Proof by Cases/Proof Rule

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Proof Rule

Proof by Cases is a valid deduction sequent in propositional logic.

As a proof rule it is expressed in the form:

If we can conclude $\phi \lor \psi$, and:
$(1): \quad$ By making the assumption $\phi$, we can conclude $\chi$
$(2): \quad$ By making the assumption $\psi$, we can conclude $\chi$
then we may infer $\chi$.


The conclusion $\chi$ does not depend upon either assumption $\phi$ or $\psi$.


It can be written:

$\displaystyle {\phi \lor \psi \quad \begin{array}{|c|} \hline \phi \\ \vdots \\ \chi \\ \hline \end{array} \quad \begin{array}{|c|} \hline \psi \\ \vdots \\ \chi \\ \hline \end{array} \over \chi} \lor_e$


Tableau Form

Let $\phi \lor \psi$ be a propositional formula in a tableau proof whose main connective is the disjunction operator.

Let $\chi$ be a propositional formula such that $\left({\phi \vdash \chi}\right)$, $\left({\psi \vdash \chi}\right)$.


Proof by Cases is invoked for $\phi \lor \psi$ as follows:

Pool:    The pooled assumptions of $\phi \lor \psi$             
The pooled assumptions of the instance of $\chi$ which was derived from the individual assumption $\phi$             
The pooled assumptions of the instance of $\chi$ which was derived from the individual assumption $\psi$             
Formula:    $\chi$             
Description:    Proof by Cases             
Depends on:    The line containing the instance of $\phi \lor \psi$             
The series of lines from where the assumption $\phi$ was made to where $\chi$ was deduced             
The series of lines from where the assumption $\psi$ was made to where $\chi$ was deduced             
Discharged Assumptions:    The assumptions $\phi$ and $\psi$ are discharged             
Abbreviation:    $\operatorname {PBC}$             


Explanation

Proof by Cases can be expressed in natural language as follows:

We are given that either $\phi$ is true, or $\psi$ is true, or both.

Suppose we make the assumption that $\phi$ is true, and from that deduce that $\chi$ has to be true.

Then suppose we make the assumption that $\psi$ is true, and from that deduce that $\chi$ has to be true.

Therefore, it has to follow that the truth of $\chi$ follows from the fact of the truth of either $\phi$ or $\psi$.


Also known as

Proof by Cases is also known as the rule of or-elimination.


Technical Note

When invoking Proof by Cases in a tableau proof, use the {{ProofByCases}} template:

{{ProofByCases|line|pool|statement|base|start1|end1|start2|end2}}

where:

line is the number of the line on the tableau proof where Proof by Cases is to be invoked
pool is the pool of assumptions (comma-separated list)
statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters
base is the line of the tableau proof where the disjunction being eliminated is situated
start1 is the start line of the block of the tableau proof upon which the demonstration of the first disjunct directly depends
end1 is the end line of the block of the tableau proof upon which the demonstration of the first disjunct directly depends
start2 is the start line of the block of the tableau proof upon which the demonstration of the second disjunct directly depends
end2 is the end line of the block of the tableau proof upon which the demonstration of the second disjunct directly depends


Sources