hcju/proofwikips huggingface dataset:
Definition:Language of Predicate Logic/Alphabet
The alphabet $\AA$ of the language of predicate logic $\LL_1$ is defined as follows: <section notitle="true" name="tc"> === Letters === The letters of $\LL_1$ are separated in three classes: * variables; * predicate or relation symbols; * function symbols. Each of these three classes is handled differently by the formal grammar of predicate logic. ==== Variables ==== The variables constitute an infinite set $\mathrm{VAR}$ of arbitrary symbols, for example: :$\mathrm{VAR} = \set {x, y, z, x_0, y_0, z_0, x_1, y_1, z_1, \ldots}$ ==== Predicate Symbols ==== The predicate symbols are a collection of arbitrary symbols. Each of these symbols is considered to be endowed with an arity (a natural number $n \in \N$). We agree to write $\PP$ for the set of predicate symbols, grouped by their arity: :$\PP = \set {\PP_0, \PP_1, \PP_2, \ldots, \PP_k, \ldots}$ The symbols in $\PP_0$ are inherited from the language of propositional logic. For example, if $P \in \PP_5$ then $P$ is a quinternary predicate symbol. ==== Function Symbols ==== The function symbols are a collection (possibly empty) of arbitrary symbols. Each of these symbols is considered to be endowed with an arity (a natural number $n \in \N$). We agree to write $\FF$ for the set of function symbols, grouped by their arity: :$\FF = \set {\FF_0, \FF_1, \ldots, \FF_k, \ldots}$ The symbols in $\FF_0$ are often called '''parameters''' or '''constants'''. Some sources write $\KK$ for the collection of '''parameters'''. === Signs === The signs of $\LL_1$ are an extension of the signs of propositional logic. They split in three classes: * connectives; * quantifiers; * punctuation. ==== Connectives ==== The '''connectives''' of $\LL_1$ comprise: {{begin-axiom}} {{axiom | ml= \land          | mo= :         | t = the conjunction sign }} {{axiom | ml= \lor         | mo= :         | t = the disjunction sign }} {{axiom | ml= \implies         | mo= :         | t = the conditional sign }} {{axiom | ml= \iff         | mo= :         | t = the biconditional sign }} {{axiom | ml= \neg         | mo= :         | t = the negation sign }} {{axiom | ml= \top         | mo= :         | t = the top sign }} {{axiom | ml= \bot         | mo= :         | t = the bottom sign }} {{end-axiom}} The symbols $\land, \lor, \implies$ and $\iff$ are called the '''binary connectives'''. The symbols $\neg$ is called a '''unary connective'''. The symbols $\top$ and $\bot$ are called the '''nullary connectives'''. ==== Quantifiers ==== The '''quantifiers''' of $\LL_1$ are: {{begin-axiom}} {{axiom | ml= \exists         | mo= :         | t = the existential quantifier sign }} {{axiom | ml= \forall         | mo= :         | t = the universal quantifier sign }} {{end-axiom}} ==== Punctuation ==== The punctuation symbols used in $\LL_1$ are: {{begin-axiom}} {{axiom | ml= (         | mo= :         | t = the left parenthesis sign }} {{axiom | ml= )         | mo= :         | t = the right parenthesis sign }} {{axiom | ml= :         | mo= :         | t = the colon }} {{axiom | ml= ,         | mo= :         | t = the comma }} {{end-axiom}} </section>
Definition:Language of Predicate Logic/Formal Grammar
Part of specifying the language of propositional logic is to provide its formal grammar. <section notitle="true" name="def"> The following rules of formation constitute a bottom-up grammar for the language of predicate logic $\LL_1$. The definition proceeds in two steps. First, we will define terms, and then well-formed formulas. === Terms === {{:Definition:Language of Predicate Logic/Formal Grammar/Term}} === Well-Formed Formulas === The WFFs of $\LL_1$ are defined by the following bottom-up grammar: {{begin-axiom}} {{axiom | n = \mathbf W ~ \PP_n         | t = If $t_1, \ldots, t_n$ are terms, and $p \in \PP_n$ is an $n$-ary predicate symbol, then $\map p {t_1, t_2, \ldots, t_n}$ is a WFF. }} {{axiom | n = \mathbf W ~ \neg         | t = If $\mathbf A$ is a WFF, then $\neg \mathbf A$ is a WFF. }} {{axiom | n = \mathbf W ~ \lor, \land, \Rightarrow, \Leftrightarrow         | t = If $\mathbf A, \mathbf B$ are WFFs and $\circ$ is one of $\lor, \land, \mathord \implies, \mathord \iff$, then $\paren {\mathbf A \circ \mathbf B}$ is a WFF }} {{axiom | n = \mathbf W ~ \forall, \exists         | t = If $\mathbf A$ is a WFF and $x$ is a variable, then $\paren {\forall x: \mathbf A}$ and $\paren {\exists x: \mathbf A}$ are WFFs. }} {{end-axiom}} </section>
Definition:Relation Segment
Let $X$ be a set. Let $R$ be a relation on $X$. Let $x \in X$. Then the '''$R$-segment of $x$''', denoted by $x^R$, is defined by: :$x^R := \left\{ {y \in X: \left({y, x}\right) \in R}\right\}$
Definition:Remainder/Real
The '''remainder of $x$ on division by $y$''' is defined as the value of $r$ in the expression: :$\forall x, y \in \R, y \ne 0: \exists! q \in \Z, r \in \R: x = q y + r, 0 \le r < \left|{y}\right|$ From the definition of the Modulo Operation: :$x \bmod y := x - y \left \lfloor {\dfrac x y}\right \rfloor$ it can be seen that the '''remainder of $x$ on division by $y$''' is defined as: : $r = x \bmod y$
Definition:Quotient (Algebra)/Real
The '''quotient of $x$ on division by $y$''' is defined as the value of $q$ in the expression: :$\forall x, y \in \R, y \ne 0: \exists! q \in \Z, r \in \R: x = q y + r, 0 \le r < \left|{y}\right|$ From the definition of the Modulo Operation: :$x \bmod y := x - y \left \lfloor {\dfrac x y}\right \rfloor = r$ it can be seen that the '''quotient of $x$ on division by $y$''' is defined as: : $q = \left \lfloor {\dfrac x y}\right \rfloor$
Definition:Increasing Mappings Satisfying Inclusion in Lower Closure
Let $R = \left({S, \preceq}\right)$ be an ordered set. Let ${\it Ids}\left({R}\right)$ be the set of all ideals in $R$. Let $L = \left({ {\it Ids}\left({R}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq\restriction_{ {\it Ids}\left({R}\right) \times {\it Ids}\left({R}\right)}$ Then '''ordered set $M$ of increasing mappings $f:R \to L$ satisfying''' $\forall x \in S: f\left({x}\right) \subseteq x^\preceq$ is defined by :$M = \left({F, \preccurlyeq}\right)$ where :$F = \left\{ {f: S \to {\it Ids}\left({R}\right): f}\right.$ is increasing mapping $\left.{\land \forall x \in S: f\left({x}\right) \subseteq x^\preceq}\right\}$ and :$\preccurlyeq$ is ordering on mappings generated by $\precsim$ where $x^\preceq$ denotes the lower closure of $x$.
Definition:Coprime/Notation
Let $a$ and $b$ be coprime integers, that is, such that $\gcd \left\{{a, b}\right\} = 1$. Then the notation $a \perp b$ is preferred on . If $\gcd \left\{{a, b}\right\} \ne 1$, the notation $a \not \!\! \mathop{\perp} b$ can be used.
Definition:Modulo Subtraction
Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$ where $\eqclass x m$ is the residue class of $x$ modulo $m$. The operation of '''subtraction modulo $m$''' is defined on $\Z_m$ as: :$\eqclass a m -_m \eqclass b m = \eqclass {a - b} m$
Definition:Congruence (Number Theory)/Notation
The relation '''$x$ is congruent to $y$ modulo $z$''', usually denoted: :$x \equiv y \pmod z$ is also frequently seen denoted as: :$x \equiv y \ \paren {\mathop {\operatorname{modulo} } z}$ Some (usually older) sources render it as: :$x \equiv y \ \paren {\mathop {\operatorname{mod.} } z}$
Definition:Permutation/Ordered Selection/Notation
The number of $r$-permutations from a set of cardinality $n$ is denoted variously: :$P_{n r}$ :${}^r P_n$ :${}_r P_n$ :${}_n P_r$ (extra confusingly) There is little consistency in the literature). On  the notation of choice is ${}^r P_n$.
Definition:Factorial/Definition 1
The '''factorial of $n$''' is defined inductively as: :$n! = \begin{cases} 1 & : n = 0 \\ n \paren {n - 1}! & : n > 0 \end{cases}$
Definition:Factorial/Definition 2
The '''factorial of $n$''' is defined as: {{begin-eqn}} {{eqn | l = n!       | r = \prod_{k \mathop = 1}^n k       | c =  }} {{eqn | r = 1 \times 2 \times \cdots \times \paren {n - 1} \times n       | c =  }} {{end-eqn}} where $\displaystyle \prod$ denotes product notation.
Definition:Gamma Function/Hankel Form
The '''{{AuthorRef|Hermann Hankel|Hankel}} form''' of the '''Gamma function''' is: :$\displaystyle \frac 1 {\Gamma \left({z}\right)} = \dfrac 1 {2 \pi i} \oint_{\mathcal H} \frac {e^t \, \mathrm d t} {t^z}$ where $\mathcal H$ is the contour starting at $-\infty$, circling the origin in an anticlockwise direction, and returning to $-\infty$. The '''Hankel form''' is valid for all $\C$.
Definition:Permanent
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. That is, let: : $\mathbf A = \begin {pmatrix}   a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\   a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\   a_{n 1} & a_{n 2} & \cdots & a_{n n} \end {pmatrix}$ Let $\lambda: \N_{>0} \to \N_{>0}$ be a permutation on $\N_{>0}$. Then the '''permanent''' of $\mathbf A$ is defined as: :$\displaystyle \sum_{\lambda} \paren {\prod_{k \mathop = 1}^n a_{k \map \lambda k} } = \sum_{\lambda} a_{1 \map \lambda 1} a_{2 \map \lambda 2} \cdots a_{n \map \lambda n}$ where: :the summation $\displaystyle \sum_\lambda$ goes over all the $n!$ permutations of $\set {1, 2, \ldots, n}$.
Definition:Constant Symbol
Let $\LL$ be a formal language (for example, the language of predicate logic $\LL_1$). A '''constant symbol''' is a letter of $\LL$ used to describe a constant. The name '''constant symbol''' is a gesture to the reader to make clear what such a symbol should (intuitively) represent in the formal language $\LL$.
Definition:Value of Term under Assignment
Let $\tau$ be a term in the language of predicate logic $\mathcal L_1$. Let $\mathcal A$ be an $\mathcal L_1$-structure. Let $\sigma$ be an assignment for $\tau$ in $\mathcal A$. Then the '''value of $\tau$ under $\sigma$''', denoted $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma}\right]$, is defined recursively by: :If $\tau = x$, with $x \in \operatorname{dom} \left({\sigma}\right)$ a variable in the domain of $\sigma$: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({x}\right) } \left[{\sigma}\right]  :=  \sigma \left({x}\right)$ :If $\tau = f \left({\tau_1, \ldots, \tau_n}\right)$ with $\tau_i$ terms and $f \in \mathcal F_n$ an $n$-ary function symbol: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({f \left({\tau_1, \ldots, \tau_n}\right) }\right) } \left[{\sigma}\right]  :=  f_{\mathcal A} \left({ \mathop{ \operatorname{val}_{\mathcal A} \left({\tau_1}\right) } \left[{\sigma}\right], \ldots, \mathop{ \operatorname{val}_{\mathcal A} \left({\tau_n}\right) } \left[{\sigma}\right] }\right)$ :where $f_{\mathcal A}$ denotes the interpretation of $f$ in $\mathcal A$. If $\tau$ contains no variables, one writes $\operatorname{val}_{\mathcal A} \left({\tau}\right)$ instead of $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma}\right]$.
Definition:Value of Formula under Assignment
Let $\mathbf A$ be a WFF in the language of predicate logic $\mathcal L_1$. Let $\mathcal A$ be an $\mathcal L_1$-structure. Let $\sigma$ be an assignment for $\mathbf A$ in $\mathcal A$. Then the '''value of $\mathbf A$ under $\sigma$''', denoted $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right]$, is defined recursively by: :If $\mathbf A = p \left({\tau_1, \ldots, \tau_n}\right)$ with $\tau_i$ terms and $p \in \mathcal P_n$ an $n$-ary predicate symbol: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({p \left({\tau_1, \ldots, \tau_n}\right) }\right) } \left[{\sigma}\right]  :=  p_{\mathcal A} \left({ \mathop{ \operatorname{val}_{\mathcal A} \left({\tau_1}\right) } \left[{\sigma}\right], \ldots, \mathop{ \operatorname{val}_{\mathcal A} \left({\tau_n}\right) } \left[{\sigma}\right] }\right)$ :where $p_{\mathcal A}$ denotes the interpretation of $p$ in $\mathcal A$ and $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau_i}\right) } \left[{\sigma}\right]$ is the value of $\tau_i$ under $\sigma$. :If $\mathbf A = \neg \mathbf B$ with $\mathbf B$ a WFF: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({\neg \mathbf B}\right) } \left[{\sigma}\right]  :=  f^\neg \left({ \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma}\right] }\right)$ :where $f^\neg$ denotes the truth function of $\neg$. :If $\mathbf A = \left({\mathbf B \circ \mathbf B'}\right)$ with $\mathbf B, \mathbf B'$ WFFs and $\circ$ one of $\land, \lor, \implies, \iff$: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B \circ \mathbf B'}\right) } \left[{\sigma}\right]  :=  f^\circ \left({ \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma}\right], \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B'}\right) } \left[{\sigma}\right] }\right)$ :where $f^\circ$ denotes the respective truth function of $\circ$. :If $\mathbf A = \left({ \exists x: \mathbf B}\right)$ with $x \in \mathrm{VAR}$ and $\mathbf B$ a WFF: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({\exists x: \mathbf B}\right) } \left[{\sigma}\right]  :=  \begin{cases}  T & \text{if for some $a \in A$, $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma + (x / a)}\right] = T$} \\ F & \text{otherwise} \end{cases}$ :where $\sigma + (x / a)$ is the extension of $\sigma$ by mapping $x$ to $a$. :If $\mathbf A = \left({ \forall x: \mathbf B}\right)$ with $x \in \mathrm{VAR}$ and $\mathbf B$ a WFF: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({\forall x: \mathbf B}\right) } \left[{\sigma}\right]  :=  \begin{cases}  T & \text{if for all $a \in A$, $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma + (x / a)}\right] = T$} \\ F & \text{otherwise} \end{cases}$ :where $\sigma + (x / a)$ is the extension of $\sigma$ by mapping $x$ to $a$. === Sentence === {{:Definition:Value of Formula under Assignment/Sentence}}
Definition:Extension of Assignment
Let $\mathcal A$ be a structure for predicate logic. Let $\sigma$ be an assignment for $\mathcal A$. Let $y \in \mathrm{VAR}$ be a variable. Let $a \in A$ be arbitrary. Then the '''extension of $\sigma$ by mapping $y$ to $a$''', denoted $\sigma + (y / a)$, is defined by: :$\forall x \in \operatorname{dom} \left({\sigma}\right) \cup \left\{{y}\right\}: \left({\sigma + (y / a)}\right)\left({x}\right) := \begin{cases} a & \text{if $x = y$} \\ \sigma \left({x}\right) & \text{otherwise} \end{cases}$ Note in particular the case where $y \in \operatorname{dom} \left({\sigma}\right)$. If $\sigma \left({y}\right) = a'$, say, then $\sigma + (y / a)$ ''overwrites'' this value to become $a$ instead.
Definition:Value of Formula under Assignment/Sentence
Let $\mathbf A$ be a sentence in the language of predicate logic. The '''value of $\mathbf A$ in $\mathcal A$''', denoted $\operatorname{val}_{\mathcal A} \left({\mathbf A}\right)$, is defined as: :$\operatorname{val}_{\mathcal A} \left({\mathbf A}\right) := \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\varnothing}\right]$ where $\varnothing$ is the empty mapping considered as an assignment for $\mathbf A$ and $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\varnothing}\right]$ is the value of $\mathbf A$ under $\varnothing$.
Definition:Assignment for Structure
Let $\mathcal L_1$ be the language of predicate logic. Let $\mathrm{VAR}$ be the collection of variables of $\mathcal L_1$. Let $\mathcal A$ be an $\mathcal L_1$-structure on a set $A$. An '''assignment for $\mathcal A$''' is a mapping $\sigma$ such that: :the codomain of $\sigma$ is $A$ :the domain of $\sigma$ is a subset of $\mathrm{VAR}$ That is, the domain of $\sigma$ contains only variables, and maps them to elements of $A$. === Assignment for Term === {{:Definition:Assignment for Structure/Term}} === Assignment for Formula === {{:Definition:Assignment for Structure/Formula}}
Definition:Structure for Predicate Logic/Formal Semantics
Let $\mathcal L_1$ be the language of predicate logic. <section notitle="true" name="tc"> {{transclude:Definition:Structure for Predicate Logic/Formal Semantics/Sentence  |section  = tc  |title    = Formal Semantics for Sentences  |header   = 3  |increase = 1  |link     = true }} {{transclude:Definition:Structure for Predicate Logic/Formal Semantics/Well-Formed Formula  |section  = tc  |title    = Formal Semantics for WFFs  |header   = 3  |increase = 1  |link     = true }} </section>
Definition:Semantic Consequence/Predicate Logic
<section notitle="true" name="tc"> Let $\mathcal F$ be a collection of WFFs of predicate logic. Then a WFF $\mathbf A$ is a '''semantic consequence''' of $\mathcal F$ : :$\mathcal A \models_{\mathrm{PL}} \mathcal F$ implies $\mathcal A \models_{\mathrm{PL}} \mathbf A$ for all structures $\mathcal A$, where $\models_{\mathrm{PL}}$ is the models relation. </section> === Notation === That $\mathbf A$ is a '''semantic consequence''' of $\mathcal F$ is denoted as: :$\mathcal F \models_{\mathrm{PL}} \mathbf A$
Definition:Structure for Predicate Logic/Formal Semantics/Sentence
Let $\mathcal L_1$ be the language of predicate logic. <section notitle="true" name="tc"> The structures for $\mathcal L_1$ can be interpreted as a formal semantics for $\mathcal L_1$, which we denote by $\mathrm{PL}$. For the purpose of this formal semantics, we consider only sentences instead of all WFFs. The structures of $\mathrm{PL}$ are said structures for $\mathcal L_1$. A sentence $\mathbf A$ is declared ($\mathrm{PL}$-)valid in a structure $\mathcal A$ : :$\operatorname{val}_{\mathcal A} \left({\mathbf A}\right) = T$ where $\operatorname{val}_{\mathcal A} \left({\mathbf A}\right)$ is the value of $\mathbf A$ in $\mathcal A$. Symbolically, this can be expressed as: :$\mathcal A \models_{\mathrm{PL}} \mathbf A$ </section>
Definition:Structure for Predicate Logic/Formal Semantics/Well-Formed Formula
Let $\mathcal L_1$ be the language of predicate logic. <section notitle="true" name="tc"> The structures for $\mathcal L_1$ can be interpreted as a formal semantics for $\mathcal L_1$, which we denote by $\mathrm{PL_A}$. The structures of $\mathrm{PL_A}$ are pairs $\left({\mathcal A, \sigma}\right)$, where: :$\mathcal A$ is a structure for $\mathcal L_1$ :$\sigma$ is an assignment for $\mathcal A$ A WFF $\mathbf A$ is declared ($\mathrm{PL_A}$-)valid in a structure $\mathcal A$ : :$\sigma$ is an assignment for $\mathbf A$ :$\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right] = T$ where $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right]$ is the value of $\mathbf A$ under $\sigma$. Symbolically, this can be expressed as one of the following: :$\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf A$ :$\mathcal A \models_{\mathrm{PL_A}} \mathbf A \left[{\sigma}\right]$ </section>
Definition:Hypergeometric Function
A '''hypergeometric function''' is an infinite series power series defined as: :$\displaystyle {}_m \operatorname F_n \left({ {a_1, \ldots, a_m} \atop {b_1, \ldots, b_n} } \, \middle \vert {\, z}\right) := \sum_{k \mathop \ge 0} \dfrac { {a_1}^{\overline k} \cdots {a_m}^{\overline k} } { {b_1}^{\overline k} \cdots {b_n}^{\overline k} } \dfrac {z^k} {k!}$ where $x^{\overline k}$ denotes the $k$th rising factorial power of $x$.
Definition:Gaussian Binomial Coefficient
Let $q \in \R_{\ne 1}$, $r \in \R$, $m \in \Z_{\ge 0}$. The '''Gaussian binomial coefficient''' is an extension of the more conventional binomial coefficient as follows: {{begin-eqn}} {{eqn | l = \binom r m_q       | o = :=       | r = \prod_{k \mathop = 0}^{m - 1} \dfrac {1 - q^{r - k} } {1 - q^{k + 1} }       | c =  }} {{eqn | r = \dfrac {\paren {1 - q^r} \paren {1 - q^{r - 1} } \cdots \paren {1 - q^{r - m + 1} } } {\paren {1 - q^m} \paren {1 - q^{m - 1} } \cdots \paren {1 - q^1} }       | c =  }} {{end-eqn}}
Definition:Multinomial Coefficient
Let $k_1, k_2, \ldots, k_m \in \Z_{\ge 0}$ be positive integers. The '''multinomial coefficient''' of $k_1, \ldots, k_m$ is defined as: :$\dbinom {k_1 + k_2 + \cdots + k_m} {k_1, k_2, \ldots, k_m} := \dfrac {\left({k_1 + k_2 + \cdots + k_m}\right)!} {k_1! \, k_2! \, \ldots k_m!}$
Definition:Multinomial Coefficient/Trinomial
The '''trinomial coefficient''' of $k_1, k_2, k_3$ is a particular case of a multinomial coefficient, defined as: :$\dbinom {k_1 + k_2 + k_3} {k_1, k_2, k_3} := \dfrac {\left({k_1 + k_2 + k_3}\right)!} {k_1! \, k_2! \, k_3!}$
Definition:Complete Proof System/Strongly Complete
$\mathscr P$ is '''strongly complete for $\mathscr M$ : :Every $\mathscr M$-semantic consequence is a $\mathscr P$-provable consequence. Symbolically, this can be expressed as the statement that, for every collection $\mathcal F$ of logical formulas, and every logical formula $\phi$ of $\mathcal L$: :$\mathcal F \models_{\mathscr M} \phi$ implies $\mathcal F \vdash_{\mathscr P} \phi$
Definition:Sound Proof System/Strongly Sound
$\mathscr P$ is '''strongly sound for $\mathscr M$''' iff: :Every $\mathscr P$-provable consequence is an $\mathscr M$-semantic consequence. Symbolically, this can be expressed as the statement that, for every collection of logical formulas $\mathcal F$, and logical formula $\phi$ of $\mathcal L$: :$\mathcal F \vdash_{\mathscr P} \phi$ implies $\mathcal F \models_{\mathscr M} \phi$
Definition:Finitely Satisfiable
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be a collection of logical formulas of $\mathcal L$. Then $\mathcal F$ is '''finitely satisfiable for $\mathscr M$''' : :For each finite subset $\mathcal F' \subseteq \mathcal F$, there is some $\mathscr M$-model $\mathcal M$ of $\mathcal F'$ That is, for each such $\mathcal F'$, there exists some structure $\mathcal M$ of $\mathscr M$ such that: :$\mathcal M \models_{\mathscr M} \mathcal F'$
Definition:Cardinality of Structure
Let $\mathcal A$ be a first-order structure. Then the '''cardinality of $\mathcal A$''', denoted $\left\vert{ \mathcal A }\right\vert$, is defined as: :$\left\vert{ \mathcal A }\right\vert := \left\vert{ A }\right\vert$ where $\left\vert{ A }\right\vert$ is the cardinality of the underlying set $A$.
Definition:Semantic Equivalence/Predicate Logic
<section notitle="true" name="tc"> Let $\mathbf A, \mathbf B$ be WFFs of predicate logic. === Definition 1 === {{:Definition:Semantic Equivalence/Predicate Logic/Definition 1|Definition 1}} === Definition 2 === {{:Definition:Semantic Equivalence/Predicate Logic/Definition 2|Definition 2}} </section>
Definition:Semantic Equivalence/Predicate Logic/Definition 1
Then $\mathbf A$ and $\mathbf B$ are '''equivalent''' : :$\mathbf A \models_{\mathrm{PL_A}} \mathbf B$ and $\mathbf B \models_{\mathrm{PL_A}} \mathbf A$ that is, each is a semantic consequence of the other. That is to say, $\mathbf A$ and $\mathbf B$ are '''equivalent''' , for all structures $\mathcal A$ and assignments $\sigma$: :$\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf A$ iff $\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf B$ where $\models_{\mathrm{PL_A}}$ denotes the models relation.
Definition:Tautology/Formal Semantics/Predicate Logic
Let $\mathbf A$ be a WFF of predicate logic. Then $\mathbf A$ is a '''tautology''' : :$\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf A$  for every structure $\mathcal A$ and assignment $\sigma$. That $\mathbf A$ is a '''tautology''' can be denoted as: :$\models_{\mathrm{PL_A}} \mathbf A$
Definition:Approximating Relation
Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $\mathcal R$ be a relation on $S$. Then $\mathcal R$ is '''approximating relation''' on $S$  :$\forall x \in S: x = \sup \left({x^{\mathcal R} }\right)$ where $x^{\mathcal R}$ denotes the $\mathcal R$-segment of $x$.
Definition:Stirling Numbers of the First Kind/Unsigned/Definition 2
'''Unsigned Stirling numbers of the first kind''' are defined as the polynomial coefficients $\displaystyle {n \brack k}$ which satisfy the equation: :$\displaystyle x^{\underline n} = \sum_k \left({-1}\right)^{n - k} {n \brack k} x^k$ where $x^{\underline n}$ denotes the $n$th falling factorial of $x$.