hcju/proofwikips huggingface dataset:
Definition:Dual Ordering
Let $\struct {S, \preceq}$ be an ordered set. Let $\succeq$ be the inverse relation to $\preceq$. That is, for all $a, b \in S$: :$a \succeq b$  $b \preceq a$ Then $\succeq$ is called the '''dual ordering of $\preceq$'''. By Dual Ordering is Ordering, it is indeed an ordering. === Dual Ordered Set === {{:Definition:Dual Ordering/Dual Ordered Set}} === Notation for Inverse Ordering === {{:Definition:Inverse Relation/Ordering Notation}} === Notation for Inverse Strict Ordering === {{:Definition:Inverse Relation/Strict Ordering Notation}}
Definition:Dual Ordering/Dual Ordered Set
The ordered set $\left({S, \succeq}\right)$ is called the '''dual ordered set''' (or just '''dual''') '''of $\left({S, \preceq}\right)$'''.
Definition:Conditional/Necessary Condition
Let $p \implies q$ be a conditional statement. Then $q$ is a '''necessary condition''' for $p$. That is, if $p \implies q$, then it is ''necessary'' that $q$ be true for $p$ to be true. This is because unless $q$ is true, $p$ can ''not'' be true.
Definition:Conditional/Sufficient Condition
Let $p \implies q$ be a conditional statement. Then $p$ is a '''sufficient condition''' for $q$. That is, if $p \implies q$, then for $q$ to be true, it is ''sufficient'' to know that $p$ is true. This is because of the fact that if you know that $p$ is true, you know enough to know also that $q$ is true.
Definition:Dual Statement (Order Theory)
Let $\left({S, \preceq}\right)$ be an ordered set. Let $\succeq$ be the dual ordering to $\preceq$. Let $\Sigma$ be any statement pertaining to $\left({S, \preceq}\right)$ (be it in natural language or a formal language). The '''dual statement of $\Sigma$''', denoted $\Sigma^*$, is the statement obtained from replacing every reference to $\preceq$ in $\Sigma$ with a reference to its dual $\succeq$. This '''dual statement''' may then be turned into a statement about $\preceq$ again by applying the equivalences on Dual Pairs (Order Theory).
Definition:Relation Conversely Compatible with Operation
Let $\left({S, \circ}\right)$ be a closed algebraic structure. Let $\mathcal R$ be a relation in $S$. Then $\mathcal R$ is '''conversely compatible with $\circ$''' : :$\forall x, y, z \in S: \left({x \circ z}\right) \mathrel {\mathcal R} \left({y \circ z}\right) \implies x \mathrel {\mathcal R} y$ :$\forall x, y, z \in S: \left({z \circ x}\right) \mathrel {\mathcal R} \left({z \circ y}\right) \implies x \mathrel {\mathcal R} y$
Definition:Relation Strongly Compatible with Operation
Let $\left({S, \circ}\right)$ be a closed algebraic structure. Let $\mathcal R$ be a relation in $S$. Then $\mathcal R$ is '''strongly compatible with $\circ$''' : :$\forall x, y, z \in S: x \mathrel {\mathcal R} y \iff \left({x \circ z}\right) \mathrel {\mathcal R} \left({y \circ z}\right)$ :$\forall x, y, z \in S: x \mathrel {\mathcal R} y \iff \left({z \circ x}\right) \mathrel {\mathcal R} \left({z \circ y}\right)$. That is,  $\mathcal R$ is compatible with $\circ$ and conversely compatible with $\circ$.
Definition:Union of Relations
Let $S$ and $T$ be sets. Let $\mathcal R_1$ and $\mathcal R_2$ be relations on $S \times T$. The '''union of $\mathcal R_1$ and $\mathcal R_2$''' is the relation $\mathcal Q$ defined by: :$\mathcal Q := \mathcal R_1 \cup \mathcal R_2$ where $\cup$ denotes set union. Explicitly, for $s \in S$ and $t \in T$, we have: :$s \mathrel{\mathcal Q} t$  $s \mathrel{\mathcal R_1} t$ or $s \mathrel{\mathcal R_2} t$ === General Definition === {{:Definition:Union of Relations/General Definition}}
Definition:Union of Relations/General Definition
Let $\mathscr R$ be a collection of relations on $S \times T$. The '''union of $\mathscr R$''' is the relation $\mathcal R$ defined by: :$\mathcal R = \displaystyle \bigcup \mathscr R$ where $\bigcup$ denotes set union. Explicitly, for $s \in S$ and $t \in T$: :$s \mathrel{\mathcal R} t$  for some $\mathcal Q \in \mathscr R$, $s \mathrel{\mathcal Q} t$
Definition:Intersection of Relations
Let $S$ and $T$ be sets. Let $\mathcal R_1$ and $\mathcal R_2$ be relations on $S \times T$. The '''intersection of $\mathcal R_1$ and $\mathcal R_2$''' is the relation $\mathcal Q$ defined by: :$\mathcal Q := \mathcal R_1 \cap \mathcal R_2$ where $\cap$ denotes set intersection. Explicitly, for $s \in S$ and $t \in T$, we have: :$s \mathrel{\mathcal Q} t$ iff both $s \mathrel{\mathcal R_1} t$ and $s \mathrel{\mathcal R_2} t$ === General Definition === {{:Definition:Intersection of Relations/General Definition}}
Definition:Intersection of Relations/General Definition
Let $\mathscr R$ be a collection of relations on $S \times T$. The '''intersection of $\mathscr R$''' is the relation $\mathcal R$ defined by: :$\mathcal R = \displaystyle \bigcap \mathscr R$ where $\bigcap$ denotes set intersection. Explicitly, for $s \in S$ and $t \in T$: :$s \mathrel{\mathcal R} t$  for all $\mathcal Q \in \mathscr R$, $s \mathrel{\mathcal Q} t$
Definition:Complement (Lattice Theory)
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a bounded lattice. Denote by $\bot$ and $\top$ the bottom and top of $S$, respectively. Let $a \in S$. Then $b \in S$ is called a '''complement of $a$''' : :$b \vee a = \top$ :$b \wedge a = \bot$ If $a$ has a unique '''complement''', it is denoted by $\neg a$. === Complemented Lattice === {{:Definition:Complemented Lattice}}
Definition:Finitely Generated Algebraic Structure
Let $\struct {A, \circ}$ be an algebraic structure. Let $\struct {A, \circ}$ have a generator which is finite. Then $\struct {A, \circ}$ is '''finitely generated'''.
Definition:Topological Group/Definition 2
Let the mapping $\psi: \left({G, \tau}\right) \times \left({G, \tau}\right) \to \left({G, \tau}\right)$ be defined as: :$\psi \left({x, y}\right) = x \odot y^{-1}$ $\left({G, \odot, \tau}\right)$ is a '''topological group''' : : $\psi$ is a continuous mapping where $\left({G, \tau}\right) \times \left({G, \tau}\right)$ is considered as $G \times G$ with the product topology.
Definition:Topological Group/Definition 1
$\left({G, \odot, \tau}\right)$ is a '''topological group''' : :$(1): \quad \odot: \left({G, \tau}\right) \times \left({G, \tau}\right) \to \left({G, \tau}\right)$ is a continuous mapping :$(2): \quad \phi: \left({G, \tau}\right) \to \left({G, \tau}\right)$ such that $\forall x \in G: \phi \left({x}\right) = x^{-1}$ is also a continuous mapping where $\left({G, \tau}\right) \times \left({G, \tau}\right)$ is considered as $G \times G$ with the product topology.
Definition:Topological Ring
Let $\struct{R, +, \circ}$ be a ring. Let $\tau$ be a topology over $R$. Then $\struct {R, +, \circ, \tau}$ is a '''topological ring''' : :$(1): \quad \struct {R, +, \tau}$ is a topological group :$(2): \quad \struct {R, \circ,\tau}$ is a topological semigroup.
Definition:Topological Field
Let $\struct {F, +, \circ}$ be a field with zero $0_F$. Let $\tau$ be a topology on $F$. Let $\struct {F, +, \circ, \tau}$ be a topological division ring. Then $\struct {F, +, \circ, \tau}$ is a '''topological field'''. That is, a topological field is a commutative topological division ring.
Definition:Field Zero
Let $\struct {F, +, \times}$ be a field. The identity for field addition is called the '''field zero''' (of $\struct {F, +, \times}$). It is denoted $0_F$ (or just $0$ if there is no danger of ambiguity).
Definition:Ray (Order Theory)
Let $\struct {S, \preccurlyeq}$ be a totally ordered set. Let $\prec$ be the reflexive reduction of $\preccurlyeq$. Let $a \in S$ be any point in $S$. === Open Ray === {{:Definition:Ray (Order Theory)/Open}} === Closed Ray === {{:Definition:Ray (Order Theory)/Closed}} === Upward-Pointing Ray === {{:Definition:Ray (Order Theory)/Upward-Pointing}} === Downward-Pointing Ray === {{:Definition:Ray (Order Theory)/Downward-Pointing}}
Definition:Complemented Lattice
Suppose that every $a \in S$ admits a complement. Then $\left({S, \vee, \wedge, \preceq}\right)$ is called a '''complemented lattice'''.
Definition:Top (Lattice Theory)
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice. === Definition 1 === {{:Definition:Top (Lattice Theory)/Definition 1}} === Definition 2 === {{:Definition:Top (Lattice Theory)/Definition 2}}
Definition:Top (Lattice Theory)/Definition 1
Let $S$ admit a greatest element $\top$. Then $\top$ is called the '''top''' of $S$.
Definition:Top (Lattice Theory)/Definition 2
Let $\wedge$ have an identity element $\top$. Then $\top$ is called the '''top''' of $S$.
Definition:Bottom (Lattice Theory)
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice. === Definition 1 === {{:Definition:Bottom (Lattice Theory)/Definition 1}} === Definition 2 === {{:Definition:Bottom (Lattice Theory)/Definition 2}}
Definition:Bottom (Lattice Theory)/Definition 1
Let $S$ admit a smallest element $\bot$. Then $\bot$ is called the '''bottom''' of $S$.
Definition:Bottom (Lattice Theory)/Definition 2
Let $\vee$ have an identity element $\bot$. Then $\bot$ is called the '''bottom''' of $S$.
Definition:Distance-Preserving Mapping
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces, pseudometric spaces, or quasimetric spaces. Let $\phi: M_1 \to M_2$ be a mapping such that: : $\forall a, b \in M_1: d_1 \left({a, b}\right) = d_2 \left({\phi \left({a}\right), \phi \left({b}\right)}\right)$ Then $\phi$ is called a '''distance-preserving mapping'''.
Definition:Connected Domain (Complex Analysis)
Let $D \subseteq \C$ be a subset of the set of complex numbers. Then $D$ is a '''connected domain'''  $D$ is open and connected.
Definition:Connected Domain (Complex Analysis)/Simply Connected Domain
Let $D \subseteq \C$ be a connected domain. Then $D$ is called a '''simply connected domain'''  $D$ is simply connected. === Simply Connectedness Requirement ===  {{:Definition:Connected Domain (Complex Analysis)/Simply Connected Domain/Requirement}}
Definition:Topologist's Sine Curve
Let $G = \set {\tuple {x, \sin \dfrac 1 x} : 0 < x \le 1}$, considered as a subset of the real number plane with the usual (Euclidean) topology $\struct {\R^2, \tau_d}$. Then the set $G \cup \set {\tuple {0, 0} }$ is called the '''Topologist's Sine Curve'''.
Definition:Staircase Contour
Let $C$ be a contour that is a concatenation of the directed smooth curves $C_1, \ldots, C_n$. For all $i \in \set {1, \ldots, n}$, let it be possible for $C_i$ be parameterized by a smooth path $\gamma_i: \closedint 0 1 \to \C$ such that either: :$\map {\gamma_i} t = z_i + t r_i$ or :$\map {\gamma_i} t = z_i + i t r_i$ for some $z_i \in \C, r_i \in \R$ for all $t \in \closedint 0 1$. Then $C$ is called a '''staircase contour'''.
Definition:Differentiable Mapping/Complex Function/Point
Let $D\subset \C$ be an open set. Let $f : D \to \C$ be a complex function. Let $z_0 \in D$ be a point in $D$. Then $f$ is '''complex-differentiable''' at $z_0$  the limit: : $\displaystyle \lim_{h \to 0} \frac {f \left({z_0+h}\right) - f \left({z_0}\right)} h$ exists as a finite number.
Definition:Differentiable Mapping/Complex Function/Region
Let $D \subseteq \C$ be an open set. Let $f: D \to \C$ be a complex function. Then $f$ is '''complex-differentiable in $D$'''  $f$ is complex-differentiable at every point in $D$.
Definition:Inequality
An '''inequality''' is a mathematical statement that two expressions relate in one of several conventional ways: : $a < b$ : $a \le b$ : $a > b$ : $a \ge b$ A statement of the form:  :$a \ne b$ may or may not be considered an '''inequality'''.
Definition:Convex Set (Vector Space)/Line Segment
The set: :$\set {t x + \paren {1 - t} y: t \in \closedint 0 1}$  is called the '''(straight) line segment joining $x$ and $y$'''. A convex set can thus be described as a set containing all '''straight line segments''' between its elements.
Definition:Boolean Lattice
=== Definition 1 === {{:Definition:Boolean Lattice/Definition 1}} === Definition 2 === {{:Definition:Boolean Lattice/Definition 2}} === Definition 3 === {{:Definition:Boolean Lattice/Definition 3}}
Definition:Boolean Lattice/Definition 1
A '''Boolean lattice''' is a complemented distributive lattice.
Definition:Boolean Lattice/Definition 2
An ordered structure $\left({S, \vee, \wedge, \preceq}\right)$ is a '''Boolean lattice''' : $(1): \quad \left({S, \vee, \wedge}\right)$ is a Boolean algebra $(2): \quad$ For all $a, b \in S$: $a \wedge b \preceq a \vee b$
Definition:Metalanguage/Formal Systems
In the context of formal systems, a '''metalanguage''' is a formal language used to specify another formal language.
Definition:Metalanguage/Object Language
The '''object language''' of a metalanguage is the language described by that metalanguage.
Definition:Metalanguage/Metasyntax
The syntax of a metalanguage is called a '''metasyntax''' of the object language of that metalanguage.
Definition:Metalanguage/Metasymbol
A '''metasymbol''' is a symbol used in a metalanguage to represent an arbitrary collation in the object language.
Definition:Theorem/Logic
A '''theorem''' in logic is a statement which can be shown to be the conclusion of a logical argument which depends on ''no'' premises except axioms. A sequent which denotes a theorem $\phi$ is written $\vdash \phi$, indicating that there are no premises. In this context, $\vdash$ is read as: :'''It is a theorem that ...'''
Definition:Theorem/Formal System
Let $\mathcal L$ be a formal language. Let $\mathscr P$ be a proof system for $\mathcal L$. A '''theorem of $\mathscr P$''' is a well-formed formula of $\mathcal L$ which can be deduced from the axioms and axiom schemata of $\mathscr P$ by means of its rules of inference. That a WFF $\phi$ is a '''theorem''' of $\mathscr P$ may be denoted as: :$\vdash_{\mathscr P} \phi$
Definition:Theorem/Mathematics
The term '''theorem''' is used throughout the whole of mathematics to mean a statement which has been proved to be true from whichever axioms relevant to that particular branch. Note that statements which are taken as axioms in one branch of mathematics may be theorems in others.
Definition:Distance/Sets/Real Numbers
Let $S, T$ be a subsets of the set of real numbers $\R$. Let $x \in \R$ be a real number. The '''distance between $x$ and $S$''' is defined and annotated $\displaystyle \map d {x, S} = \map {\inf_{y \mathop \in S} } {\map d {x, y} }$, where $\map d {x, y}$ is the distance between $x$ and $y$. The '''distance between $S$ and $T$''' is defined and annotated $\displaystyle \map d {S, T} = \map {\inf_{\substack {x \mathop \in S \\ y \mathop \in T} } } {\map d {x, y} }$.
Definition:Distance/Sets/Metric Spaces
Let $M = \left({A, d}\right)$ be a metric space. Let $x \in A$. Let $S, T$ be subsets of $A$. The '''distance between $x$ and $S$''' is defined and annotated $\displaystyle d \left({x, S}\right) = \inf_{y \mathop \in S} \left({d \left({x, y}\right)}\right)$. The '''distance between $S$ and $T$''' is defined and annotated $\displaystyle d \left({S, T}\right) = \inf_{\substack{x \mathop \in S \\ y \mathop \in T}} \left({d \left({x, y}\right)}\right)$.
Definition:Bounded Lattice/Definition 3
Let $\left({S, \wedge, \vee, \preceq}\right)$ be a lattice. Let $S$ be bounded in $\left({S,\preceq}\right)$. Then $\left({S, \wedge, \vee, \preceq}\right)$ is a '''bounded lattice'''.
Definition:Uniquely Complemented Lattice
Let $\left({S,\wedge,\vee,\preceq}\right)$ be a complemented lattice. If each element of $S$ has only one complement, then $\left({S,\wedge,\vee,\preceq}\right)$ is a '''uniquely complemented lattice'''. That is, a uniquely complemented lattice is a lattice in which each element has exactly one complement.
Definition:Boolean Algebra/Definition 3
A '''Boolean algebra''' is an algebraic structure $\struct {S, \vee, \wedge}$ such that: {{begin-axiom}} {{axiom | n = \text {BA} 0         | lc=         | t = $S$ is closed under both $\vee$ and $\wedge$ }} {{axiom | n = \text {BA} 1         | lc=         | t = Both $\vee$ and $\wedge$ are commutative }} {{axiom | n = \text {BA} 2         | lc=         | t = Both $\vee$ and $\wedge$ distribute over the other }} {{axiom | n = \text {BA} 3         | lc=         | t = Both $\vee$ and $\wedge$ have identities $\bot$ and $\top$ respectively }} {{axiom | n = \text {BA} 4         | lc=         | t = $\forall a \in S: \exists \neg a \in S: a \vee \neg a = \top, a \wedge \neg a = \bot$ }} {{end-axiom}}
Definition:Boolean Algebra/Definition 1
{{:Definition:Boolean Algebra/Axioms/Definition 1}}
Definition:Degenerate Boolean Algebra
Let $\struct {S, \vee, \wedge, \neg}$ be a Boolean algebra. Then $\struct {S, \vee, \wedge, \neg}$ is said to be '''degenerate'''  $S$ is a singleton.
Definition:Exponential Function/Real/Inverse of Natural Logarithm
Consider the natural logarithm $\ln x$, which is defined on the open interval $\openint 0 {+\infty}$. From Logarithm is Strictly Increasing: :$\ln x$ is strictly increasing. From Inverse of Strictly Monotone Function: :the inverse of $\ln x$ always exists. The inverse of the natural logarithm function is called the '''exponential function''', which is denoted as $\exp$. Thus for $x \in \R$, we have: :$y = \exp x \iff x = \ln y$
Definition:Exponential Function/Real/Extension of Rational Exponential
Let $e$ denote Euler's number. Let $f: \Q \to \R$ denote the real-valued function defined as: :$f \left({ x }\right) = e^x$ That is, let $f \left({ x }\right)$ denote $e$ to the power of $x$, for rational $x$. Then $\exp : \R \to \R$ is defined to be the unique continuous extension of $f$ to $\R$. $\exp \left({ x }\right)$ is called the '''exponential of $x$'''.
Definition:Exponential Function/Real/Limit of Sequence
The '''exponential function''' can be defined as the following limit of a sequence: :$\exp x := \displaystyle \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$
Definition:Exponential Function/Real/Sum of Series
The '''exponential function''' can be defined as a power series: :$\exp x := \displaystyle \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$
Definition:Exponential Function/Real/Differential Equation
The '''exponential function''' can be defined as the unique solution $y = \map f x$ to the first order ODE: :$\dfrac {\d y} {\d x} = y$ satisfying the initial condition $\map f 0 = 1$.
Definition:Logical Not/Truth Table
The characteristic truth table of the negation operator $\neg p$ is as follows: :$\begin{array}{|c||c|} \hline p & \neg p \\ \hline \F & \T \\ \T & \F \\ \hline \end{array}$
Definition:Truth Value/Aristotelian Logic
In Aristotelian logic, a statement can be either true or false, and there is no undefined, in-between value. Whether it is true or false is called its '''truth value'''. Note that a statement's '''truth value''' may change depending on circumstances. Thus, the statement: :''It is currently raining on the grass outside my window'' has the truth value false, whereas it had the truth value true last week. The statement: :''I am listening to Shostakovich's 4th symphony'' is currently true, but that will last only for the next twenty minutes or so as I type. The '''truth values''' true and false are usually represented in one of two ways: :$\T$ for true and $\F$ for false; :$1$ for true and $0$ for false. There are advantages for both notations. In particular, the second lends itself to extending the discipline of logic into that of probability theory.
Definition:Conjunction/Truth Table
The characteristic truth table of the logical conjunction operator $p \land q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \land q \\ \hline \F & \F & \F \\ \F & \T & \F \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$
Definition:Compactification
Let $\left({X, \tau_1}\right)$ be a topological space. Let $\left({Y, \tau_2}\right)$ be a compact space. Let $f: X \to Y$ be a topological embedding. Let $f \left({X}\right)$ be everywhere dense in $Y$. Then either $f$ or $\left({Y, \tau_2}\right)$ may be called a '''compactification''' of $\left({X, \tau_1}\right)$.
Definition:Disjunction/Truth Table
The characteristic truth table of the logical disjunction operator $p \lor q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \lor q \\ \hline \F & \F & \F \\ \F & \T & \T \\ \T & \F & \T \\ \T & \T & \T \\ \hline \end{array}$
Definition:Biconditional/Truth Table
The characteristic truth table of the biconditional operator $p \iff q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \iff q \\ \hline \F & \F & \T \\ \F & \T & \F \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$
Definition:Exclusive Or/Truth Table
The characteristic truth table of the exclusive or operator $p \oplus q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \oplus q \\ \hline \F & \F & \F \\ \F & \T & \T \\ \T & \F & \T \\ \T & \T & \F \\ \hline \end{array}$
Definition:Conditional/Truth Table
The characteristic truth table of the conditional (implication) operator $p \implies q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \implies q \\ \hline \F & \F & \T \\ \F & \T & \T \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$
Definition:Logical NAND/Truth Table
The characteristic truth table of the logical NAND operator $p \uparrow q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \uparrow q \\ \hline F & F & T \\ F & T & T \\ T & F & T \\ T & T & F \\ \hline \end{array}$
Definition:Logical NOR/Truth Table
The characteristic truth table of the logical NOR operator $p \downarrow q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \downarrow q \\ \hline F & F & T \\ F & T & F \\ T & F & F \\ T & T & F \\ \hline \end{array}$
Definition:Series/General
Let $\left({S, \circ}\right)$ be a semigroup. Let $\left \langle {a_n} \right \rangle$ be a sequence in $S$. Informally, a '''series''' is what results when an infinite product is taken of $\left \langle {a_n} \right \rangle$: :$\displaystyle s := \sum_{n \mathop = 1}^\infty a_n = a_1 \circ a_2 \circ a_3 \circ \cdots$ Formally, a '''series''' is a sequence in $S$.
Definition:Series/Number Field
The '''series''' is what results when $\sequence {a_n}$ is summed to infinity: :$\displaystyle \sum_{n \mathop = 1}^\infty a_n = a_1 + a_2 + a_3 + \cdots$
Definition:Series/Tail
Let $N \in \N$. The expression $\displaystyle \sum_{n \mathop = N}^\infty a_n$ is known as a '''tail''' of the '''series''' $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.
Definition:Conditional/Semantics of Conditional
$p \implies q$ can be stated thus: * '''''If'' $p$ is true ''then'' $q$ is true.''' * '''$q$ is true ''if'' $p$ is true.''' * '''(The truth of) $p$ ''implies'' (the truth of) $q$.''' * '''(The truth of) $q$ ''is implied by'' (the truth of) $p$.''' * '''$q$ ''follows from'' $p$.''' * '''$p$ is true ''only if'' $q$ is true.''' The latter one may need some explanation. $p$ can be either true or false, as can $q$. But if $q$ is false, and $p \implies q$, then $p$ can not be true. Therefore, $p$ can be true ''only if'' $q$ is also true, which leads us to our assertion. * '''$p$ is true ''therefore'' $q$ is true.''' * '''$p$ is true ''entails'' that $q$ is true.''' * '''$q$ is true ''because'' $p$ is true.''' * '''$p$ ''may'' be true ''unless'' $q$ is false.''' * '''''Given that'' $p$ is true, $q$ is true.''' * '''$q$ is true ''whenever'' $p$ is true.''' * '''$q$ is true ''provided that'' $p$ is true.''' * '''$q$ is true ''in case'' $p$ is true.''' * '''$q$ is true ''assuming that'' $p$ is true.''' * '''$q$ is true ''on the condition that'' $p$ is true.'''
Definition:Conditional/Language of Conditional/Weak
In a conditional $p \implies q$, the statement $q$ is '''weaker''' than $p$.
Definition:Conditional/Language of Conditional/Strong
In a conditional $p \implies q$, the statement $p$ is '''stronger''' than $q$.
Definition:Conditional/Superimplicant
In a conditional $p \implies q$, the statement $p$ is '''superimplicant''' to $q$.
Definition:Conditional/Subimplicant
In a conditional $p \implies q$, the statement $q$ is '''subimplicant''' to $p$.