Demonstratio Rationum Logicae Classicae

SUMMARIUM
In hac lectione exhibentur variae technicae logicae classicae ad coniunctiones et disiunctiones introducendas atque removendas, praeterea regula tertii exclusi et regula contradictionis, quae etiam praeceptum explosionis appellatur. Explicatur quoque technica probationis per casus et reductio ad absurdum, ambae in demonstrationibus mathematicis ac logicis utilissimae. Unaquaeque technica formaliter exhibetur et demonstratio gradatim proposita datur ad intellegendum. Si logicam propositionum penitius explorare cupis et facultatem theoremata demonstrandi augere vis, haec lectio tibi valde proderit.

PROPOSITA DISCENDI:

Intelligere rationem quae subest technicis coniunctionis et disiunctionis introducendae et removendae.
Intelligere proprietatem tertii exclusi sive tautologiam (TAU) in logica classica.
Intelligere regulam contradictionis (CON) sive praeceptum explosionis in logica classica.
Intelligere technicam disiunctionum removendarum (∨-eliminatio3) in logica classica.
Intelligere technicam probationum per casus (CAS) in logica classica.
Intelligere technicam reductionis ad absurdum (absurdum) in logica classica.
Adhibere scientiam diversarum technicarum logicae classicae ad quaestiones et demonstrationes difficiles solvendas.

INDEX
INTRODUCTIO ET REMOTIO CONIUNCTIONUM ET DISIUNCTIONUM
∨-INTRODUCTIO
∨-ELIMINATIO
∧-INTRODUCTIO
∧-ELIMINATIO
TECHNICAE CONTRADICTIONUM ET TAUTOLOGIARUM
REGULA TERTII EXCLUSI SIVE TAUTOLOGIA (TAU)
REGULA CONTRADICTIONIS SIVE PRAECEPTUM EXPLOSIONIS
∨-ELIMINATIO3
PROBATIONES PER CASUS (CAS)
REDUCTIO AD ABSURDUM (ABSURDUM)

Introductio et Remotio Coniunctionum et Disiunctionum

Una ex technicis logicae classicae est introductio et remotio connectorum et disiunctorum. Quamquam hae technicae modo quodam intuitivo adhibentur, earum iustificatio non est omnino trivialis, sed ex regulis logicae propositionum, quas in lectionibus prioribus iam demonstravimus, deduci possunt. Formaliter, technicae introductionis et remotiones connectorum et disiunctorum sunt hae:

∨-Introductio	$\{\alpha \} \vdash (\alpha \vee \beta)$
∨-Remotio	$\{(\alpha\vee\beta), \neg\alpha \} \vdash\beta$
∧-Introductio	$\{\alpha.\beta \} \vdash(\alpha \wedge \beta)$
∧-Remotio	$\{(\alpha \wedge \beta) \} \vdash \alpha$

Et earum demonstrationes ex logica propositionum infra ostenduntur:

∨-Introductio

(1)	$\{\alpha\} \vdash \alpha$	; Praemissa
(2)	$\{\alpha\} \vdash( \alpha \rightarrow (\neg \beta \rightarrow \alpha))$	; A1, Mon
(3)	$\{\alpha\} \vdash (\neg \beta \rightarrow \alpha)$	; MP(1,2)
(4)	$\boxed{\{\alpha\} \vdash (\beta \vee \alpha)}$	; $\rightarrow$ -Definitio(3)

∨-Remotio

(1)	$\{(\alpha \vee \beta), \neg\alpha\}\vdash (\alpha \vee\beta)$	; Praemissa
(2)	$\{(\alpha \vee \beta), \neg\alpha\}\vdash \neg\alpha$	; Praemissa
(3)	$\{(\alpha \vee \beta), \neg\alpha\}\vdash (\neg \alpha \rightarrow \beta)$	; $\rightarrow$ -Definitio (1)
(4)	$\boxed{\{(\alpha \vee \beta), \neg\alpha\}\vdash \beta}$	; MP(2,3)

∧-Introductio

(1)	$\{(\neg\alpha \vee \neg \beta), \neg\neg\beta\} \vdash \neg\alpha$	; $\vee$ -Remotio
(2)	$\{\neg\neg\beta\} \vdash ((\neg\alpha \vee \neg \beta) \rightarrow \neg\alpha)$	; TD(1)
(3)	$\{\neg\neg\beta\} \vdash (\neg \neg\alpha \rightarrow \neg (\neg\alpha \vee \neg \beta))$	; CPI(2))
(4)	$\vdash (\neg\neg\beta \rightarrow (\neg \neg\alpha \rightarrow \neg (\neg\alpha \vee \neg \beta)))$	; TD(3)
(5)	$\{\alpha, \beta \} \vdash (\neg\neg\beta \rightarrow (\neg \neg\alpha \rightarrow \neg (\neg\alpha \vee \neg \beta)))$	; Monotonia ×2 (4)
(6)	$\{\alpha, \beta \} \vdash \beta$	; Praemissa
(7)	$\{\alpha, \beta \} \vdash \neg\neg\beta$	; DN(6)
(8)	$\{\alpha, \beta \} \vdash (\neg \neg\alpha \rightarrow \neg (\neg\alpha \vee \neg \beta))$	; MP(7,5)
(9)	$\{\alpha, \beta \} \vdash \alpha$	; Praemissa
(10)	$\{\alpha, \beta \} \vdash \neg\neg\alpha$	; DN(9)
(11)	$\{\alpha, \beta \} \vdash \neg (\neg\alpha \vee \neg \beta)$	; MP(10,8)
(12)	$\boxed{\{\alpha, \beta \} \vdash (\alpha \wedge \beta)}$	; $\wedge$ -Definitio(11)

∧-Remotio

(1)	$\{(\alpha \wedge \beta)\} \vdash (\alpha \wedge \beta)$	; Praemissa
(2)	$\{\neg \alpha\} \vdash (\neg \alpha \vee \neg\beta)$	; $\vee$ -Introductio
(3)	$\vdash (\neg \alpha \rightarrow (\neg \alpha \vee \neg\beta))$	; TD(2)
(4)	$\vdash (\neg(\neg \alpha \vee \neg\beta) \rightarrow \alpha)$	; CPI(3))
(5)	$\vdash ( ( \alpha \wedge \beta) \rightarrow \alpha)$	; $\wedge$ -Definitio(4)
(6)	$\{(\alpha \wedge \beta)\} \vdash ( ( \alpha \wedge \beta) \rightarrow \alpha)$	; Monotonia(5)
(7)	$\boxed{\{(\alpha \wedge \beta)\} \vdash \alpha}$	; MP(1,6)

Technicae Contradictionum et Tautologiarum

Regula Tertii Exclusi sive Tautologia (tau)

Alia ex notabilioribus proprietatibus logicae classicae est proprietas tertii exclusi (tertium non datur). Haec statuit quod, si duae propositiones habentur quarum una alteram negat, tunc necessario una earum vera esse debet; sive aliter dicendo, coniunctio duarum propositionum quarum una alteram negat, necessario tautologiam constituit. Formaliter, hoc exprimitur sic scribendo:

$\vdash (\neg\alpha \vee\alpha)$

Et demonstratio eius facile obtineri potest.

(1)	$\{\alpha\}\vdash \alpha$	; Praemissa
(2)	$\vdash (\alpha \rightarrow \alpha)$	; TD(1)
(3)	$\boxed{\vdash (\neg \alpha \vee \alpha)}$	; ex (2) quia $(\alpha \rightarrow \beta) := (\neg \alpha \vee \beta)$

Alia forma huius principii enuntiandi est lex non contradictionis, quae affirmat propositionem non posse simul vera et falsa esse, quae formaliter exprimitur per:

$\vdash \neg(\neg\alpha \wedge \alpha)$

Haec proprietas demonstratione non eget, non quia ipsa per se evidens est, sed quia directe ex definitione coniunctionis super principio tertii exclusi deducitur.

Regula Contradictionis sive Principium Explosionis

Alia nota proprietas logicae classicae est principium explosionis, quod plerumque hac sententia exprimitur: “ex praemissis contradictoriis quidlibet concludi potest”. Formulatio eius exhiberi solet qualibet ex sequentibus formis:

$\{(\neg\alpha \wedge \alpha)\}\vdash \beta$

$\{\alpha, \neg\alpha\}\vdash \beta$

Demonstratio huius regulae est simplex:

(1)	$\{\alpha ,\neg\alpha\} \vdash \neg\alpha$	; Praemissa
(2)	$\{\alpha ,\neg\alpha\} \vdash (\neg\alpha \vee \beta)$	; $\vee$ -Introductio
(3)	$\{\alpha ,\neg\alpha\} \vdash (\alpha \rightarrow \beta)$	; $\rightarrow$ -Definitio(2)
(4)	$\{\alpha ,\neg\alpha\} \vdash \alpha$	; Praemissa
(5)	$\boxed{\{\alpha ,\neg\alpha\} \vdash \beta}$	; MP(4,3)

∨-Remotio3

Modus ponens scribi potest duobus modis distinctis. Una ex formis quas iam novimus est $\{\alpha,(\alpha \rightarrow \beta)\}\vdash \beta$ . Altera autem paulo minus nota est:

$\{\alpha\}\vdash\beta \; \wedge \; \vdash \alpha \; \Longrightarrow \; \vdash \beta$

In hac forma secunda innixi, videre possumus extensionem huius regulae quam ∨-Remotio3 appellamus, quia similis est simplificationi quae ex disiunctione oritur. Haec regula declarat: si $\gamma$ ex $\alpha$ et ex $\beta$ separatim deduci potest, et insuper disiunctio inter $\alpha$ et $\beta$ est theorema, tunc $\gamma$ est etiam theorema. Hoc formaliter sic exprimimus:

$\{\alpha\}\vdash\gamma\; \wedge \; \{\beta\}\vdash\gamma \; \wedge \; \vdash (\alpha \vee \beta)$ $\Longrightarrow$ $\vdash \gamma$

Demonstratio huius technicae logicae classicae haec est:

(1)	$\boxed{\alpha \vdash \gamma}$	; Praemissa
(2)	$\boxed{\beta \vdash \gamma}$	; Praemissa
(3)	$\boxed{\vdash (\alpha \vee \beta)}$	; Praemissa
(4)	$\vdash (\alpha \rightarrow \gamma)$	; TD(1)
(5)	$\vdash (\beta \rightarrow \gamma)$	; TD(2)
(6)	$\vdash (\neg \gamma \rightarrow \neg \alpha)$	; CPI(4)
(7)	$\vdash (\neg \gamma \rightarrow \neg \beta)$	; CPI(5)
(8)	$\{\neg \gamma \}\vdash \neg \alpha$	; RTD(6)
(9)	$\{\neg \gamma\}\vdash \neg \beta$	; RTD(7)
(10)	$\{\neg \gamma\}\vdash (\neg \alpha \wedge \neg \beta)$	; $\wedge$ -Introductio(8,9)
(11)	$\vdash (\neg \gamma \rightarrow (\neg \alpha \wedge \neg \beta))$	; TD(10)
(12)	$\vdash (\neg(\neg \alpha \wedge \neg \beta)\rightarrow \gamma )$	; CPI(11)
(13)	$(A \wedge B) := \neg(\neg A \vee \neg B)$	; $\wedge$ – Definitio
(14)	$\neg(A \wedge B) := \neg\neg(\neg A \vee \neg B)$	; Utraque pars negatur in (13)
(15)	$\neg(\neg\alpha \wedge \neg\beta) := \neg\neg(\neg\neg\alpha \vee \neg\neg\beta)$	; Substituendo $A:=\neg\alpha$ et $B:=\neg\beta$ in (14)
(16)	$\neg(\neg\alpha \wedge \neg\beta) \dashv \vdash (\alpha \vee \beta)$	; DN(15)
(17)	$\vdash ((\alpha \vee \beta) \rightarrow \neg(\neg\alpha \wedge \neg\beta) )$	; TD(16)
(17)	$\vdash ((\alpha \vee \beta) \rightarrow \gamma )$	; SH(17,12)
(18)	$\boxed{ \vdash \gamma}$	; MP(3,17)

Probationes per Casus (cas)

Alia ex technicis logicae classicae est probatio per casus. Si expressio $\beta$ tam ex alia expressione $\alpha$ quam ex eius negatione inferri potest, tunc expressio $\beta$ est necessario theorema. Hoc formaliter repraesentatur scriptione: $\alpha \vdash \beta \; \wedge \; \neg\alpha \vdash \beta$ $\Longrightarrow$ $\vdash \beta$ . Huius demonstratio est haec:

$\begin{array}{rll} (1) & \alpha \vdash \beta &; Praemissa\\ (2) & \neg \alpha \vdash \beta &; Praemissa \\ (3) & \vdash \alpha \vee \neg\alpha &; TAU \\ (4) & \vdash \beta &; \vee-Remotio3(1,2,3) \end{array}$

Reductio ad Absurdum (absurdo)

Una ex technicis logicae classicae maxime adhibitis in demonstrationibus, praesertim in mathematicis, est reductio ad absurdum. Haec consistit in hoc: si ex expressione $\alpha$ deducitur contradictio (scilicet affirmatio et eius negatio), tunc negatio $\alpha$ est tautologia. Formaliter exprimitur hoc modo: $\{\alpha\}\vdash \beta \; \wedge \; \{\alpha\}\vdash \neg\beta$ $\Longrightarrow$ $\vdash \neg\alpha$ . Et hoc demonstrari potest per sequentem ratiocinationem:

(1)	$\boxed{\{\alpha\}\vdash \beta}$	; Praemissa
(2)	$\boxed{\{\alpha\}\vdash \neg\beta}$	; Praemissa
(3)	$\vdash (\alpha \rightarrow \beta)$	; TD(1)
(4)	$\vdash (\alpha \rightarrow \neg\beta)$	; TD(2)
(5)	$\vdash (\neg \beta \rightarrow \neg \alpha)$	; CPI(3)
(6)	$\vdash (\beta \rightarrow \neg \alpha)$	; CPI(4)
(7)	$\{\neg \beta \}\vdash \neg \alpha$	; RTD(5)
(8)	$\{\beta \}\vdash \neg \alpha$	; RTD(6)
(9)	$\boxed{\vdash \neg \alpha}$	; CAS(7,8)