Tutorial Contents Tutorial Five: Predicate Calculus: Tableaux - The $xj rule - Bad Mistakes and Technical Mistakes - Ticking - Finished Tableaux - Sequents - Soundness and Completeness - Proof of a syntactic sequent: example
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The "xj rule

The rule is:

 

The rationale for this rule is that "$xj" is equivalent to ""xj".

 

Note that the rule applies only to formulae which begin with """.

 

The $xj rule

 

The rule is:

 

The rationale for this rule is that ""xj" is equivalent to "$xj".

 


Note that the rule applies only to formulae which begin with "$".


 

Example: a proof that the following argument is valid:

 

Fa

[$xFx"xGx]

So, Gb

 

The counterexample set is {Fa, [$xFx"xGx], Gb}

 

 

The tableau closes; so the counterexample set is inconsistent; so the argument is valid.

 

The following is an incorrect attempt to apply the rule:

 

 

The mistake consists in the application of the rule to a formula which does not begin with "$x".

 

The following is correct:

 

 

 

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