Lecture 1-03.01.2017

　　1.1 Universal algebra

　　Defination: An algebrical vocabulary $\Omega$ consists of :

• A denumberable none-empty set $V=\{...,x,y,z\}$(variables).
• A set for $F_{n}$ for each $n\ge 1$.
• A set $F_{0}$.　　

　　Defination: An $\Omega- structure$consists of

• A nonempty set $A$
• $\forall \omega \in A ,\omega_{A}:A^{n}\rightarrow A$(n-ary map).

　　Example:  A group $G$:

　　$|F_{0}|=1$:　uinte:$e$

　　$|F_{1}|=1$: $  ()^{-1}:G\rightarrow G $

　　$|F_{2}|=1$: $  \times :G\times G \rightarrow G $

　　$|F_{n}|=0 , n\ge 3$ 

　　Defination: A homomorphism $f:\left( A, \left(\omega_{A}\right)_{\omega}\right) \longrightarrow \left( B, \left(\omega_{B}\right)_{\omega}\right) $ is a map 

　　$f:A\longrightarrow B$ such that $\forall \omega $, the below gragh  communicate:

　　$$\begin{array}[c]{ccc}
　　A^{n}&\stackrel{\omega_{A}}{\longrightarrow}&A\\
　　\Big\downarrow\scriptstyle{f^{n}}&&\Big\downarrow\scriptstyle{f}\\
　　B^{n}&\stackrel{\omega_{B}}{\longrightarrow}&B
　　\end{array}$$　

　　Example: Free $\Omega-structure$ generated by $V$,denote by $\mathcal{L}(V)$.

　　elements of $\mathcal{L}(V)$:$terms$.

• each variable is a term
• if $\omega \in F_{n}$, and if $t_{1},\cdots ,t_{n}$ are terms,then $\omega( t_{1},\cdots ,t_{n})$ is a term.

$$\begin{array}[c]{ccccc}
V &\xrightarrow{i}&\mathcal{L}(V)\\
&\searrow{f}&\Big\downarrow\\
&&(A,(\omega{A})_{\omega})
\end{array}$$

　　In most time, we take$\left( A, \left(\omega_{A}\right)_{\omega}\right) =\mathbb{2}=\{0,1\}$

　　1.2  The language of Sentential(Proposition）Logic

　　Example: $\Omega-structure$ : $A=\{0,1\}$(true:$1$,false:$0$)

　　$\neg 0=1,\neg 1=0$,

$0\vee 0=0,0\vee 1=1\vee 0=1\vee 1=1$,

$0\wedge 0=1\wedge 0=0\wedge 1=0,1\wedge 1=1$

,$0\rightarrow 0=0\rightarrow 1=1\rightarrow 1=1,1\rightarrow 0=0$.

　　Let $X$ be the set of some sentence symbol and denote the set of all sentences generated by $X$ and these five function $\neg,\vee,\wedge,\rightarrow,\leftrightarrow$ by $P(X)$.

　　Defination:(Trueth Assignments) A truth assignment on $P(X)$ is a homomorphism $\bar{v}:\  P(X)\longrightarrow 2$ which is extended by $v:\ X\longrightarrow 2$.

$$\begin{array}[c]{ccccc}
X &\xrightarrow{i}&P(X)\\
&\searrow{v}&\Big\downarrow\\
&&2
\end{array}$$

\end{array}$$

　　Defination: 

1. A truth assignment $\bar{v}:\ P(X)\longrightarrow 2$ satisfies $\tau \in P(X)$ if $\bar{v}(\tau)=1$,then $\bar{v}(\neg \tau)=0$.
2. $\Sigma \subseteq P(X)$ sententially implies $\tau \in P(X)$ if for any truth assignments $v:\ X\longrightarrow 2$ if $\bar{v}(\alpha)=1$ for all $\alpha \in \Sigma$, then $\bar{v}(\tau)=1$.Denote by $\Sigma \models \tau$.　　　
3. $\emptyset \models \tau$(i.e. $\forall v: \ X\longrightarrow 2,\ \bar{v}(\tau)=1$).

　　Tautology: $\tau$ is said to be a tautology if and only if $\models \tau$.

　　Hilbert definded these three types tautology which is now said to be axiom:

• $\mathscr{A}_{1}:\  \alpha \rightarrow (\beta \rightarrow \alpha)$.
• $\mathscr{A}_{2}:\  (\alpha \rightarrow (\beta \rightarrow \gamma))\rightarrow (\alpha \rightarrow \beta)\rightarrow (\alpha \rightarrow \gamma)$.
• $\mathscr{A}_{3}:\  \neg \neg \alpha \rightarrow \alpha$.　

　　Let $\Omega\ =\{\perp,\rightarrow\}$ where $\perp$ is consistant and $\rightarrow$ is a binary operation. Then $2=\{0,1\}$ is a $\Omega-structure$.

Notes on Logic-lecture 1