Formal Logic

Split the brief and the ver­bose ex­po­si­tion into two lenses.

For­mal logic makes our rea­son­ing pre­cise by pro­vid­ing for­mal sys­tems in which to carry out rea­son­ing. Th­ese sys­tems are speci­fied by a proof the­ory and of­ten a cor­re­spond­ing model the­ory. Either of these the­o­ries will take many differ­ent forms and pre­sen­ta­tions, the im­por­tant part is that they are rigor­ous and clearly defined.

For­mal sys­tems of logic are not con­cerned with the con­tent of an ar­gu­ment, but only with the form.

For ex­am­ple, con­sider the fol­low­ing ar­gu­ments.

\( \begin{array}{l} \text{If Socrates is a man, then Socrates is mortal.} \\ \text{Socrates is a man.} \\\hline \text{Therefore, Socrates is mortal.} \end{array} \)

\( \begin{array}{l} \text{If the moon is ruled by the Mouse King, then the moon is made of cheese.} \\ \text{The moon is ruled by the Mouse King.} \\\hline \text{Therefore, the moon is made of cheese.} \end{array} \)

For­mal logic can not tell you that the lat­ter ar­gu­ment is about non-sense. What it will tell you is about the form of the ar­gu­ment and whether that’s valid and it so hap­pens that both of these ar­gu­ments “look” the same.

Let \(S\) stand for “Socrates is a man”, \(O\) stand for “Socrates is mor­tal”, \(M\) stand for “the moon is ruled by the Mouse King”, and \(C\) for “the moon is made of cheese”.

Then these ar­gu­ments can be seen as fol­lows.

\( \begin{array}{l} \text{If \)S\( then \)O\(.} \\ \text{\)S\(} \\\hline \text{Therefore, \)O\(.} \end{array} \)

\( \begin{array}{l} \text{If \)M\( then \)C\(.} \\ \text{\)M\(} \\\hline \text{Therefore, \)C\(.} \end{array} \)

This makes it ap­par­ent that the ar­gu­ments pro­ceed in the same way. Then if we in­tro­duce the sym­bol \(\rightarrow\) to mean “If \(A\), then \(B\)” in \(A \rightarrow B\), and the sym­bol \(\therefore\) for “there­fore”, the form if these ar­gu­ments is writ­ten as

\( \begin{array}{l} A \rightarrow B \\ A \\\hline \therefore B \end{array} \)

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  • Logic

    Logic is the study of cor­rect ar­gu­ments.