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Modal logic fixups (#363)
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* truth-at-w.tex: Add missing w

* truth-in-model.tex: Add missing "in" to "we are interested which"

* schemas.tex: Add missing w
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davidvandebunte authored Feb 20, 2024
1 parent 6ce0abe commit 66d0133
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W$ be arbitrary. To show that a conditional is true at a world we
assume the antecedent is true to show that consequent is true as
well. In this case, let $\mSat{M}{\Box(!A \lif !B)}[w]$ and
$\mSat{M}{\Box !A}[w]$. We need to show $\mSat{M}{\Box !B}$. So let
$\mSat{M}{\Box !A}[w]$. We need to show $\mSat{M}{\Box !B}[w]$. So let
$w'$ be arbitrary such that $Rww'$. Then by the first assumption
$\mSat{M}{!A \lif !B}[w']$ and by the second assumption
$\mSat{M}{!A}[w']$. It follows that $\mSat{M}{!B}[w']$. Since $w'$
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\begin{proof}
\begin{enumerate}
\item $\mSat{M}{\lnot\Diamond\lnot !A}[w]$ iff $\mSat/{M}{\Diamond\lnot
!A}$ by definition of $\mSat{M}{}[w]$. $\mSat{M}{\Diamond\lnot
!A}[w]$ by definition of $\mSat{M}{}[w]$. $\mSat{M}{\Diamond\lnot
!A}[w]$ iff for some $w'$ with $Rww'$, $\mSat{M}{\lnot
!A}[w']$. Hence, $\mSat/{M}{\Diamond\lnot !A}[w]$ iff for all
$w'$ with $Rww'$, $\mSat/{M}{\lnot !A}[w']$. We also have
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\olsection{Truth in a Model}

Sometimes we are interested which !!{formula}s are true at every world
Sometimes we are interested in which !!{formula}s are true at every world
in a given model. Let's introduce a notation for this.

\begin{defn}
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