束値論理
latice valued logic。廣域論理 (global logic)
$ 1:=\max{\cal L}
$ 0:=\min{\cal L}
$ \varphiを命題とし、$ \llbracket\varphi\rrbracketをその眞理値とする
眞理値$ \llbracket~\rrbracketは$ \cal Lを値とする
廣域含意$ \xrightarrow{\square}
$ \llbracket\varphi\xrightarrow{\square}\psi\rrbracket=1iff.$ \llbracket\varphi\rrbracket\le\llbracket\psi\rrbracket
$ \llbracket\varphi\xrightarrow{\square}\psi\rrbracket=0iff.$ \llbracket\varphi\rrbracket\cancel\le\llbracket\psi\rrbracket
略記
$ \topiff.$ \varphi\xrightarrow{\square}\varphi
$ \botiff.$ \neg\top
$ \neg\varphiiff.$ \varphi\xrightarrow{\square}\bot
$ \varphi\xleftrightarrow{\square}\psiiff.$ (\varphi\xrightarrow{\square}\psi)\land(\psi\xrightarrow{\square}\varphi)
$ \square\varphiiff.$ (\varphi\xrightarrow{\square}\varphi)\xrightarrow{\square}\varphi
規則 N (必然化規則)$ \frac{\top\vdash\varphi}{\top\vdash\square\varphi}に當たる
$ \lozenge\varphiiff.$ \neg(\neg\square\varphi)
? 樣相論理と見做すとどの公理系に當たるか
公理 K$ \square(\varphi\to\psi)\to(\square\varphi\to\square\psi)は?
$ \square閉 ($ \square-closed)
以下の論理式は$ \square閉であると言ふ
$ \varphi\xrightarrow{\square}\psiの形の論理式は$ \square閉である
$ \square閉な論理式$ \varphi,$ \psiに對して、$ \varphi\land\psi,$ \varphi\lor\psi,$ \neg\varphiは$ \square閉である
$ aを自由變數とする論理式$ \varphi(a)が$ \square閉ならば、$ \forall x\varphi(x),$ \exist x\varphi(x)も$ \square閉である
$ \square閉な論理式の眞理値は$ 1または$ 0となる
記號
$ \Gamma,\Delta,\Pi,\Lambda,\dotsは論理式の有限列
$ \bar\varphi,\bar\psi,\dotsは$ \square閉な論理式
$ \bar\Gamma,\bar\Delta,\bar\Pi,\bar\Lambda,\dotsは$ \square閉な論理式の有限列
公理
$ \varphi\vdash\varphi
構造規則
弱化 (增 W)$ \frac{\Gamma\vdash\Delta}{\varphi,\Gamma\vdash\Delta},$ \frac{\Gamma\vdash\Delta}{\Gamma\vdash\Delta,\varphi} .
縮約 (減 C)$ \frac{\varphi,\varphi,\Gamma\vdash\Delta}{\varphi,\Gamma\vdash\Delta},$ \frac{\Gamma\vdash\Delta,\varphi,\varphi}{\Gamma\vdash\Delta,\varphi} .
轉置 (換 P)$ \frac{\Gamma,\varphi,\psi,\Pi\vdash\Delta}{\Gamma,\psi,\varphi,\Pi\vdash\Delta},$ \frac{\Gamma\vdash\Delta,\varphi,\psi,\Lambda}{\Gamma\vdash\Delta,\psi,\varphi,\Lambda} .
cut (三段論法)$ \frac{\Gamma\vdash\bar\Delta,\varphi\quad\varphi,\Pi\vdash\Lambda}{\Gamma,\Pi\vdash\bar\Delta,\Lambda},$ \frac{\Gamma\vdash\Delta,\varphi\quad\varphi,\bar\Pi\vdash\Lambda}{\Gamma,\bar\Pi\vdash\Delta,\Lambda},$ \frac{\Gamma\vdash\Delta,\bar\varphi\quad\bar\varphi,\Pi\vdash\Lambda}{\Gamma,\Pi\vdash\Delta,\Lambda}
論理規則
$ \land
$ \frac{\varphi,\Gamma\vdash\Delta}{\varphi\land\psi,\Gamma\vdash\Delta},$ \frac{\psi,\Gamma\vdash\Delta}{\varphi\land\psi,\Gamma\vdash\Delta}
.
$ \frac{\Gamma\vdash\bar\Delta,\varphi\quad\Gamma\vdash\bar\Delta,\psi}{\Gamma\vdash\bar\Delta,\varphi\land\psi},$ \frac{\Gamma\vdash\Delta,\bar\varphi\quad\Gamma\vdash\Delta,\bar\psi}{\Gamma\vdash\Delta,\bar\varphi\land\bar\psi}
$ \lor
$ \frac{\varphi,\bar\Gamma\vdash\Delta\quad\psi,\bar\Gamma\vdash\Delta}{\varphi\lor\psi,\bar\Gamma\vdash\Delta},$ \frac{\bar\varphi,\Gamma\vdash\Delta\quad\bar\psi,\Gamma\vdash\Delta}{\bar\varphi\lor\bar\psi,\Gamma\vdash\Delta}
.
$ \frac{\Gamma\vdash\Delta,\varphi}{\Gamma\vdash\Delta,\varphi\lor\psi},$ \frac{\Gamma\vdash\Delta,\psi}{\Gamma\vdash\Delta,\varphi\lor\psi}
$ \neg
$ \frac{\Gamma\vdash\Delta,\bar\varphi}{\neg\bar\varphi,\Gamma\vdash\Delta}
.
$ \frac{\bar\varphi,\Gamma\vdash\Delta}{\Gamma\vdash\Delta,\neg\bar\varphi}
$ \xrightarrow{\square}
$ \frac{\Gamma\vdash\bar\Delta,\varphi\quad\psi,\bar\Pi\vdash\Lambda}{(\varphi\xrightarrow{\square}\psi),\Gamma,\bar\Pi\vdash\bar\Delta,\Lambda}
.
$ \frac{\varphi,\bar\Gamma\vdash\bar\Delta,\psi}{\bar\Gamma\vdash\bar\Delta,(\varphi\xrightarrow{\square}\psi)},$ \frac{\bar\varphi,\Gamma\vdash\Delta,\bar\psi}{\Gamma\vdash\Delta,(\bar\varphi\xrightarrow{\square}\bar\psi)}
$ \forall
$ \frac{\varphi[t/x],\Gamma\vdash\Delta}{\forall x\varphi(x),\Gamma\vdash\Delta}
.
$ \frac{\Gamma\vdash\bar\Delta,\varphi[a/x]}{\Gamma\vdash\bar\Delta,\forall x\varphi(x)} ,$ \frac{\Gamma\vdash\Delta,\bar\varphi[a/x]}{\Gamma\vdash\Delta,\forall x\bar\varphi(x)}
$ \exist
$ \frac{\varphi[a/x],\bar\Gamma\vdash\Delta}{\exist x\varphi(x),\bar\Gamma\vdash\Delta} ,$ \frac{\bar\varphi[a/x],\Gamma\vdash\Delta}{\exist x\bar\varphi(x),\Gamma\vdash\Delta}
.
$ \frac{\Gamma\vdash\Delta,\varphi[t/x]}{\Gamma\vdash\Delta,\exist x\varphi(x)}
定理
$ \square\varphi\vdash\varphi
$ \frac{\bar\Gamma\vdash\varphi}{\bar\Gamma\vdash\square\varphi}
束値宇宙 (latice valued univerce)$ V^{\cal L}
$ {V^{\cal L}}_0:=\varnothing
$ {V^{\cal L}}_\alpha:=\{u|\beta<\alpha,u:{V^{\cal L}}_\beta\to{\cal L}\}
$ V^{\cal L}:=\bigcup_{\alpha\in{\rm On}}{V^{\cal L}}_\alpha
眞理値の割り當て
$ \llbracket\varphi\land\psi\rrbracket:=\llbracket\varphi\rrbracket\land\llbracket\psi\rrbracket=\min\{\llbracket\varphi\rrbracket,\llbracket\psi\rrbracket\}
$ \llbracket\varphi\lor\psi\rrbracket:=\llbracket\varphi\rrbracket\lor\llbracket\psi\rrbracket=\max\{\llbracket\varphi\rrbracket,\llbracket\psi\rrbracket\}
$ \llbracket\varphi\xrightarrow{\square}\psi\rrbracket:=\llbracket\varphi\rrbracket\xrightarrow{\square}\llbracket\psi\rrbracket=\begin{cases}1 & \llbracket\varphi\rrbracket\le\llbracket\psi\rrbracket \\ 0 & \llbracket\varphi\rrbracket\cancel\le\llbracket\psi\rrbracket\end{cases}
$ \llbracket\neg\varphi\rrbracket:=\neg\llbracket\varphi\rrbracket=\begin{cases}1 & \llbracket\varphi\rrbracket=0 \\ 0 & \llbracket\varphi\rrbracket=1\end{cases}
$ \llbracket\forall x~\varphi(x)\rrbracket:=\bigwedge\{\llbracket\varphi(\alpha)\rrbracket|\alpha\in V^{\cal L}\}
$ \llbracket\exist x~\varphi(x)\rrbracket:=\bigvee\{\llbracket\varphi(\alpha)\rrbracket|\alpha\in V^{\cal L}\}