Propositional Logic(2)

 

We'll use truth value symbols $T$ for $True$, $F$ for $False$.

 

about truth values, a truth assignment for v for a sentence symbols set S is a function $$v : S \rightarrow \{F,T\}$$ now let $\bar{S}$ be the set of all wffs(remind induction principle for wffs). then extension $\bar{v}$ of $v$ is a function $\bar{v} : \bar{S} \rightarrow {F,T}$ which meets below conditions.

 

for any $A \in S$, $\bar{v}(A)=v(A)$

 

and for any $\alpha, \beta$ in $\bar{S}$

 

 

 

 

 

 

 

now define logical definitions more rigorously. we'll call a truth assignment $v$ satisfies $\varphi$ if and only if $\bar{v}(\varphi)=T$. using this definition, introduce the concept of tautology.

 

Let $\Sigma$ be a set of wffs, and $\tau$ a wff. then $\Sigma$ tautologically implies $\tau$ if and only if every truth assignment for sentence symbols in $\Sigma$ and $\tau$ that satisfies every member of $\Sigma$ also satisfies $\tau$. In symbol, we'll write this concept as $\Sigma \models \tau$.

 

$\emptyset$ is special case for this. one can easily check $\emptyset \models \tau$ if and only if every truth assignment for sentence symbols in $\tau$ satisfies $\tau$(use vacuous truth). we'll shorten notation for this as $\models \tau$ and call $\tau$ a tautology.

 

another special case is when $\Sigma$ is a singleton set. we write $\varphi\models\tau$ for$\{\varphi\}\models\tau$. and for a case $\varphi\models\tau$ and $\tau\models\varphi$, we say $\tau$ and $\varphi$ are tautologically equivalent.

 

p.s.

implication을 수식으로 나타내면 다음과 같습니다.
$\Sigma\models\tau$ iff for every $v$ for $S$, $\bar{v}(\Sigma)=\{T\} \rightarrow \bar{v}(\tau)=T$ where $v$ is a truth assignment for a set $S$ of sentence symbols of $\Sigma$ and $\tau$.