''' $\RR$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \text {RS} 1_3         | lc= Empty Set:         | m = \varnothing \in \RR }} {{axiom | n = \text {RS} 2_3         | lc= Closure under Set Difference:         | q = \forall A, B \in \RR         | m = A \setminus B \in \RR }} {{axiom | n = \text {RS} 3_3         | lc= Closure under Disjoint Union:         | q = \forall A, B \in \RR         | m = A \cap B = \O \implies A \cup B \in \RR }} {{end-axiom}}
Definition:Sigma-Ring/Definition 1
A '''$\sigma$-ring''' is a ring of sets which is closed under countable unions. That is, a ring of sets $\Sigma$ is a '''$\sigma$-ring''' : :$\ds A_1, A_2, \ldots \in \Sigma \implies \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma$
Definition:Sigma-Ring/Definition 2
A '''$\sigma$-ring''' $\Sigma$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \text {SR} 1         | lc= Empty Set:         | m = \O \in \Sigma }} {{axiom | n = \text {SR} 2         | lc= Closure under Set Difference:         | q = \forall A, B \in \Sigma         | m = A \setminus B \in \Sigma }} {{axiom | n = \text {SR} 3         | lc= Closure under Countable Unions:         | q = \forall A_n \in \Sigma: n = 1, 2, \ldots         | m = \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma }} {{end-axiom}}
Definition:Formal Semantics
A '''formal semantics''' for $\mathcal L$ comprises: * A collection of objects called '''structures'''; * A notion of '''validity''' of $\mathcal L$-WFFs in these structures. Often, a '''formal semantics''' provides these by using a lot of auxiliary definitions.
Definition:Sigma-Algebra/Definition 4
Let $X$ be a set. A '''$\sigma$-algebra''' $\Sigma$ over $X$ is an algebra of sets which is closed under countable unions.
Definition:Sigma-Algebra/Definition 1
Let $X$ be a set. A '''$\sigma$-algebra''' $\Sigma$ over $X$ is a system of subsets of $X$ with the following properties: {{begin-axiom}} {{axiom | n = \text {SA} 1         | lc= Unit:         | m = X \in \Sigma }} {{axiom | n = \text {SA} 2         | lc= Closure under Complement:         | q = \forall A \in \Sigma         | m = \relcomp X A \in \Sigma }} {{axiom | n = \text {SA} 3         | lc= Closure under Countable Unions:         | q = \forall A_n \in \Sigma: n = 1, 2, \ldots         | m = \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma }} {{end-axiom}}
Definition:Logical Language
Let $\mathcal L$ be a formal language used in symbolic logic. Then $\mathcal L$ is called a '''logical language'''.
Definition:Tautology
A '''tautology''' is a statement which is ''always true'', independently of any relevant circumstances that could theoretically influence its truth value. It is epitomised by the form: :$p \implies p$ that is: :'''if $p$ is true then $p$ is true.''' An example of a "relevant circumstance" here is the truth value of $p$. The archetypal '''tautology''' is symbolised by $\top$, and referred to as Top.
Definition:Formal Semantics/Valid
Part of specifying a formal semantics $\mathscr M$ for $\LL$ is to define a notion of '''validity'''. Concretely, a precise meaning needs to be assigned to the phrase: :"The $\LL$-WFF $\phi$ is '''valid''' in the $\mathscr M$-structure $\MM$." It can be expressed symbolically as: :$\MM \models_{\mathscr M} \phi$
Definition:Formal Semantics/Structure
Part of specifying a formal semantics $\mathscr M$ for $\LL$ is to specify '''structures''' $\MM$ for $\mathscr M$. A '''structure''' can in principle be any object one can think of. However, to get a useful formal semantics, the '''structures''' should support a meaningful definition of validity for the WFFs of $\LL$. It is common that '''structures''' are sets, often endowed with a number of relations or functions.
Definition:Model (Logic)
Let $\mathscr M$ be a formal semantics for a logical language $\LL$. Let $\MM$ be a structure of $\mathscr M$. === Model of Logical Formula === {{:Definition:Model (Logic)/Logical Formula}} === Model of Set of Logical Formulas === {{:Definition:Model (Logic)/Set of Logical Formulas}}
Definition:Sigma-Algebra/Definition 3
A '''$\sigma$-algebra''' $\Sigma$ is a $\sigma$-ring with a unit.
Definition:Boolean Interpretation/Formal Semantics
The boolean interpretations for $\mathcal L_0$ can be interpreted as a formal semantics for $\mathcal L_0$, which we denote by $\mathrm{BI}$. The structures of $\mathrm{BI}$ are the boolean interpretations. A WFF $\phi$ is declared ($\mathrm{BI}$-)valid in a boolean interpretation $v$ iff: :$v \left({\phi}\right) = T$ Symbolically, this can be expressed as: :$v \models_{\mathrm{BI}} \phi$
Definition:Model (Logic)/Logical Formula
Let $\phi$ be a logical formula of $\LL$. Then $\MM$ is a '''model of $\phi$''' : :$\MM \models_{\mathscr M} \phi$ that is, if $\phi$ is valid in $\MM$.
Definition:Model (Logic)/Set of Logical Formulas
Let $\FF$ be a set of logical formulas of $\mathcal L$. Then $\MM$ is a '''model of $\FF$''' : :$\MM \models_{\mathscr M} \phi$ for every $\phi \in \FF$ that is, if it is a model of every logical formula $\phi \in \FF$.
Definition:Sound Proof System
Let $\LL$ be a logical language. Let $\mathscr P$ be a proof system for $\LL$, and let $\mathscr M$ be a formal semantics for $\LL$. Then $\mathscr P$ is said to be '''sound for $\mathscr M$''' : :Every $\mathscr P$-theorem is an $\mathscr M$-tautology. Symbolically, this can be expressed as the statement that, for every logical formula $\phi$ of $\LL$: :$\vdash_{\mathscr P} \phi$ implies $\models_{\mathscr M} \phi$ === Strongly Sound Proof System === {{:Definition:Sound Proof System/Strongly Sound}}
Definition:Complete Proof System
Let $\LL$ be a logical language. Let $\mathscr P$ be a proof system for $\LL$, and let $\mathscr M$ be a formal semantics for $\LL$. Then $\mathscr P$ is said to be '''complete for $\mathscr M$''' : :Every $\mathscr M$-tautology is a $\mathscr P$-theorem. Symbolically, this can be expressed as the statement that, for every logical formula $\phi$ of $\LL$: :$\models_{\mathscr M} \phi$ implies $\vdash_{\mathscr P} \phi$ === Strongly Complete Proof System === {{:Definition:Complete Proof System/Strongly Complete}}
Definition:Unsatisfiable/Formula
A logical formula $\phi$ of $\LL$ is '''unsatisfiable for $\mathscr M$''' : :$\phi$ is valid in none of the structures of $\mathscr M$ That is, for all structures $\MM$ of $\mathscr M$: :$\MM \not\models_{\mathscr M} \phi$
Definition:Sigma-Ring/Definition 3
A '''$\sigma$-ring''' $\Sigma$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \text {SR} 1'         | lc= Empty Set:         | m = \O \in \Sigma }} {{axiom | n = \text {SR} 2'         | lc= Closure under Set Difference:         | q = \forall A, B \in \Sigma         | m = A \setminus B \in \Sigma }} {{axiom | n = \text {SR} 3'         | lc= Closure under Countable Disjoint Unions:         | q = \forall A_n \in \Sigma: n = 1, 2, \ldots         | m = \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma }} {{end-axiom}}
Definition:Sigma-Algebra/Definition 2
Let $X$ be a set. A '''$\sigma$-algebra''' $\Sigma$ over $X$ is a system of subsets of $X$ with the following properties: {{begin-axiom}} {{axiom | n = \text {SA} 1'         | lc= Unit:         | m = X \in \Sigma }} {{axiom | n = \text {SA} 2'         | lc= Closure under Complement:         | q = \forall A \in \Sigma         | m = \relcomp X A \in \Sigma }} {{axiom | n = \text {SA} 3'         | lc= Closure under Countable Disjoint Unions:         | q = \forall A_n \in \Sigma: n = 1, 2, \ldots         | m = \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma }} {{end-axiom}}
Definition:Satisfiable/Formula
A logical formula $\phi$ of $\mathcal L$ is '''satisfiable for $\mathscr M$''' iff: :$\phi$ is valid in some structure $\mathcal M$ of $\mathscr M$ That is, there exists some structure $\mathcal M$ of $\mathscr M$ such that: :$\mathcal M \models_{\mathscr M} \phi$
Definition:Falsifiable/Formula
A logical formula $\phi$ of $\mathcal L$ is '''falsifiable for $\mathscr M$''' iff: :$\phi$ is not valid in some structure $\mathcal M$ of $\mathscr M$ That is, there exists some structure $\mathcal M$ of $\mathscr M$ such that: :$\mathcal M \not\models_{\mathscr M} \phi$
Definition:Satisfiable
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. === Satisfiable Formula === {{:Definition:Satisfiable/Formula}} === Satisfiable Set of Formulas === {{:Definition:Satisfiable/Set of Formulas}} === Satisfiable for Boolean Interpretations === {{:Definition:Satisfiable/Boolean Interpretations}}
Definition:Satisfiable/Set of Formulas
A collection $\mathcal F$ of logical formulas of $\mathcal L$ is '''satisfiable for $\mathscr M$''' iff: :There is some $\mathscr M$-model $\mathcal M$ of $\mathcal F$ That is, there exists some structure $\mathcal M$ of $\mathscr M$ such that: :$\mathcal M \models_{\mathscr M} \mathcal F$
Definition:Falsifiable
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. === Falsifiable Formula === {{:Definition:Falsifiable/Formula}} === Falsifiable Set of Formulas === {{:Definition:Falsifiable/Set of Formulas}} === Falsifiable for Boolean Interpretations === {{:Definition:Falsifiable/Boolean Interpretations}}
Definition:Falsifiable/Set of Formulas
A collection $\mathcal F$ of logical formulas of $\mathcal L$ is '''falsifiable for $\mathscr M$''' iff: :Some $\mathscr M$-structure $\mathcal M$ is not a model of $\mathcal F$ That is, there exists some structure $\mathcal M$ of $\mathscr M$ such that: :$\mathcal M \not\models_{\mathscr M} \mathcal F$
Definition:Unsatisfiable/Set of Formulas
A collection $\FF$ of logical formulas of $\LL$ is '''unsatisfiable for $\mathscr M$''' : :There is no $\mathscr M$-model $\MM$ for $\FF$ That is, for all structures $\MM$ of $\mathscr M$: :$\MM \not \models_{\mathscr M} \FF$
Definition:Rooted Tree/Child Node/Grandchild Node
A child of a child node of a node $t$ can be referred to as a '''grandchild node''' of $t$. In terms of the parent mapping $\pi$ of $T$, a '''grandchild node''' of $t$ is a node $s$ such that: :$\pi \left({\pi \left({s}\right)}\right) = t$
Definition:Propositional Tableau/Graphical Representation
The conditions by which a propositional tableau may be identified, respectively constructed, can be graphically represented like so: ::{| style="text-align: center" |- | $\begin{xy}\xymatrix @R=1em @C=3em{  \ar@{.}[d] & \ar@{.}[d] & \ar@{.}[d] & \ar@{.}[d] \\  *++{\neg \neg \mathbf A} \ar@{.}[d] &  *++{\mathbf A \land \mathbf B} \ar@{.}[d] &  *++{\neg \left({\mathbf A \lor \mathbf B}\right)} \ar@{.}[d] &  *++{\neg \left({\mathbf A \implies \mathbf B}\right)} \ar@{.}[d] \\  *++{t} \ar@{-}[d] &  *++{t} \ar@{-}[d] &  *++{t} \ar@{-}[d] &  *++{t} \ar@{-}[d] \\  \mathbf A &  \mathbf A \ar@{=}[d] &  \neg \mathbf A \ar@{=}[d] &  \mathbf A \ar@{=}[d] \\   & \mathbf B & \neg \mathbf B & \neg \mathbf B \\  \boxed{\neg\neg} & \boxed \land & \boxed{\neg\lor} & \boxed{\neg\implies} }\end{xy}$ |- | $\begin{xy}\xymatrix @R=1em @C=1em{  & \ar@{.}[d] & & & & & & \ar@{.}[d] & & & & & & \ar@{.}[d] \\  & *++{\mathbf A \lor \mathbf B} \ar@{.}[d] &  & & & &  & *++{\neg \left({\mathbf A \land \mathbf B}\right)} \ar@{.}[d] & & & & &  & *++{\mathbf A \implies \mathbf B} \ar@{.}[d] \\  & *++{t} \ar@{-}[dl] \ar@{-}[dr] & & & & &  & *++{t} \ar@{-}[dl] \ar@{-}[dr] & & & & &  & *++{t} \ar@{-}[dl] \ar@{-}[dr] \\  \mathbf A & & \mathbf B & & & &  \neg \mathbf A & & \neg \mathbf B & & & &  \neg \mathbf A & & \mathbf B \\  & \boxed{\lor} & & & & & & \boxed{\neg \land} & & & & & & \boxed \implies  }\end{xy}$ |- | $\begin{xy}\xymatrix @R=1em @C=0.7em{  & \ar@{.}[d] & & & & & & \ar@{.}[d] \\  & *++{\mathbf A \iff \mathbf B} \ar@{.}[d] &  & & & &  & *++{\neg \left({\mathbf A \iff \mathbf B}\right)} \ar@{.}[d] \\  & *++{t} \ar@{-}[dl] \ar@{-}[dr] & & & & &  & *++{t} \ar@{-}[dl] \ar@{-}[dr] \\  *++{\mathbf A \land \mathbf B} & & *++{\neg \mathbf A \land \neg \mathbf B} & & & &  *++{\mathbf A \land \neg \mathbf B} & & *++{\neg \mathbf A \land \mathbf B} \\  & \boxed{\iff} & & & & & & \boxed{\neg \iff} }\end{xy}$ |}
Definition:Projective Geometry
'''Projective Geometry''' is the field of geometry concerned with properties which are invariant under projective transformations.
Definition:Analytic Geometry/Plane
'''Plane analytic geometry''' is the branch of analytic geometry that makes an algebraic study of the real number plane $\R^2$.
Definition:Analytic Geometry/Solid
'''Solid analytic geometry''' is the branch of analytic geometry that makes an algebraic study of the  real vector space $\R^3$.
Definition:Collinear Points
Three or more points which belong to the same line are '''collinear'''.
Definition:Concurrent Lines
A set of $3$ or more (straight) lines through a single point are '''concurrent'''.
Definition:Coplanar Points
Four or more points which belong to the same plane are '''coplanar'''.
Definition:Collinear Planes
A set of planes which share the same (straight) line are '''collinear'''.
Definition:Contradictory Branch
Let $T$ be a labeled tree for propositional logic. Let $\Gamma$ be a branch of $T$. Then $\Gamma$ is a '''contradictory branch''' iff, for some WFF of propositional logic $\mathbf A$, both $\mathbf A$ and $\neg \mathbf A$ occur along $\Gamma$.
Definition:Tableau Confutation
Let $\mathbf H$ be a set of WFFs of propositional logic. A '''tableau confutation''' of $\mathbf H$ is a propositional tableau $T$ with root $\mathbf H$ such that every branch of $T$ is contradictory.
Definition:Tableau Proof (Propositional Tableaus)
Let $\mathbf H$ be a set of WFFs of propositional logic. Let $\mathbf A$ be a WFF. A '''tableau proof''' of $\mathbf A$ from $\mathbf H$ is a tableau confutation of $\mathbf H \cup \set {\neg \mathbf A}$. This definition also applies when $\mathbf H = \varnothing$. Then a '''tableau proof''' of $\mathbf A$ is a tableau confutation of $\set {\neg \mathbf A}$. If there exists a '''tableau proof''' of $\mathbf A$ from $\mathbf H$, one can write: :$\mathbf H \vdash_{\mathrm{PT} } \mathbf A$ Specifically, the notation: :$\vdash_{\mathrm{PT} } \mathbf A$ means that there exists a '''tableau proof''' of $\mathbf A$. === Proof System === {{:Definition:Tableau Proof (Propositional Tableaus)/Proof System}}
Definition:Consistent (Logic)
Let $\LL$ be a logical language. Let $\mathscr P$ be a proof system for $\LL$. === Proof System === {{:Definition:Consistent (Logic)/Proof System}} === Set of Formulas === {{:Definition:Consistent (Logic)/Set of Formulas}}
Definition:Provable Consequence
Let $\mathscr P$ be a proof system for a formal language $\mathcal L$. Let $\mathcal F$ be a collection of WFFs of $\mathcal L$. Denote with $\mathscr P \left({\mathcal F}\right)$ the proof system obtained from $\mathscr P$ by adding all the WFFs from $\mathcal F$ as axioms. Let $\phi$ be a theorem of $\mathscr P \left({\mathcal F}\right)$. Then $\phi$ is called a '''provable consequence''' of $\mathcal F$, and this is denoted as: :$\mathcal F \vdash_{\mathscr P} \phi$ Note in particular that for $\mathcal F = \varnothing$, this notation agrees with the notation for a $\mathscr P$-theorem: :$\vdash_{\mathscr P} \phi$
Definition:Monotone Sequence of Sets
Let $X$ be a set. Let $\mathcal S \subseteq \powerset X$ be a collection of subsets of $X$. A '''monotone sequence of sets (in $\mathcal S$)''' is a sequence $\sequence {A_n}_{n \mathop \in \N}$ in $\mathcal S$, such that either: :$\forall n \in \N: A_n \subseteq A_{n + 1}$ or: :$\forall n \in \N: A_n \supseteq A_{n + 1}$ That is, such that $\sequence  {A_n}_{n \mathop \in \N}$ is either increasing or decreasing
Definition:Inconsistent (Logic)
Let $\LL$ be a logical language. Let $\mathscr P$ be a proof system for $\LL$. A collection $\FF$ of logical formulas is '''inconsistent for $\mathscr P$''' : :For every logical formula $\phi$, $\FF \vdash_{\mathscr P} \phi$. That is, ''every'' logical formula $\phi$ is a provable consequence of $\FF$.
Definition:Proof System/Formal Proof
Let $\phi$ be a WFF of $\mathcal L$. A '''formal proof of $\phi$''' in $\mathscr P$ is a collection of axioms and rules of inference of $\mathscr P$ that leads to the conclusion that $\phi$ is a theorem of $\mathscr P$. The term '''formal proof''' is also used to refer to specific presentations of such collections. For example, the term applies to tableau proofs in natural deduction.
Definition:Tarski's Geometry
Tarski's geometry is an axiomatic treatment of geometry.  Unless specified otherwise,  will use the term Tarski's geometry to be a formal systematic treatment of geometry containing only: :$(1):\quad$ The language and axioms of first-order logic, and the disciplines preceding it :$(2):\quad$ The undefined terms of Tarski's Geometry :$(3):\quad$ Tarski's Axioms of Geometry.
Definition:Boolean Lattice/Definition 3
A '''Boolean lattice''' is a bounded lattice $\left({S, \vee, \wedge, \preceq, \bot, \top}\right)$ together with a unary operation $\neg$ called '''complementation''', subject to: $(1): \quad$ For all $a, b \in S$, $a \preceq \neg b$  $a \wedge b = \bot$ $(2): \quad$ For all $a \in S$, $\neg \neg a = a$.
Definition:Symmetric Relation/Definition 1
$\RR$ is '''symmetric''' : :$\tuple {x, y} \in \RR \implies \tuple {y, x} \in \RR$