# Szerkesztő:Thuluviel/term

## Szokások

### Kalkulusok

#### Elsőrendű kalkulus Csirmaz Lászlónál

(${\displaystyle \scriptstyle {\varphi \to \psi \iff _{def}(\lnot \varphi )\vee \psi }}$ ) ${\displaystyle \scriptstyle {\begin{array}{cl}(A1)&\varphi \vee \varphi \to \varphi \\(A2)&\varphi \to \varphi \vee \psi \\(A3)&\varphi \vee \psi \to \psi \vee \varphi \\(A4)&(\varphi \to \psi )\to (\theta \vee \varphi \to \theta \vee \psi )\\&{\text{ vegul ha minden }}k\in K(t){\text{ kifejezesre és }}x{\text{ valtozojelre, ha a }}\varphi [x/k]{\text{ helyettesites megengedett,}}\\(A5)&\varphi [x/k]\to \exists x\varphi \\(E1)&x=x\\(E2)&x=y\to y=x\\(E3)&x=y\land y=z\to x=z\\(E4)&x_{0}=y_{0}\land \dots \land x_{n-1}=y_{n-1}\to (r(x_{0},\dots ,x_{n-1})\to r(y_{0},\dots ,y_{n-1}))\\(E5)&x_{0}=y_{0}\land \dots \land x_{n-1}=y_{n-1}\to f(x_{0},\dots ,x_{n-1})\to r(y_{0},\dots ,y_{n-1})\\(K1)&\varphi \to \psi ,\varphi |\psi \\(K2)&\varphi \to \psi |(\exists x\varphi )\to \psi \\\end{array}}}$

#### Elsőrendű kalkulus E. Szabó Lászlónál

${\displaystyle \scriptstyle {\begin{array}{cl}(PC1)&(\phi \to (\psi \to \phi ))\\(PC2)&((\phi \to (\psi \to chi))\to ((\phi \to \psi )\to (\psi \to \chi ))\\(PC3)&((\lnot \phi \to \lnot \psi )\to (\psi \to \phi ))\\(PC4)&(\forall x(\phi \to \psi )\to (\phi \to \forall x\psi )){\text{, ha }}x{\text{ nem fordul elo szabadon }}\varphi {\text{-ben.}}\\(PC5)&(\forall x\phi \to \phi ){\text{, ha }}x{\text{ nem fordul elo szabadon }}\varphi {\text{-ben.}}\\(PC6)&(\forall x\phi (x)\to \phi (t)){\text{ feltéve, hogy a }}t{\text{ terminus szabad }}x{\text{-re nezve }}\phi (x){\text{-ben.}}\\(MP)&\phi {\text{-bol és }}(\phi \to \psi ){\text{-bol kovetkezik }}\psi \\(G)&\phi {\text{-bol kovetkezik }}\forall x\phi \\(E1)&E(x,x)\\(E2)&E(t,s)\to E(f^{n}(u_{1},u_{2},\dots ,t,\dots u_{n}),f^{n}(u_{1},u_{2},\dots ,s,\dots ,u_{n}))\\(E3)&E(t,s)\to (\phi (u_{1},u_{2},\dots ,t,\dots u_{n})\to \phi (u_{1},u_{2},\dots ,s,\dots ,u_{n}))\\\end{array}}}$

#### Elsőrendű kalkulus Ferenczi Miklósnál

${\displaystyle \scriptstyle {\begin{array}{cl}(i)&\alpha \to \beta \to \alpha \\(ii)&(\alpha \to \beta \to \gamma )\to (\alpha \to \beta )\to \alpha \to \gamma \\(iii)&(\lnot \alpha \to \beta )\to (\lnot \alpha \to \lnot \beta )\to \alpha \\(iv)&\forall x(\alpha \to \beta )\to (\forall x\alpha \to \forall x\beta )\\(v)&\alpha \to \forall x\alpha {\text{, ha }}x{\text{ nem szabad }}\alpha {\text{-ban}}\\(vi)&\forall x\alpha \to \alpha (y/t)\\(vii)&(t=t)\\(viii)&t_{1}=t_{n+1}\to \dots t_{n}=t_{2n}\to f(t_{1},\dots ,t_{n})=f(t_{n+1},\dots ,t_{2n})\\(ix)&t_{1}=t_{n+1}\to \dots t_{n}=t_{2n}\to R(t_{1},\dots ,t_{n})=f(t_{n+1},\dots ,t_{2n})\\(x)&{\text{az }}(i)-(ix){\text{-ben foglalt semak osszes lehetséges altalanositasai,}}\\&{\text{ ahol egy }}\alpha {\text{ sema egy lehetseges altalanositasan vagy generalizaciojan}}\\&{\text{ ertjuk az }}\forall x\alpha {\text{ semat, ahol }}x{\text{ tetszoleges individuumvaltozo.}}\\(MP)&{\frac {\alpha ,\alpha \to \beta }{\beta }}\\\end{array}}}$

#### Elsőrendű kalkulus Komjáth Péternél

${\displaystyle \scriptstyle {\begin{array}{cl}1.&\alpha \to (\beta \to \alpha )\\2.&(\alpha \to (\beta \to \gamma ))\to ((\alpha \to \beta )\to (\alpha \to \gamma ))\\3.&(\lnot \beta \to \lnot \alpha )\to (\lnot \beta \to \alpha )\to \beta \\4.&\forall v_{i}\varphi \to \varphi _{v_{i}}[t]{\text{, ha a helyettesites megengedett}}\\5.&(\forall v_{i}(\varphi \to \psi ))\to \varphi \to \forall v_{i}\psi ){\text{, ha }}v_{i}\notin V(\varphi )\\6.&v_{i}=v_{i}\quad (i=0,1,...)\\7.&(v_{i}=v_{j})\to (v_{j}=v_{i})\quad (i=0,1,...)\\8.&((v_{i}=v_{j})\land (v_{j}=v_{k}))\to (v_{i}=v_{k})\quad (i,j,k=0,1,...)\\9.&((v_{i_{1}}=v_{j_{1}})\land \cdots \land (v_{i_{n}}=v_{j_{n}}))\to (R(v_{i_{1}},\dots ,v_{i_{n}})\to R(v_{j_{1}},\dots ,v_{j_{n}}))\\10.&((v_{i_{1}}=v_{j_{1}})\land \cdots \land (v_{i_{n}}=v_{j_{n}}))\to (f(v_{i_{1}},\dots ,v_{i_{n}})=f(v_{j_{1}},\dots ,v_{j_{n}}))\\(MP)&\alpha ,\alpha \to \beta \vdash \beta \\(GEN)&\phi \vdash \forall v_{i}\phi ,\quad (i=0,1,...)\\\end{array}}}$

#### Elsőrendű kalkulus Pásztorné Varga Katalinnál és Várterész Magdánál

${\displaystyle \scriptstyle {\begin{array}{cl}(B1)&A\supset (B\supset A)\\(B2)&(A\supset (B\supset C))\supset ((A\supset B)\supset (A\supset C))\\(B3)&(\lnot A\supset B)\supset ((\lnot A\supset \lnot B)\supset A)\\(B4)&\forall xA\supset [A(x\parallel t)]\\(B5)&(\forall x(A\supset B)\supset (\forall xA\supset \forall xB))\\(B6)&(A\supset (\forall x\cdot A){\text{, ahol }}x\notin Par(A)\\(B7)&{\text{ a }}(B1)-(B6){\text{ semak altalanositasai}}\\(MP)&\{A\supset B,A\}\vdash B\\\end{array}}}$

#### Elsőrendű kalkulus Ruzsa Imrénél

${\displaystyle \scriptstyle {\begin{array}{cl}(A1)&(A\supset (B\supset A))\\(A2)&((A\supset (B\supset C))\supset ((A\supset B)\supset (A\supset C)))\\(A3)&((\sim A\supset \sim B)\supset (B\supset A))\\(A4)&(\forall x\cdot A\supset A_{x}^{t})\\(A5)&(\forall x(A\supset B)\supset (\forall x\cdot A\supset \forall x\cdot B))\\(A6)&(A\supset (\forall x\cdot A){\text{- ha }}A{\text{-ban }}x{\text{-nek nincs szabad elofordulasa}}\\(A7)&(\sigma =\sigma )\\(A8)&(x=y\supset (A_{z}^{x}=A_{z}^{y}))\\(MP)&\{A,A\supset B\}\vdash B\\\end{array}}}$