Logica propositionalis

Logica propositionalis, in logica et mathematica, dicitur cognitionis ratio quae praecipue ad argumentorum analyses spectat.

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Argumentum quoddam in varias propositiones dividitur; quae appellantur praemisa et deductiones seu conclusiones, quae verae aut falsae possunt esse; quae in denotatione formali litteris A, B, C, etc., repraesentantur et coniunguntur per sex operationes logicas, invicem symbolis repraesentatas, quae sunt: condicionale (${\displaystyle \to }$), bicondicionale (${\displaystyle \leftrightarrow }$), coniugens (${\displaystyle \land }$), disiugens (${\displaystyle \lor }$), excludens (${\displaystyle \veebar }$), negans (${\displaystyle \neg }$), et implicans (${\displaystyle \vdash }$).

Argumentum, cum legibus logicae propositionalis paret, validum esse dicitur; et argumentum validum, cum omnia praemisa eius sunt vera, verum dicitur.

Definitio operationum

Condicionalis

Propositio condicionalis dicit: "Si ${\displaystyle p}$, tunc ${\displaystyle q}$". Si ${\displaystyle p}$ et ${\displaystyle q}$ praemisa proposita sunt, scimus:

• si ${\displaystyle p}$ verum est et ${\displaystyle q}$ verum, ${\displaystyle p}$${\displaystyle \to }$${\displaystyle q}$ verum esse;
• si ${\displaystyle p}$ verum est et ${\displaystyle q}$ falsum, ${\displaystyle p}$${\displaystyle \to }$${\displaystyle q}$ falsum esse;
• si ${\displaystyle p}$ falsum est et ${\displaystyle q}$ verum, ${\displaystyle p}$${\displaystyle \to }$${\displaystyle q}$ verum esse;
• si ${\displaystyle q}$ falsum est et ${\displaystyle q}$ falsum, ${\displaystyle p}$${\displaystyle \to }$${\displaystyle q}$ verum esse.

Bicondicionalis

Propositio bicondicionalis dicit: "Si ${\displaystyle p}$, tunc ${\displaystyle q}$ si et solum si ${\displaystyle q}$, tunc ${\displaystyle p}$." Scimus igitur:

• si ${\displaystyle p}$ et ${\displaystyle q}$ vera sunt aut ${\displaystyle p}$ et ${\displaystyle q}$ falsa sunt, ${\displaystyle p}$ ${\displaystyle \leftrightarrow }$ ${\displaystyle q}$ verum esse;
• si ${\displaystyle p}$ falsum est sed ${\displaystyle q}$ verum est aut ${\displaystyle p}$ verum est et ${\displaystyle q}$ falsum est, ${\displaystyle p}$ ${\displaystyle \leftrightarrow }$ ${\displaystyle q}$ falsum esse.

Coniunctio

Propositio coniunctiva dicit: "${\displaystyle p}$ et ${\displaystyle q}$ sunt vera."

• si ${\displaystyle p}$ et ${\displaystyle q}$ vera sunt, ${\displaystyle p}$${\displaystyle \land }$${\displaystyle q}$ verum esse;
• si ${\displaystyle p}$ aut ${\displaystyle q}$ falsum est, ${\displaystyle p}$${\displaystyle \land }$${\displaystyle q}$ falsum esse.

Disiunctio

Propositio disiunctiva dicit: "${\displaystyle p}$ vel ${\displaystyle q}$ verum est."

• si ${\displaystyle p}$ vel ${\displaystyle q}$ verum est, ${\displaystyle p}$${\displaystyle \lor }$${\displaystyle q}$ verum esse;
• si ${\displaystyle p}$ falsum est et ${\displaystyle q}$ falsum est, ${\displaystyle p}$${\displaystyle \lor }$${\displaystyle q}$ falsum esse.

Negatio

Propositio negativa dicit: "non ${\displaystyle p}$." Si ${\displaystyle p}$ propositio est, scimus,

• si ${\displaystyle p}$ verum est, ${\displaystyle \neg }$${\displaystyle p}$ falsum esse;
• si ${\displaystyle p}$ falsum est, ${\displaystyle \neg }$${\displaystyle p}$ verum esse.

Leges logicae propositionalis

Argumentum, cum legibus paret, validum esse dicitur. Quae leges in tabula infra ostenduntur.

Leges
Nomen	Symbola	Dictum
Modus ponens	${\displaystyle ((p\to q)\land p)\vdash q}$	Si ${\displaystyle p}$ tunc ${\displaystyle q}$; ${\displaystyle p}$; ergo ${\displaystyle q}$
Modus tollens	${\displaystyle ((p\to q)\land \neg q)\vdash \neg p}$	Si ${\displaystyle p}$ tunc ${\displaystyle q}$; non ${\displaystyle q}$; ergo non ${\displaystyle p}$
Syllogismus hypotheticus	${\displaystyle ((p\to q)\land (q\to r))\vdash (p\to r)}$	Si ${\displaystyle p}$ tunc ${\displaystyle q}$; si ${\displaystyle q}$ tunc ${\displaystyle r}$; ergo, si ${\displaystyle p}$ tunc ${\displaystyle r}$
Syllogismus disiunctivus	${\displaystyle ((p\lor q)\land \neg p)\vdash q}$	Aut ${\displaystyle p}$ aut ${\displaystyle q}$; non ${\displaystyle p}$; ergo, ${\displaystyle q}$
Dilemma constructivum	${\displaystyle ((p\to q)\land (r\to s)\land (p\lor r))\vdash (q\lor s)}$	Si ${\displaystyle p}$ tunc ${\displaystyle q}$; et si ${\displaystyle r}$ tunc ${\displaystyle s}$; sed ${\displaystyle p}$ vel ${\displaystyle r}$; ergo ${\displaystyle q}$ vel ${\displaystyle s}$
Dilemma destructivum	${\displaystyle ((p\to q)\land (r\to s)\land (\neg q\lor \neg s))\vdash (\neg p\lor \neg r)}$	Si ${\displaystyle p}$ tunc ${\displaystyle q}$; et si ${\displaystyle r}$ tunc ${\displaystyle s}$; sed non ${\displaystyle q}$ vel non ${\displaystyle s}$; ergo non ${\displaystyle p}$ vel non ${\displaystyle r}$
Dilemma bidirectionale	${\displaystyle ((p\to q)\land (r\to s)\land (p\lor \neg s))\vdash (q\lor \neg r)}$	Si ${\displaystyle p}$ tunc ${\displaystyle q}$; et si ${\displaystyle r}$ tunc ${\displaystyle s}$; sed ${\displaystyle p}$ vel non ${\displaystyle s}$; ergo ${\displaystyle q}$ vel non ${\displaystyle r}$
Simplificatio	${\displaystyle (p\land q)\vdash p}$	${\displaystyle p}$ et ${\displaystyle q}$ sunt vera; ergo ${\displaystyle p}$ est verum
Coniunctio	${\displaystyle p,q\vdash (p\land q)}$	${\displaystyle p}$ et ${\displaystyle q}$ sunt vera singula; ergo vera sunt una
Additio	${\displaystyle p\vdash (p\lor q)}$	${\displaystyle p}$ est verum; ergo disiunctio (${\displaystyle p}$ aut ${\displaystyle q}$) est vera
Compositio	${\displaystyle ((p\to q)\land (p\to r))\vdash (p\to (q\land r))}$	Si ${\displaystyle p}$ tunc ${\displaystyle q}$; et si ${\displaystyle p}$ tunc ${\displaystyle r}$; ergo si ${\displaystyle p}$ est verum tunc ${\displaystyle q}$ et ${\displaystyle r}$ sunt vera
Theorema De Morgan (I)	${\displaystyle \neg (p\land q)\vdash (\neg p\lor \neg q)}$	Negare (et ${\displaystyle p}$ et ${\displaystyle q}$) idem valet ac (non ${\displaystyle p}$ vel non ${\displaystyle q}$)
Theorema De Morgan (2)	${\displaystyle \neg (p\lor q)\vdash (\neg p\land \neg q)}$	Negare (aut ${\displaystyle p}$ aut ${\displaystyle q}$) idem valet ac (non ${\displaystyle p}$ et non ${\displaystyle q}$)
Commutatio (1)	${\displaystyle (p\lor q)\vdash (q\lor p)}$	(aut ${\displaystyle p}$ aut ${\displaystyle q}$) idem valet ac (aut ${\displaystyle q}$ aut ${\displaystyle p}$)
Commutatio (2)	${\displaystyle (p\land q)\vdash (q\land p)}$	(et ${\displaystyle p}$ et ${\displaystyle q}$) idem valet ac (et ${\displaystyle q}$ et ${\displaystyle p}$)
Commutatio (3)	${\displaystyle (p\leftrightarrow q)\vdash (q\leftrightarrow p)}$	(${\displaystyle p}$ idem esse ac ${\displaystyle q}$) idem valet ac (${\displaystyle q}$ esse idem ac ${\displaystyle p}$)
Associatio (1)	${\displaystyle (p\lor (q\lor r))\vdash ((p\lor q)\lor r)}$	aut ${\displaystyle p}$ aut (aut ${\displaystyle q}$ aut ${\displaystyle r}$) idem valet ac (aut ${\displaystyle p}$ aut ${\displaystyle q}$) aut ${\displaystyle r}$
Associatio (2)	${\displaystyle (p\land (q\land r))\vdash ((p\land q)\land r)}$	et ${\displaystyle p}$ et (et ${\displaystyle q}$ et ${\displaystyle r}$) idem valet ac (et ${\displaystyle p}$ et ${\displaystyle q}$) et ${\displaystyle r}$
Distributio (1)	${\displaystyle (p\land (q\lor r))\vdash ((p\land q)\lor (p\land r))}$	et ${\displaystyle p}$ et (aut ${\displaystyle q}$ aut ${\displaystyle r}$) idem valet ac aut (et ${\displaystyle p}$ et ${\displaystyle q}$) at (et ${\displaystyle p}$ et ${\displaystyle r}$)
Distributio (2)	${\displaystyle (p\lor (q\land r))\vdash ((p\lor q)\land (p\lor r))}$	aut ${\displaystyle p}$ aut (et ${\displaystyle q}$ et ${\displaystyle r}$) idem valet ac et (aut ${\displaystyle p}$ aut ${\displaystyle q}$) et (aut ${\displaystyle p}$ aut ${\displaystyle r}$)
Negatio duplex	${\displaystyle p\vdash \neg \neg p}$	${\displaystyle p}$ idem valet ac (non ${\displaystyle p}$) negare
Transpositio	${\displaystyle (p\to q)\vdash (\neg q\to \neg p)}$	Si ${\displaystyle p}$ tunc ${\displaystyle q}$ idem valet ac si non ${\displaystyle q}$ tunc non ${\displaystyle p}$
Implicatio materialis	${\displaystyle (p\to q)\vdash (\neg p\lor q)}$	Si ${\displaystyle p}$ tunc ${\displaystyle q}$ idem valet ac aut non ${\displaystyle p}$ aut ${\displaystyle q}$
Aequivalentia materialis (1)	${\displaystyle (p\leftrightarrow q)\vdash ((p\to q)\land (q\to p))}$	(${\displaystyle p}$ idem ac ${\displaystyle q}$ esse) significat quod (si ${\displaystyle p}$ verum est, tunc ${\displaystyle q}$ verum est) et (si ${\displaystyle q}$ verum est, tunc ${\displaystyle p}$ verum est)
Aequivalentia materialis (2)	${\displaystyle (p\leftrightarrow q)\vdash ((p\land q)\lor (\neg p\land \neg q))}$	(${\displaystyle p}$ idem ac ${\displaystyle q}$ valere) significat aut (et ${\displaystyle p}$ et ${\displaystyle q}$ vera esse) aut (et ${\displaystyle p}$ et ${\displaystyle q}$ falsa esse)
Aequivalentia materialis (3)	${\displaystyle (p\leftrightarrow q)\vdash ((p\lor \neg q)\land (\neg p\lor q))}$	(${\displaystyle p}$ idem ac ${\displaystyle q}$ valere) significat, et (aut ${\displaystyle p}$ aut non ${\displaystyle q}$ verum esse) et (aut non ${\displaystyle p}$ aut ${\displaystyle q}$ verum ese)
Exportatio	${\displaystyle ((p\land q)\to r)\vdash (p\to (q\to r))}$	De (si et ${\displaystyle p}$ et ${\displaystyle q}$ vera sunt, tunc ${\displaystyle r}$ verum est) possumus demonstrare (si ${\displaystyle q}$ verum est, tunc ${\displaystyle r}$ verum est, si ${\displaystyle p}$ verum est)
Importatio	${\displaystyle (p\to (q\to r))\vdash ((p\land q)\to r)}$
Tautologia (1)	${\displaystyle p\vdash (p\lor p)}$	${\displaystyle p}$ esse verum idem valet ac aut ${\displaystyle p}$ esse verum aut ${\displaystyle p}$ essev verum
Tautologia (2)	${\displaystyle p\vdash (p\land p)}$	${\displaystyle p}$ esse verum idem valet ac et ${\displaystyle p}$ esse verum et ${\displaystyle p}$ esse verum
Tertium non datur	${\displaystyle \vdash (p\lor \neg p)}$	aut ${\displaystyle p}$ aut non ${\displaystyle p}$ verum est
Lex contradictionis	${\displaystyle \vdash \neg (p\land \neg p)}$	quod et ${\displaystyle p}$ et non ${\displaystyle p}$ esse falsum, verum est

Nexus interni

• Algebra Booleeana
• Associativitas (mathematica)
• Formula bene formata
• Logica
• Logica mathematica
• Mathematica discreta
• Ratio propositionum
• Theoria copiarum

Bibliographia

• Brown, Frank Markham. 2003. Boolean Reasoning: The Logic of Boolean Equations. Editio 1a. Norwell, Massachusettae. Editio 2a. Mineolae Novi Eboraci: Dover Publications.
• Chang, C. C., et H. J. Keisler. 1973. Model Theory. Amstelodami, Hollandiae Septentrionalis.
• Hofstadter, Douglas. 1979. Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books. ISBN 978-0-46-502656-2.
• Kohavi, Zvi. 1970, 1978. Switching and Finite Automata Theory. McGraw–Hill.
• Korfhage, Robertus R. 1974. Discrete Computational Structures. Novi Eboraci: Academic Press.
• Lambek, J., et P. J. Scott. 1986. Introduction to Higher Order Categorical Logic. Cantabrigiae: Cambridge University Press
• Mendelson, Elliot. 1964. Introduction to Mathematical Logic. D. Van Nostrand Company.

mathematica

Haec stipula ad mathematicam spectat. Amplifica, si potes!