Metatheoria: introdvctio

Antequam vnitatem incipias, pvnctvm vnde proficisceris metire. Brevis est sui ipsivs probatio: non aestimatvr, et eivs exitvs in solo tvo navigatro servatvr.

1. Qvaestio 1Ex vnitatibvs prioribvsQvid significat $\Gamma \models \varphi$?

   Omnem aestimationem qvae omnes formvlas classis $\Gamma$ satisfacit etiam $\varphi$ satisfacere.

   Exstare dedvctionem formvlae $\varphi$ ex $\Gamma$.

   Aliqvam aestimationem simvl $\Gamma$ et $\varphi$ satisfacere.

   $\varphi$ ex vna sola formvla classis $\Gamma$ seqvi.

2. Qvaestio 2Ex vnitatibvs prioribvsQvod ex seqventibvs envntiatis aeqvivalet ipsi $\Gamma \models \varphi$?

   $\Gamma \cup \{\neg\varphi\}$ insatisfactibilem esse.

   $\Gamma \cup \{\varphi\}$ satisfactibilem esse.

   $\Gamma \cup \{\neg\varphi\}$ satisfactibilem esse.

   $\neg\varphi$ insatisfactibilem esse.

3. Qvaestio 3Ex vnitatibvs prioribvsSi aestimatio $v$ formvlae $\alpha$ valorem $1$ tribvit, qvem valorem tribvit ipsi $(\alpha \downarrow \beta)$?

   $0$, qvicvmqve sit valor ipsivs $\beta$.

   Pendet ex $\beta$: valet $1$ si $\beta$ valet $0$.

   $1 - v(\beta)$.

   $1$, qvia vna componentivm vera est.

4. Qvaestio 4Ex vnitatibvs prioribvsVariabiles formvlae $\varphi$ svnt $v_{2}$ et $v_{7}$. Dvae aestimationes $v$ et $w$ eosdem valores ipsis $v_{2}$ et $v_{7}$ tribvvnt, et valores oppositos omnibvs ceteris variabilibvs. Si $v(\varphi) = 1$, qvid de $w(\varphi)$ dici potest?

   $w(\varphi) = 1$.

   Nihil, qvia $v$ et $w$ in infinitis variabilibvs differvnt.

   $w(\varphi) = 0$, qvia $w$ fere omnes valores ipsivs $v$ invertit.

   $w(\varphi) = 1$ solvm si $v$ et $w$ etiam in $v_{1}, \ldots, v_{6}$ congrvvnt.

5. Qvaestio 5Ex vnitatibvs prioribvsSit $\Gamma$ classis infinita formvlarvm cvm $\Gamma \vdash \varphi$. Qvid affirmari potest?

   Aliqvam svbclassem finitam classis $\Gamma$ dedvcere $\varphi$.

   $\varphi$ theorema esse.

   Dedvctionem omnibvs formvlis classis $\Gamma$ vti.

   Nihil sine theoremate compactitatis, qvia $\Gamma$ infinita est.

6. Qvaestio 6Ex vnitatibvs prioribvsScitvr $\Gamma \cup \{p\} \vdash \chi$ et $\Gamma \cup \{\neg p\} \vdash \chi$. Qvid demonstratio per casvs conclvdere sinit?

   $\Gamma \vdash \chi$.

   Nihil, donec sciatvr vtrvm $\Gamma$ dedvcat $p$ an dedvcat $\neg p$.

   $\Gamma \vdash (p \leftrightarrow \chi)$.

   Solvm $\Gamma \cup \{p, \neg p\} \vdash \chi$.

7. Qvaestio 7Systema cvrsvs firmvm est. Qvid haec proprietas affirmat?

   Pro omni classe $\Gamma$, si $\Gamma \vdash \varphi$, tvnc $\Gamma \models \varphi$.

   Si $\Gamma \models \varphi$, tvnc $\Gamma \vdash \varphi$.

   Non exstare formvlam $\varphi$ cvm $\vdash \varphi$ et $\vdash \neg\varphi$.

   Omnem formvlam veram theorema esse.

8. Qvaestio 8Sit $v$ aestimatio qvae valorem $1$ ipsi $p$ et valorem $0$ ipsi $q$ tribvit. Qvid affirmat lemma Kalmarii, cvm variabilibvs $p$ et $q$, pro formvla $(p \leftrightarrow q)$?

   $\{p, \neg q\} \vdash \neg(p \leftrightarrow q)$.

   $\{p, \neg q\} \vdash (p \leftrightarrow q)$.

   $\{p, q\} \vdash \neg(p \leftrightarrow q)$.

   $\vdash \neg(p \leftrightarrow q)$.

9. Qvaestio 9Potestne systema dedvctivvm firmvm esse qvin completvm sit?

   Ita: systema Łvkasiewiczii sine regvla SDN firmvm est neqve tavtologiam $((v_{1} \downarrow v_{2}) \rightarrow \neg v_{1})$ dedvcit.

   Minime: firmitas et completvdo altera alteram implicant.

   Minime: systema cvivs axiomata tavtologiae svnt omnes tavtologias dedvcit.

   Ita, sed solvm si systema inconsistens est.

10. Qvaestio 10Qvid affirmat theorema compactitatis?

    Classem formvlarvm satisfactibilem esse si et solvm si omnis eivs svbclassis finita satisfactibilis est.

    Classem formvlarvm satisfactibilem esse si et solvm si omnis eivs formvla satisfactibilis est.

    Omnem classem infinitam formvlarvm satisfactibilem esse.

    Classem satisfactibilem esse si aliqva eivs svbclassis finita est.

11. Qvaestio 11Sit $\Gamma$ classis infinita formvlarvm cvm $\Gamma \models \varphi$. Qvid compactitas praestat?

    $\Gamma_{0} \models \varphi$ pro aliqva svbclasse finita $\Gamma_{0}$ classis $\Gamma$.

    $\varphi$ ex vna sola formvla classis $\Gamma$ seqvi.

    $\varphi$ ex omni svbclasse finita classis $\Gamma$ seqvi.

    $\varphi$ validam esse.

12. Qvaestio 12Prima lectio completvdinem pro classibvs finitis praemissarvm demonstrat. Cvr finitas dedvctionis non svfficit ad eam in classem infinitam $\Gamma$ extendendam?

    Qvia finitas de $\Gamma \vdash \varphi$ loqvitvr, et qvod habetvr est $\Gamma \models \varphi$: oportet vt conseqventia, non dedvctio, ad partem finitam redigatvr, et hoc est compactitas.

    Immo svfficit: si $\Gamma \models \varphi$, finitas praebet $\Gamma_{0}$ finitam cvm $\Gamma_{0} \models \varphi$.

    Qvia lemma Kalmarii pro formvlis cvm infinitis variabilibvs non valet.

    Qvia completvdo pro classibvs infinitis praemissarvm falsa est.

13. Qvaestio 13Qvae est resolvens clavsvlarvm $\{p, \neg q, r\}$ et $\{q, s\}$?

    $\{p, r, s\}$.

    $\{p, \neg q, r, q, s\}$.

    $\{p, r\}$.

    Resolventem non habent, qvia nvllvm litterale commvnicant.

14. Qvaestio 14Vt per resolvtionem decernatvr vtrvm $\{(p \vee q), \neg p\} \models q$, qvam classem clavsvlarvm refvtare conaris?

    $\{\{p, q\}, \{\neg p\}, \{\neg q\}\}$.

    $\{\{p, q\}, \{\neg p\}, \{q\}\}$.

    $\{\{p, q\}, \{\neg p\}\}$.

    $\{\{\neg p, \neg q\}, \{p\}, \{\neg q\}\}$.

15. Qvaestio 15Potestne clavsvla vacva $\Box$ per resolvtionem ex classe $\{\{p, q\}, \{\neg p, \neg q\}\}$ derivari?

    Minime: classis satisfactibilis est et resolvtio firma est.

    Ita, vna sola illatione, ambo paria complementaria simvl tollendo.

    Ita, dvabvs illationibvs: primvm in $p$, deinde in $q$.

    Minime, qvia nevtra clavsvla singvlaris est.